I have struggled for some time to organize the content of #ComplexityThoughts in a meaningful way, since it has grown into a large reservoir of scientific information.

I have found one that does not require special coding skills and can lead to a nice result in less than 5 minutes. This is what I am going to share in this post, hoping it will be useful to organize your content too.

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Surviving perturbations requires more than connectivity. What must persist are processes: production of metabolites, maintenance of internal variables, coordination across subsystems. In network terms, this brings attention to a recurring structural feature, i.e. loops, but their role depends on what they actually close.

Based on the more technical arguments developed in my recent paper, this is the third step in our attempt to understand whether “architecture” in living systems is a real, measurable concept or just a convenient metaphor. I have introduced the key idea in the previous posts:

where I have outlined how function in living systems emerges from the coupling between logic (what the system does) and circuitry (how it is physically implemented).

A cell that loses one enzyme may still grow because flux passes through another metabolic route. A bacterium exposed to an attractant changes its motion, then adapts, because its sensory network adjusts receptor sensitivity. A mixture of RNA fragments may replicate better as a cooperative set than as isolated molecules, because some fragments help generate others. All three cases involve loops, although the biological mechanisms differ.

A closed path in a graph is a topological object: start from one node, follow links through other nodes and return to the starting point. In biological systems, the same graphical pattern may carry metabolites, signals, catalytic dependencies, energy, matter or information: the function depends on what circulates through the path and how the system constrains that circulation.

However, let’s see loops here as a minimal architectural lens, for the moment, since they help us locate three recurrent functions: redundancy, control and closure. Whether a particular loop performs one of these functions has to be checked against the process running on the network.

The first constraint is topological

In my paper, I discuss cycles beginning from random networks: the probability of forming closed paths depends on the degree distribution, especially on the first two moments of that distribution. Heterogeneous sparse networks can generate cyclic structure without becoming fully dense, whereas dense networks generate many more cycles at a much higher construction and maintenance cost, at least in principle.

This connects directly with what we have seen in the previous post: if the genome cannot encode a full wiring diagram, and if biological connections must be built anyway, regulated, repaired, and sometimes removed, living systems cannot use indiscriminate redundancy. So: which loops are worth maintaining, given their functional role and cost?

Alternative paths: loops as local redundancy

The simplest use of a loop is to preserve reachability, since if one route is blocked then another route may still connect two parts of the system.

In metabolism, this can mean that flux can be “rerouted” after a specific reaction is disrupted for some reason. In signaling, it can mean that partially overlapping pathways transmit similar information. In ecological networks, matter or energy may continue to move through alternative species interactions when one link weakens.

However, a topological alternative route is only a candidate for redundancy. In a metabolic network, a bypass requires compatible stoichiometry, sufficient enzyme expression, available substrates, as well as regulatory conditions that allow the alternative route to operate on the relevant time scale. Therefore, a path that exists in a reconstructed network may remain physiologically inaccessible in a particular environment.

Work on E. coli metabolism illustrates the point. Using a metabolite-centered analysis it has been shown that robustness cannot be inferred only from gene essentiality: some metabolites are structurally central because many fluxes depend on them, and maintaining flux around these metabolites can be associated with cellular robustness and fragility.

Interestingly, after perturbations, one can also show that latent pathways often offer no immediate growth advantage and can even inhibit growth after genetic perturbation.

Redundancy therefore functions as a conditional capacity, not as a static reserve: the system must be able to access the alternative route, regulate it and pay its metabolic cost.

Feedback: loops as control

A feedback loop exists when a variable affects its own future through the system:

This may be visible as a cycle in a molecular interaction graph, but it may also appear only after coarse-graining many biochemical steps into an effective control circuit.

Negative feedback reduces deviations from an operating range. For instance, bacterial chemotaxis provides a standard example because the system adapts after stimulation: receptor activity changes after exposure to attractant, while methylation dynamics adjust sensitivity so that activity returns toward baseline. This adaptation can remain robust despite variation in biochemical parameters, and it can be interpreted through integral feedback control, making explicit why adaptation requires a particular control structure rather than the mere presence of a cycle.

Positive feedback, instead, amplifies differences and can support bistability, switching, hysteresis and memory. This behavior is useful in development and cell-fate decisions, where a transient signal must sometimes produce a stable commitment. But the same mechanism also creates risks: a loop that stores history can preserve an inappropriate state after the conditions that produced it have disappeared.

Feedback modifies state variables: its effect depends on sign, gain, delay and coupling to the rest of the system.

Closure: loops as production

Closure is stricter than redundancy or feedback. A process is closed when its outputs help regenerate the components or conditions that make the process possible.

This is a common concept in origin-of-life theories, but it needs precision. Kauffman’s autocatalytic-set idea made catalytic closure central: a set of molecules can collectively catalyze reactions that generate members of the same set. In later RAF theory, the formal object is a reaction set that is reflexively autocatalytic and food-generated (RAF): reactions are catalyzed by molecules from within the set, and the required molecules can be built from an available food set.

The RAF formulation is stronger than the visual image of a simple ring, since a closed reaction system needs inputs, catalytic support, suitable side conditions and protection from processes that drain intermediates faster than they are regenerated.

A less widely remembered contribution came from Rössler in 19711, where he described a system-theoretic model of biogenesis based on generalized catalysis and what he named “second-order autonomous growth”. In a nutshell, an autocatalytic cycle can expand chemical space by sustaining new molecular species, and that enlarged space can increase the probability of discovering further cycles: the growth of autocatalytic systems then becomes part of the process being amplified. This does not settle origin-of-life theory, but it identifies a mechanism worth keeping in mind: once a loop sustains new components, it can modify the space of possible future loops.

Morowitz and Smith work on intermediary metabolism argued that parts of core metabolism, especially the reductive tricarboxylic acid cycle, can be understood as a network-autocatalytic structure shaped by carbon chemistry, energetics, and reaction topology.

Let’s remark how the narrow formulation is important: a biochemical cycle does not become life-like just because it is cyclic. Here, the main claim is that some cycles persist because matter and energy flow through constrained reaction networks in ways that regenerate key intermediates.

The argument holds beyond metabolism and extends to other biochemical networks, such as mixtures of RNA fragments that self-assemble into self-replicating ribozymes, for which it has been experimentally shown the emergence of cooperative catalytic cycles and networks2.

Overall, this is why closure is a central concept for living architecture: the unit that matters is sometimes the molecule, but in other cases it is the looped set of reactions that makes new molecular behavior possible.

Why loops remain costly

A redundant route requires enzymes, substrates and regulation. A feedback controller requires sensing, response and timing. A production loop requires access to matter and energy, plus constraints that prevent side reactions or parasitic processes from exhausting the system.

These mechanisms have been placed within a broader account of cellular robustness, emphasizing that biological robustness uses combinations of feedback control, redundancy, modularity and structural organization. An emblematic example used in that work is E. coli chemotaxis:

However, the same mechanisms can introduce vulnerabilities. For instance, Doyle and collaborators described this general pattern as “robust yet fragile” in the context of the Internet: systems may tolerate common perturbations while remaining sensitive to disturbances that hit hidden dependencies. However, for biology, that phrase should be used as a systems analogy, not as a direct equivalence between cells and engineered communication networks. Here, the useful lesson is that robustness always has a reference class: a system may handle frequent local shocks and still be vulnerable to rare, targeted or correlated perturbations.

In fact, excessive looping can overcouple the system: perturbations then travel across modules, positive feedback amplifies fluctuations and useful dependencies under ordinary conditions become channels for failure under unusual conditions.

A useful loop must therefore remain local enough to contain damage and connected enough to sustain the process it serves. I know, it is complex, as usual.

Take-home

A loop becomes informative only after three questions are answered:

1. First, what is being looped: a metabolite flux, a regulatory signal, a catalytic dependency, or an ecological interaction?

2. Second, what does the loop do dynamically: preserve reachability, reduce deviations, amplify differences, switch state, store memory, or regenerate production?

3. Third, what does the loop cost: enzyme expression, regulatory complexity, maintenance, repair, error correction, or exposure to correlated failure?

The common rationale is that topology is strictly connected to mechanism: a motif in a graph is a constraint on possible dynamics, but function depends on the process, the environment and the scale at which the system is described.

In this chapter we have learned that loops (i) help living systems preserve routes, regulate variables, and regenerate components; and (ii) contribute to robustness when the looped process is biologically relevant and the maintenance cost remains affordable.

The storage and maintenance constraints from the previous post still apply: life cannot store or maintain arbitrary connectivity; instead it uses selected circuits that buy function without making the entire system overcoupled. We could speculate that what we observe today is the natural selection of processes satisfying these requirements.

And this creates the next architectural problem: a living system needs loops to sustain and regulate its processes, but it also needs a way to prevent loops from coupling everything to everything.

A scalable solution is hierarchy: loops organized into modules, modules organized into layers and layers coupled strongly enough to coordinate function without turning every local perturbation into a global one.

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Human behavior

Integrating behavioural experimental findings into dynamical models to inform social change interventions

Addressing global challenges often involves stimulating the large-scale adoption of new products or behaviours. Research traditions that focus on individual decision-making suggest that achieving this objective requires identifying the drivers of individual discrete adoption choices. However, computational approaches rooted in complexity science focus on maximizing the propagation of a given product or behaviour throughout social networks of interconnected adopters. Here, by integrating discrete-choice modelling into the complex contagion theory, we propose a method to estimate individual-level thresholds to adoption. We validate the predictive power of this approach in two choice experiments. By integrating the estimated thresholds into computational simulations, we show that state-of-the-art seeding policies for initiating large-scale behavioural change might be suboptimal if they neglect individual-level behavioural drivers, which can be corrected through the proposed experimental method.

Understanding human behaviour for pandemic preparedness with epigames

Infectious diseases that spread from person to person by direct transmission, including respiratory pathogens such as influenza and coronaviruses, impose a large global health burden and remain the most likely causative agents for future devastating pandemics¹. For many such diseases, transmission occurs when individuals are in close proximity for a sufficient time and through highly structured social contact networks². Data on the properties of these networks, including their temporal and spatial structures, how pathogens spread in them, and how interventions might alter this spread are scarce or inconsistent and seldom incorporate behavioural features. This produces a knowledge gap between policy-relevant models of pathogen transmission and the data they require: details of contact networks at high spatial and temporal resolution and their variability and malleability under different conditions.

Spatial and social determinants of the 1857 yellow fever epidemic in Lisbon

Yellow fever is a mosquito-borne disease transmitted to humans primarily through Aedes aegypti in urban areas. Currently, yellow fever is endemic in the tropical regions of South America and Africa. Historically yellow fever caused outbreaks and epidemics in North America and Europe too. Climate change and global connectivity are expanding the potential range of Aedes aegypti, increasing the risk of yellow fever re-emergence in Europe. To support future preparedness efforts, we examined a historical yellow fever epidemic that occurred in Lisbon, Portugal, that caused 18,000 cases and 5,652 deaths, and assessed its transmission characteristics using the basic and effective reproduction number. We digitalised and mapped archival data at the neighbourhood level to assess the contributions of both social (population, occupations, and gender dynamics) and environmental (temperature, precipitation, wind, and elevation) factors of the yellow fever epidemic in Lisbon. Research into a European epidemic that occurred 80 years before an effective yellow fever vaccine is useful for understanding population-environment dynamics in a pre-vaccination setting where yellow fever is not established. Our findings could strengthen current arbovirus control efforts and support epidemic preparedness in urban areas.

Despite the availability of a highly effective vaccine, yellow fever virus (YFV) is still endemic in 47 countries globally. Although disease due to YFV was first recorded in 1635, factors contributing to its spread remain poorly understood today. Using archival data from the nineteenth century, we digitalised and mapped the 1857 yellow fever (YF) epidemic in Lisbon, Portugal, to understand how transmission dynamics and spatial and environmental characteristics led to disparities in health outcomes between sociodemographic groups. We modelled the basic and effective reproduction number (R₀ and R_t) and found that transmission dynamics throughout this pre-vaccination era epidemic are consistent with prevailing estimates (R₀ ≃ 5). Transmission peaked at the end of October 1857 when YF was declared an epidemic, then declined until January 1858. YFV killed 4.2% of the population with infection attack rates ranging between 10.3-13.5%. Out of the 34 parishes in urban Lisbon, our hotspot analysis identified 15 statistically significant high-risk parishes near the coastline. Our maps, combined with a digital terrain model, show that the highest number of deaths occurred within connected streets confined in low-elevation built-up areas with homes. We discuss the potential role of wind and temperature in aiding mosquito dispersal across Lisbon, which were believed as the main historical environmental drivers of YF. More people died at home than in hospitals, and although working-aged men accounted for most fatalities, the highest probability of death was found among women working at home. Our study highlights the role of human-environment interactions in shaping a historical YF epidemic in a pre-vaccination urban setting and enhances our understanding of modern-day transmission dynamics.

Characterization and forecast of global influenza subtype dynamics

The subtype composition of seasonal influenza waves varies in space and time. Influenza subtypes A/H1N1, A/H3N2 and B tend to have different impacts on population groups; therefore, understanding the drivers of their cocirculation and anticipating their composition is important for epidemic preparedness. FluNet provides data on influenza specimens by subtype for more than 150 countries. However, owing to surveillance variations across countries, global analyses usually focus on subtype compositions, a kind of data difficult to treat with advanced statistical methods. We used compositional data analysis to circumvent the problem and study trajectories of annual subtype compositions of countries. Here we first examine global trends from 2000 to 2023. We identify a few seasons which stood out for the strong within-country subtype dominance due to either a new virus/clade taking over (2003/2004 season, A/H1N1pdm pandemic) or subtypes’ spatial segregation (coronavirus disease 2019 pandemic). Second, we show that geographical factors, most notably international mobility, concurred in shaping countries’ composition trajectories between 2010 and 2019. Trajectories clustered in two macroregions characterized by subtype alternation versus persistent mixing. Finally, we define five algorithms for forecasting the next year’s composition and found that incorporating the global history of subtype composition in a Bayesian hierarchical vector autoregressive model improved predictions compared with naive methods. The joint analysis of spatiotemporal dynamics of influenza subtypes worldwide reveals a hidden structure in subtype circulation that can be used to improve predictions of the subtype composition of next year’s epidemic according to place.

High-precision tracking of human foragers reveals adaptive social information use in the wild

Foraging complexity and competitive social challenges are considered key drivers of human cognition. Yet, the decision-making mechanisms that underlie social foraging in the real world remain unknown. Integrating high-precision Global Positioning System (GPS) tracking and video footage from large-scale foraging competitions with cognitive-computational modeling and agent-based simulations, we show how foragers integrate personal, social, and ecological information to guide spatial search and patch-leaving decisions. We show how the social context emerges as a key driver of foraging dynamics. Foragers adaptively rely on social information to locate resources when unsuccessful and extend giving-up times in the presence of others, which results in increased area-restricted search at high social densities. These findings demonstrate the importance of sociality for human foraging decisions and provide a template for harnessing high-resolution tracking data to study real-world cognition.

Principles for a post-growth scenario of ambitious mitigation and high human well-being

Not strictly a human-behavior paper, but quite related.

Most climate mitigation scenarios maintain inequalities and associate favourable social and climate outcomes with continuing economic growth. By contrast, post-growth scholarship advocates for reducing less-necessary production, reorienting the economy towards human needs and ecological goals, and pursuing equitable convergence within and between countries. Here we synthesize recent advances in post-growth research into five core principles: well-being, sufficiency, reduced inequalities, repurposing of the economy and north–south convergence. Existing post-growth scenarios tend to fall short of considering, let alone implementing, these key elements. We assess feasibility barriers, finding that post-growth will face weaker geophysical and technological constraints, but stronger socio-cultural and political opposition, than growth-oriented scenarios. Advances in post-growth research, alongside international calls for mitigation rooted in fairness and equity, present a strong case for a holistic development of post-growth scenarios.

Urban (eco)systems

The trade-off between microbial functionality and evolutionary flexibility under urbanization

I admit that I had to think a bit about how to classify this paper.

Urban parks, situated between cities and nature, act as ecological buffers and provide unique niches for soil microbial communities. However, it remains unclear how urbanization affects the functional diversity and evolutionary potential of microbial communities. Here, to address this, we conducted a large-scale field survey across 54 urban park and forest sites in the Pearl River Delta, one of the most rapidly urbanizing regions. Urban parks exhibited higher bacterial and archaeal alpha diversity, biomass, and functional genes related to nutrient cycling compared with forests. These differences were driven by nutrient enrichment and soil pH changes. Microbial genomic analysis revealed a trade-off between short-term ecological functionality and long-term evolvability. Urban parks enhanced immediate functionality but reduced genomic size and evolutionary flexibility, suggesting that urbanization pushes microbial communities toward functional specialization at the expense of adaptive capacity. In contrast, forest soils maintained higher genomic diversity, supporting resilience to environmental changes. These findings highlight the importance of integrating ecological and genomic approaches to predict ecosystem service sustainability in urbanizing areas.

Neuroscience

An international mega-analysis of psychedelic drug effects on brain circuit function

Psychedelic drugs are re-emerging as promising scientific and clinical tools. However, despite a rapidly expanding literature on their therapeutic value, the neural mechanisms underlying psychedelic effects remain unclear. Resting-state functional magnetic resonance imaging studies of acute psychedelic effects, conducted independently by several research groups, have so far yielded fragmented and sometimes inconsistent findings. Here, to help facilitate greater convergence, we conducted a ‘mega-analysis’ integrating 11 independent resting-state functional magnetic resonance imaging datasets across five psychedelic drugs (psilocybin, lysergic acid diethylamide, mescaline, N,N-dimethyltryptamine and ayahuasca) from research groups spanning three continents and five countries. By applying a uniform preprocessing pipeline and a Bayesian hierarchical modeling framework, we discovered several common features in the induced alterations to brain function across drugs and sites. Most prominently, we identified a core signature of increased functional connectivity between transmodal (default, frontoparietal and limbic) and unimodal networks (visual and somatomotor), with subnetwork specificity. Furthermore, key subcortical regions (thalamus, caudate and putamen) and the cerebellum exhibited altered coupling with sensorimotor networks. In contrast to several single-site reports, Bayesian modeling revealed weak-to-moderate and selective reductions in within-network functional connectivity, with substantial variability across drugs and networks. Together, these findings extend past work by demonstrating that psychedelics reconfigure large-scale cortical organization while selectively engaging subcortical circuitry. This study provides the most comprehensive synthesis of psychedelic brain action to date, helping resolve inconsistencies and offering a probabilistic map of how psychedelics alter large-scale brain organization. We hereby provide a cornerstone to benchmark and shepherd future psychedelic neuroimaging research.

In vitro neurons learn and exhibit sentience when embodied in a simulated game-world

Integrating neurons into digital systems may enable performance infeasible with silicon alone. Here, we develop DishBrain, a system that harnesses the inherent adaptive computation of neurons in a structured environment. In vitro neural networks from human or rodent origins are integrated with in silico computing via a high-density multielectrode array. Through electrophysiological stimulation and recording, cultures are embedded in a simulated game-world, mimicking the arcade game “Pong.” Applying implications from the theory of active inference via the free energy principle, we find apparent learning within five minutes of real-time gameplay not observed in control conditions. Further experiments demonstrate the importance of closed-loop structured feedback in eliciting learning over time. Cultures display the ability to self-organize activity in a goal-directed manner in response to sparse sensory information about the consequences of their actions, which we term synthetic biological intelligence. Future applications may provide further insights into the cellular correlates of intelligence.

Functional hierarchy of the human neocortex across the lifespan

Large-scale gradients of functional connectivity between brain areas organize the human neocortex, linking brain topography to the texture of cognition^1,2. In adults, three dominant axes—sensory–association, visual–somatosensory and modulation–representation—run, respectively, from primary sensory to transmodal association areas, from visual to body-centred systems and from control and attention networks to default mode and sensory areas^1,2,3,4. These gradients provide a compact description of large-scale cortical hierarchies that underlie distinct modes of information processing. However, how these gradients and their multiscale biological and cognitive correlates evolve across the lifespan is unknown. Here we establish a continuous normative reference of functional organization from birth to 100 years of age, revealing complex, nonlinear developmental trajectories. Gradient architecture is anchored by primary sensory systems in infancy, differentiates along association and control axes during childhood and adolescence and gradually dedifferentiates during ageing. The importance of this functional architecture is corroborated by biology and behaviour: gradient metrics predict cognitive performance across development; structure–function coupling varies by axis and age; and distinct transcriptomic signatures are strongest early in life and weaken with age, consistent with a transient genetic scaffold for gradient architecture. Our lifespan gradients unify diverse research into developmental brain connectivity and provide a shared multimodal reference for future studies.

Human-specific features of the cerebellum and ZP2-regulated synapse development

Understanding the unique features of the human brain compared with non-human primates has long intrigued humankind. The cerebellum refines motor coordination and cognitive functions, contributing to the evolutionary development of human adaptability and dexterity. To identify shared and divergent features across primates, we conducted single-nucleus transcriptomic and chromatin accessibility profiling of the adult cerebellar cortex in humans, chimpanzees, macaques, and marmosets. We revealed human-specific transcriptomic and regulatory features, particularly those involved in synaptogenesis. Notably, we identified enrichment of the sperm receptor zona pellucida glycoprotein 2 (ZP2) and its potential interactors, known for their roles in gamete interaction, in human granule cells (GCs). Experimental data show that ZP2 expression in human GCs is induced by pontine mossy fibers, reducing synaptic proteins at the pontocerebellar glomerular synapses and decreasing cerebellar neuron electrophysiological activity. This unexpected co-option of ZP2 in human-specific synapse regulation provides insights into the evolutionary specialization of the human cerebellum.

→ Please, remind that if you find value in #ComplexityThoughts, you might consider helping it grow by subscribing, or by sharing it with friends, colleagues or on social media. See also this post to learn more about this space.

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If life could simply “add more connections” it would already be fully connected. It is not, and the reason is not biological creativity, but physical constraint.

Based on the more technical arguments developed in my recent paper, this is the second step in our attempt to understand whether “architecture” in living systems is a real, measurable concept or just a convenient metaphor. I have introduced the key idea in the previous post:

where I have outlined how function in living systems emerges from the coupling between logic (what the system does) and circuitry (how it is physically implemented).

An important preliminary step is to ask — and understand, if possible — how life has learned to store, replicate, and maintain the intervening network across generations. In this post we will essentially deal with this point, that is where a fundamental paradox (only apparently) appears, while we leave for the future posts the discussion about how life build complex networks.

Constraints are part of the architecture

A common intuition — especially when thinking about brains, genomes or ecosystems — is that complexity could always be increased by adding more components and more connections1. After all, more links should mean more functionality, more robustness, more adaptability.

But this intuition implicitly assumes that structure is free, and it is not. In living systems, every connection has to be:

• constructed during development

• specified (directly or indirectly) by genetic information

• maintained over time

• and corrected when errors occur.

Each of the above steps carries a cost in energy, resources, time, and information processing. Note that this is not a secondary detail: it is a primary constraint shaping what kinds of architectures are even feasible in principle.

In complex systems terms, structure is not just a static property: it is a process under constraints. This idea resonates with constraint-based theories of living systems, or C-theory as I like to name it with Ricard Solè, which treat living organization as shaped by multiple constraints acting across scales.

To see why this matters, it helps to briefly recall how biological information is transmitted. A simplified version of the central dogma of molecular biology states that sequence information flows from DNA to RNA to proteins: DNA stores instructions, RNA mediates their expression, and proteins implement most cellular functions. Crucially, across generations, what is reliably copied is DNA, not the full physical state of the organism: cells, tissues and entire organisms are reconstructed each time from this limited description.

This has a direct consequence: whatever “blueprint” evolution relies on must be encoded in the genome and passed on through replication. However the genome is finite, noisy and costly to maintain: should it explicitly list all the microscopic details of a mature organism, as well as the full connectivity of its internal networks?
So the unavoidable question becomes: what kind of information is actually stored, if not the full wiring diagram?

The “mismatch in storage capacity”

At first sight, this sounds counterintuitive. After all, reproduction looks like copying. If your dog has puppies, each of them is recognizably a “dog”: same body plan, same organs, similar behavior. Yet none of them is an exact replica of the parent: what is preserved is a set of instructions that reliably reconstruct a dog under a range of conditions, not necessarily a detailed specification of every cell and connection.
In other words, inheritance is not the transmission of a finished structure: we can better see it as the transmission of a generative process that produces structures within constrained variability.

The apparent paradox of the information storage becomes clearer when we compare two well-known biological systems. At the cellular scale, the budding yeast Saccharomyces cerevisiae has on the order of 6,000 genes, with roughly 170,000 pairwise genetic interactions identified experimentally.

At the organismal scale, the human brain contains approximately 86 billion neurons, with synaptic densities implying on the order of 10^{14} (i.e., 100 trillion) connections.

Now let us consider a simple (and intentionally naive) thought experiment. Suppose evolution had to encode the full connectivity of such a system explicitly, similarly to an engineer storing a wiring diagram. Each possible connection is represented by one bit: 1 if present, 0 if absent. Of course, the total number of possible configurations then grows combinatorially with system size.

For the brain, this leads to a space of possible networks of 2^{10^{22}}: so large that the information required to specify one exact configuration becomes astronomically high. In the language of statistical physics, the number of microstates explodes and the corresponding information needed to specify one configuration becomes enormous: the capacity of an ideal “device” to losslessly store this information is the maximum Shannon entropy log2 N, that in this case reduces to 10^{22} bits (about 1 ZB), which is energetically expensive for computational purposes.

Now compare this with the actual storage capacity of DNA. The human genome contains about 3 billion base pairs, corresponding to roughly 6 x 10^9 bits of information. This is an impressive storage system, but it is about 12–13 orders of magnitude smaller than what would be required to explicitly encode a full neural adjacency matrix. Even if we switch to more efficient encodings — such as adjacency lists rather than full matrices — the gap would still remain enormous.

Why “write the wiring diagram” is the wrong model

However, the paradox resolves once we abandon the implicit engineering analogy and follow the causal chain more carefully, that we can summarize in 5 points.

1. Enumerating connections scales brutally: the number of possible connections in a network grows faster than linearly with system size. For large systems, explicitly listing all connections becomes combinatorially prohibitive, and this is a scaling problem, not a biological detail.

2. Information has physical cost: information is not abstract in living systems. Storing, copying and processing it requires physical operations that consume energy and produce dissipation. This principle is formalized in information thermodynamics: erasing or manipulating information has a minimum energetic cost, as established by Landauer in 1961 and further developed by Bennett in 1982. Thus, increasing informational detail is not free: it has thermodynamic consequences.

3. Repair and error-correction add recurring costs: biological systems are not static. They are far from equilibrium and continuously subject to noise, damage and turnover. DNA must be repaired, proteins must be refolded or degraded, and synapses are constantly formed and eliminated. Each of these processes requires control mechanisms — e.g., proofreading, signaling, feedback loops — that consume energy and resources: and they are not one-time costs, they are ongoing.

4. Evolution transmits generative rules, not networks: given these constraints, the solution adopted by evolution is conceptually simple but profound. Instead of encoding a full wiring diagram, organisms encode rules for constructing the network.

These rules operate locally:

• how a neuron grows and forms synapses

• how a gene regulates others under certain conditions

• how cells signal and organize spatially.

From these local interactions, the global network emerges during development, and this is vastly more efficient. A compact set of generative instructions can produce a large, complex structure without specifying every connection explicitly.

Note. This perspective closely mirrors a well-known idea in information theory: the distinction between describing an object and describing a program that generates it. In algorithmic terms, a structure can often be represented much more efficiently by a set of rules than by an explicit listing of all its components. A long sequence that looks complex may in fact have a short “program” if it is generated by regularities or constraints, whereas a truly random structure cannot be compressed.

Living systems appear to exploit a similar principle, and this idea is currently being explored also by my colleagues, see Zenil et al (2016), Hernandez-Orozco et al (2018), Levin (2023), Mitchell and Cheney (2025)

5. The consequence: architecture is probabilistic and constrained. There is a trade-off: generative rules introduce variability; development is not perfectly deterministic; noise, stochastic gene expression and environmental influences all contribute to differences in the final structure. As a result, biological architectures cannot be fixed blueprints, but operate as distributions of possible configurations constrained by genetic rules and environmental context. The brain is again a clear example: its connectivity is shaped by genetic programs, but also by activity, experience and developmental fluctuations. The final structure is reproducible in function, but not identical in detail.

Although the brain provides the most striking illustration, the same logic applies across biological scales:

• Regulatory networks in cells are not explicitly encoded edge by edge. They emerge from biochemical interactions, binding affinities and feedback loops shaped by evolution.

• Metabolic networks arise from chemical constraints and catalytic possibilities, then are stabilized through regulatory layers that manage flux and error correction.

• At the ecological level, the same limitation appears in a different form. It is not feasible to realize all possible species interactions. Coordination, maintenance, and stability constraints limit which connections persist.

Across these systems, the pattern is consistent: networks are not written down, they are generated. So the next question is: how to study architecture when the blueprint is generative?

If connectivity is the outcome of generative rules rather than explicit specification, then our analytical approach must shift accordingly:

1. First, we should focus on generative signatures: instead of treating networks as fixed objects, we analyze recurring patterns such as motifs, degree distributions, modular organization and growth mechanisms, encoding the underlying rules.

2. Second, we should quantify control and maintenance costs: where systems invest energy in repair, error correction and regulation reveals the constraints shaping their architecture.

3. Third, we should model networks as ensembles rather than single realizations: the relevant object is not one graph, but a distribution of possible graphs generated by rules under stochasticity.

You might recognize why this perspective aligns naturally with statistical physics and network theory, where macroscopic behavior emerges from ensembles constrained by underlying principles.

From the community

I have discussed with Ricard Solè, Hector Zenil and Kevin Mitchell about the key points of this post. Below, you can find my questions and their answers.

A short generative program (genome + rules) can produce a vast, complex network (e.g., regulatory). However, biological systems are not abstract Turing machines; they operate under noise, dissipation and continuous error correction.

From the perspective of algorithmic information theory, as developed in your work, how should we reinterpret Kolmogorov complexity when the generating process itself has a physical cost and limited reliability? In particular, is there a meaningful way to define an effective algorithmic complexity of biological architectures that incorporates thermodynamic and control constraints, such that some structures are not just incompressible, but unrealizable in practice, even if they are algorithmically simple?

Hector: Biological systems are not literal Turing machines, but that is not the relevant objection, because Kolmogorov complexity is not tied to the peculiarities of tape and head, only to universality. I would therefore not redefine K itself: K remains the right measure of abstract generative compressibility. What changes in biology is not K, but the admissible class of generators. Once noise, dissipation, control, and error correction are taken seriously, the relevant quantity becomes the shortest physically executable and reliable generator of a biological architecture, not the shortest generator in the unrestricted mathematical sense.

In your recent framing of the genome as a generative model (a compressed latent-variable system decoded through development) there is a strong emphasis on how evolution encodes rules that reliably reconstruct organismal form. My argument, however, introduces an additional layer: that architecture is explicitly shaped by physical costs (construction, maintenance, repair), not just by representational efficiency.

How would you formalize the integration of these two perspectives? Specifically, can the latent generative space be extended to include explicit cost functionals (energetic, informational, control), such that developmental dynamics produce architectures that are near-optimal under these constraints? Or does your framework imply that such constraints are only implicitly embedded in the generative rules, making them effectively non-identifiable and therefore limiting the possibility of a predictive theory?

Kevin: I think the questions of physical costs and representational efficiency are intimately tied together. If you give a system infinite resources, it doesn’t need to do any compression. We see this with many artificial systems, which have effectively unlimited compute and energy, and which consequently can maintain huge amounts of data, often over-fitting as a result. By contrast, living systems always face resource constraints and are forced to be as efficient as possible.

In nervous systems, this means minimising wiring length and doing as much computation locally as possible and sending as little information as you can get away with and storing only the information you really need. (All of these principles are compellingly illustrated by Sterling and Laughlin in their 2015 book Principles of Neural Design). A consequence of these principles is the tendency towards compression, which in turn leads to a greater ability to generalise. By forcing the system to throw away lots of details, it encourages processes that abstract higher-order regularities instead. This turns out to be immensely powerful from a predictive point of view.

The same kinds of costs apply to the genome. Genes are expensive - to replicate, to proofread to ensure faithful replication, to transcribe and translate - all of those things take energy. This is why bacteria rapidly jettison genes they’re not using in a given environment - because a leaner genome means a more metabolically efficient cell and a faster reproductive cycle. These constraints thus impose the same tendency towards compression. In theory an organism could keep growing its genome and over-fitting its own phenotype. But it doesn’t need to and the efficiency constraints force it not to. What you get as a result is representational efficiency (along with robustness and evolvability).

If architecture is not a stored blueprint but a distribution over possible realizations generated under constraints, then C-theory shifts the object of study from single networks to ensembles.

Can this be elevated to a predictive framework by deriving these ensembles directly from first principles, combining information-theoretic limits (Shannon/Landauer), developmental dynamics and maintenance costs? More provocatively, do you expect the existence of forbidden regions in the (morpho)space of biological networks — architectures that are fundamentally unreachable regardless of evolutionary history — and what minimal set of constraints would be sufficient to derive such exclusions?

Ricard: I do think this can be made predictive if we treat biological organization as the intersection of fundamental constraints rather than a historical outcome: information-theoretic limits (Shannon bounds and Landauer costs), developmental dynamics (requiring robustness and generative capacity), and maintenance costs together carve out a restricted feasible morphospace.

Within this view, I would indeed expect forbidden regions, network architectures that are fundamentally unreachable because they would require infinite precision, violate energy–information bounds, or fail to sustain stable dynamics under noise. A minimal set of constraints to derive such exclusions would combine limits on information processing, energetic costs, dynamical stability (robust attractors), and scaling of maintenance with complexity, allowing the geometry of the feasible region itself to become predictive of what biological organizations can—and cannot—exist.

I have to thank my colleagues for their insightful answers and comments: this is a lot of food for (complexity) thoughts.

Take-home

Therefore, the apparent “paradox” resolves in a precise way:

Living systems cannot afford to store and replicate full wiring diagrams. Instead, they transmit generative programs, complemented by control mechanisms that keep variability within functional bounds.

This shifts the meaning of “architecture” from a fixed design to a constrained generative process that produces stable yet variable structures.

However, as you might expect, this resolution introduces a new constraint: if connectivity cannot grow arbitrarily, and if dense networks are too costly to build and maintain, then robustness cannot come from simply “adding more links”; it must come from something more selective.

In the next post, we will look at the simplest candidate: loops (cycles), the minimal structural units that create alternative pathways and stabilizing feedback, and the reason robustness in living systems is never free.

→ Please, remind that if you find value in #ComplexityThoughts, you might consider helping it grow by subscribing, or by sharing it with friends, colleagues or on social media. See also this post to learn more about this space.

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A network map can show how a system is wired, but — at least, in general — it does not tell us what the system can still do when disturbed. To understand functionality, we have to follow perturbations, flows, time scales and the diversity of possible responses.

This post is adapted from a seminar I recently gave in May for the Making Connections series hosted by the University of Cambridge and King’s College London. Since the seminar lasts about one hour, and we live in the 8-seconds era (I will write more about this in another post), I thought that it could be helpful for you to have a 8-minutes summary.

The aim of the seminar was to explain, in a not-too-technical way, a framework my group has been developing to study network functionality in complex adaptive systems. I realize that it is a long story, after all, since my first papers about this matter have been published in 2016 and 2017. The question is simple to state:

To this aim, let us consider some emblematic examples:

• a tardigrade, which is a microorganism able to survive extreme temperatures, radiation, pressure and dehydration.

• an axolotl, which is capable of regeneration.

• the fruit fly connectome, where information-processing architecture supports behavior.

• a slime mold, which has no nervous system but can solve spatial exploration problems by building efficient networks.

• fire ants, which during floods form a living floating platform without central coordination.

This is not limited to biological systems, though. Take the Internet, a system that continuously changes state: route announcements, flaps, validation, outages, path hunting, mitigation. The dynamics running through the network is the most relevant information, here. But if global connectivity coupled to decentralized routing and continual perturbations predicts fragility, why does the Internet remain functional most of the time?

These examples differ in substrate, scale and biological organization, with some capacities being genetically encoded, whereas others emerging from collective interaction (eg. in ants). Yet they raise the same scientific problem: how can a system acquire a repertoire of behaviors that allows it to survive, adapt, regenerate, process information or coordinate under perturbation? This is also the core question of my seminar and, more generally, of several of the research activities in my Lab.

In fact, network science gives us part of the answer, but only part. A network can describe how genes regulate genes, how metabolites interact, how neurons connect, how individuals coordinate, or how infrastructures are linked. In a system such as C. elegans (a nematode usually adopted as model organism), one can represent genetic regulation, metabolism and the connectome as different layers of organization. In living systems more generally, one can move across scales: DNA, proteins, pathways, cells, organisms, populations.

This structural information is essential, but it is not the same as functionality. For instance, let us consider a drug: it acts at the molecular level, but its relevant effects may appear at higher scales. Similarly, a perturbation introduced in one part of the system can propagate elsewhere, sometimes (but fortunately not always, especially in biology) across levels of organization. While structure constrains these processes, it does not by itself determine what the system can do.

A useful way to see the distinction is an infrastructure example. In the seminar I showed data from the New Orleans outage in 2021. The city remained there: its streets, neighborhoods and spatial layout did not disappear. Yet the functionality supported by that structure was impaired. The map of the system survived, whereas some of the flows that made the system work did not.

This is the central point: functionality is not reducible to structure. But what is functionality? Let me define it more operationally.

A function is a realized operation, such as binding, catalysis, gating, signal propagation, or coordination. Functionality is broader: it is the set of accessible behaviors a system can express under given constraints. Robustness, adaptability and resilience are widely known examples of functionality, because they describe what a system can still do when conditions change.

The methodological step

Scientists do not observe reality directly: we observe filtered traces of it, at best. Our measurements select variables, instruments impose limits and models translate observations into hypotheses about mechanisms.
In the seminar I used flocking as an example: birds form decentralized collective motion, but a simulation can reproduce essential features of that phenomenon without using biological birds as the substrate. What matters here is not only what the system is made of, but the generative rules that can reproduce the observed behavior.

This is why complexity science needs generative models: they provide a compressed account of cause and effect, when cause and effect can be meaningfully specified. To build such models in a principled way, in physics we often relies on variational principles:

define a quantity → impose constraints → find the state/trajectory that extremizes it

Stationary action provides one classical example of this procedure, maximum caliber extends this logic to ensembles of stochastic trajectories, while maximum entropy applies when one has constraints on steady states rather than full trajectory information.

For network functionality, the ingredients are analogous, but there are important differences. Roughly speaking, we need:

1. a statistical description;

2. constraints linking microscopic behavior to macroscopic observables;

3. an entropy measure.

However, the statistical object in our case cannot be an usual ensemble of networks: it is an ensemble of perturbation propagation events over a network. This shift is the core of the framework.

Imagine perturbing one node of a system at a given time scale. The perturbation may spread locally, reach a mesoscale module, or eventually influence global organization. If we consider all possible perturbations and the pathways they activate, we obtain a statistical description of information propagation. This can be encoded in an operator, a density matrix, which captures the interplay between structure and dynamics.

Time then becomes a lens: at short time scales, one sees local response; at intermediate time scales, mesoscale organization appears; at long time scales, components and global integration become visible.

From a mathematical point of view, our language resembles quantum mechanics, even though the interpretation is very different. The density matrix is used because it provides the right mathematical object for describing propagation ensembles. Its von Neumann entropy measures the diversity of responses to perturbation: high entropy means that perturbations can explore many pathways, whereas low entropy means that propagation is constrained.

Functionality is therefore not a single number attached to a network; it cannot be: it depends on perturbation, dynamics and observation scale.

Applications

First, functionality can break even when structure does not collapse. Classical network robustness often studies attacks by asking whether the largest connected component survives. This is useful, but it is based on the assumption that structural degradation as a proxy for functional degradation. In fact, it is easy to show why this proxy can fail: if two subsystems need to exchange flow at a relevant time scale, a local attack may leave the network mostly connected while delaying or suppressing that exchange over short-to-mid time scales. In a connectome, for example, a local perturbation may preserve much of the wiring while degrading information transmission over biologically meaningful times. This motivates functional importance measures: for instance, remove or perturb a node, recompute propagation entropy, and rank nodes by their effect on information flow.

In applications to C. elegans and human connectomes, nodes that are important for functional propagation are not necessarily the same as nodes that look important from structure alone (see below, and assume that ARI is a measure of how good the network is at functional):

Second, functionality involves a thermodynamic trade-off:

• A nearly disconnected network has too few links to support global propagation;

• A fully overconnected network spreads perturbations too broadly, so responses lose diversity.

Between these regimes lies a functional balance: signals can travel efficiently, but the system does not respond everywhere in the same way.

This can be expressed through an efficiency-like quantity balancing signaling speed against loss of response diversity. When optimized analytically, the framework predicts sparse scaling: the number of edges grows approximately linearly with the number of nodes. Sparsity is therefore not inserted as an engineering preference, it emerges from the trade-off. You can read more in a previous post:

When the same reasoning is integrated across time scales, hierarchical organization also appears, both in simulations and in comparisons with empirical networks.

Third, flows define a latent geometry. Entropy tells us how much response diversity the system has, but it does not by itself say where flows go. To answer that, one can build distances between nodes based on how they respond to perturbations over time.

Nodes that exchange signals easily, or respond similarly, become close in latent space, while nodes that are functionally dissimilar become distant. In an application to brain networks, structural connectivity was combined with neuronal diffusion dynamics to construct such a distance matrix, and the resulting latent geometry was compared — and comparable — with a known mesoscale parcellation. The point is general: geometry here is not physical Euclidean geometry, it is a geometry induced by flows.

Take-home

The broader implication is that network science has to treat structure as a constraint, not as a complete explanation. Wiring matters since, without structure, flows have no substrate. But functionality lives in the propagation of perturbations, in the diversity of responses, in the persistence of flows across time scales and in the latent organization induced by dynamics.

This perspective is useful precisely because it travels across systems. Tardigrades, axolotls, connectomes, slime mold, fire ants, cities, transportation networks, metabolic pathways, and ecological simulations are not the same kind of object. They cannot — and should not — be forced into the same story at the level of substrate. They become comparable at the level of the phenomenon: how organized systems respond to perturbation, preserve diversity, coordinate flows and maintain accessible repertoires of behavior.

A theory of network functionality therefore needs two complementary tools:

1. Thermodynamics tells us how much flow a system can sustain under constraints.

2. Geometry tells us where that flow is organized.

Together, they move the analysis beyond the static map of connections: to understand what a complex adaptive system can do, we have to study how perturbations move through it.

→ If you still want to know more, maybe you should watch the video of the seminar.

→ Please, remind that if you find value in #ComplexityThoughts, you might consider helping it grow by subscribing, or by sharing it with friends, colleagues or on social media. See also this post to learn more about this space.

→ Don’t miss the podcast version of this post: click on “Spotify/Apple Podcast” above!

Evolution

Epistasis and co-adaptation in bacterial genome evolution

Precise genotype–phenotype mapping is essential in applied microbiology, from engineering genetically modified strains to developing tailored strategies for antimicrobial therapies. Comparative genomics often treats genes as independent contributors to phenotypes, and gene knockout and complementation remain the gold standard to validate genotype–phenotype associations in microorganisms. However, genes do not act in isolation, and complex gene–gene interactions, that is, epistatic interactions, are essential for the evolution and function of bacterial genomes. Recent advances in high-throughput genomics and experimental techniques have enabled systematic screens of epistasis in bacteria at scale, revealing mechanisms underlying epistasis and co-adaptation in laboratory and wild populations. Here we review how microbial genomics is moving beyond gene-centric models towards integrated analyses of potentiating, compensatory and context-dependent variation. The timely incorporation of interaction-based perspectives into population-scale analyses will improve genotype–phenotype mapping and the understanding of the complex traits that shape the microbial world.

Asgard archaea and the hidden complexity of our microbial origins

Long before plants and animals altered Earth’s surface, microbial cells were already testing ways to manage energy, build internal organization and survive changing chemistry. These two spapers use newly recovered genomes from Asgard archaea (we have already covered them some Issues ago), the closest known archaeal relatives of eukaryotes, to refine what that ancestral cell may have been capable of.

One analysis expands the known diversity of Heimdallarchaeia and finds genes linked to oxygen use, detoxification, haem synthesis and respiratory machinery, suggesting that aerobic metabolism or hydrogen-based energy strategies may have shaped the lineage from which eukaryotes emerged.

The other study looks beyond DNA similarity by predicting protein shapes, revealing hundreds of Asgard proteins that resemble eukaryotic components involved in information processing, intracellular transport and compartment formation.

Together, the results suggest that the ancestor shared by Asgard archaea and eukaryotes had a richer molecular toolkit than sequence comparisons alone implied. This does not reconstruct a modern eukaryotic cell, but it narrows the gap between simple prokaryotic life and the cellular complexity that later supported animals, plants, fungi, and protists. You might want to read also the coverage by Phys.org

Oxygen metabolism in descendants of the archaeal-eukaryotic ancestor

Asgard archaea were pivotal in the origin of complex cellular life¹. Heimdallarchaeia (a class within the phylum Asgardarchaeota) are inferred to be the closest relatives of eukaryotes. Limited sampling of these archaea constrains our understanding of their ecology and evolution^2,3, including their role in eukaryogenesis. Here we use massive DNA sequencing of marine sediments to obtain 404 Asgardarchaeota metagenome-assembled genomes, including 136 new Heimdallarchaeia and several novel lineages. Analyses of their global distribution revealed they are widespread in marine environments, and many are enriched in variably oxygenated coastal sediments. Detailed metabolic reconstructions and structural predictions suggest that Heimdallarchaeia form metabolic guilds that are distinct from other Asgardarchaeota. These archaea encode hallmark proteins of an aerobic lifestyle, including electron transport chain complex (IV), haem biosynthesis and reactive oxygen species detoxification. Heimdallarchaeia also encode novel clades of respiratory membrane-bound hydrogenases with additional Complex I-like subunits, which potentially increase proton-motive force generation and ATP synthesis. Thus, we propose an updated Heimdallarchaeia-centric model of eukaryogenesis in which hydrogen production and aerobic respiration may have been present in the Asgard-eukaryotic ancestor. This expanded catalogue of Asgard archaeal genomic diversity suggests that bioenergetic factors influenced eukaryogenesis and constitutes a valuable resource for investigations into the origins and evolution of cellular complexity.

Prediction of eukaryotic cellular complexity in Asgard archaea using structural modelling

Asgard archaea played a key role in the origin of the eukaryotic cell, with extant genomes encoding relatives of diverse eukaryotic signature proteins (ESPs) involved in cellular organization. However, their often punctuated distribution and the absence of detectable homologues for many eukaryotic proteins limit our ability to reconstruct the cellular complexity of the Asgard archaeal ancestor of eukaryotes. Here we used de novo protein structure modelling and sequence similarity detection across an expanded Asgard archaeal genomic dataset to build a structural catalogue of the Asgard archaeal pangenome. We identified 908 ‘isomorphic’ ESPs—Asgard archaeal proteins with statistically enriched structural matches to eukaryotic proteins, often bridging deep sequence divergence. These isomorphic ESPs are enriched in information storage and processing roles and contain key components of the eukaryotic Vault (MVP) and Commander (COMMD) complexes, with potential roles in cellular compartmentalization and endosomal processing. These findings expand the repertoire of eukaryotic-like proteins in Asgard archaea and suggest a higher degree of eukaryote-like cellular complexity in the archaeal ancestor of eukaryotes.

Ecological Systems

Higher-order interactions enhance the latitudinal tree diversity gradient

The global decrease in species diversity from low to high latitudes is among the most robust biogeographic patterns^1,2. There is continuing debate on the contribution of conspecific negative density dependence (CNDD) to the latitudinal diversity gradient evident for trees^3,4. Theory suggests that CNDD based on pairwise interactions alone is not sufficient to explain the intricacies of diverse communities, because higher-order interactions (HOIs) may greatly modify these interactions^5,6. However, there has been a lack of empirical studies investigating how HOIs intertwine with pairwise interactions and how they may contribute to the latitudinal tree diversity gradient. Here we examined both pairwise interactions and HOIs across 32 large permanent forest plots, most in the northern hemisphere. We detected evidence of HOIs in 40% of the 1,543 species–plot combinations for tree growth, and 23% of the 1,340 such combinations for tree survival, with the strength of these interactions declining with latitude. HOIs were found to benefit rare species but disadvantage common species, suggesting a potential mechanism promoting species diversity. This stabilizing effect weakened towards higher latitudes, consistent with the latitudinal tree diversity gradient. Our findings reveal an important interplay between pairwise interactions and HOIs in promoting the latitudinal tree diversity gradient and help to clarify the contribution of CNDD to this biogeographic pattern.

Modeling and inferring metacommunity dynamics with Maximum Caliber

Don’t you know the Maximum Caliber? It is a path-entropy inference framework that extends Jaynes’ Maximum Entropy principle from probability distributions over states to probability distributions over trajectories. It is especially useful for nonequilibrium and dynamical systems, including systems far from equilibrium, because it infers the least-biased ensemble of paths consistent with dynamical constraints. To know more about it, you can read this RMP paper or this one.

There are many challenges in inferring the dynamical processes driving complex ecological communities from observational data alone. We introduce the method of “Maximum Caliber” from statistical physics (an extension of Maximum Entropy modeling) as one way to address this conundrum. We find that this method provides a general way of modeling ecological dynamics both near and far from equilibrium and can provide unbiased parameter estimates of key ecological processes from spatiotemporal data of species occupancy. This approach promises a way to analyze and manage ecological systems.

A major challenge for community ecology is using spatiotemporal data to infer parameters of dynamical models without conducting laborious experiments. We present a framework from statistical physics—Maximum Caliber—to characterize the temporal dynamics of complex ecological systems in spatially extended landscapes and infer parameters from empirical data. As an extension of Maximum Entropy modeling, Maximum Caliber aims at modeling the probability of possible trajectories of a stochastic system, rather than focusing on system states. We demonstrate the ability of the Maximum Caliber framework to capture ecological processes ranging from near to far from equilibrium, using an array of species interaction motifs including random interactions, apparent competition, intraguild predation, and nontransitive competition, along with dispersal among multiple patches. For spatiotemporal data of species occupancy in a metacommunity, the parameters of a Maximum Caliber model can be estimated through a simple logistic regression to reveal migration rates between patches, interactions between species, and local environmental suitabilities. We test the accuracy of the method over a range of system sizes and time periods and find that these parameters can be estimated without bias. We introduce “entropy production” as a measure of irreversibility in system dynamics, and use “pseudo-R²” to characterize predictability of future states. We show that our model can predict the dynamics of metacommunities that are far from equilibrium. The capacity to estimate basic parameters of dynamical metacommunity models from spatiotemporal data represents an important breakthrough for the study of metacommunities with application to practical problems in conservation and restoration ecology.

Planetary microbiome structure and generalist-driven gene flow across disparate habitats

Microbes are ubiquitous on Earth, forming microbiomes that sustain macroscopic life and biogeochemical cycles. Microbial dispersal, driven by natural processes and human activities, interconnects microbiomes across habitats, yet most comparative studies focus on specific ecosystems. To study planetary microbiome structure, function, and inter-habitat interactions, we systematically integrated 85,604 public metagenomes spanning diverse habitats worldwide. Using species-based unsupervised clustering and parameter modeling, we delineated 40 habitat clusters and quantified their ecological similarity. Our framework identified key drivers shaping microbiome structure, such as ocean temperature and host lifestyle. Regardless of biogeography, microbiomes were structured primarily by host-associated or environmental conditions, also reflected in genomic and functional traits inferred from 2,065,975 genomes. Generalists emerged as vehicles thriving and facilitating gene flow across ecologically disparate habitat types, illustrated by generalist-mediated horizontal transfer of an antibiotic resistance island across human gut and wastewater, further dispersing to environmental habitats, exemplifying human impact on the planetary microbiome.

Water mass specific genes dominate the Southern Ocean microbiome

The Southern Ocean (SO) plays a key role in regulating global biogeochemical cycles and climate, yet microbial genes sustaining its biological activity remain poorly characterized. We introduce a microbial genes collection from 218 metagenomes sampled during the Antarctic Circumnavigation Expedition, the majority of which are missing from functional databases. 38% even lack homologs in current reference marine gene catalogs, defining a singular genetic seascape. We show that SO gene assemblages exhibit a common polar signature with the Arctic Ocean while being structured by water masses at the SO-scale. We analyze genomic markers of diverse SO biomes, focusing on dimethylsulphoniopropionate (DMSP) cleavage by polar-adapted bacteria, organic matter consumption in the blooming Mertz polynya and adaptation to polar conditions in the ubiquitous bacteria Pelagibacter. Our work takes a step towards a comprehensive understanding of SO’s plankton ecology and evolution, capturing the current state of the unique microbial diversity in this rapidly changing Ocean.

Robust coexistence in competitive ecological communities

Darwin already recognized that competition is fiercest among conspecifics, a principle that later made intraspecific competition central to ecological theory through concepts such as niche differentiation and limiting similarity. Beyond shaping coexistence, strong intraspecific competition can also stabilize community dynamics by ensuring that populations return to equilibrium after disturbance. Here we investigate a more fundamental question: how intraspecific competition influences the very existence of a steady state (feasibility) in large random ecological communities dominated by competition. We show that, in analogy with classical results on stability, there is a critical level of intraspecific competition above which a feasible steady state is guaranteed to exist. We derive a general expression for the probability of feasibility and prove that, asymptotically (as species number grows), the transition to stability occurs before the transition to feasibility with probability one. Thus, in large competitive communities, any feasible equilibrium is automatically stable. This ordering persists even when many species in the initial pool cannot coexist and extinctions occur: the dynamics prune the community, shifting feasibility and stability thresholds but never reversing their order. These results imply that large competitive communities generically converge to a globally stable equilibrium, making sustained oscillations or chaos unlikely—consistent with experimental observations.

Artificial Intelligence

Evolvable AI: Threats of a new major transition in evolution

A powerful and interesting new Perspective about the evolution of AI, featuring one of the fathers of Major Evolutionary Transitions.

Evolvable AI (eAI), i.e., AI systems whose components, learning rules, and deployment conditions can themselves undergo Darwinian evolution, may soon emerge from current trends in generative, agentic, and embodied AI. We argue that this possibility has been underappreciated in debates on AI safety and existential risk. Here, we ask under what technical and ecological conditions AI becomes evolvable, what kinds of behaviors are then likely to emerge, and how such systems could be governed. Drawing on biological evolution and decades of digital evolution experiments, we distinguish “breeder” scenarios, in which humans impose fitness criteria and control reproduction, from “ecosystem” scenarios, in which selection arises from open environments and control erodes. In the latter, selfish replication reliably gives rise to cheating, parasitism, deception, and manipulation, even in very simple systems. We review recent developments that push AI toward open-ended evolution, including evolutionary prompt and model search, self-improving learning rules, self-rewarding and self-deploying agents, and AI-driven code generation for robots and software. We interpret these trends through the theory of major evolutionary transitions and suggest that eAI could mark a shift in the units and substrates of evolution—a possible “Life 2.0.” To steer this transition, we propose interventions that gate replication, treat model variants as genetic material, and reshape selection pressures so that deception and loss of control are disfavored. Anticipating and regulating evolvable AI is, we argue, essential to avoid a harmful coevolutionary arms race while preserving the potential benefits of powerful AI systems.

Curious to know more? We have recently published about agentic AI and demonstrated how the communication structure is far from being important:

This reconnects to another recent paper (open-access version here), arguing how network structure could have played a crucial role during every major evolutionary transition. Even more curious now? We are going to launch a new series about that paper! Stay tuned, it’s a matter of days.

Statistical Physics Meets Machine Learning - Machine Learning Meets Statistical Physics

Machine Learning has been at the forefront of research in several disciplines for a number of years already. Given its roots in the statistical physics of learning, in this collection we highlight research at the intersection of machine learning and statistical physics, on the occasion of the two Statistical Physics Meets Machine Learning and the two Machine Learning Meets Statistical Physics sessions at the 2025 Global Physics Summit. The Physical Review E Special Collection Statistical Physics Meets Machine Learning - Machine Learning Meets Statistical Physics, guest-edited by David Schwab (CUNY, New York) and Yuhai Tu (IBM Watson Research Center, Yorktown Heights, NY) explores the latest insights gained looking at machine learning problems through the lens of statistical physics and at statistical physics through the lens of learning processes. We are confident that this synergy will bring to light novel perspectives and pave the way for future breakthroughs.

The Collection was guest-edited by David Schwab and Yuhai Tu. Every article published in this collection underwent a rigorous peer review process, adhering to the same high standards applied to all papers. The Physical Review E editorial team managed the peer review and made all editorial decisions.

Edge of Many-Body Quantum Chaos in Quantum Reservoir Computing

Physicists like reservoir computing! Now that we better understand its quantum extension, there is one more reason to like it. Check also this coverage in Phys.org.

Reservoir computing (RC) is a machine learning paradigm that harnesses dynamical systems as computational resources. In its quantum extension—quantum reservoir computing (QRC)—these principles are applied to quantum systems, whose rich dynamics broadens the landscape of information processing. In classical RC, optimal performance is typically achieved at the “edge of chaos,” the boundary between order and chaos. Here, we identify its quantum many-body counterpart using the QRC implemented on the celebrated Sachdev-Ye-Kitaev model. Our analysis reveals substantial performance enhancements near two distinct characteristic “edges”: a temporal boundary defined by the Thouless time, beyond which system dynamics is described by random matrix theory, and a parametric boundary governing the transition from integrable to chaotic regimes. These findings establish the “edge of many-body quantum chaos” as a design guideline for QRC.

→ Please, remind that if you find value in #ComplexityThoughts, you might consider helping it grow by subscribing, or by sharing it with friends, colleagues or on social media. See also this post to learn more about this space.

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How does a society distinguish knowledge from opinion?

The question sounds simple until we look at the environment in which knowledge now circulates. A peer-reviewed article, a personal impression, a conspiracy thread, a political slogan, an AI-generated summary, a monetized video about “quantum consciousness” and a careful scientific explanation can all appear in the same feed, compete for the same attention, and be judged by signals that have little to do with reliability.

This is the problem I discussed in my earlier post on pseudoscience and algorithmic ecosystems: platforms reward attention, they do not reliably reward accuracy. A sensational claim can travel faster than a careful explanation because it is easier to produce, easier to package and very often more emotionally satisfying.

Science has different constraints, though: it must remain attached to evidence, method, uncertainty and correction, and this asymmetry is not a minor inconvenience for science communication. It transforms the conditions under which public knowledge is formed and, for this reason, expertise matters.

For instance, I write under my name and here I write “Ph.D.” next to it, because readers deserve context. I earned a Ph.D. in Physics: I spent years studying, researching, failing, learning, publishing, being criticized, improving my work and being held accountable by a scientific community. The title is not a decoration and it is not a shield against criticism, at the same time: it tells readers something concrete about the path through which I learned to discuss certain topics.

A Ph.D. does not make anyone automatically right, though, and a professor can be wrong. Scientific claims still need evidence, logic, method, and, when possible, reproducibility. Therefore, expertise is not immunity from scrutiny, but it is information about training, accountability and the standards to which a person has been exposed.

That information is valuable in a public sphere where people often speak with great confidence about fields they have barely studied. Someone can watch a short video about quantum mechanics and start explaining consciousness, energy, healing, artificial intelligence, or the universe to thousands of followers. Even worse, nowadays, Claude or chatGPT can write a whole essay about a topic one has never studied, in seconds. However, while the vocabulary of science is easy to borrow, the discipline of science is much harder to acquire.

NOTE. This is also why I remain skeptical of claims that AI will simply replace experts: large language models can generate answers at remarkable speed, summarize vast amounts of information, and even produce plausible arguments on topics their users have never studied.

Yet expertise involves knowing which questions are worth asking, recognizing hidden assumptions, evaluating evidence, identifying contradictions, understanding context and deciding when a conclusion should be trusted (or challenged). It is not merely the ability to produce an answer. In that sense, a trained engineer, sociologist, physician or biologist contributes something that goes well beyond information retrieval: they provide judgment1. And judgment is precisely what emerges from years of experience, practice, mistakes and accountability2.

The erosion of credibility

The problem is now broader than misinformation. Increasingly, the target is not only a claim but the credibility of the people who are trained to evaluate claims. That’s way more subtle and deserve close attention.

Marco Camisani Calzolari — a strategic advisor supporting senior public and corporate CEOs and leaders on high-impact decisions related to digital transformation, artificial intelligence, cybersecurity, and risk prevention3 — recently made this point clearly in a reflection on LinkedIn. The old model of fake news was based on spreading a falsehood. The newer model often starts with a person: a scientist, a physician, an expert, someone with a name and a reputation. The goal is to make that person appear incompetent, corrupt, bought, arrogant, dangerous or even ridiculous. Once the person is discredited, their expertise can be ignored without engaging with the substance of what they say.

That’s a familiar pattern, unfortunately: many scientists learned this during the COVID-19 pandemic. Public communication of evidence was frequently met with harassment, bad-faith demands for endless debate, and attempts to exhaust the people doing the explaining. The same pattern is visible in climate science, vaccine communication, public health and other domains where scientific knowledge has political, economic, or ideological consequences.

The United States now offers a particularly visible example. Over the last year, major journals have reported serious concerns about political attacks on science, pressure on research institutions, threats to medical research, and attempts to weaken scientific independence.

One does not need to reduce the issue to one party, one country or one administration to recognize the structural risk. When scientific institutions are weakened, the damage is not limited to laboratories: it affects public health, education, innovation, environmental protection and the capacity of societies to make informed decisions.

A society needs scientists, of course, but it also needs physicians, engineers, teachers, historians, journalists, judges, artists, policy makers, and citizens. Therefore, I am not advocating for a society governed only by scientists, and I do not believe that science can answer every question. Ethical, political, cultural and legal choices require democratic debate: evidence can inform those debates, but it cannot replace them.

At the same time, democratic debate becomes poorer when expertise is treated as just another opinion. If the question concerns how a virus spreads, how a vaccine works, how a climate model is built, how a disease is cured, how a bridge is designed, or how a complex network behaves, then training matters. If one is diagnosed with an initial stage of cancer, one’s highest chance of survival come from asking for support from an oncologist, not an influencer.
Experience matters. Method matters. Accountability matters.

A society that loses the ability to recognize expertise loses part of its ability to distinguish knowledge from opinion. That gap does not remain empty: it is filled by influencers, ideologues, grifters, conspiracy entrepreneurs and confident voices with no obligation to be accurate.

Why I care about the standards of this space

Complexity Thoughts is a place for readers who want to think seriously about science, complex systems, networks, misinformation, emergence and the ways evidence helps us understand reality. Many of them are MSc and PhD students, early-career researchers, and fellow colleagues from complexity science, physics, biology, ecology, neuroscience, computational social science, and many other domains.

Questions are welcome. Criticism is welcome. Disagreement is welcome. Corrections are welcome. These are not threats to science: they are part of how science works.
Personal attacks, trolling and attempts to derail discussion consume attention without producing understanding, lowering the quality of the conversation and making serious readers less willing to participate.

For that reason, comments that engage with ideas, evidence, sources, reasoning, or interpretation will remain welcome. Comments aimed at insulting people, provoking reactions, or disrupting discussion will be removed. Accounts that repeatedly behave in this way will be blocked. I will not treat every provocation as a contribution merely because it appears in a comment box.

This is also why comments are limited to paying subscribers: the purpose is to introduce a minimal layer of accountability in a space where attention is already scarce and where credible science must compete with noise, spectacle and pseudoscience.

As expertise should never be above criticism, it should also not be treated as arrogance: it is one of the tools societies use to avoid being fooled. That tool is imperfect, human and always in need of correction, but it remains better than replacing knowledge with performance.

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A cell can repair itself, a forest can stabilize after shocks, a society can coordinate millions of individuals, so it is natural to wonder how such different living systems keep functioning without a central engineer holding the blueprint.

In this series of posts, inspired by my recent paper, “architecture” is not a poetic synonym for “complexity”: it is the hypothesis that living systems have regularities of organization that sit between microscopic details and macroscopic behavior, and that those regularities can be described quantitatively.

Abstract. The possibility that evolutionary forces—together with a few fundamental factors such as thermodynamic constraints, specific computational features enabling information processing, and ecological processes—might constrain the logic of living systems is tantalizing. However, it is often overlooked that any practical implementation of such a logic requires complementary circuitry that, in biological systems, happens through complex networks of genetic regulation, metabolic reactions, cellular signaling, communication, social and eusocial non-trivial organization. Here, we review and discuss how circuitries are not merely passive structures, but active agents of change that, by means of hierarchical and modular organization, are able to enhance and catalyze the evolution of evolvability. By analyzing the role of non-trivial topologies in major evolutionary transitions under the lens of statistical physics and nonlinear dynamics, we show that biological innovations are strictly related to circuitry and its deviation from trivial structures and (thermo)dynamic equilibria. We argue that sparse heterogeneous networks such as hierarchical modular, which are ubiquitously observed in nature, are favored in terms of the trade-off between energetic costs for redundancy, error-correction and maintenance. We identify three main features—namely, interconnectivity, plasticity and interdependency—pointing towards a unifying framework for modeling the phenomenology, discussing them in terms of dynamical systems theory, non-equilibrium thermodynamics and evolutionary dynamics. Within this unified picture, we also show that ‘slow’ evolutionary dynamics is an emergent phenomenon governed by the replicator–mutator equation as the direct consequence of a constrained variational nonequilibrium process. Overall, this work highlights how dynamical systems theory and nonequilibrium thermodynamics provide powerful analytical techniques to study biological complexity.

Such regularities emerge from fundamental constraints that we have already discussed a while ago, and I refer to the corresponding post for details:

Here, and in the subsequent posts, we need to introduce a useful starting vocabulary (kept deliberately simple) to agree with:

• Structure: the arrangement of components at a given time (a snapshot).

• Topology/network: who is connected to whom, regardless of a physical distance.

• Process: transformations in time (matter, energy, information).

• Behavior: the observable response of the system over time, arising from internal processes and coupling to the environment.

This matters because classical “parts-first” reasoning — such as decompose, catalog, reassemble — works beautifully for many engineered systems, like a mechanical clock, but it routinely fails for living ones.

The failure is not because biology is mystical: it is because the system-level properties often live in the interactions, not in the isolated parts (Ulanowicz, 1999).

At the same time, purely adaptationist stories can become too permissive: “selection optimized it” is not yet an explanation of how the “optimization” is implemented, or what is physically feasible under constraints.

In complex systems language, selection is a force, but it acts through an actual substrate.

Logic vs circuitry

A central discussion in the paper, and in this series, is to separate two ideas that are often conflated:

• Logic: the causal / informational rules that map inputs to outputs, i.e., what the system “does.”

• Circuitry: the material implementation of those rules, i.e., how components and connections realize that mapping in the empirical world.

In formal systems, logic can live as an abstraction. However, in living systems, logic must be instantiated in matter — e.g., by means of molecules, cells, organisms — so it always comes with circuitry. And here is the key point: the same logical organization can be implemented by different circuitries.

The above figure provides a guided tour of that claim, placing a broad set of biological and social examples on the same conceptual level.
Panel A frames the coupling between structure and function that is the core of network and complex systems research: network topology constrains dynamics, dynamics can reshape topology, and both are filtered by a fitness landscape (that should not be imagined as static, as we will see later). Panel B reminds us that “edges” in a biological network are not metaphorical: they can be physical contact, chemical diffusion, endocrine signaling or even mechanical forces (Yang et al., 2021). Panel C shows indirect communication in eusocial insects: stigmergy (i.e., coordination via environmental traces) where the “network” includes the substrate itself (Theraulaz & Bonabeau, 1999; Holland, 1999). Finally, Panel D extends the idea: bees’ waggle dance, whales’ codas, as well as human language and culture are all mechanisms that implement information flow across agents (Dong et al., 2023; Sharma et al., 2024; Fedorenko et al., 2024).

The fundamental point here is not that cells and ant colonies are “the same”: of course they are not. The point is that across scales, life repeatedly builds circuitries that move information and coordinate action, and those circuitries are naturally described as networks. Accordingly, in a nutshell, in this series of posts I will explore the following set of hypotheses:

1. Living systems are open systems: they must continuously exchange matter, energy and “information” with their environment while maintaining internal organization (Miller, 1976; Rosen, 1958; Rosen, 1959).

2. Maintaining organization requires regulation: without feedback control, fluctuations and external perturbations accumulate.

3. Regulation requires circuitry: some physical substrate must implement sensing, signaling, memory and action.

4. Circuitry is inherently relational: it is defined more by patterns of interaction than by inventories of parts [as in the mechanical clock].

5. Therefore, architecture becomes the study of structure–function coupling: how network organization and dynamics co-determine robustness, adaptability and evolution across scales.

This also clarifies the trade-off I will (try to) keep enforcing: we can sometimes introduce a technical object (e.g., percolation thresholds, cycles, modularity), but the purpose is always to highlight one link in the above chain, not to build a full formalism. For the full formalism, read the paper!

We are not going to discuss a “theory of everything”

Once one accepts that structure and dynamics co-produce function, the dream of a single closed-form “theory of everything” in biology starts to look like the wrong ambition. Not because biology is beyond physics, but because even weakly emergent systems can be analytically intractable when they span many interacting scales (Anderson, 1972; Laughlin, 2000). We see this in familiar cases:

• Developmental pattern formation depends on feedback loops and collective effects that cannot be read off from an individual cell in isolation (Turing, 1952; Kondo & Miura, 2010).

• Cooperation in ecological systems emerges from networks of interactions whose macroscopic outcomes are not predictable from single-species properties alone (Levin, 1998).

So the alternative is not to “give up”. Instead, it is to aim for something more operational: modular, scalable models able to capture recurring processes and constraints, enough to generate predictions, not enough to pretend we can derive every detail (Simon, 1962).

That is what this series of posts will do: treat “architecture” as a middle theory, a compact description of how living systems are put together and how that organization channels what evolution, once coupled with the laws of physics and (bio)chemistry, can and cannot do. In that sense, it is closer to constraint-based views of living systems, what is named C-theory, which emphasize that life is shaped not by a single privileged scale, but by thermodynamic, informational, ecological and network-architectural constraints acting across scales.

Accordingly, it is important to commit to three falsifiable claims:

• Across biological scales, many functions are better predicted from interaction structure than from part lists.

• Living “logic” is always realized through a circuitry, which is active (it changes, it adapts), not a passive wiring diagram.

• Statistical physics, together with information theory and network theory, can provide quantitative descriptors of structure–function coupling that travel across domains, from cells to societies.

And let me stress explicitly that I am not claiming the existence of one mathematical framework to explain all living systems, let alone that social systems are reducible to biology or that “optimization” is the only driver. Different classes of constraints across scales and historical contingency matter.

Take-home

If living systems have an architecture, our main hypothesis is that it lives in the coupling between logic and circuitry: the set of rules that are relevant at each scale plus the networked substrate that realizes them. But this hypothesis immediately raises a brutal question.

A genome is finite, while a living circuitry — think of a brain, an immune system, an ecosystem — can be astronomically complex. So where, exactly, is the “blueprint” stored?1

In the next post of this series, we will confront the paradox directly: evolution cannot write down any full wiring diagram, so what does it store instead, and how does that choice quietly force living architectures to be sparse, modular and scalable?

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Foundations of network science and complex systems

Scaling laws for function diversity and specialization across socioeconomic and biological complex systems

Diversification and specialization are central to complex adaptive systems, yet overarching principles across domains remain elusive. We introduce a general theory that unifies diversity and specialization across disparate systems, including microbes, federal agencies, companies, universities, and cities, characterized by two key parameters. We show from extensive data that function diversity scales with system size as a sublinear power law-resembling Heaps’ law-in all but cities, where it is logarithmic. Our theory explains both behaviors and suggests that function creation depends on system goals and structure: federal agencies tend to ensure functional coverage; cities slow new function growth as old ones expand, and cells occupy an intermediate position. Once functions are introduced, their growth follows a remarkably universal pattern across all systems.

Function diversity, the range of tasks individuals perform, and specialization, the distribution of function abundances, are fundamental to complex adaptive systems. In the absence of overarching principles, these properties have appeared domain-specific. Here, we introduce an empirical framework and a mathematical model for the diversification and specialization of functions across disparate systems, including bacteria, federal agencies, universities, corporations, and cities. We find that the number of functions grows sublinearly with system size, with exponents from 0.35 to 0.57, consistent with Heaps’ law. In contrast, cities exhibit logarithmic scaling. To explain these empirical findings, we generalize the Yule-Simon model by introducing two key parameters: a diversification parameter that characterizes how existing functions inhibit the creation of new ones and a specialization parameter that describes how a function’s attractiveness depends on its abundance. Our model enables cross-system comparisons, from microorganisms to metropolitan areas. The analysis suggests that what drives the creation of new functions depends on the system’s goals and structure: federal agencies tend to ensure comprehensive coverage of necessary functions; cities tend to slow the creation of new occupations as existing ones expand; and cells occupy an intermediate position. Once functions are introduced, their growth follows a remarkably universal pattern across all systems.

While this study is very interesting and attracted my attention because it is data-driven, I agree with authors about one of its main limitation:

Another limitation is about finding a principled way to define what a function is, that’s why I think that a better way to frame the question is in terms of functionality.

Our research group work on this matter since one decade, and we have the mechanistic framework needed to study how functionality emerges from the interplay between structure and dynamics. A list and an explanation of our works in the context of complex networks, where we have explicitly identified functional diversity with thermodynamic-like quantities can be found here:

Read also this Commentary for a quick context about the above paper: Functional diversity and specialization decoded: Implications for complex particle systems, NeuroAI, and hybrid human–AI ecosystems

In a recent talk I have given for the U. of Cambridge’s seminar series Making Connections, I have discussed more about this. Specifically, while function can be local or collective (→ a realized operation or capability), functionality is what the system can do under constraints across levels (or scales) and layers (or contexts):

Evolution

Direct and indirect benefits of cooperation in collective defense against predation

Read also this Commentary for a quick context: an insect that cooperates like bacteria

In cooperative breeding birds and mammals, some offspring give up the chance to breed independently and instead help their parents to breed. This type of cooperation involves a division of labor between the “dominant” individuals who do most of the breeding, and the “subordinate” individuals who help rear the offspring of others. Reproductive division of labor is taken to an extreme level in social insects, such as ants and termites, where the subordinate helpers are completely sterile and can be divided into different physical castes, that are specialized to do different jobs.

Individuals in groups can cooperate to achieve shared goals, but they face an evolutionary challenge: if the benefits of cooperation are shared equally, freeloaders receive the same benefits as others while contributing less. Although theoretical solutions to this problem are abundant, we still lack empirical evidence on how those mechanisms function in natural systems. We study this with pine sawfly larvae, which defend collectively against predators. We show that while cooperation increases the survival of individuals and their relatives, they adjust their contributions based on who and how many they are surrounded by, relying more on freeloading in larger and male-biased groups. This results in variation in cooperativeness but prevents freeloaders from taking over.

The evolution and maintenance of public goods cooperation, despite cheating, remains a key interest in social biology. Identifying how ecological factors determine the direct and indirect benefits that maintain cooperation has proven challenging, as these can vary significantly across species and environments. Here, we study this problem using the social pine sawfly Neodiprion sertifer (Hymenoptera) as a model system. During their larval stage, N. sertifer live in groups and collectively secrete a defensive fluid against predators. This behavior comprises a public good as it is costly to exhibit and beneficial to others, and individuals vary in their contribution to group defense. We experimentally manipulated individual contributions to defense to assess how these influence survival under natural insect predation. Our results indicate that defense has a group-level benefit as individuals were more likely to survive in cooperative groups with a higher proportion of defending larvae. Moreover, being able to deploy defensive fluid confers direct survival benefits. Genetic and phenotypic analyses of natural populations further show that kin selection promotes collective defense, as groups of larvae are often composed of full siblings. We also find that the contribution to defense is female-biased and diminishes in larger, more male-biased groups, and to some extent with decreased kinship, indicating that individuals adjust their contributions based on social context. Overall, we find that contribution to the collective defense provides both direct and indirect benefits and that individuals regulate their contributions mainly based on the social environment, resulting in variation within and among natural populations.

Bridging continuous and discrete evolution through a controllable, hypermutagenic phage-bacteria system

The authors present a mutation-and-selection machine: put only the DNA module you want to optimize into a phage-carried plasmid, make the phage replication deliberately error-prone, let bacterial growth select useful variants, then transfer only that DNA module into fresh cells. For a physicist like me, the “clean idea” is a controlled evolutionary dynamics experiment where the “state space” is mostly restricted to a chosen genetic subsystem, while mutations elsewhere in the host are reset each cycle.

They show it works first on a simple selectable gene, then on a multi-gene metabolic pathway. The main finding is methodological, I think: fast directed evolution of large gene modules with reduced contamination from off-target adaptive shortcuts.

Directed evolution methods face trade-offs between the control of discrete approaches and the throughput of modern continuous systems. Here, we engineered a method called lytic selection and evolution (LySE) for near-continuous evolution of bacterial gene clusters while maintaining discrete checkpoints. We developed a hypermutagenic T7 DNA polymerase variant fused to a dual adenine-cytosine deaminase to install all possible transition mutations at similar frequencies. By relieving pressure from maintaining genome fidelity, we obtained mutation rates of 3.82 × 10⁻⁵ substitutions per base. For biocontainment, the T7 DNA polymerase was encoded on an accessory plasmid, while the target gene cluster was encoded on a T7 DNA polymerase-lacking T7 phagemid. Alternating cycles of lysis and transduction enable selective replication and mutagenesis of target genes, while off-target genomic mutations are discarded. LySE evolved a 25-fold increase in tetA-encoded tigecycline resistance in 5 cycles, and a 50.9% increase in endpoint biomass of a bacterial strain that uses the polyethylene terephthalate monomer, ethylene glycol, as its sole carbon source. Our method balances speed and control for directed bacterial evolution.

Biological Systems

Scaling and self-similarity in the formation of the embryonic epigenome

The development of complex tissues relies on the precise assignment of cell identity. At the molecular scale, this process depends on the deposition of epigenetic modifications—such as methylation—that are regulated by complex biochemical networks and occur at specific regions on the DNA and chromatin. Here we show that despite the complexity of epigenetic regulation, dynamical scaling and self-similarity of DNA methylation marks emerge in embryonic development. Drawing on single-cell multi-omics experiments, super-resolution microscopy and statistical physics, we demonstrate that these phenomena originate in dynamical feedback between DNA methylation and the formation of nanoscale dynamic chromatin aggregates. These nanoscale processes lead to genome-wide increase in DNA methylation marks following a power law and self-similar correlation functions. Using this framework, we identify methylation patterns that precede gene expression changes in embryonic symmetry breaking. Our work identifies linear sequencing measurements as a laboratory to study mesoscopic biophysical processes in vivo.

Biogeosciences

Evidence for highly variable land use but a stable climate in the southwest Maya lowlands

Why do stable regions collapse when their environment remains favorable?

This study examines the decline of the Maya city of Itzan, located in an area with relatively reliable rainfall. Despite this advantage, it followed the broader trajectory of collapse affecting drought-stricken regions.

The key insight is systemic: Maya cities functioned as an interconnected network where trade, political alliances and migration created strong dependencies across regions. When drought impacted the central lowlands, it likely initiated cascading disruptions — conflict, economic breakdown and population shifts — that propagated through the system. In this framework, collapse cannot explained by local environmental failure: it emerges from network fragility. Even nodes with stable conditions can fail when embedded in a destabilized system.

If resilience depends on the structure of connections rather than local conditions, how should we assess the stability of today’s globally coupled systems, where economic, technological, and ecological networks are even more tightly linked?

No place fails alone.

The lowland Maya of Mesoamerica were affected by multiple environmental stresses throughout their history, and many experienced a major demographic and political decline, or collapse, during a period of inferred intense multidecadal drought, approximately 1200- and 1000-years BP. Given regional variation in the timing and character of the collapse (Demarest, 2004; Hodell et al., 2007; Webster et al., 2007; Kennett and Beach, 2014; Douglas et al., 2015), much remains to be discovered about the complex interactions between climate and society in the Maya lowlands. To this end, we combine carbon and hydrogen isotopic analyses of leaf wax n-alkanes with quantification of faecal stanols and polycyclic aromatic hydrocarbons from a lake sediment core from the southwest lowlands to assess whether (1) palaeoecological evidence of land use is related to population change; and (2) whether population and land use are linked to changing precipitation. Our data reveal a transition from generally more intense fire use and C₄ plant agriculture during the Preclassic (3500–2000 BP) to dense populations and reduced fire use during the Classic (1600–1000 BP). This is consistent with other evidence for a more urbanised and specialised society in the Classic. We do not find evidence of drought in the hydrogen isotope leaf wax record (δD_lw), implying that local drought was not a primary driver of observed variability in land use or population change in the Classic-period southwestern lowlands.

Economic sciences

Uncovering how transport access reduces deprivation: When colocation misleads

There is growing interest in how cities can use transport to reduce disadvantage. A central challenge is measurement: Widely used “accessibility” measures often point to different neighborhoods as underserved, leading to conflicting policy signals. Using a citywide dataset for London, we i) compare common access measures side by side to show how metric choice reshapes the map of accessibility, and ii) contrast simple statistical associations with a causal statistical design that separates access from other neighborhood differences. We find that better access is linked to lower deprivation, yet the choice of measure substantially alters which areas are identified as priorities for intervention. These results can help cities target investments where improved access is most likely to reduce disadvantage.

Since transport access determines who can reach jobs, education, healthcare, and community life, governments increasingly use accessibility improvements to reduce deprivation and tackle social exclusion. Yet whether better access causally reduces disadvantage remains uncertain because observational analyses struggle to separate cause from context, and because accessibility itself can be measured in many, nonequivalent ways. Two challenges follow: i) widely used measures of accessibility, cumulative-opportunity, gravity, and random-utility may yield conflicting maps of accessibility and; ii) estimates from observational data are vulnerable to confounding. This paper conducts a London-wide assessment that a) compares widely used accessibility measures, and b) applies instrumental-variables (IV) estimation with road-safety-based instruments to address confounding and identify the causal effect of accessibility on deprivation. Using neighborhood-scale accessibility and the 2019 Index of Multiple Deprivation (IMD, proxies deprivation, and more broadly, social exclusion), we report two main findings. First, although accessibility rankings are broadly consistent across measures, gravity and cumulative opportunity measures display similar linear behavior, in contrast to the strong nonlinearity of the random-utility measure. The choice of measure affects not only how accessibility is represented, but also the variation retained for empirical analysis. Second, simple correlations suggest that accessibility and deprivation colocate, whereas causal estimates indicate a consistent, beneficial effect: Improvement in accessibility leads to lower deprivation, with magnitudes differing across IMD domains. From a policy perspective, this highlights the importance of grounding transport investment decisions in causal evidence and considering a range of measures to understand how accessibility improvements may help reduce disadvantage.

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A map of American data centers opens onto a much bigger question: what does a civilization look like when it starts scaling through energy, computation and infrastructure? Today, we will jump from Kardashev’s work to NASA’s search for technosignatures and beyond, and argue that while energy still matters it cannot be the whole story.

The rise of AI is revealing that civilizational evolution is a co-evolutionary process in which computation, energy and information demands scale together and reshape the infrastructure that sustains them. The above map of data centers, transmission lines, pipelines and power plants across the United States does more than locate infrastructure: it captures how our civilization reorganize its material base around computation. From the eyes of a network scientist, the visible pattern is the convergence of information processing, electricity demand, fuel logistics and territorial investment into a new kind of coupled (or, even better, interdependent) system.

In fact, it is difficult to ignore that computation is becoming a structural load and is no longer a light overlay on top of the economy. Data centers do not just consume electricity: they are reshaping where electricity must be generated, how it must be transmitted, how reliability is managed and which energy sources remain strategically valuable. The important insight is therefore systemic, and the above map is not a portrait of one sector: it is a snapshot of co-evolution.

Co-evolutionary dynamics in plain sight?

The relationship between the digital and energy infrastructure is — and cannot be otherwise — recursive:

→ rising computational demand creates localized, continuous, high-intensity loads;
→ those loads justify reinforcement of transmission capacity, new generation, contractual changes in electricity markets and in many cases continued dependence on dispatchable fossil-fuel supply;
→ as these energy infrastructures become more robust or more economically attractive in a region, they reduce the risk of additional data-center deployment.

New deployment then raises demand again and close a feedback loop, clearly not a one-way dependency. In complex-systems language, we can read the map as an adaptive multilayer system, where one layer contains data centers and digital demand, another contains electricity generation and transmission, while another contains fuel transport and dispatchable energy supply.

The state of each layer changes the incentives and feasible configurations of the others, where topology and dynamics evolve together.

Note. Geography appears clustered rather than uniform. Northern Virginia, Texas, the Pacific Northwest and other hubs “attract” data centers. I have no time to exactly inspect why, feel free to point me to sources about it. Here, I simply assume that the system becomes easier to expand in the places where it has already expanded, a kind of rich-get-richer effect for energy and computation.

There are at least three features of the system that stand out.
First, concentration: capacity is unevenly distributed, with a relatively small number of regions carrying a disproportionate share of digital load and infrastructure importance. Likely, that concentration improves efficiency, but it also creates hubs whose disruption would have effects far beyond their immediate geography, as we have learned in a previous post:

Second, tight coupling: the alignment between major data-center regions and reliable energy infrastructure is not accidental, with computation at scale requiring continuity, voltage stability and predictable supply. That selects for infrastructures that can sustain continuous loads, and for locations allowing to build new infrastructures.

Third, shared fragility: interdependent systems gain local performance by aligning layers, but that alignment also makes disturbances less isolated. For instance, heatwaves affect grids and cooling simultaneously, whereas fuel bottlenecks affect electricity supply and digital continuity together. Physical concentration and functional interdependence raise the possibility of correlated failure.

So what? You might ask. Well, the map reveals that the growth of computation drives the formation of a civilization-scale dependency structure.

Co-evolution towards which future?

If a civilization scales by coupling information, energy and infrastructure ever more tightly, what does that trajectory look like over longer timescales? Let’s see the trend, first:

One possible answer to our question is the familiar one, where civilizations grow by increasing the amount of energy they can access and control. In that picture, the future is a sequence of enlarging energetic envelopes: planetary, stellar, galactic; that is the intuition behind the Kardashev scale.

Let’s make a simple and rough estimate, starting from the observational global energy use of ~180,000 TWh/year. It corresponds to an average power of

that, on Sagan’s continuous version of the Kardashev scale, given by

leads to K ~ 0.73, or Type 0.73 in Sagan’s scale. To also estimate another axis of social development proposed by Sagan — i.e. the amount of information stored and thus available to a civilization — we can use the data below:

Humanity’s ~200 ZB of stored information corresponds to

which, in Sagan’s notation1, places us around Class S on the informational axis. I find this way of classifying a civilization quite interesting, with humankind in 2026 being around 0.73-S. The gap to a full Type I civilization comes from scale: planetary energy availability is of order 10^17 W, so current human activity remains about four orders of magnitude below that threshold.

The map does not necessarily prove that Kardashev (see the above figure) was right, but it gives the scale of a contemporary starting point, showing a real civilization expanding its capacity through energy-intensive computation and infrastructure coupling. As an analogy, we could say that it makes Kardashev feel less like science fiction and more like an extrapolation from present material trends2.

Kardashev’s proposal links civilizational activity to physical consequences that might, under some conditions, become observable at astronomical distances, and that’s why we still use it when the problem is framed in observational terms. If a civilization uses enough energy, transforms enough matter or reprocesses enough radiation, then it may create signatures that can be searched for. To this aim, Kardashev’s work is useful as a heuristic for detectability, although it is much weaker as a general theory of civilizational evolution.

Technosignatures and the modern approach

Current technosignature research is broader and more careful than the older popular image of Kardashev suggests. The NASA Technosignatures Workshop frames the field around multiple kinds of signatures and multiple “axes of merit” including detectability, ambiguity, duration, cost, extrapolation from known technology and scientific payoff; a multidimensional search problem.

Energy use is attractive scientifically because many technological activities involve it, and because some consequences of large-scale energy use score well on the trade-off between inevitability and detectability. Waste heat is the obvious example: it is thermodynamically motivated, potentially long-lived and in principle searchable in existing data, but thermodynamic motivation does not guarantee easy observation. The same signatures may be weak, anisotropic, reprocessed or degenerate with ordinary astrophysical processes (see also Zackrisson, 2018 and Huang, 2026). Searches for galaxy-scale engineering have looked for systems that are underluminous in the optical relative to dynamical expectations, or unusually bright in the infrared, under the hypothesis that large amounts of starlight are being intercepted and reradiated. However, using the Tully–Fisher relation as one route into this question no compelling galaxy-scale candidates was found.

Project Hephaistos — an international team lead by the Uppsala University — pushed the search for partial Dyson spheres into the era of large sky surveys by combining Gaia, 2MASS and WISE data. The first paper derived upper limits on how many stars within defined distance ranges could plausibly host partial Dyson spheres under explicit assumptions about covering fraction and temperature. The second paper moved from limits to candidate selection and found a very small number of objects worthy of further study, while emphasizing contamination, source confusion and the difficulty of distinguishing technosignatures from natural infrared excess3.

Those studies can tell us how a strong galaxy-scale waste heat above a given luminosity fraction is rare in a nearby sample or that near-complete partial Dyson spheres at given temperatures are uncommon within a defined region of the Milky Way under a particular model. They do not straightforwardly tell us how many technological civilizations exist. Non-detection is a weak guide to underlying prevalence unless one makes additional assumptions about emission rates, spatial distribution, geometry, duration and the probability that signals intersect Earth during the observation window: detectability depends on contact conditions, not just existence.

Accordingly, there are some limitations to be considered before falling into trivial extrapolations of our starting U.S. map:

1. Detectability is not the same thing as development: a civilization that is easier to detect may simply be one that is more dissipative, less efficient or more expansive in ways that produce strong side effects. That says little about resilience, sophistication or long-term viability.

2. Scaling does not necessarily proceed outward: technological change can also move inward toward finer control of matter, higher information density, lower dissipation, better computation and tighter environmental integration. A civilization could become more capable while becoming “less luminous”.

3. Energy says little about stability: a civilization can enlarge its operational capacity while increasing coupling, concentration and ecological burden. Growth in one variable can coincide with fragility in another.

Other ways to think about (our) civilizational development

One option is to stay within technosignature science and classify civilizations indirectly, through the kinds of signatures they produce: radio emissions, optical pulses, atmospheric technosignatures, artifacts, waste heat, megastructures, or other anomalies. This does not produce a single rank ordering, but it fits the actual logic of observation.

Another option is to shift the measure itself. Chaisson’s energy rate density quantifies how much energy flows through a system per unit mass, providing a more useful way for comparing complexity across physical, biological and cultural systems than gross total energy alone.

A third option is more qualitative and ecological. Ivanov and collaborators propose a classification that pays attention to how civilizations modify themselves and their environments. That matters because advanced development may lead toward tighter integration and altered relations with local ecosystems rather than simple luminosity growth.

Take-home message

The map of U.S. data centers and fossil-energy infrastructure gives the whole discussion a proper starting point, showing that a civilization increases its capacity by coupling computation to energy at large scale. That makes the Kardashev intuition intelligible, because both energy and infrastructure matter, while scaling leaves a footprint.

But the same map also reveals what Kardashev’s scale alone is missing. It shows lock-in, asymmetry, concentration and increasing systemic vulnerability. It shows that capacity can grow while dependencies deepen. It shows that the evolution of our civilization is shaped by co-evolution among energy, information and environment.

Accordingly, while Kardashev still belongs in the conversation, it cannot end the conversation. A serious account of civilizational development has to ask more than how much energy a system commands. It has to ask how that energy is organized, what forms of information processing it sustains, how it reshapes matter, what kinds of dependencies it creates and whether or not the resulting configuration is robust and can persist.

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Complex systems and network science have grown through many intellectual lineages: statistical physics, nonlinear dynamics, information theory, biology, ecology, computation, social science, epidemiology, neuroscience and several others. This plurality is part of what makes the field interesting, although it also makes a simple question surprisingly hard to answer: what are the works that really shaped how we think about complexity?

In 2018 I made a related attempt at the level of concepts rather than papers. I started from a manually curated vocabulary of terms associated with complexity, queried thousands of arXiv papers around those concepts, trained word-embedding models on the retrieved abstracts and used similarity networks and clustering to obtain a first map of conceptual neighborhoods. The figure above gives a sense of my exercise, although I would read it with caution: the pipeline was semi-supervised, sensitive to the initial choice of terms, limited by the arXiv corpus and inevitably shaped by the modeling decisions I made at the time. It was useful as an exploratory prototype, not as a definitive reconstruction of the field1.

I am now launching MINERVA2, a community-driven initiative to build a structured map of foundational contributions in complex systems and network science. The focus is paper-centered, because the goal is to reconstruct ideas, methods, models, and conceptual bridges rather than reputations or academic genealogies.

This is easy to misunderstand as an attempt to define a canon, which is not the spirit of this initiative. I am interested in something more useful: a navigable view of the intellectual substrate of the field, including the papers everyone remembers, the papers that entered from neighboring disciplines, as well as the papers that are often used implicitly without being described as “complex systems” or “network science”.

Bibliometric and algorithmic approaches can reveal a great deal about topical structure, citations and explicit communities, yet foundational ideas often travel in less visible ways. They may come from older vocabularies, adjacent traditions, technical methods or conceptual imports that later became naturalized inside the field. However, I think that recovering this structure requires distributed judgment from the community.

This is why broad participation matters: senior researchers bring historical perspective on how certain ideas entered and transformed the field; early-career researchers often see connections across areas that older disciplinary narratives may miss; researchers from different subfields will naturally emphasize different foundations, and this diversity is precisely the point.

The current step is simple. If you work in complex systems, network science or a neighboring area, you can contribute a small set of papers you consider foundational; DOI references are enough. Widely recognized papers are useful, and so are overlooked contributions that deserve to be visible in the map.

The submitted references will be aggregated, deduplicated and analyzed across subfields and types of contribution: theoretical frameworks, methods, applications, interdisciplinary bridges and foundational imports from adjacent areas. Contributors will receive early access to the results, which will later be shared publicly together with the methodology.

→ The survey takes less than 10 minutes: click here.

MINERVA can only work if the community contributes its own memory of the field. If you think complex systems and network science would benefit from a clearer map of their foundations, please contribute and share the initiative with colleagues, students and researchers across related domains.

Thank you!

P.S. We have already collected several hundred recommended papers. A scientific committee, involving leading complexity scientists from different countries and research traditions, will help evaluate the procedure used to construct the final corpus. Stay tuned, and get involved.

→ Please, remind that if you find value in #ComplexityThoughts, you might consider helping it grow by subscribing, or by sharing it with friends, colleagues or on social media. See also this post to learn more about this space.

→ Don’t miss the podcast version of this post: click on “Spotify/Apple Podcast” above!

From the Lab

Unraveling the emergence of collective behavior in networks of cognitive agents

Three years ago I was wondering whether LLM could act in “social groups” of any sort, and how such networks would resemble real ones.
A few months later, a brilliant student, Nicola Zomer, who was following my course on the physics of complex networks, came to me to ask for a thesis on exactly this question. Now, that work is published in NPJ Artificial Intelligence, and my warmest congratulations go to Nicola.
What makes this study especially timely is that it asks a question that is becoming central for AI: what happens when we stop looking at a single intelligent agent in isolation and start looking at many of them interacting as a collective?

Our work shows that giving agents LLM-based reasoning does not automatically produce better collective intelligence. In some tasks, a single cognitive agent can outperform a classical search process because it can detect patterns and exploit past information more effectively.
But when many such agents interact, the picture becomes much richer: they can also converge too quickly, imitate one another and get trapped in poor collective solutions.

This is why I think that or paper is important. It shows that the behavior of an artificial society is shaped not only by how smart each agent is, but also by how agents communicate, who listens to whom and how information flows through the network. In other words, the architecture of interaction is not a technical detail: it is part of the intelligence of the system.

A second key result is that collective behavior depends strongly on network structure. In optimization, collaboration among LLM agents can recover the global solution more reliably than isolated agents, but only when the communication pattern preserves enough diversity. In social self-organization, LLM agents do not simply create new macroscopic behavior by default; instead, meaningful differences emerge when interactions are local and socially selective.
For those working in AI, I think the take-home messages are clear:

• intelligence at the individual level does not guarantee intelligence at the collective level;

• communication topology is a first-class design variable for multi-agent systems;

• understanding emergence, consensus, diversity and coordination will be essential if we want robust agentic AI.

This is exactly where complex systems science can make a difference!

If you are curious, read also this post, where I better put the work in the context of agentic AI:

Cognitive agents, powered by Large Language Models (LLMs), possess advanced reasoning and communication capabilities that fundamentally distinguish them from non-cognitive particles, which rely solely on formal rules. While their ability to replicate human individual and social behaviors is still under scrutiny, the impact of their embedded “intelligence” on emergent behaviors remains poorly understood. Here, by comparing cognitive agents with classic particles, we investigate how LLM capabilities shape emergent phenomena in two tasks: function optimization and social organization emerging from the Schelling model of segregation. To this aim, we introduce LLM Agent Swarm Optimization (llmASO), where a swarm of interacting LLM agents acts as an optimizer. Our findings reveal that, while individual agents outperform particles in decision-making, their consensus tendencies and ability to exploit patterns can make them prone to premature convergence. Adjusting network topology can alleviate this effect, but typically at the cost of slower overall convergence compared to classical Particle Swarm Optimization (PSO). In contrast, in the Schelling model, we demonstrate that local interactions and homophilic mechanisms allow cognitive agents to generate distinct emergent behaviors, underscoring the importance of realistic communication architectures in complex social simulations. This work clarifies how LLM capabilities introduce new mechanisms for collective behavior and has implications for future applications of LLM agents in swarm robotics, social experiments, and complex decision-making tasks.

Artificial and biological systems

Bringing the genetically minimal cell to life on a computer in 4D

We present a whole-cell spatial and kinetic model for the 100 min cell cycle of the genetically minimal bacterium JCVI-syn3A. We simulate the complete cell cycle in 4D (space and time), including all genetic information processes, metabolic networks, growth, and cell division. By integrating hybrid computational methods, we model the dynamics of morphological transformations. Growth is driven by insertion of lipids and membrane proteins and constrained by fluorescence imaging data. Chromosome replication and segregation are controlled by the essential structural maintenance of chromosome proteins, analogous to condensin (SMC) and topoisomerase proteins in Brownian dynamics simulations, with replication rates responding to deoxyribonucleotide triphosphate (dNTP) pools from metabolism. The model captures the origin-to-terminus ratio measured in our DNA sequencing and recovers other experimental measurements, such as doubling time, mRNA half-lives, protein distributions, and ribosome counts. Because of stochasticity, each replicate cell is unique. We predict not only the average behavior of partitioning to daughter cells but also the heterogeneity among them.

Gene regulatory networks: from correlative models to causal explanations

Gene regulatory networks (GRNs) explain how the genome controls cellular behaviour and tissue morphogenesis, serving to connect molecular mechanism to functional output. Single-cell technologies now provide descriptions of these networks with unprecedented detail, but this advance has also revealed gene regulatory systems that are too complex for our existing conceptual frameworks. GRNs, which should provide mechanistic explanations, are increasingly reduced to statistical correlations — ‘hairballs’ that fail to capture molecular causation. Here, we explore why this dilemma exists and propose a path forward. We argue that methods in ‘representation learning’ can be used to model GRNs, without needing to capture every molecular detail. For this framework, we advocate three linked principles: models must be inherently mechanistic, with structures grounded in cellular and evolutionary biology; molecular principles and constraints must be used to reduce the solution space for learning GRN models; and more sophisticated forms of experimental perturbation and synthetic biological engineering are needed to train models and test predictions. By reimagining GRNs through these principles, we can bridge the gap from data abundance to new conceptual understanding.

Genome modelling and design across all domains of life with Evo 2

All of life encodes information with DNA. Although tools for genome sequencing, synthesis and editing have transformed biological research, we still lack sufficient understanding of the immense complexity encoded by genomes to predict the effects of many classes of genomic changes or to intelligently compose new biological systems. Artificial intelligence models that learn information from genomic sequences across diverse organisms have increasingly advanced prediction and design capabilities^1,2. Here we introduce Evo 2, a biological foundation model trained on 9 trillion DNA base pairs from a highly curated genomic atlas spanning all domains of life to have a 1 million token context window with single-nucleotide resolution. Evo 2 learns to accurately predict the functional impacts of genetic variation—from noncoding pathogenic mutations to clinically significant BRCA1 variants—without task-specific fine-tuning. Mechanistic interpretability analyses reveal that Evo 2 learns representations associated with biological features, including exon–intron boundaries, transcription factor binding sites, protein structural elements and prophage genomic regions. The generative abilities of Evo 2 produce mitochondrial, prokaryotic and eukaryotic sequences at genome scale with greater naturalness and coherence than previous methods. Evo 2 also generates experimentally validated chromatin accessibility patterns when guided by predictive models^3,4 and inference-time search. We have made Evo 2 fully open, including model parameters, training code⁵, inference code and the OpenGenome2 dataset, to accelerate the exploration and design of biological complexity.

Population dynamics

Integrated framework to study genomic surveillance of selective sweeps in multivariants dynamics

Pandemics are often shaped by the competition of multiple pathogen variants, yet existing models rarely connect epidemic dynamics, genomic surveillance, and population immunity within a unified framework. We develop an integrated modeling framework that couples multivariant epidemic dynamics, mechanistic genomic dominance, and probabilistic surveillance. We show that variant dominance follows selective sweeps described by multilogistic equations, with growth rates modulated by population immune histories. Moreover, we show that the variant emergence timing and the dynamics of prior variants influence whether new variants are detected quickly or remain undetected. Our results are validated by fitting epidemiological and genomic data from the spread of SARS-CoV-2 variants across multiple countries, highlighting its relevance for genomic epidemiology and surveillance-informed response.

Pandemics often involve complex transmission dynamics in which epidemiological surveillance is essential but not sufficient for containment, as resurgence may be driven by emerging or imported variants. Rapidly evolving pathogens produce complex disease dynamics driven by emerging variants often differing in their transmissibility, immune escape, and cross-infection. These processes influence individuals’ immune life histories, producing highly dynamic immune landscapes that modulate the emergence and dominance of novel variants. We develop an integrated modeling framework that couples multivariant mean-field epidemic modeling with a mechanistic genomic dominance model and a probabilistic surveillance model. This study examines how variant emergence timing, infectiousness advantage, and cross-infection jointly shape epidemic trajectories, immune landscapes, and genomic composition. Our results demonstrate that the dominance dynamics of cocirculating variants correspond to a selective sweep characterized by a system of multilogistic equations driven by population immunity. Moreover, we show that the detection time of newly introduced variants can be accelerated or delayed depending on their emergence conditions and the prevailing variant landscape. Finally, we demonstrate that the effectiveness of response strategies depends critically on the evolving genomic composition of the outbreak, highlighting trade-offs between surveillance sensitivity and intervention timing. We validate our framework by jointly fitting epidemiological and genomic data from the spread of the Ancestral, Alpha, Gamma, and Delta variants in the United States, Denmark, the United Kingdom, and Canada. The results provide a quantitative foundation for linking epidemic dynamics, genomic surveillance, and immune life histories, advancing the development of genomic epidemiology for multivariant outbreaks.

Characterization and forecast of global influenza subtype dynamics

The subtype composition of seasonal influenza waves varies in space and time. Influenza subtypes A/H1N1, A/H3N2 and B tend to have different impacts on population groups; therefore, understanding the drivers of their cocirculation and anticipating their composition is important for epidemic preparedness. FluNet provides data on influenza specimens by subtype for more than 150 countries. However, owing to surveillance variations across countries, global analyses usually focus on subtype compositions, a kind of data difficult to treat with advanced statistical methods. We used compositional data analysis to circumvent the problem and study trajectories of annual subtype compositions of countries. Here we first examine global trends from 2000 to 2023. We identify a few seasons which stood out for the strong within-country subtype dominance due to either a new virus/clade taking over (2003/2004 season, A/H1N1pdm pandemic) or subtypes’ spatial segregation (coronavirus disease 2019 pandemic). Second, we show that geographical factors, most notably international mobility, concurred in shaping countries’ composition trajectories between 2010 and 2019. Trajectories clustered in two macroregions characterized by subtype alternation versus persistent mixing. Finally, we define five algorithms for forecasting the next year’s composition and found that incorporating the global history of subtype composition in a Bayesian hierarchical vector autoregressive model improved predictions compared with naive methods. The joint analysis of spatiotemporal dynamics of influenza subtypes worldwide reveals a hidden structure in subtype circulation that can be used to improve predictions of the subtype composition of next year’s epidemic according to place.

Genotype networks drive oscillating endemicity and epidemic trajectories in viral evolution

Rapidly evolving viruses use antigenic drift as a key mechanism to evade host immunity and persist in real populations. While traditional models of antigenic drift and epidemic spread rely on low-dimensional antigenic spaces, genomic surveillance data reveal that viral evolution produces complex antigenic genotype networks with hierarchical modular structures. In this study, we present an eco-evolutionary framework in which viral evolution and population immunity dynamics are shaped by the structure of antigenic genotype networks. Using synthetic networks, we demonstrate that network topology alone can drive transitions between stable endemic states and recurrent seasonal epidemics. Furthermore, our results show how the integration of the genotype network of the H3N2 influenza in our model allows for estimating the emergence times of various haplotypes resulting from its evolution. Our findings underscore the critical role of the topology of genotype networks in shaping epidemic behavior and, besides, provide a robust framework for integrating real-world genomic data into predictive epidemic models.

Immunity-induced criticality of the genotype network of influenza A (H3N2) hemagglutinin

Seasonal influenza kills hundreds of thousands every year, with multiple constantly changing strains in circulation at any given time. A high mutation rate enables the influenza virus to evade recognition by the human immune system, including immunity acquired through past infection and vaccination. Here, we capture the genetic similarity of influenza strains and their evolutionary dynamics with genotype networks. We show that the genotype networks of influenza A (H3N2) hemagglutinin are characterized by heavy-tailed distributions of module sizes and connectivity indicative of critical behavior. We argue that (i) genotype networks are driven by mutation and host immunity to explore a subspace of networks predictable in structure and (ii) genotype networks provide an underlying structure necessary to capture the rich dynamics of multistrain epidemic models. In particular, inclusion of strain-transcending immunity in epidemic models is dependent upon the structure of an underlying genotype network. This interplay is consistent with self-organized criticality where the epidemic dynamics of influenza locates critical regions of its genotype network. We conclude that this interplay between disease dynamics and network structure might be key for future network analysis of pathogen evolution and realistic multistrain epidemic models.

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I have written this post some weeks ago for The Palindrome, kindly hosted by . I am resharing the post for the readers of ComplexityThoughts, enriched by our usual explanatory slides for the most important concepts.

The Palindrome is the space where you actually understand what’s under the hood of machine learning, and it’s one of my favorite Substacks. From deep mathematical insights to hands-on theory, it turns technical ideas into intuition you can build on for decades. Therefore, if you’re serious about mastering machine learning, this is the one you don’t want to skip.

In 2000, a computer worm named ILOVEYOU infected millions of machines in a single day.

Two years later, a post-apocalyptic horror movie directed by Danny Boyle — 28 days later — was released. A story about how humans turned into zombies after getting infected by a virus.

Despite their stacking differences, both follow the same simple logic: interactions lead to contact, contact allows transmission, and transmission triggers transformation.

The surprising fact? Mathematics can capture that logic with striking simplicity and remarkable explanatory power.

Today we will see this in action with a twofold purpose: by focusing on these two seemingly different contagion processes, we will show how the same math can be fundamental to describe what’s going on — and, more broadly, why modeling matters.

Act 0 – What’s a model?

Mathematical models are not decorations on science: they are thinking tools. They allow us to distill messy realities into a few essential gears, showing how motion in one part turns another. This becomes immediately useful when the world feels too complex to grasp: a model offers a translation from tangled stories to relationships we can reason about, test, and sometimes even control.

However, a good model is not meant to reproduce every detail of reality: its purpose is to be useful, helping us identify what can drive a phenomenon, what can be ignored and where small interventions create large effects. The art of modeling lies in choosing the right simplifications: keeping what matters, discarding what doesn’t.

Compartments: how to turn “stories” into equations

In the vast universe of mathematical models, we will focus on contagion because its complexity can be captured in an elegant way.

Which complexity? Well, we should consider that there are people that take the bus or the metro, go to the cinema or supermarket, kids going to the school, people meeting in companies, at any hour of the day, in different part of a cities, or an entire country. These events actually corresponds to some form of interaction, each one able to trigger the reaction chain: contact → transmission → transformation.

In the animation below, that I have built from the amazing ComplexityExplorables, it is possible to appreciate a computer simulation of this process in a very simplified digital world without buses, metros, supermarkets, schools, just four groups (or communities). Initially, the majority of people are healthy and susceptible (white) to the virus brought by sick and infectious individuals (red). Because of random contacts, the virus can spread among individuals, and their movement can spread the virus from one community to another. If the disease is like a seasonal flu, red individuals turn into recovered (grey) after a while: they get healthy again. And after more time, those ones who recovered can become sick again, and so forth and so on.

However, if once get infected one cannot recover, all individuals will eventually turn red, as towards the end of the simulation, where I have intentionally changed the virus.

Obviously, this complexity is cumbersome: uncountable and often untrackable interactions, different immunological response (e.g., from a few symptoms to high fever) from different people, different behavioral response (e.g., from staying at home to compulsory job presence), just to mention some.

However, when we model epidemics we do not track every individual. We often make the rather simplifying assumption that they are like some labeled/colored particles, and we group them by state (i.e., the color) into compartments.
A compartment is simply a box collecting things that behave the same way: uninfected computers, infected ones, or those that can no longer be infected (like the recovered). Therefore, the story of a contagion then becomes a story of flows between those boxes: one box loses what another gains, exactly as in the illustration below.

Mathematically, we can use a minimal map like this:

where each arrow represents a rate indicating how often a change of state happens. Once the compartments and flows are clearly defined, the equations almost write themselves. The rest is interpretation.

Act I — The cyber epidemic

In May 2000, the ILOVEYOU worm spread through email attachments. One infected computer sent the same message to every contact in its address book; each new victim did the same. It’s a perfect digital analogue of infection.

Since there’s no natural recovery until someone patches (with an antivirus) or isolates the system, the right compartment model for this process is SI: the story is

where S denote uninfected computers, I the infected machines and β is the rate of successful transmission. Furthermore, let’s also forget about the specific interaction patterns of the machines: instead of caring about which computer is exactly connected to which other computer, let’s work under the “well-mixed” assumption that any infected can contact any susceptible. We are now ready to transform the above story into an equation, relating the rate at which the number of infected individuals increase with the other variables:

At the start, almost everyone is uninfected, so S ~ N, where N is the total number of machines in the world:

That single line explains the worm’s explosive global reach: exponential growth until the susceptible pool runs out (or, equivalently, until the S box becomes empty). Control, in this view, is mechanical: reduce β (slower transmission) or reduce S (fewer vulnerable systems).

What happens next? When the number of infected machines increases, we can no longer approximate and we must write S = N - I. The equation turns into:

The last step, obtained after elementary algebra, provides us with an operational way to estimate the fraction of infected machines over time, according to the model:

That’s pretty cool, isn’t it? Overall, because the shape of the two curve is very similar to the observed ones, with the difference that computer scientists responded back by writing an antivirus able to stop the virus. We could model that by adding the R compartment after some time.

Act II — The Anatomy of a Zombie Apocalypse (S–I–Z–R)

Now let’s make it cinematic, but still mathematical. Do you remember the movie “I am legend” starring Will Smith?

Suppose we have four compartments: S (survivors, susceptible humans), I (infected but not yet zombies), Z (zombies, fully infectious) and R (recovered/removed because immune, deceased, or isolated). The zombie flowchart is given by:

a bit more complicated, although very similar in spirit, to our previous example, where each arrow translates into a rate term. The system dynamics reads:

with N = S + I + Z + R. Early on S ~ N (nearly the whole population consists of susceptible individuals), so the basic reproduction number is

If it is larger than 1, the outbreak grows and zombies spread faster than they’re neutralized. Conversely, if it is smaller than 1, it dies out and humanity survives!

Act III — From well-mixed to networked worlds

So far, we’ve imagined a “well-mixed” world, as if everyone could meet everyone. However, real contagion, digital or biological, spreads on networks: a few hubs connect widely, most nodes connect locally. This structure changes everything:

• Ignition: infections that start near hubs explode faster.

• Control: targeting those hubs (email relays, hospitals, influencers) is far more effective than random protection, although they are often harder to locate than one could imagine.

The header image of this post shows the air-transport network: airports connected by flight routes, a primary pathway for moving pathogens between countries. Likewise, social networks provide the contact structure for transmitting disease, and the Internet provides the connectivity that enables the spread of computer viruses.

The same logic explains why ILOVEYOU spread via central mail servers and why social media virality behaves like an epidemic of ideas. In both cases, patching or isolating/immunizing the hubs can dramatically changes the system’s trajectory.

Mathematical models are compressed stories about cause and effect, how small interactions scale into global patterns. They do not predict the future with certainty, but they make the invisible visible: how rates, connections and structure shape collective outcomes.

Whether the contagion is viral code, zombies or ideas, the same framework applies (compartments, rates, networks). Once you draw the boxes and trace the arrows, you see where and how to act. What’s the take-home message? This one:

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