Complexity Thoughts: Issue #57
Unraveling complexity: building knowledge, one paper at a time
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Foundations of network science and complex systems
Network Reconstruction via the Minimum Description Length Principle
In a recent post I have discussed the many (hidden) assumptions behind the “model-free” analysis of complex systems with a network structure.
In this paper just published by Tiago Peixoto, it is shown how one can explicitly include modeling assumptions into a very powerful inferential pipeline, using the principle of Minimum Description Length to guide network reconstruction and model selection even when only indirect data is available.
A fundamental problem associated with the task of network reconstruction from dynamical or behavioral data consists in determining the most appropriate model complexity in a manner that prevents overfitting and produces an inferred network with a statistically justifiable number of edges and their weight distribution. The status quo in this context is based on 𝐿1 regularization combined with cross-validation. However, besides its high computational cost, this commonplace approach unnecessarily ties the promotion of sparsity, i.e., abundance of zero weights, with weight “shrinkage.” This combination forces a trade-off between the bias introduced by shrinkage and the network sparsity, which often results in substantial overfitting even after cross-validation. In this work, we propose an alternative nonparametric regularization scheme based on hierarchical Bayesian inference and weight quantization, which does not rely on weight shrinkage to promote sparsity. Our approach follows the minimum description length principle, and uncovers the weight distribution that allows for the most compression of the data, thus avoiding overfitting without requiring cross-validation. The latter property renders our approach substantially faster and simpler to employ, as it requires a single fit to the complete data, instead of many fits for multiple data splits and choice of regularization parameter. As a result, we have a principled and efficient inference scheme that can be used with a large variety of generative models, without requiring the number of reconstructed edges and their weight distribution to be known in advance. In a series of examples, we also demonstrate that our scheme yields systematically increased accuracy in the reconstruction of both artificial and empirical networks. We highlight the use of our method with the reconstruction of interaction networks between microbial communities from large-scale abundance samples involving on the order of 10^4–10^5 species and demonstrate how the inferred model can be used to predict the outcome of potential interventions and tipping points in the system.
Evolution
Experimental evolution of evolvability
Can the capacity to evolve be selected by natural selection for greater ability to evolve? Barnett et al. designed experiments in which lineages of bacteria cycled between two selective environments (see the Perspective by Kussell). In response to changing conditions, a multistep evolutionary dynamic emerged in which selection first acts to elevate transcription rates at a single regulatory gene. An increase in frameshift mutations occurs at the same locus that also allows hitchhiking of secondary, possibly adaptive mutations. This phenomenon provides an evolutionary mechanism for increasing the capacity for evolvability toward specific adaptive outcomes. —Caroline Ash
INTRODUCTION. The capacity to generate adaptive variation is critical for long-term evolutionary success. However, the extent to which natural selection directly favors enhanced evolvability remains debated. Although studies with microbes show that mutants with elevated genome-wide mutation rates can be selected, a deeper question persists: Can natural selection structure genetic and developmental systems to bias mutations toward adaptive outcomes? This hypothesis challenges the traditional view of evolution as a “blind” process fueled by random variation, which amplifies traits beneficial in the present without regard for future contingencies.
RATIONALE. Mutation being biased toward adaptive outcomes challenges conventional perspectives but aligns with the logic of natural selection acting on lineages. Across changing environments, lineages capable of rapid adaptation are more likely to survive and replace those less able. If competing lineages, because of their varying genetic architecture, tend to generate phenotypic variation in different ways, then those with tendencies that are more conducive to an adaptive response in a given environment will be favored. Provided the same environmental challenges recur over time, an iterative process of selection can take place, potentially refining the capacity to adapt. To test this idea, we designed an experiment where lineages of bacteria competed to repeatedly achieve, through mutation, phenotypes optimal for growth under two alternating conditions. Lineages that failed to evolve the target phenotype within a set time went extinct and were replaced by successful lineages. This birth-death dynamic created conditions for selection to refine the ability of lineages to evolve between phenotypic states.
RESULTS. During the course of a 3-year selection experiment, involving identification and ordering of more than 500 mutations, a lineage emerged that was capable of rapid mutational transitions between alternate phenotypic states through localized hypermutation. The mutable locus arose through a multistep evolutionary process: Initial mutations targeted a wide range of genes but eventually focused on a single regulator. A series of mutations that alternately activated and inactivated function of the regulatory gene then followed. A subset of these inactivating mutations were compensated for by mutations that increased transcription and, concomitantly, frameshift mutation rate. The overall effect was to promote, through slipped-strand mispairing, the duplication, and then further amplification, of a heptanucleotide sequence. This process led the locus-specific mutation rate to increase ~10,000-fold. In turn, the resulting frameshift mutations enabled reversible phenotypic changes through expansion and contraction of the heptanucleotide sequence, mirroring the contingency loci of pathogenic bacteria. Lineages with the hypermutable locus exhibited enhanced evolvability to altered rates of environmental change and were more likely to acquire additional adaptive mutations, highlighting an unanticipated evolutionary advantage of localized hypermutability.
CONCLUSION. Our study demonstrates how selection can incorporate evolutionary history into the genetic architecture of a single cell, giving rise to a hypermutable locus that appears to anticipate environmental change, thereby accelerating adaptive evolution. This was possible only as an outcome of selection working at two levels. Whereas individual-level selection repeatedly drove cell populations between the same two phenotypic states, the genetic underpinnings of these phenotypes were free to diverge, fueling an exploration of evolutionary potential, the consequences of which only emerged on the timescale of lineages. Ultimately, this exploration generated the variation necessary for construction and cumulative refinement of a lineage-level adaptive trait. More generally, our experiment clarifies the conditions by which evolvability can itself evolve adaptively and highlights the importance of this process for microbial pathogens.
How the spatial arrangement of a population shapes its evolutionary dynamics has been of long-standing interest in population genetics. Most previous studies assume a small number of demes or symmetrical structures that, most often, act as well-mixed populations. Other studies use network theory to study more heterogeneous spatial structures, however they usually assume small, regular networks, or strong constraints on the strength of selection considered. Here we build network generation algorithms, conduct evolutionary simulations and derive general analytic approximations for probabilities of fixation in populations with complex spatial structure. We build a unifying evolutionary theory across network families and derive the relevant selective parameter, which is a combination of network statistics, predictive of evolutionary dynamics. We also illustrate how to link this theory with novel datasets of spatial organization and use recent imaging data to build the cellular spatial networks of the stem cell niches of the bone marrow. Across a wide variety of parameters, we find these networks to be strong suppressors of selection, delaying mutation accumulation in this tissue. We also find that decreases in stem cell population size also decrease the suppression strength of the tissue spatial structure.
Ecosystems
A travelling-wave strategy for plant–fungal trade
What's a mycorrhizal network?
They are intricate underground systems formed by the symbiotic relationships between fungi and the roots of trees. This partnership, termed “mycorrhiza”, involves fungi extending their thread-like structures called mycelium into tree roots to exchange essential resources. Trees, especially older and larger “hub” trees, produce excess sugars through photosynthesis, which fungi absorb for nourishment: in return, fungi provide the trees with critical soil nutrients and water.
These networks effectively connect numerous trees within forests, enabling them to communicate by transferring nutrients, aiding seedling growth and even sending distress signals when threatened. Understanding these subterranean interactions, often referred to as the “wood wide web” can inform conservation strategies aimed at preserving forest health and resilience.
But where the network is emerging from? Well, you might check also this video:
For nearly 450 million years, mycorrhizal fungi have constructed networks to collect and trade nutrient resources with plant roots1,2. Owing to their dependence on host-derived carbon, these fungi face conflicting trade-offs in building networks that balance construction costs against geographical coverage and long-distance resource transport to and from roots3. How they navigate these design challenges is unclear4. Here, to monitor the construction of living trade networks, we built a custom-designed robot for high-throughput time-lapse imaging that could track over 500,000 fungal nodes simultaneously. We then measured around 100,000 cytoplasmic flow trajectories inside the networks. We found that mycorrhizal fungi build networks as self-regulating travelling waves—pulses of growing tips pull an expanding wave of nutrient-absorbing mycelium, the density of which is self-regulated by fusion. This design offers a solution to conflicting trade demands because relatively small carbon investments fuel fungal range expansions beyond nutrient-depletion zones, fostering exploration for plant partners and nutrients. Over time, networks maintained highly constant transport efficiencies back to roots, while simultaneously adding loops that shorten paths to potential new trade partners. Fungi further enhance transport flux by both widening hyphal tubes and driving faster flows along ‘trunk routes’ of the network5. Our findings provide evidence that symbiotic fungi control network-level structure and flows to meet trade demands, and illuminate the design principles of a symbiotic supply-chain network shaped by millions of years of natural selection.
Neuroscience
So, I think that the next two papers should be read in parallel! Is higher-order organization needed or not?
On the one hand, one paper highlights that:
Pairwise interactions are major contributors to entropy production at the macroscale
Contributions from high-order interactions to non-equilibrium dynamics are relatively minor
Entropy production of pairs of brain regions characterizes cognitive states
from which we should deduce that pairwise interactions are the ones that mostly characterize cognitive states. On the other hand, one paper suggests that:
Reinforcement of prior expectations during an auditory recognition task is facilitated through a hierarchy of irreversible higher-order interactions in the brain, an observation that we link to both the mechanisms of predictive coding and the hierarchical structure of the auditory system
To some extent, the two papers can be seen as complementary:
The first paper tells us that the brain’s non-equilibrium dynamics are largely determined by pairwise interactions.
The second papers suggests that exploring higher-order interactions can uncover a hierarchical organization that is critical for specific cognitive functions.
Taken together: brain dynamics requires appreciating the dominant role of pairwise interactions and the functional significance of the subtle, yet important, higher-order interactions?
Information processing in the human brain can be modeled as a complex dynamical system operating out of equilibrium with multiple regions interacting nonlinearly. Yet, despite extensive study of the global level of nonequilibrium in the brain, quantifying the irreversibility of interactions among brain regions at multiple levels remains an unresolved challenge. Here, we present the Directed Multiplex Visibility Graph Irreversibility framework, a method for analyzing neural recordings using network analysis of time-series. Our approach constructs directed multilayer graphs from multivariate time-series where information about irreversibility can be decoded from the marginal degree distributions across the layers, which each represents a variable. This framework is able to quantify the irreversibility of every interaction in the complex system. Applying the method to magnetoencephalography recordings during a long-term memory recognition task, we quantify the multivariate irreversibility of interactions between brain regions and identify the combinations of regions which showed higher levels of nonequilibrium in their interactions. For individual regions, we find higher irreversibility in cognitive versus sensorial brain regions while for pairs, strong relationships are uncovered between cognitive and sensorial pairs in the same hemisphere. For triplets and quadruplets, the most nonequilibrium interactions are between cognitive–sensorial pairs alongside medial regions. Combining these results, we show that multilevel irreversibility offers unique insights into the higher-order, hierarchical organization of neural dynamics from the perspective of brain network dynamics.
Non-equilibrium whole-brain dynamics arise from pairwise interactions
The human brain is a complex system of multiple neural elements that interact at different orders (pairwise, triplets, etc.), displaying non-equilibrium processes from the neuronal scale to the whole-brain scale. Here, we study how non-equilibrium dynamics of large-scale brain activity is driven by the interaction of its constituent elements at different orders. We hypothesize that the interactions generating non-equilibrium dynamics at the macroscopic brain scale are typically pairwise, with higher-order dependences playing a diminishing role. By expanding the entropy production into a sequence of orders of interactions, we find that pairwise interactions contribute dominantly. In light of this finding, we demonstrate that it is possible to characterize non-equilibrium brain dynamics using the interactions of pairs of macroscopic brain regions rather than complex interactions involving three or more regions. Furthermore, we propose that the entropy production of pairs of brain regions is a sensitive indicator for characterizing task-induced brain states.
Human brain dynamics are shaped by rare long-range connections over and above cortical geometry
A fundamental topological principle is that the container always shapes the content. In neuroscience, this translates into how the brain anatomy shapes brain dynamics. From neuroanatomy, the topology of the mammalian brain can be approximated by local connectivity, accurately described by an exponential distance rule (EDR). The compact, folded geometry of the cortex is shaped by this local connectivity, and the geometric harmonic modes can reconstruct much of the functional dynamics. However, this ignores the fundamental role of the rare long-range (LR) cortical connections, crucial for improving information processing in the mammalian brain, but not captured by local cortical folding and geometry. Here, we show the superiority of harmonic modes combining rare LR connectivity with EDR (EDR+LR) in capturing functional dynamics (specifically LR functional connectivity and task-evoked brain activity) compared to geometry and EDR representations. Importantly, the orchestration of dynamics is carried out by a more efficient manifold made up of a low number of fundamental EDR+LR modes. Our results show the importance of rare LR connectivity for capturing the complexity of functional brain activity through a low-dimensional manifold shaped by fundamental EDR+LR modes.
Oldies but goldies
From genes to phenotype: dynamical systems and evolvability
A paper by Pere Alberch, from 1991! He was looking in the correct direction before many others.
