Complexity Thoughts: Issue #60
Unraveling complexity: building knowledge, one paper at a time
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From the Lab
Critical behavior of epidemics depending on the interplay between temporal scales and human behavior
The study asks how epidemic dynamics emerge from interdependent human behaviors, using an epidemic model with distinct time scales for mobility, contacts, and recovery. The critical threshold curve exhibits a phase transition in parameter space, with a metacritical point marking when the epidemic threshold becomes mobility-dependent. Our model identifies how to balance contact and mobility cuts to mitigate epidemics. With this paper, we dramatically generalize the findings of a previous study from the lab, where we have studied the critical behavior in interdependent spatial spreading processes with distinct characteristic time scales,
with a focus on reaction-diffusion dynamics, from a pure particle-based approach. In this new study, we go beyond that from multiple perspectives.
The spatial spread of infectious diseases is ruled by two processes: the disease progression and human behavior, which comprises human contacts and human mobility. A standard modeling approach is to define an epidemiological model upon a metapopulation network under some restrictive assumptions, such as that all processes happen at the same time scale, dispersal is diffusive, and agents are indistinguishable with respect to age and behavioral features. Although several models relax at least one of those assumptions, providing new insights about an epidemic, rarely are they relaxed all at once. Here, we introduce a model whose equations explicitly contain two parameters that encode the ratios between the time scales of the recovery, contact, and mobility processes, while simultaneously accounting for the age structure of the agents, different mobility layers, and different social settings for contacts. Furthermore, to reflect more closely real-world scenarios, we consider two settings: a diffusion-based one in which agents disperse like particles to spread the epidemics, effectively changing the starting population size far from and at equilibrium, and a force-based one where spread happens without changing the population of patches far from equilibrium. For both spreading frameworks, we study the regimes under which they provide distinct results and the critical properties of the epidemic process, finding that the curve of the critical points, which separate increasing or decreasing trends in the number of infected individuals – as well as other macroscopic observables of interest such as the attack rate – in the space of the two scale parameters exhibits a critical point itself. Complementing spatially-oriented strategies, the proposed approaches can contribute to the design of effective non-pharmaceutical interventions by focusing on the mutual influence of temporal scales, dispersal dynamics, and human behavior. In particular, our study allows for an optimal tuning of mobility and contact restrictions based on the characteristics of the disease and its spatial distribution in the early phase of spreading.
Foundations of network science and complex systems
Multiscale Field Theory for Network Flows
Network flows are pervasive, including the movement of people, transportation of goods, transmission of energy, and dissemination of information; they occur on a range of empirical interconnected systems, from designed infrastructure to naturally evolved networks. Despite the broad spectrum of applications, because of their domain-specific nature and the inherent analytic complexity, a comprehensive theory of network flows is lacking. We introduce a unifying treatment for network flows that considers the fundamental properties of packet symmetries, conservation laws, and routing strategies. For example, electrons in power grids possess interchangeability symmetry, unlike packages sent by postal mail, which are distinguishable. Likewise, packets can be conserved, such as cars in road networks, or dissipated, such as Internet packets that time out. We introduce a hierarchy of analytical field-theoretic approaches to capture the different scales of complexity required. Mean-field analysis uncovers the nature of the transition through which flow becomes unsustainable upon unchecked growth of demand. Mesoscopic field theory accurately accounts for complicated network structures, packet symmetries, and conservation laws and yet is capable of admitting closed-form solutions. Finally, the full-scale field theory allows us to study routing strategies ranging from random diffusion to shortest path. Our theoretical results indicate that flow bottlenecks tend to be near sources for interchangeable packets and near sinks for distinguishable ones, and that dissipation hinders the maximum sustainable throughput for interchangeable packets but can enhance throughput for distinguishable packets. Finally, we showcase the flexibility of our multiscale theory by applying it in two distinct domains of road networks and the C. elegans neuronal network. Our work paves the way for a more unifying and comprehensive theory of network flows.
Recovery coupling in multilayer networks
The increased complexity of infrastructure systems has resulted in critical interdependencies between multiple networks—communication systems require electricity, while the normal functioning of the power grid relies on communication systems. These interdependencies have inspired an extensive literature on coupled multilayer networks, assuming a hard interdependence, where a component failure in one network causes failures in the other network, resulting in a cascade of failures across multiple systems. While empirical evidence of such hard failures is limited, the repair and recovery of a network requires resources typically supplied by other networks, resulting in documented interdependencies induced by the recovery process. In this work, we explore recovery coupling, capturing the dependence of the recovery of one system on the instantaneous functional state of another system. If the support networks are not functional, recovery will be slowed. Here we collected data on the recovery time of millions of power grid failures, finding evidence of universal nonlinear behavior in recovery following large perturbations. We develop a theoretical framework to address recovery coupling, predicting quantitative signatures different from the multilayer cascading failures. We then rely on controlled natural experiments to separate the role of recovery coupling from other effects like resource limitations, offering direct evidence of how recovery coupling affects a system’s functionality.
Network isolators inhibit failure spreading in complex networks
In our daily lives, we rely on the proper functioning of supply networks, from power grids to water transmission systems. A single failure in these critical infrastructures can lead to a complete collapse through a cascading failure mechanism. Counteracting strategies are thus heavily sought after. In this article, we introduce a general framework to analyse the spreading of failures in complex networks and demostrate that not only decreasing but also increasing the connectivity of the network can be an effective method to contain damages. We rigorously prove the existence of certain subgraphs, called network isolators, that can completely inhibit any failure spreading, and we show how to create such isolators in synthetic and real-world networks. The addition of selected links can thus prevent large scale outages as demonstrated for power transmission grids.
(Explainable) AI-based methods and applications
I recommend to read the following papers in the given order, since they are chronologically related. We have seen some of them in a past Issue, but consistency is important!
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
Understanding dynamic constraints and balances in nature has facilitated rapid development of knowledge and enabled technology, including aircraft, combustion engines, satellites, and electrical power. This work develops a novel framework to discover governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. There are many critical data-driven problems, such as understanding cognition from neural recordings, inferring climate patterns, determining stability of financial markets, predicting and suppressing the spread of disease, and controlling turbulence for greener transportation and energy. With abundant data and elusive laws, data-driven discovery of dynamics will continue to play an important role in these efforts.
Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.
Data-driven discovery of coordinates and governing equations
Governing equations are essential to the study of physical systems, providing models that can generalize to predict previously unseen behaviors. There are many systems of interest across disciplines where large quantities of data have been collected, but the underlying governing equations remain unknown. This work introduces an approach to discover governing models from data. The proposed method addresses a key limitation of prior approaches by simultaneously discovering coordinates that admit a parsimonious dynamical model. Developing parsimonious and interpretable governing models has the potential to transform our understanding of complex systems, including in neuroscience, biology, and climate science.
The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam’s razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom deep autoencoder network to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. This method places the discovery of coordinates and models on an equal footing.
A Bayesian machine scientist to aid in the solution of challenging scientific problems
Closed-form, interpretable mathematical models have been instrumental for advancing our understanding of the world; with the data revolution, we may now be in a position to uncover new such models for many systems from physics to the social sciences. However, to deal with increasing amounts of data, we need “machine scientists” that are able to extract these models automatically from data. Here, we introduce a Bayesian machine scientist, which establishes the plausibility of models using explicit approximations to the exact marginal posterior over models and establishes its prior expectations about models by learning from a large empirical corpus of mathematical expressions. It explores the space of models using Markov chain Monte Carlo. We show that this approach uncovers accurate models for synthetic and real data and provides out-of-sample predictions that are more accurate than those of existing approaches and of other nonparametric methods.
Learning interpretable dynamics of stochastic complex systems from experimental data
Complex systems with many interacting nodes are inherently stochastic and best described by stochastic differential equations. Despite increasing observation data, inferring these equations from empirical data remains challenging. Here, we propose the Langevin graph network approach to learn the hidden stochastic differential equations of complex networked systems, outperforming five state-of-the-art methods. We apply our approach to two real systems: bird flock movement and tau pathology diffusion in brains. The inferred equation for bird flocks closely resembles the second-order Vicsek model, providing unprecedented evidence that the Vicsek model captures genuine flocking dynamics. Moreover, our approach uncovers the governing equation for the spread of abnormal tau proteins in mouse brains, enabling early prediction of tau occupation in each brain region and revealing distinct pathology dynamics in mutant mice. By learning interpretable stochastic dynamics of complex systems, our findings open new avenues for downstream applications such as control.