Sandpile cascades on oscillator networks: The BTW model meets Kuramoto.

Cascading failures abound in complex systems and the Bak-Tang-Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a "cascade of larger cascades," which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.

Homophily Based on Few Attributes Can Impede Structural Balance.

Homophily between agents and structural balance in connected triads of agents are complementary mechanisms thought to shape social groups leading to, for instance, consensus or polarization. To capture both processes in a unified manner, we propose a model of pair and triadic interactions. We consider N fully connected agents, where each agent has G underlying attributes, and the similarity between agents in attribute space (i.e., homophily) is used to determine the link weight between them. For structural balance we use a triad-updating rule where only one attribute of one agent is changed intentionally in each update, but this also leads to accidental changes in link weights and even link polarities. The link weight dynamics in the limit of large G is described by a Fokker-Planck equation from which the conditions for a phase transition to a fully balanced state with all links positive can be obtained. This "paradise state" of global cooperation is, however, difficult to achieve requiring G>O(N^{2}) and p>0.5, where the parameter p captures a willingness for consensus. Allowing edge weights to be a consequence of attributes naturally captures homophily and reveals that many real-world social systems would have a subcritical number of attributes necessary to achieve structural balance.

Why understanding multiplex social network structuring processes will help us better understand the evolution of human behavior.

Social scientists have long appreciated that relationships between individuals cannot be described from observing a single domain, and that the structure across domains of interaction can have important effects on outcomes of interest (e.g., cooperation; Durkheim, 1893). One debate explicitly about this surrounds food sharing. Some argue that failing to find reciprocal food sharing means that some process other than reciprocity must be occurring, whereas others argue for models that allow reciprocity to span domains in the form of trade (Kaplan and Hill, 1985.). Multilayer networks, high-dimensional networks that allow us to consider multiple sets of relationships at the same time, are ubiquitous and have consequences, so processes giving rise to them are important social phenomena. The analysis of multi-dimensional social networks has recently garnered the attention of the network science community (Kivelä et al., 2014). Recent models of these processes show how ignoring layer interdependencies can lead one to miss why a layer formed the way it did, and/or draw erroneous conclusions (Górski et al., 2018). Understanding the structuring processes that underlie multiplex networks will help understand increasingly rich data sets, giving more accurate and complete pictures of social interactions.

Interdependent Network Recovery Games.

Recovery of interdependent infrastructure networks in the presence of catastrophic failure is crucial to the economy and welfare of society. Recently, centralized methods have been developed to address optimal resource allocation in postdisaster recovery scenarios of interdependent infrastructure systems that minimize total cost. In real-world systems, however, multiple independent, possibly noncooperative, utility network controllers are responsible for making recovery decisions, resulting in suboptimal decentralized processes. With the goal of minimizing recovery cost, a best-case decentralized model allows controllers to develop a full recovery plan and negotiate until all parties are satisfied (an equilibrium is reached). Such a model is computationally intensive for planning and negotiating, and time is a crucial resource in postdisaster recovery scenarios. Furthermore, in this work, we prove this best-case decentralized negotiation process could continue indefinitely under certain conditions. Accounting for network controllers' urgency in repairing their system, we propose an ad hoc sequential game-theoretic model of interdependent infrastructure network recovery represented as a discrete time noncooperative game between network controllers that is guaranteed to converge to an equilibrium. We further reduce the computation time needed to find a solution by applying a best-response heuristic and prove bounds on ε-Nash equilibrium, where ε depends on problem inputs. We compare best-case and ad hoc models on an empirical interdependent infrastructure network in the presence of simulated earthquakes to demonstrate the extent of the tradeoff between optimality and computational efficiency. Our method provides a foundation for modeling sociotechnical systems in a way that mirrors restoration processes in practice.

Koopman operator and its approximations for systems with symmetries.

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, often described by complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces a block diagonal structure in operator approximations, revealing hidden organization. Practically, if the symmetries are known, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise.

Maximizing synchronizability of duplex networks.

We study the synchronizability of duplex networks formed by two randomly generated network layers with different patterns of interlayer node connections. According to the master stability function, we use the smallest nonzero eigenvalue and the eigenratio between the largest and the second smallest eigenvalues of supra-Laplacian matrices to characterize synchronizability on various duplexes. We find that the interlayer linking weight and linking fraction have a profound impact on synchronizability of duplex networks. The increasingly large inter-layer coupling weight is found to cause either decreasing or constant synchronizability for different classes of network dynamics. In addition, negative node degree correlation across interlayer links outperforms positive degree correlation when most interlayer links are present. The reverse is true when a few interlayer links are present. The numerical results and understanding based on these representative duplex networks are illustrative and instructive for building insights into maximizing synchronizability of more realistic multiplex networks.

Complex Dynamical Networks Constructed with Fully Controllable Nonlinear Nanomechanical Oscillators.

Control of the global parameters of complex networks has been explored experimentally in a variety of contexts. Yet, the more difficult prospect of realizing arbitrary network architectures, especially analog physical networks that provide dynamical control of individual nodes and edges, has remained elusive. Given the vast hierarchy of time scales involved, it also proves challenging to measure a complex network's full internal dynamics. These span from the fastest nodal dynamics to very slow epochs over which emergent global phenomena, including network synchronization and the manifestation of exotic steady states, eventually emerge. Here, we demonstrate an experimental system that satisfies these requirements. It is based upon modular, fully controllable, nonlinear radio frequency nanomechanical oscillators, designed to form the nodes of complex dynamical networks with edges of arbitrary topology. The dynamics of these oscillators and their surrounding network are analog and continuous-valued and can be fully interrogated in real time. They comprise a piezoelectric nanomechanical membrane resonator, which serves as the frequency-determining element within an electrical feedback circuit. This embodiment permits network interconnections entirely within the electrical domain and provides unprecedented node and edge control over a vast region of parameter space. Continuous measurement of the instantaneous amplitudes and phases of every constituent oscillator node are enabled, yielding full and detailed network data without reliance upon statistical quantities. We demonstrate the operation of this platform through the real-time capture of the dynamics of a three-node ring network as it evolves from the uncoupled state to full synchronization.

Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators.

Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest-neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first analytically identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators, we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators, we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture.

Suppressing cascades of load in interdependent networks.

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak-Tang-Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.

Microtransition cascades to percolation.

We report the discovery of a discrete hierarchy of microtransitions occurring in models of continuous and discontinuous percolation. The precursory microtransitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase transitions in complex systems.

What you have, not who you know: food-enhanced social capital and changes in social behavioural relationships in a non-human primate.

Social network position in non-human primates has far-reaching fitness consequences. Critically, social networks are both heterogeneous and dynamic, meaning an individual's current network position is likely to change due to both intrinsic and extrinsic factors. However, our understanding of the drivers of changes in social network position is largely confined to opportunistic studies. Experimental research on the consequences of in situ, controlled network perturbations is limited. Here we conducted a food-based experiment in rhesus macaques to assess whether allowing an individual the ability to provide high-quality food to her group changed her social behavioural relationships. We considered both her social network position across five behavioural networks, as well as her dominance and kin interactions. We found that gaining control over a preferential food resource had far-reaching social consequences. There was an increase in both submission and aggression centrality and changes in the socio-demographic characteristics of her agonistic interaction partners. Further, we found that her grooming balance shifted in her favour as she received more grooming than she gave. Together, these results provide a novel, preliminary insight into how in situ, experimental manipulations can modify social network position and point to broader network-level shifts in both social capital and social power.

Controlling self-organizing dynamics on networks using models that self-organize.

Controlling self-organizing systems is challenging because the system responds to the controller. Here, we develop a model that captures the essential self-organizing mechanisms of Bak-Tang-Wiesenfeld (BTW) sandpiles on networks, a self-organized critical (SOC) system. This model enables studying a simple control scheme that determines the frequency of cascades and that shapes systemic risk. We show that optimal strategies exist for generic cost functions and that controlling a subcritical system may drive it to criticality. This approach could enable controlling other self-organizing systems.

Multi-scale phase separation by explosive percolation with single-chromatin loop resolution.

The 2 m-long human DNA is tightly intertwined into the cell nucleus of the size of 10 µm. The DNA packing is explained by folding of chromatin fiber. This folding leads to the formation of such hierarchical structures as: chromosomal territories, compartments; densely-packed genomic regions known as Topologically Associating Domains (TADs), or Chromatin Contact Domains (CCDs), and loops. We propose models of dynamical human genome folding into hierarchical components in human lymphoblastoid, stem cell, and fibroblast cell lines. Our models are based on explosive percolation theory. The chromosomes are modeled as graphs where CTCF chromatin loops are represented as edges. The folding trajectory is simulated by gradually introducing loops to the graph following various edge addition strategies that are based on topological network properties, chromatin loop frequencies, compartmentalization, or epigenomic features. Finally, we propose the genome folding model - a biophysical pseudo-time process guided by a single scalar order parameter. The parameter is calculated by Linear Discriminant Analysis of chromatin features. We also include dynamics of loop formation by using Loop Extrusion Model (LEM) while adding them to the system. The chromatin phase separation, where fiber folds in 3D space into topological domains and compartments, is observed when the critical number of contacts is reached. We also observe that at least 80% of the loops are needed for chromatin fiber to condense in 3D space, and this is constant through various cell lines. Overall, our in-silico model integrates the high-throughput 3D genome interaction experimental data with the novel theoretical concept of phase separation, which allows us to model event-based time dynamics of chromatin loop formation and folding trajectories.

Explosive percolation with multiple giant components.

We generalize the random graph evolution process of Bohman, Frieze, and Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms, 25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete graph, are considered one at a time and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function asymptotically approaching the value α=1/2. We show that multiple giant components appear simultaneously in a strongly discontinuous percolation transition and remain distinct. Furthermore, tuning the value of α determines the number of such components with smaller α leading to an increasingly delayed and more explosive transition. The location of the critical point and strongly discontinuous nature are not affected if only edges which span components are sampled.

Analysis of early host responses for asymptomatic disease detection and management of specialty crops.

The rapid and unabated spread of vector-borne diseases within US specialty crops threatens our agriculture, our economy, and the livelihood of growers and farm workers. Early detection of vector-borne pathogens is an essential step for the accurate surveillance and management of vector-borne diseases of specialty crops. Currently, we lack the tools that would detect the infectious agent at early (primary) stages of infection with a high degree of sensitivity and specificity. In this paper, we outline a strategy for developing an integrated suite of platform technologies to enable rapid, early disease detection and diagnosis of huanglongbing (HLB), the most destructive citrus disease. The research has two anticipated outcomes: i) identification of very early, disease-specific biomarkers using a knowledge base of translational genomic information on host and pathogen responses associated with early (asymptomatic) disease development; and ii) development and deployment of novel sensors that capture these and other related biomarkers and aid in presymptomatic disease detection. By combining these two distinct approaches, it should be possible to identify and defend the crop by interdicting pathogen spread prior to the rapid expansion phase of the disease. We believe that similar strategies can also be developed for the surveillance and management of diseases affecting other economically important specialty crops.

Local cluster aggregation models of explosive percolation.

We introduce perhaps the simplest models of graph evolution with choice that demonstrate discontinuous percolation transitions and can be analyzed via mathematical evolution equations. These models are local, in the sense that at each step of the process one edge is selected from a small set of potential edges sharing common vertices and added to the graph. We show that the evolution can be accurately described by a system of differential equations and that such models exhibit the discontinuous emergence of the giant component. Yet they also obey scaling behaviors characteristic of continuous transitions, with scaling exponents that differ from the classic Erdos-Rényi model.

Mutualistic networks emerging from adaptive niche-based interactions.

Mutualistic networks are vital ecological and social systems shaped by adaptation and evolution. They involve bipartite cooperation via the exchange of goods or services between actors of different types. Empirical observations of mutualistic networks across genres and geographic conditions reveal correlated nested and modular patterns. Yet, the underlying mechanism for the network assembly remains unclear. We propose a niche-based adaptive mechanism where both nestedness and modularity emerge simultaneously as complementary facets of an optimal niche structure. Key dynamical properties are revealed at different timescales. Foremost, mutualism can either enhance or reduce the network stability, depending on competition intensity. Moreover, structural adaptations are asymmetric, exhibiting strong hysteresis in response to environmental change. Finally, at the evolutionary timescale we show that the adaptive mechanism plays a crucial role in preserving the distinctive patterns of mutualism under species invasions and extinctions.

A multiplex centrality metric for complex social networks: sex, social status, and family structure predict multiplex centrality in rhesus macaques.

Members of a society interact using a variety of social behaviors, giving rise to a multi-faceted and complex social life. For the study of animal behavior, quantifying this complexity is critical for understanding the impact of social life on animals' health and fitness. Multilayer network approaches, where each interaction type represents a different layer of the social network, have the potential to better capture this complexity than single layer approaches. Calculating individuals' centrality within a multilayer social network can reveal keystone individuals and more fully characterize social roles. However, existing measures of multilayer centrality do not account for differences in the dynamics and functionality across interaction layers. Here we validate a new method for quantifying multiplex centrality called consensus ranking by applying this method to multiple social groups of a well-studied nonhuman primate, the rhesus macaque. Consensus ranking can suitably handle the complexities of animal social life, such as networks with different properties (sparse vs. dense) and biological meanings (competitive vs. affiliative interactions). We examined whether individuals' attributes or socio-demographic factors (sex, age, dominance rank and certainty, matriline size, rearing history) were associated with multiplex centrality. Social networks were constructed for five interaction layers (i.e., aggression, status signaling, conflict policing, grooming and huddling) for seven social groups. Consensus ranks were calculated across these five layers and analyzed with respect to individual attributes and socio-demographic factors. Generalized linear mixed models showed that consensus ranking detected known social patterns in rhesus macaques, showing that multiplex centrality was greater in high-ranking males with high certainty of rank and females from the largest families. In addition, consensus ranks also showed that females from very small families and mother-reared (compared to nursery-reared) individuals were more central, showing that consideration of multiple social domains revealed individuals whose social centrality and importance might otherwise have been missed.

Cascading failures in scale-free interdependent networks.

Large cascades are a common occurrence in many natural and engineered complex systems. In this paper we explore the propagation of cascades across networks using realistic network topologies, such as heterogeneous degree distributions, as well as intra- and interlayer degree correlations. We find that three properties, scale-free degree distribution, internal network assortativity, and cross-network hub-to-hub connections, are all necessary components to significantly reduce the size of large cascades in the Bak-Tang-Wiesenfeld sandpile model. We demonstrate that correlations present in the structure of the multilayer network influence the dynamical cascading process and can prevent failures from spreading across connected layers. These findings highlight the importance of internal and cross-network topology in optimizing robustness of interconnected systems.