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Explosive transitions induced by interdependent contagion-consensus dynamics in multiplex networks
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Abstract
We introduce a model to study the delicate relation between the spreading of information and the formation of opinions in social systems. For this purpose, we propose a two-layer multiplex network model in which consensus dynamics takes place in one layer while information spreading runs across the other one. The two dynamical processes are mutually coupled by considering that the control parameters that govern the dynamical evolution of the state of the nodes inside each layer depend on the dynamical states at the other layer. In particular, we explore the scenario in which consensus is favored by the common adoption of information while information spreading is boosted between agents sharing similar opinions. Numerical simulations together with some analytical results point out that, when the coupling between the dynamics of the two layers is strong enough, a double explosive transition, i.e. an explosive transition both for consensus dynamics and for the information spreading appears. Such explosive transitions lead to bi-stability regions in which the consensus-informed stated and the disagreement-ignorant states are both stable solutions.
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We propose a theoretical framework for the study of spreading processes in structured metapopulations, with heterogeneous agents, subjected to different recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze classical epidemic models within this framework and obtain an excellent agreement with extensive Monte Carlo simulations. This agreement allows us to derive analytical expressions of the epidemic threshold and to face the challenge of characterizing a real multiplex metapopulation, the city of Medellín in Colombia, where different recurrent mobility patterns are observed depending on the socioeconomic class of the agents. Our framework allows us to unveil the geographical location of those patches that trigger the epidemic state at the critical point. A careful exploration reveals that social mixing between classes and mobility crucially determines these critical patches and, more importantly, it can produce abrupt changes of the critical properties of the epidemic onset.
The containment of epidemic spreading is a major challenge in science. Vaccination, whenever available, is the best way to prevent the spreading, because it eventually immunizes individuals. However, vaccines are not perfect, and total immunization is not guaranteed. Imperfect immunization has driven the emergence of anti-vaccine movements that totally alter the predictions about the epidemic incidence. Here, we propose a mathematically solvable mean-field vaccination model to mimic the spontaneous adoption of vaccines against influenza-like diseases, and the expected epidemic incidence. The results are in agreement with extensive Monte Carlo simulations of the epidemics and vaccination co-evolutionary processes. Interestingly, the results reveal a non-monotonic behavior on the vaccination coverage, that increases with the imperfection of the vaccine and after decreases. This apparent counterintuitive behavior is analyzed and understood from stability principles of the proposed mathematical model.
Multilayer and multiplex networks represent a good proxy for the description of social phenomena where social structure is important and can have different origins. Here, we propose a model of opinion competition where individuals are organized according to two different structures in two layers. Agents exchange opinions according to the Abrams–Strogatz model in each layer separately and opinions can be copied across layers by the same individual. In each layer a different opinion is dominant, so each layer has a different absorbing state. Consensus in one opinion is not the only possible stable solution because of the interaction between the two layers. A new mean field solution has been found where both opinions coexist. In a finite system there is a long transient time for the dynamical coexistence of both opinions. However, the system ends in a consensus state due to finite size effects. We analyze sparse topologies in the two layers and the existence of positive correlations between them, which enables the coexistence of inter-layer groups of agents sharing the same opinion.
From critical infrastructure, to physiology and the human brain, complex systems rarely occur in isolation. Instead, the functioning of nodes in one system often promotes or suppresses the functioning of nodes in another. Despite advances in structural interdependence, modeling interdependence and other interactions between dynamic systems has proven elusive. Here we define a broadly applicable dynamic dependency link and develop a general framework for interdependent and competitive interactions between general dynamic systems. We apply our framework to studying interdependent and competitive synchronization in multi-layer oscillator networks and cooperative/competitive contagions in an epidemic model. Using a mean-field theory which we verify numerically, we find explosive transitions and rich behavior which is absent in percolation models including hysteresis, multi-stability and chaos. The framework presented here provides a powerful new way to model and understand many of the interacting complex systems which surround us.
We introduce a framework to intertwine dynamical processes of different nature, each with its own distinct network topology, using a multilayer network approach. As an example of collective phenomena emerging from the interactions of multiple dynamical processes, we study a model where neural dynamics and nutrient transport are bidirectionally coupled in such a way that the allocation of the transport process at one layer depends on the degree of synchronization at the other layer, and vice versa. We show numerically, and we prove analytically, that the multilayer coupling induces a spontaneous explosive synchronization and a heterogeneous distribution of allocations, otherwise not present in the two systems considered separately. Our framework can find application to other cases where two or more dynamical processes such as synchronization, opinion formation, information diffusion, or disease spreading, are interacting with each other.
The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multi-layered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavour in mathematics and physics, and has potential applications to several societally relevant topics, such as power grids engineering and neural dynamics. We propose a general framework to assess stability of the synchronized state in networks with multiple interaction layers, deriving a necessary condition that generalizes the Master Stability Function approach. We validate our method applying it to a network of R\"ossler oscillators with a double layer of interactions, and show that highly rich phenomenology emerges. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely due to the true multi-layer structure of the interactions amongst the units in the network.
We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.
The new concept of multilevel network is introduced in order to embody some topological properties of complex systems with structures in the mesoscale which are not completely captured by the classical models. This new model, which generalizes the hyper-network and hyper-structure models, fits perfectly with several real-life complex systems, including social and public transportation networks. We present an analysis of the structural properties of the multilevel network, including the clustering and the metric structures. Some analytical relationships amongst the efficiency and clustering coefficient of this new model and the corresponding parameters of the underlying network are obtained. Finally some random models for multilevel networks are given to illustrate how different multilevel structures can produce similar underlying networks and therefore that the mesoscale structure should be taken into account in many applications.
Explosive synchronization (ES) is nowadays a hot topic of interest in
nonlinear science and complex networks. So far, it is conjectured that ES is
rooted in the setting of specific microscopic correlation features between the
natural frequencies of the networked oscillators and their effective coupling
strengths. We show that ES, in fact, is far more general, and can occur in
adaptive and multilayer networks also in the absence of such correlation
properties. Precisely, we first report evidence of ES in the absence of
correlation for networks where a fraction f of the nodes have links adaptively
controlled by a local order parameter, and then we extend the study to a
variety of two-layer networks with a fraction f of their nodes coupled each
other by means of dependency links. In this latter case, we even show that ES
sets in, regardless of the differences in the frequency distribution and/or in
the topology of connections between the two layers. Finally, we provide a
rigorous, analytical, treatment to properly ground all the observed scenario,
and to facilitate the understanding of the actual mechanisms at the basis of ES
in real-world systems.
We study synchronization of N oscillators indirectly coupled through a
medium which is inhomogeneous and has its own dynamics. The system is
formalized in terms of a multilayer network, where the top layer is made of
disconnected oscillators and the bottom one, modeling the medium, consists of
oscillators coupled according to a given topology. The different dynamics of
the medium and the top layer is accounted by including a frequency mismatch
between them. We show a novel regime of synchronization as intra-layer
coherence does not necessarily require inter-layer coherence. This regime
appears under mild conditions on the bottom layer: arbitrary topologies may be
considered, provided that they support synchronization of the oscillators of
the medium. The existence of a density-dependent threshold as in quorum-sensing
phenomena is also demonstrated.
Explosive synchronization and explosive percolation are currently two independent phenomena occurring in complex networks, where the former takes place in dynamical phase space while the latter in configuration space. It has been revealed that the mechanism of EP can be explained by the Achlioptas process, where the formation of a giant component is controlled by a suppressive rule. We here introduce an equivalent suppressive rule for ES. Before the critical point of ES, the suppressive rule induces the presence of multiple, small sized, synchronized clusters, while inducing the abrupt formation of a giant cluster of synchronized oscillators at the critical coupling strength. We also show how the explosive character of ES degrades into a second-order phase transition when the suppressive rule is broken. These results suggest that our suppressive rule can be considered as a dynamical counterpart of the Achlioptas process, indicating that ES and EP can be unified into a same framework.
We present the analysis of the interrelation between two processes accounting for the spreading of an epidemic, and the information awareness to prevent its infection, on top of multiplex networks. This scenario is representative of an epidemic process spreading on a network of persistent real contacts, and a cyclic information awareness process diffusing in the network of virtual social contacts between the same individuals. The topology corresponds to a multiplex network where two diffusive processes are interacting affecting each other. The analysis using a microscopic Markov chain approach reveals the phase diagram of the incidence of the epidemics and allows us to capture the evolution of the epidemic threshold depending on the topological structure of the multiplex and the interrelation with the awareness process. Interestingly, the critical point for the onset of the epidemics has a critical value (metacritical point) defined by the awareness dynamics and the topology of the virtual network, from which the onset increases and the epidemics incidence decreases.
Most real and engineered systems include multiple subsystems and layers of
connectivity, and it is important to take such features into account to try to
improve our understanding of these systems. It is thus necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community detection,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.
A network representation is useful for describing the structure of a large
variety of complex systems. However, most real and engineered systems have
multiple subsystems and layers of connectivity, and the data produced by such
systems is very rich. Achieving a deep understanding of such systems
necessitates generalizing "traditional" network theory, and the newfound deluge
of data now makes it possible to test increasingly general frameworks for the
study of networks. In particular, although adjacency matrices are useful to
describe traditional single-layer networks, such a representation is
insufficient for the analysis and description of multiplex and time-dependent
networks. One must therefore develop a more general mathematical framework to
cope with the challenges posed by multi-layer complex systems. In this paper,
we introduce a tensorial framework to study multi-layer networks, and we
discuss the generalization of several important network descriptors and
dynamical processes --including degree centrality, clustering coefficients,
eigenvector centrality, modularity, Von Neumann entropy, and diffusion-- for
this framework. We examine the impact of different choices in constructing
these generalizations, and we illustrate how to obtain known results for the
special cases of single-layer and multiplex networks. Our tensorial approach
will be helpful for tackling pressing problems in multi-layer complex systems,
such as inferring who is influencing whom (and by which media) in multichannel
social networks and developing routing techniques for multimodal transportation
systems.
An imperative condition for the functioning of a power-grid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of a power grid is stable, and use this condition to identify tunable parameters of the generators that are determinants of spontaneous synchronization. Our analysis gives rise to an approach to specify parameter assignments that can enhance synchronization of any given network, which we demonstrate for a selection of both test systems and real power grids. Because our results concern spontaneous synchronization, they are relevant both for reducing dependence on conventional control devices, thus offering an additional layer of protection given that most power outages involve equipment or operational errors, and for contributing to the development of "smart grids" that can recover from failures in real time.
We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.
We study evolutionary game dynamics on structured populations in which individuals take part in several layers of networks of interactions simultaneously. This multiplex of interdependent networks accounts for the different kind of social ties each individual has. By coupling the evolutionary dynamics of a Prisoner's Dilemma game in each of the networks, we show that the resilience of cooperative behaviors for extremely large values of the temptation to defect is enhanced by the multiplex structure. Furthermore, this resilience is intrinsically related to a non-trivial organization of cooperation across the network layers, thus providing a new way out for cooperation to survive in structured populations.
Since roughly a decade ago, network science has focused among others on the problem of how the spreading of diseases depends on structural patterns. Here, we contribute to further advance our understanding of epidemic spreading processes by proposing a nonperturbative formulation of the heterogeneous mean-field approach that has been commonly used in the physics literature to deal with this kind of spreading phenomena. The nonperturbative equations we propose have no assumption about the proximity of the system to the epidemic threshold, nor any linear approximation of the dynamics. In particular, we first develop a probabilistic description at the node level of the epidemic propagation for the so-called susceptible-infected-susceptible family of models, and after we derive the corresponding heterogeneous mean-field approach. We propose to use the full extension of the approach instead of pruning the expansion to first order, which leads to a nonperturbative formulation that can be solved by fixed-point iteration, and used with reliability far away from the epidemic threshold to assess the prevalence of the epidemics. Our results are in close agreement with Monte Carlo simulations, thus enhancing the predictive power of the classical heterogeneous mean-field approach, while providing a more effective framework in terms of computational time.
Explosive collective phenomena have attracted much attention since the discovery of an explosive percolation transition. In this Letter, we demonstrate how an explosive transition shows up in the synchronization of scale-free networks by incorporating a microscopic correlation between the structural and the dynamical properties of the system. The characteristics of the explosive transition are analytically studied in a star graph reproducing the results obtained in synthetic networks. Our findings represent the first abrupt synchronization transition in complex networks and provide a deeper understanding of the microscopic roots of explosive critical phenomena.
We use the annealed formulation of complex networks to study the dynamical behavior of disease spreading on both static and adaptive networked systems. This unifying approach relies on the annealed adjacency matrix, representing one network ensemble, and allows to solve the dynamical evolution of the whole network ensemble all at once. Our results accurately reproduce those obtained by extensive numerical simulations showing a large improvement with respect to the usual heterogeneous mean-field formulation. Moreover, by means of the annealed formulation we derive a new heterogeneous mean-field formulation that correctly reproduces the epidemic dynamics.
Many epidemic processes in networks spread by stochastic contacts among their connected vertices. There are two limiting cases widely analyzed in the physics literature, the so-called contact process (CP) where the contagion is expanded at a certain rate from an infected vertex to one neighbor at a time, and the reactive process (RP) in which an infected individual effectively contacts all its neighbors to expand the epidemics. However, a more realistic scenario is obtained from the interpolation between these two cases, considering a certain number of stochastic contacts per unit time. Here we propose a discrete-time formulation of the problem of contact-based epidemic spreading. We resolve a family of models, parameterized by the number of stochastic contact trials per unit time, that range from the CP to the RP. In contrast to the common heterogeneous mean-field approach, we focus on the probability of infection of individual nodes. Using this formulation, we can construct the whole phase diagram of the different infection models and determine their critical properties. Comment: 6 pages, 4 figures. Europhys Lett (in press 2010)
Human behaviour plays an important role in the spread of infectious diseases, and understanding the influence of behaviour on the spread of diseases can be key to improving control efforts. While behavioural responses to the spread of a disease have often been reported anecdotally, there has been relatively little systematic investigation into how behavioural changes can affect disease dynamics. Mathematical models for the spread of infectious diseases are an important tool for investigating and quantifying such effects, not least because the spread of a disease among humans is not amenable to direct experimental study. Here, we review recent efforts to incorporate human behaviour into disease models, and propose that such models can be broadly classified according to the type and source of information which individuals are assumed to base their behaviour on, and according to the assumed effects of such behaviour. We highlight recent advances as well as gaps in our understanding of the interplay between infectious disease dynamics and human behaviour, and suggest what kind of data taking efforts would be helpful in filling these gaps.
Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks.
Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.
Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a ‘mixing network’. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks—and how these deviate from the random-mixing norm—have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures.
Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.
The understanding of emergent collective phenomena in natural and social systems has driven the interest of scientists from different disciplines during decades. Among these phenomena, the synchronization of a set of interacting individuals or units has been intensively studied because of its ubiquity in the natural world. In this Letter, we show how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks, driving the process towards a certain global synchronization degree following different paths. The dependence of the dynamics on the coupling strength and on the topology is unveiled. This study provides a new perspective and tools to understand this emerging phenomena.
Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems. Comment: 58 pages, 22 figures. Final version published in Reviews of Modern Physics
Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in com- plex networks' structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the network's structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks' dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and de- bates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.
Despite the large number of studies on synchronization, the hypothesis that interactions bear a cost for involved individuals has been considered seldom. The introduction of costly interactions leads, instead, to the formulation of a dichotomous scenario in which an individual may decide to cooperate and pay the cost in order to get synchronized with the rest of the population. Alternatively, the same individual can decide to free ride, without incurring in any cost, waiting that others get synchronized to her state. The emergence of synchronization may thus be seen as the byproduct of an evolutionary game in which individuals decide their behavior according to the benefit/cost ratio they accrue in the past. We study the onset of cooperation/synchronization in networked populations of Kuramoto oscillators and report how topology is essential in order for cooperation to thrive. We display also how different classes of topology foster differently synchronization both at a microscopic and macroscopic level.
The strength of weak ties is that they tend to be long - they connect socially distant locations, allowing information to diffuse rapidly. The authors test whether this "strength of weak ties" generalizes from simple to complex contagions. Complex contagions require social affirmation from multiple sources. Examples include the spread of high-risk social movements, avant garde fashions, and unproven technologies. Results show that as adoption thresholds increase, long ties can impede diffusion. Complex contagions depend primarily on the width of the bridges across a network, not just their length. Wide bridges are a characteristic feature of many spatial networks, which may account in part for the widely observed tendency for social movements to diffuse spatially.
We develop a theoretical framework for the study of epidemic-like social
contagion in large scale social systems. We consider the most general setting
in which different communication platforms or categories form multiplex
networks. Specifically, we propose a contact-based information spreading model,
and show that the critical point of the multiplex system associated to the
active phase is determined by the layer whose contact probability matrix has
the largest eigenvalue. The framework is applied to a number of different
situations, including a real multiplex system. Finally, we also show that when
the system through which information is disseminating is inherently multiplex,
working with the graph that results from the aggregation of the different
layers is flawed.
Despite tendencies toward convergence, differences between individuals and groups continue to exist in beliefs, attitudes, and behavior. An agent-based adaptive model reveals the effects of a mechanism of convergent social influence. The actors are placed at fixed sites. The basic premise is that the more similar an actor is to a neighbor, the more likely that that actor will adopt one of the neighbor's traits. Unlike previous models of social influence or cultural change that treat features one at a time, the proposed model takes into account the interaction between different features. The model illustrates how local convergence can generate global polarization. Simulations show that the number of stable homogeneous regions decreases with the number of features, increases with the number of alternative traits per feature, decreases with the range of interaction, and (most surprisingly) decreases when the geographic territory grows beyond a certain size.
Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics.
Many systems take the form of networks, including the Internet, distribution and transport networks, neural networks, food webs, and social networks. The characterization and modeling of these systems has proved amenable to treatment using techniques drawn from statistical and computational physics, and has as a result attracted considerable attention in the physics literature in recent years. In this paper the author reviews some of the interesting issues in this area and recounts some recent work on these issues by himself and by others.
Modern network-like systems are usually coupled in such a way that
failures in one network can affect the entire system. In
infrastructures, biology, sociology, and economy, systems are
interconnected and events taking place in one system can propagate to
any other coupled system. Recent studies on such coupled systems show
that the coupling increases their vulnerability to random failure.
Properties for interdependent networks differ significantly from those
of single-network systems. In this article, these results are reviewed
and the main properties discussed.
Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences. Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com/
Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, we discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, as well as the interplay between topology and the network's robustness against failures and attacks. Comment: 54 pages, submitted to Reviews of Modern Physics
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