From scale-free to Erdos-Rényi networks

Deep-learning reconstruction of complex dynamical networks from incomplete data

Reconstructing complex networks and predicting the dynamics are particularly challenging in real-world applications because the available information and data are incomplete. We develop a unified collaborative deep-learning framework consisting of three modules: network inference, state estimation, and dynamical learning. The complete network structure is first inferred and the states of the unobserved nodes are estimated, based on which the dynamical learning module is activated to determine the dynamical evolution rules. An alternating parameter updating strategy is deployed to improve the inference and prediction accuracy. Our framework outperforms baseline methods for synthetic and empirical networks hosting a variety of dynamical processes. A reciprocity emerges between network inference and dynamical prediction: better inference of network structure improves the accuracy of dynamical prediction, and vice versa. We demonstrate the superior performance of our framework on an influenza dataset consisting of 37 US States and a PM2.5 dataset covering 184 cities in China.

Decision Making in Networks: A Model of Awareness Raising

This work investigates how interpersonal interactions among individuals could affect the dynamics of awareness raising. Even though previous studies on mathematical models of awareness in the decision making context demonstrate how the level of awareness results from self-observation impinged by optimal decision selections and external uncertainties, an explicit accounting of interaction among individuals is missing. Here we introduce for the first time a theoretical mathematical framework to evaluate the effect on individual awareness exerted by the interaction with neighbor agents. This task is performed by embedding the single agent model into a graph and allowing different agents to interact by means of suitable coupling functions. The presence of the network allows, from a global point of view, the emergence of diffusion mechanisms for which the population tends to reach homogeneous attractors, and, among them, the one with the highest level of awareness. The structural and behavioral patterns, such as the initial levels of awareness and the relative importance the individual assigns to their own state with respect to others’, may drive real actors to stress effective actions increasing individual and global awareness.

Parameterizing network graph heterogeneity using a modified Weibull distribution

We present a simple method to quantitatively capture the heterogeneity in the degree distribution of a network graph using a single parameter σ . Using an exponential transformation of the shape parameter of the Weibull distribution, this control parameter allows the degree distribution to be easily interpolated between highly symmetric and highly heterogeneous distributions on the unit interval. This parameterization of heterogeneity also recovers several other canonical distributions as intermediate special cases, including the Gaussian, Rayleigh, and exponential distributions. We then outline a general graph generation algorithm to produce graphs with a desired amount of heterogeneity. The utility of this formulation of a heterogeneity parameter is demonstrated with examples relating to epidemiological modeling and spectral analysis.

Conserved Control Path in Multilayer Networks

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that form multilayer networks. In this study, we describe a graph pattern called the conserved control path, which allows us to model a common control structure among different types of relations. We present a practical conserved control path detection method (CoPath), which is based on a maximum-weighted matching, to determine the paths that play the most consistent roles in controlling signal transmission in multilayer networks. As a pragmatic application, we demonstrate that the control paths detected in a multilayered pan-cancer network are statistically more consistent. Additionally, they lead to the effective identification of drug targets, thereby demonstrating their power in predicting key pathways that influence multiple cancers.

Effects of topological characteristics on rhythmic states of the D-dimensional Kuramoto model in complex networks

Synchronization is a ubiquitous phenomenon in engineering and natural ecosystems. While the dynamics of synchronization modeled by the Kuramoto model are commonly studied in two dimensions and the state of dynamic units is characterized by a scalar angle variable, we studied the Kuramoto model generalized to D dimensions in the framework of a complex network and utilized the local synchronous order parameter between the agent and its neighbors as the controllable variable to adjust the coupling strength. Here, we reported that average connectivity of networks affects the time-dependent, rhythmic, cyclic state. Importantly, we found that the level of heterogeneity of networks governs the rhythmic state in the transition process. The analytical treatment for observed scenarios in a D-dimensional Kuramoto model at D=3 was provided. These results offered a platform for a better understanding of time-dependent swarming and flocking dynamics in nature.

Assortativity-induced explosive synchronization in a complex neuronal network

In this study, we consider a scale-free network of nonidentical Chialvo neurons, coupled through electrical synapses. For sufficiently strong coupling, the system undergoes a transition from completely out of phase synchronized to phase synchronized state. The principal focus of this study is to investigate the effect of the degree of assortativity over the synchronization transition process. It is observed that, depending on assortativity, bistability between two asymptotically stable states allows one to develop a hysteresis loop which transforms the phase transition from second order to first order. An expansion in the area of hysteresis loop is noticeable with increasing degree-degree correlation in the network. Our study also reveals that effective frequencies of nodes simultaneously go through a continuous or sudden transition to the synchronized state with the corresponding phases. Further, we examine the robustness of the results under the effect of network size and average degree, as well as diverse frequency setup. Finally, we investigate the dynamical mechanism in the process of generating explosive synchronization. We observe a significant impact of lower degree nodes behind such phenomena: in a positive assortative network the low degree nodes delay the synchronization transition.

Markovian approach to tackle the interaction of simultaneous diseases

The simultaneous emergence of several abrupt disease outbreaks or the extinction of some serotypes of multistrain diseases are fingerprints of the interaction between pathogens spreading within the same population. Here, we propose a general and versatile benchmark to address the unfolding of both cooperative and competitive interacting diseases. We characterize the explosive transitions between the disease-free and the epidemic regimes arising from the cooperation between pathogens and show the critical degree of cooperation needed for the onset of such abrupt transitions. For the competing diseases, we characterize the mutually exclusive case and derive analytically the transition point between the full-dominance phase, in which only one pathogen propagates, and the coexistence regime. Finally, we use this framework to analyze the behavior of the former transition point as the competition between pathogens is relaxed.

Effective approach to epidemic containment using link equations in complex networks

Epidemic containment is a major concern when confronting large-scale infections in complex networks. Many studies have been devoted to analytically understand how to restructure the network to minimize the impact of major outbreaks of infections at large scale. In many cases, the strategies are based on isolating certain nodes, while less attention has been paid to interventions on the links. In epidemic spreading, links inform about the probability of carrying the contagion of the disease from infected to susceptible individuals. Note that these states depend on the full structure of the network, and its determination is not straightforward from the knowledge of nodes' states. Here, we confront this challenge and propose a set of discrete-time governing equations that can be closed and analyzed, assessing the contribution of links to spreading processes in complex networks. Our approach allows a scheme for the containment of epidemics based on deactivating the most important links in transmitting the disease. The model is validated in synthetic and real networks, yielding an accurate determination of epidemic incidence and critical thresholds. Epidemic containment based on link deactivation promises to be an effective tool to maintain functionality of networks while controlling the spread of diseases, such as disease spread through air transportation networks.

Synchronization invariance under network structural transformations

Synchronization processes are ubiquitous despite the many connectivity patterns that complex systems can show. Usually, the emergence of synchrony is a macroscopic observable; however, the microscopic details of the system, as, e.g., the underlying network of interactions, is many times partially or totally unknown. We already know that different interaction structures can give rise to a common functionality, understood as a common macroscopic observable. Building upon this fact, here we propose network transformations that keep the collective behavior of a large system of Kuramoto oscillators invariant. We derive a method based on information theory principles, that allows us to adjust the weights of the structural interactions to map random homogeneous in-degree networks into random heterogeneous networks and vice versa, keeping synchronization values invariant. The results of the proposed transformations reveal an interesting principle; heterogeneous networks can be mapped to homogeneous ones with local information, but the reverse process needs to exploit higher-order information. The formalism provides analytical insight to tackle real complex scenarios when dealing with uncertainty in the measurements of the underlying connectivity structure.

A systems biology and proteomics-based approach identifies SRC and VEGFA as biomarkers in risk factor mediated coronary heart disease

Coronary heart disease (CHD) is the most common cause of death worldwide. The burden of CHD increases with risk factors such as smoking, hypertension, obesity and diabetes. Several studies have demonstrated the association of these classical risk factors with CHD. However, the mechanisms of these associations remain largely unclear due to the complexity of disease pathophysiology and the lack of an integrative approach that fails to provide a definite understanding of molecular linkage. To overcome these problems, we propose a novel systems biology approach that relates causative genes, interactomes and pathways to elucidate the risk factors mediating the molecular mechanisms and biomarkers for feasible diagnosis. The literature was mined to retrieve the causative genes of each risk factor and CHD to construct protein interactomes. The interactomes were examined to identify 298 common molecular signatures. The common signatures were mapped to the tissue network to synthesize a sub-network consisting of 82 proteins. Further, the dissection of the sub-network provides functional modules representing a diverse range of molecular functions, including the AKT/p13k, MAPK and wnt pathways. Also, the prioritization of functional modules identifies SRC, VEGFA and HIF1A as potential candidate markers. Further, we validate these candidates with the existing markers CRP, NOS3 and VCAM1 in the serum of 63 individuals, 33 with CHD and 30 controls, using ELISA. SRC, VEGFA, H1F1A, CRP and NOS3 were significantly altered in patients compared to controls. These results support the utility of these candidate markers for the diagnosis of CHD. Overall, our molecular observations indicate the influence of risk factors in the pathophysiology of CHD and identify serum markers for diagnosis.

A new method to reduce the number of time delays in a network

Time delays may cause dramatic changes to the dynamics of interacting oscillators. Coupled networks of interacting dynamical systems can have unexpected behaviours when the signal between the vertices are time delayed. It has been shown for a very general class of systems that the time delays can be rearranged as long as the total time delay over the constitutive loops of the network is conserved. This fact allows to reduce the number of time delays of the problem without loss of information. There is a theoretical lower bound for this number that can be numerically improved if the time delays are commensurable. Here we propose a formulation of the problem and a numerical method to even further reduce the number of time delays in a network.

Effects of behavioral patterns and network topology structures on Parrondo's paradox

A multi-agent Parrondo's model based on complex networks is used in the current study. For Parrondo's game A, the individual interaction can be categorized into five types of behavioral patterns: the Matthew effect, harmony, cooperation, poor-competition-rich-cooperation and a random mode. The parameter space of Parrondo's paradox pertaining to each behavioral pattern, and the gradual change of the parameter space from a two-dimensional lattice to a random network and from a random network to a scale-free network was analyzed. The simulation results suggest that the size of the region of the parameter space that elicits Parrondo's paradox is positively correlated with the heterogeneity of the degree distribution of the network. For two distinct sets of probability parameters, the microcosmic reasons underlying the occurrence of the paradox under the scale-free network are elaborated. Common interaction mechanisms of the asymmetric structure of game B, behavioral patterns and network topology are also revealed.

Synchronization in networks with multiple interaction layers

The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multilayered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavor in mathematics and physics and has potential applications in several socially relevant topics, such as power grid engineering and neural dynamics. We propose a general framework to assess the 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 by applying it to a network of Rössler oscillators with a double layer of interactions and show that highly rich phenomenology emerges from this. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely arising from the true multilayer structure of the interactions among the units in the network.

Erosion of synchronization: Coupling heterogeneity and network structure

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

Amplitude dynamics favors synchronization in complex networks

In this paper we study phase synchronization in random complex networks of coupled periodic oscillators. In particular, we show that, when amplitude dynamics is not negligible, phase synchronization may be enhanced. To illustrate this, we compare the behavior of heterogeneous units with both amplitude and phase dynamics and pure (Kuramoto) phase oscillators. We find that in small network motifs the behavior crucially depends on the topology and on the node frequency distribution. Surprisingly, the microscopic structures for which the amplitude dynamics improves synchronization are those that are statistically more abundant in random complex networks. Thus, amplitude dynamics leads to a general lowering of the synchronization threshold in arbitrary random topologies. Finally, we show that this synchronization enhancement is generic of oscillators close to Hopf bifurcations. To this aim we consider coupled FitzHugh-Nagumo units modeling neuron dynamics.

Layer-layer competition in multiplex complex networks

The coexistence of multiple types of interactions within social, technological and biological networks has moved the focus of the physics of complex systems towards a multiplex description of the interactions between their constituents. This novel approach has unveiled that the multiplex nature of complex systems has strong influence in the emergence of collective states and their critical properties. Here we address an important issue that is intrinsic to the coexistence of multiple means of interactions within a network: their competition. To this aim, we study a two-layer multiplex in which the activity of users can be localized in each of the layers or shared between them, favouring that neighbouring nodes within a layer focus their activity on the same layer. This framework mimics the coexistence and competition of multiple communication channels, in a way that the prevalence of a particular communication platform emerges as a result of the localization of user activity in one single interaction layer. Our results indicate that there is a transition from localization (use of a preferred layer) to delocalization (combined usage of both layers) and that the prevalence of a particular layer (in the localized state) depends on the structural properties.

Effective centrality and explosive synchronization in complex networks

Synchronization of networked oscillators is known to depend fundamentally on the interplay between the dynamics of the graph's units and the microscopic arrangement of the network's structure. We here propose an effective network whose topological properties reflect the interplay between the topology and dynamics of the original network. On that basis, we are able to introduce the effective centrality, a measure that quantifies the role and importance of each network's node in the synchronization process. In particular, in the context of explosive synchronization, we use such a measure to assess the propensity of a graph to sustain an irreversible transition to synchronization. We furthermore discuss a strategy to induce the explosive behavior in a generic network, by acting only upon a fraction of its nodes.

Effects of degree correlations on the explosive synchronization of scale-free networks

We study the organization of finite-size, large ensembles of phase oscillators networking via scale-free topologies in the presence of a positive correlation between the oscillators' natural frequencies and the network's degrees. Under those circumstances, abrupt transitions to synchronization are known to occur in growing scale-free networks, while the transition has a completely different nature for static random configurations preserving the same structure-dynamics correlation. We show that the further presence of degree-degree correlations in the network structure has important consequences on the nature of the phase transition characterizing the passage from the phase-incoherent to the phase-coherent network state. While high levels of positive and negative mixing consistently induce a second-order phase transition, moderate values of assortative mixing, such as those ubiquitously characterizing social networks in the real world, greatly enhance the irreversible nature of explosive synchronization in scale-free networks. The latter effect corresponds to a maximization of the area and of the width of the hysteretic loop that differentiates the forward and backward transitions to synchronization.

Explosive synchronization with partial degree-frequency correlation

Networks of Kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of their corresponding vertices exhibit so-called explosive synchronization behavior, which is now under intensive investigation. Here we study and discuss explosive synchronization in a situation that has not yet been considered, namely when only a part, typically a small part, of the vertices is subjected to a degree-frequency correlation. Our results show that in order to have explosive synchronization, it suffices to have degree-frequency correlations only for the hubs, the vertices with the highest degrees. Moreover, we show that a partial degree-frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree-frequency correlation would not allow it. We perform a mean-field analysis and our conclusions were corroborated by exhaustive numerical experiments for synthetic networks and also for the undirected and unweighed version of a typical benchmark biological network, namely the neural network of the worm Caenorhabditis elegans. The latter is an explicit example where partial degree-frequency correlation leads to explosive synchronization with hysteresis, in contrast with the fully correlated case, for which no explosive synchronization is observed.

Erosion of synchronization in networks of coupled oscillators

We report erosion of synchronization in networks of coupled phase oscillators, a phenomenon where perfect phase synchronization is unattainable in steady state, even in the limit of infinite coupling. An analysis reveals that the total erosion is separable into the product of terms characterizing coupling frustration and structural heterogeneity, both of which amplify erosion. The latter, however, can differ significantly from degree heterogeneity. Finally, we show that erosion is marked by the reorganization of oscillators according to their node degrees rather than their natural frequencies.

Experimental synchronization of chaos in a large ring of mutually coupled single-transistor oscillators: phase, amplitude, and clustering effects

In this paper, experimental evidence of multiple synchronization phenomena in a large (n = 30) ring of chaotic oscillators is presented. Each node consists of an elementary circuit, generating spikes of irregular amplitude and comprising one bipolar junction transistor, one capacitor, two inductors, and one biasing resistor. The nodes are mutually coupled to their neighbours via additional variable resistors. As coupling resistance is decreased, phase synchronization followed by complete synchronization is observed, and onset of synchronization is associated with partial synchronization, i.e., emergence of communities (clusters). While component tolerances affect community structure, the general synchronization properties are maintained across three prototypes and in numerical simulations. The clusters are destroyed by adding long distance connections with distant notes, but are otherwise relatively stable with respect to structural connectivity changes. The study provides evidence that several fundamental synchronization phenomena can be reliably observed in a network of elementary single-transistor oscillators, demonstrating their generative potential and opening way to potential applications of this undemanding setup in experimental modelling of the relationship between network structure, synchronization, and dynamical properties.

Robustness to noise in synchronization of complex networks

In this report, we investigate dynamical robustness of a complex network to noise injected through one of its nodes. We focus on synchronization of coupled nonlinear systems and, as a special instance, we address the classical consensus protocol for linear integrators. We establish an exact closed-form expression of the synchronization error for the consensus protocol and an approximate result for chaotic units. While structural robustness is known to be significantly affected by attacks targeted to network hubs, our results posit that dynamical robustness is controlled by both the topology of the network and the dynamics of the units. We provide examples where hubs perform better or worse than isolated nodes.

Variation of critical point of aging transition in a networked oscillators system

In this work, we study the variation of critical point in aging transition in a networked system consisting of both active and inactive oscillators. By theoretical analysis and numerical simulations, we show that the critical point of aging transition actually is determined by the (normalized) cross links between active and inactive subpopulations of oscillators. This reveals how specific configuration of active and inactive oscillators in the network can lead to the variation of transition point. In particular, we investigate how different strategies of targeted inactivation influence the transition point based on the theory. Our results theoretically explain why the low-degree nodes are crucial regarding dynamical robustness in such systems.

Degree-dependent network growth: from preferential attachment to explosive percolation

We present a simple model of network growth and solve it by writing the dynamic equations for its macroscopic characteristics such as the degree distribution and degree correlations. This allows us to study carefully the percolation transition using a generating functions theory. The model considers a network with a fixed number of nodes wherein links are introduced using degree-dependent linking probabilities pk. To illustrate the techniques and support our findings using Monte Carlo simulations, we introduce the exemplary linking rule pk∝k-α, with α between -1 and +∞. This parameter may be used to interpolate between different regimes. For negative α, links are most likely attached to high-degree nodes. On the other hand, in case α>0, nodes with low degrees are connected and the model asymptotically approaches a process undergoing explosive percolation.

Analysis of remote synchronization in complex networks

A novel regime of synchronization, called remote synchronization, where the peripheral nodes form a phase synchronized cluster not including the hub, was recently observed in star motifs [Bergner et al., Phys. Rev. E 85, 026208 (2012)]. We show the existence of a more general dynamical state of remote synchronization in arbitrary networks of coupled oscillators. This state is characterized by the synchronization of pairs of nodes that are not directly connected via a physical link or any sequence of synchronized nodes. This phenomenon is almost negligible in networks of phase oscillators as its underlying mechanism is the modulation of the amplitude of those intermediary nodes between the remotely synchronized units. Our findings thus show the ubiquity and robustness of these states and bridge the gap from their recent observation in simple toy graphs to complex networks.

Explosive synchronization in weighted complex networks

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. Given a set of phase oscillators networking with a generic wiring of connections and displaying a generic frequency distribution, we show how combining dynamical local information on frequency mismatches and global information on the graph topology suggests a judicious and yet practical weighting procedure which is able to induce and enhance explosive, irreversible, transitions to synchronization. We report extensive numerical and analytical evidence of the validity and scalability of such a procedure for different initial frequency distributions, for both homogeneous and heterogeneous networks, as well as for both linear and nonlinear weighting functions. We furthermore report on the possibility of parametrically controlling the width and extent of the hysteretic region of coexistence of the unsynchronized and synchronized states.

Parrondo's games based on complex networks and the paradoxical effect

Parrondo’s games were first constructed using a simple tossing scenario, which demonstrates the following paradoxical situation: in sequences of games, a winning expectation may be obtained by playing the games in a random order, although each game (game A or game B) in the sequence may result in losing when played individually. The available Parrondo’s games based on the spatial niche (the neighboring environment) are applied in the regular networks. The neighbors of each node are the same in the regular graphs, whereas they are different in the complex networks. Here, Parrondo’s model based on complex networks is proposed, and a structure of game B applied in arbitrary topologies is constructed. The results confirm that Parrondo’s paradox occurs. Moreover, the size of the region of the parameter space that elicits Parrondo’s paradox depends on the heterogeneity of the degree distributions of the networks. The higher heterogeneity yields a larger region of the parameter space where the strong paradox occurs. In addition, we use scale-free networks to show that the network size has no significant influence on the region of the parameter space where the strong or weak Parrondo’s paradox occurs. The region of the parameter space where the strong Parrondo’s paradox occurs reduces slightly when the average degree of the network increases.

Reexamination of explosive synchronization in scale-free networks: the effect of disassortativity

Previous work [J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011)] has reported that explosive synchronization can be achieved in heterogeneous networks with a microscopic correlation between the structural and dynamical properties of the networks. This phenomenon, however, cannot be observed in all heterogeneous networks even if this structure-dynamics correlation is preserved. It is therefore of particular interest to identify the general topological factors that can induce the first order synchronization transition and to understand the underlying mechanisms. Here we investigate this issue using the scenario of the smooth transformation from homogeneous Erdös-Rènyi networks to heterogeneous Barabàsi-Albert networks. Specifically, we scrutinize how local and global properties of the network change during this process, and how these properties are associated with the emergence of explosive synchronization. We find that the local degree-degree correlation in the network contributes primarily to explosive synchronization, other than the global topological property or starlike subgraphs. We furthermore demonstrate that the degree of disassortative mixing also has a great effect in the presence of explosive synchronization.

Network-based criterion for the success of cooperation in an evolutionary prisoner's dilemma

We consider an evolutionary prisoner's dilemma on a random network. We introduce a simple quantitative network-based parameter and show that it effectively predicts the success of cooperation in simulations on the network. The criterion is shown to be accurate on a variety of networks with degree distributions ranging from regular to Poisson to scale free. The parameter allows for comparisons of random networks regardless of their underlying topology. Finally, we draw analogies between the criterion for the success of cooperation introduced here and existing criteria in other contexts.

Reciprocal interactions out of congestion-free adaptive networks

In this paper we study the jamming transition in complex adaptive networks. We introduce an adaptation mechanism that modifies the weight of the communication channels in order to delay the congestion onset. Adaptivity takes place locally as it is driven by each node of the network. Surprisingly, regardless of the structural properties of the backbone topology, e.g., its degree distribution, the adaptive network can reach optimal functioning provided it allows a reciprocal distribution of the weights. Under this condition, the optimal functioning is achieved through an extensive network reshaping ending up in a highly reciprocal weighted network whose critical onset of congestion is delayed up to its maximum possible value. We also show that, for a given network, the reciprocal weighting obtained from adaptation produce optimal static configurations.

Local synchronization in complex networks of coupled oscillators

We investigate the effects that network topology, natural frequency distribution, and system size have on the path to global synchronization as the overall coupling strength between oscillators is increased in a Kuramoto network. In particular, we study the scenario recently found by Gómez-Gardeñes et al. [Phys. Rev. E 73, 056124 (2006)] in which macroscopic global synchronization emerges through a process whereby many small synchronized clusters form, grow, and merge, eventually leading to a macroscopic giant synchronized component. Our main result is that this scenario is robust to an increase in the number of oscillators or a change in the distribution function of the oscillators' natural frequencies, but becomes less prominent as the number of links per oscillator increases.

The dynamics of network coupled phase oscillators: an ensemble approach

We consider the dynamics of many phase oscillators that interact through a coupling network. For a given network connectivity we further consider an ensemble of such systems where, for each ensemble member, the set of oscillator natural frequencies is independently and randomly chosen according to a given distribution function. We then seek a statistical description of the dynamics of this ensemble. Use of this approach allows us to apply the recently developed ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] to the marginal distribution of the ensemble of states at each node. This, in turn, results in a reduced set of ordinary differential equations determining these marginal distribution functions. The new set facilitates the analysis of network dynamics in several ways: (i) the time evolution of the reduced system of ensemble equations is much smoother, and thus numerical solutions can be obtained much faster by use of longer time steps; (ii) the new set of equations can be used as a basis for obtaining analytical results; and (iii) for a certain type of network, a reduction to a low dimensional description of the entire network dynamics is possible. We illustrate our approach with numerical experiments on a network version of the classical Kuramoto problem, first with a unimodal frequency distribution, and then with a bimodal distribution. In the latter case, the network dynamics is characterized by bifurcations and hysteresis involving a variety of steady and periodic attractors.

Explosive synchronization transitions in scale-free networks

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.

Evolution of microscopic and mesoscopic synchronized patterns in complex networks

Previous studies about synchronization of Kuramoto oscillators in complex networks have shown how local patterns of synchronization emerge differently in homogeneous and heterogeneous topologies. The main difference between the paths to synchronization in both topologies is rooted in the growth of the largest connected component of synchronized nodes when increasing the coupling between the oscillators. Nevertheless, a recent study focusing on this same phenomenon has claimed the contrary, stating that the statistical distribution of synchronized clusters for both types of networks is similar. Here we provide extensive numerical evidences that confirm the original claims, namely, that the microscopic and mesoscopic dynamics of the synchronized patterns indeed follow different routes.

Comment on "Cooperation in an evolutionary prisoner's dilemma on networks with degree-degree correlations"

In a recent publication [Phys. Rev. E 80, 026105 (2009)10.1103/PhysRevE.80.026105], Devlin and Treloar characterized the effects of heterogeneity and degree-degree correlation on cooperation in correlated networks by using standardized variance and a correlated extension of standardized variance. We show that some statements made by them are problematic by considering the relation between these two types of standardized variance with Pearson correlation coefficient.

Dynamical organization towards consensus in the Axelrod model on complex networks

We analyze the dynamics toward cultural consensus in the Axelrod model on scale-free networks. By looking at the microscopic dynamics of the model, we are able to show how culture traits spread across different cultural features. We compare the diffusion at the level of cultural features to the growth of cultural consensus at the global level, finding important differences between these two processes. In particular, we show that even when most of the cultural features have reached macroscopic consensus, there are still no signals of globalization. Finally, we analyze the topology of consensus clusters both for global culture and at the feature level of representation.

Cooperation in an evolutionary prisoner's dilemma on networks with degree-degree correlations

We study the effects of degree-degree correlations on the success of cooperation in an evolutionary prisoner's dilemma played on a random network. When degree-degree correlations are not present, the standardized variance of the network's degree distribution has been shown to be an accurate analytical measure of network heterogeneity that can be used to predict the success of cooperation. In this paper, we use a local-mechanism interpretation of standardized variance to give a generalization to graphs with degree-degree correlations. Two distinct mechanisms are shown to influence cooperation levels on these types of networks. The first is an intrinsic measurement of base-line heterogeneity coming from the network's degree distribution. The second is the increase in heterogeneity coming from the degree-degree correlations present in the network. A strong linear relationship is found between these two parameters and the average cooperation level in an evolutionary prisoner's dilemma on a network.

Diversity of rationality affects the evolution of cooperation

By modifying the Fermi updating rule, we present the diversity of individual rationality to the evolutionary prisoner's dilemma game, and our results shows that this diversity heavily influences the evolution of cooperation. Cluster-forming mechanism of cooperators can either be highly enhanced or severely deteriorated by different distributions of rationality. Slight change in the rationality distribution may transfer the whole system from the global absorbing state of cooperators to that of defectors. Based on mean-field argument, quantitative analysis of the stability of cooperative clusters reveals the critical role played by agents with moderate degree values in the evolution of the whole system. The inspiration from our work may provide us a deeper comprehension toward some social phenomena.

Evolution of cooperation through the heterogeneity of random networks

We use the standardized variance (nu_{st}) of the degree distribution of a random network as an analytic measure of its heterogeneity. We show that nu_{st} accurately predicts, quantitatively, the success of cooperators in an evolutionary prisoner's dilemma. Moreover, we show how the generating functional expression for nu_{st} suggests an intrinsic interpretation for the heterogeneity of the network that helps explain local mechanisms through which cooperators thrive in heterogeneous populations. Finally, we give a simple relationship between nu_{st} , the cooperation level, and the epidemic threshold of a random network that reveals an appealing connection between epidemic disease models and the evolutionary prisoner's dilemma.

How scale-free networks and large-scale collective cooperation emerge in complex homogeneous social systems

We study how heterogeneous degree distributions and large-scale collective cooperation in social networks emerge in complex homogeneous systems by a simple local rule: learning from the best in both strategy selections and linking choices. The prisoner's dilemma game is used as the local dynamics. We show that the social structure may evolve into single-scale, broad-scale, and scale-free (SF) degree distributions for different control parameters. In particular, in a relatively strong-selfish parameter region the SF property can be self-organized in social networks by dynamic evolutions and these SF structures help the whole node community to reach a high level of cooperation under the poor condition of a high selfish intention of individuals.

Synchronizability determined by coupling strengths and topology on complex networks

We investigate in depth the synchronization of coupled oscillators on top of complex networks with different degrees of heterogeneity within the context of the Kuramoto model. In a previous paper [Phys. Rev. Lett. 98, 034101 (2007)], we unveiled how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks. Here, we provide more evidence on this phenomenon, extending the previous work to networks that interpolate between homogeneous and heterogeneous topologies. We also introduce details of the path towards synchronization for the evolution of clustering in the synchronized patterns. Finally, we investigate the synchronization of networks with modular structure and conclude that, in these cases, local synchronization is first attained at the most internal level of organization of modules, progressively evolving to the outer levels as the coupling constant is increased. The present work introduces parameters that are proved to be useful for the characterization of synchronization phenomena in complex networks.

Dynamical organization of cooperation in complex topologies

In this Letter, we study how cooperation is organized in complex topologies by analyzing the evolutionary (replicator) dynamics of the prisoner's dilemma, a two-player game with two available strategies, defection and cooperation, whose payoff matrix favors defection. We show that, asymptotically, the population is partitioned into three subsets: individuals that always cooperate (pure cooperators), always defect (pure defectors), and those that intermittently change their strategy. In fact, the size of the later set is the biggest for a wide range of the "stimulus to defect" parameter. While in homogeneous random graphs pure cooperators are grouped into several clusters, in heterogeneous scale-free (SF) networks they always form a single cluster containing the most connected individuals (hubs). Our results give further insights into why cooperation in SF networks is enhanced.

Paths to synchronization on complex networks

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.