Chaotic dynamics on large networks

Many systems in nature are governed by a large number of agents that interact nonlinearly through complex feedback loops. When the networks are sufficiently large and interconnected, they typically exhibit self-organization and chaos. This paper examines the prevalence and degree of chaos on large unweighted recurrent networks of ordinary differential equations with sigmoidal nonlinearities and unit coupling. The largest Lyapunov exponent is used as the signature and measure of the chaos, and the study includes the effects of damping, asymmetries in the distribution of coupling strengths, network symmetry, and sparseness of connections. Minimum conditions and optimal network architectures are determined for the existence of chaos. The results have implications for the design of social and other networks in the real world in which weak chaos is deemed desirable or as a way of understanding why certain networks might exist on "the edge of chaos."     

Opinion dynamics on directed small-world networks

In this paper, we investigate the self-affirmation effect on formation of public opinion in a directed small-world social network. The system presents a non-equilibrium phase transition from a consensus state to a disordered state with coexistence of opinions. The dynamical behaviors are very sensitive to the density of long-range-directed interactions and the strength of self-affirmation. When the long-range-directed interactions are sparse and individual generally does not insist on his/her opinion, the system will display a continuous phase transition, in the opposite case with strong self-affirmation and dense long-range-directed interactions, the system does not display a phase transition. Between those two extreme cases, the system undergoes a discontinuous phase transition.     

Traffic dynamics on layered complex networks

To minimize traffic congestion, understanding how traffic dynamics depend on network structure is necessary. Many real-world complex systems can be described as multilayer structures. In this paper, we introduce the idea of layers to establish a traffic model of two-layer complex networks. By comparing different two-layer complex networks based on random and scale-free networks, we find that the physical layer is much more important to the network capacity of two-layer complex networks than the logical layer. Two-layer complex networks with a homogeneous physical topology are found to be more tolerant to congestion. Moreover, simulation results show that the heterogeneity of logical and physical topologies makes the packet-delivery process of two-layer networks more efficient in the free-flow state, without the occurrence of traffic congestion.     

The linear noise approximation for reaction-diffusion systems on networks

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays travelling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.     

Synchronization of coupled chaotic dynamics on networks

We review some recent work on the synchronization of coupled dynamical systems on a variety of networks. When nodes show synchronized behaviour, two interesting phenomena can be observed. First, there are some nodes of the floating type that show intermittent behaviour between getting attached to some clusters and evolving independently. Secondly, two different ways of cluster formation can be identified, namely self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings.     

Topological stabilization for synchronized dynamics on networks

A general scheme is proposed and tested to control the symmetry breaking instability of a homogeneous solution of a spatially extended multispecies model, defined on a network. The inherent discreteness of the space makes it possible to act on the topology of the inter-nodes contacts to achieve the desired degree of stabilization, without altering the dynamical parameters of the model. Both symmetric and asymmetric couplings are considered. In this latter setting the web of contacts is assumed to be balanced, for the homogeneous equilibrium to exist. The performance of the proposed method are assessed, assuming the Complex Ginzburg-Landau equation as a reference model. In this case, the implemented control allows one to stabilize the synchronous limit cycle, hence time-dependent, uniform solution. A system of coupled real Ginzburg-Landau equations is also investigated to obtain the topological stabilization of a homogeneous and constant fixed point.     

Adaptive synchronization of dynamics on evolving complex networks

We study the problem of synchronizing a general complex network by means of an adaptive strategy in the case where the network topology is slowly time varying and every node receives at each time only one aggregate signal from the set of its neighbors. We introduce an appropriately defined potential that each node seeks to minimize in order to reach or maintain synchronization. We show that our strategy is effective in tracking synchronization as well as in achieving synchronization when appropriate conditions are met.     

Synchronized state of coupled dynamics on time-varying networks

We consider synchronization properties of coupled dynamics on time-varying networks and the corresponding time-average network. We find that if the different Laplacians corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand for noncommuting Laplacians the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.     

Cross-link-governed dynamics of biopolymer networks

Recent experiments show that networks of stiff biopolymers cross-linked by transient linker proteins exhibit complex stress relaxation, enabling network flow at long times. We present a model for the dynamics controlled by cross-links in such networks. We show that a single microscopic time scale for cross-linker unbinding leads to a broad spectrum of macroscopic relaxation times and a shear modulus G ～ ω(1/2) for low frequencies ω. This model quantitatively describes the measured rheology of actin networks cross-linked with α-actinin-4 over more than four decades in frequency.     

Modeling dynamics of information networks

We propose an information-based model for network dynamics in which imperfect information leads to networks where the different vertices have widely different numbers of edges to other vertices, and where the topology has hierarchical features. The possibility to observe scale-free networks is linked to a minimally connected system where hubs remain dynamic.     

Agreement dynamics of finite-memory language games on networks

We propose a Finite-Memory Naming Game (FMNG) model with respect to the bounded rationality of agents or finite resources for information storage in communication systems. We study its dynamics on several kinds of complex networks, including random networks, small-world networks and scale-free networks. We focus on the dynamics of the FMNG affected by the memory restriction as well as the topological properties of the networks. Interestingly, we found that the most important quantity, the convergence time of reaching the consensus, shows some non-monotonic behaviors by varying the average degrees of the networks with the existence of the fastest convergence at some specific average degrees. We also investigate other main quantities, such as the success rate in negotiation, the total number of words in the system and the correlations between agents of full memory and the total number of words, which clearly explain the nontrivial behaviors of the convergence. We provide some analytical results which help better understand the dynamics of the FMNG. We finally report a robust scaling property of the convergence time, which is regardless of the network structure and the memory restriction.     

The rigid molecule approximation in lattice dynamics

The theory of the lattice dynamics of molecular crystals with and without the rigid molecule approximation is developed and the two compared. To make the two compatible, the equilibrium conditions for the internal molecular dimensions are replaced in the rigid molecule approximation by equations representing the constrained constant values of the molecular dimensions expanded in vibrational coordinates.     

Agreement dynamics on interaction networks with diverse topologies

We review the behavior of a recently introduced model of agreement dynamics, called the "Naming Game." This model describes the self-organized emergence of linguistic conventions and the establishment of simple communication systems in a population of agents with pairwise local interactions. The mechanisms of convergence towards agreement strongly depend on the network of possible interactions between the agents. In particular, the mean-field case in which all agents communicate with all the others is not efficient, since a large temporary memory is requested for the agents. On the other hand, regular lattice topologies lead to a fast local convergence but to a slow global dynamics similar to coarsening phenomena. The embedding of the agents in a small-world network represents an interesting tradeoff: a local consensus is easily reached, while the long-range links allow to bypass coarsening-like convergence. We also consider alternative adaptive strategies which can lead to faster global convergence.     

Complex networks in climate dynamics

Complex network theory provides a powerful framework to statistically investigate the topology of local and non-local statistical interrelationships, i.e. teleconnections, in the climate system. Climate networks constructed from the same global climatological data set using the linear Pearson correlation coefficient or the nonlinear mutual information as a measure of dynamical similarity between regions, are compared systematically on local, mesoscopic and global topological scales. A high degree of similarity is observed on the local and mesoscopic topological scales for surface air temperature fields taken from AOGCM and reanalysis data sets. We find larger differences on the global scale, particularly in the betweenness centrality field. The global scale view on climate networks obtained using mutual information offers promising new perspectives for detecting network structures based on nonlinear physical processes in the climate system.     

Social dynamics with peer support on heterogeneous networks

Human behavior often exhibits a scheme in which individuals adopt indifferent, neutral, or radical positions on a given topic. The mechanisms leading to community formation are strongly related with social pressure and the topology of the contact network. Here, we discuss an approach to model social behavior which accounts for the protection by alike peers proportional to their relative abundance in the closest neighborhood. We explore the ensuing non-linear dynamics emphasizing the role of the specific structure of the social network, modeled by scale-free graphs. We find that both coexistence of opinions and consensus on the default position are possible stationary states of the model. In particular, we show how these states critically depend on the heterogeneity of the social network and the specific distribution of external control elements.     

Random recurrent neural networks dynamics

This paper is a review dealing with the study of large size random recurrent neural networks. The connection weights are varying according to a probability law and it is possible to predict the network dynamics at a macroscopic scale using an averaging principle. After a first introductory section, the section 2 reviews the various models from the points of view of the single neuron dynamics and of the global network dynamics. A summary of notations is presented, which is quite helpful for the sequel. In section 3, mean-field dynamics is developed. The probability distribution characterizing global dynamics is computed. In section 4, some applications of mean-field theory to the prediction of chaotic regime for Analog Formal Random Recurrent Neural Networks (AFRRNN) are displayed. The case of AFRRNN with an homogeneous population of neurons is studied in section 4.1. Then, a two-population model is studied in section 4.2. The occurrence of a cyclo-stationary chaos is displayed using the results of [16]. In section 5, an insight of the application of mean-field theory to IF networks is given using the results of [9].