Naming game on small-world networks with geographical effects

The naming game model characterizes the main evolutionary features of languages or more generally of communication systems. Very recently, the combination of complex networks and the naming game has received much attention and the influences of various topological properties on the corresponding dynamical behavior have been widely studied. In this paper, we investigate the naming game on small-world geographical networks. The small-world geographical networks are constructed by randomly adding links to two-dimensional regular lattices, and it is found that the convergence time is a nonmonotonic function of the geographical distance of randomly added shortcuts. This phenomenon indicates that, although a long geographical distance of the added shortcuts favors consensus achievement, too long a geographical distance of the added shortcuts inhibits the convergence process, making it even slower than the moderates.     

Agreement Dynamics of Memory-Based Naming Game with Forgetting Curve of Ebbinghaus

We propose a memory-based naming game （MBNG） model which is like some previous opinion propagation models. with two-state variables in full-connected networks, It is found that this model is deeply affected by the memory decision parameter, and then its dynamical behaviour can be partly analysed by using numerical simulation and analytical argument. We also report a modified MBNG model with the forgetting curve of Ebbinghaus in the memory. With deletion of one parameter in the MBNG model, it can converge to success rate S（t） = 1 and the average sum E（t） is decided by the size of network N.     

Coevolutionary extremal dynamics on gasket fractal

We considered a Bak-Sneppen model on a Sierpinski gasket fractal. We calculated the avalanche size distribution and the distribution of distances between subsequent minimal sites. To observe the temporal correlations of the avalanche, we estimated the return time distribution, the first-return time, and the all-return time distribution. The avalanche size distribution follows the power law, P(s)∼s−τ, with the exponent τ=1.004(7). The distribution of jumping sites also follows the power law, P(r)∼r−π, with the critical exponent π=4.12(4). We observe the periodic oscillation of the distribution of the jumping distances which originated from the jumps of the level when the minimal site crosses the stage of the fractal. The first-return time distribution shows the power law, Pf(t)∼t−τf, with the critical exponent τf=1.418(7). The all-return time distribution is also characterized by the power law, Pa(t)∼t−τa, with the exponent τa=0.522(4). The exponents of the return time satisfy the scaling relation τf+τa=2 for τf?2.     

Relaxation dynamics of aftershocks after large volatility shocks in the SSEC index

The relaxation dynamics of aftershocks after large volatility shocks are investigated based on two high-frequency data sets of the Shanghai Stock Exchange Composite (SSEC) index. Compared with previous relevant work, we have defined main financial shocks based on large volatilities rather than large crashes. We find that the occurrence rate of aftershocks with the magnitude exceeding a given threshold for both daily volatility (constructed using 1-minute data) and minutely volatility (using intra-minute data) decays as a power law. The power-law relaxation exponent increases with the volatility threshold and is significantly greater than 1. Taking financial volatility as the counterpart of seismic activity, the power-law relaxation in financial volatility deviates remarkably from the Omori law in Geophysics.     

Robustness of heterogeneous complex networks

In this paper we study the robustness of heterogeneous preferential attachment networks. The robustness of a network measures its structural tolerance to the random removal of nodes and links. We numerically analyze the influence of the affinity parameters on a set of ensemble-averaged robustness metrics. We show that the presence of heterogeneity does not fundamentally alter the smooth nature of the fragmentation process of the models. We also show that a moderate level of locality translates into slight improvements in the robustness metrics, which prompts us to conjecture an evolutionary argument for the existence of real networks with power-law scaling in their connectivity and clustering distributions.     

Local affinity in heterogeneous growing networks

In this paper we present a study of the influence of local affinity in heterogeneous preferential attachment (PA) networks. Heterogeneous PA models are a generalization of the Barabási-Albert model to heterogeneous networks, where the affinity between nodes biases the attachment probability of links. Threshold models are a class of heterogeneous PA models where the affinity between nodes is inversely related to the distance between their states. We propose a generalization of threshold models where network nodes have individual affinity functions, which are then combined to yield the affinity of each potential interaction. We analyze the influence of the affinity functions in the topological properties averaged over a network ensemble. The network topology is evaluated through the distributions of connectivity degrees, clustering coefficients and geodesic distances. We show that the relaxation of the criterion of a single global affinity still leads to a reasonable power-law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions. We also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks.     

Accelerating networks: Effects of preferential connections

Networks are commonly observed structures in complex systems with interacting and interdependent parts that self-organize. For nonlinearly growing networks, when the total number of connections increases faster than the total number of nodes, the network is said to accelerate. We propose a systematic model for the dynamics of growing networks represented by distribution kinetics equations. We define the nodal-linkage distribution, construct a population dynamics equation based on the association-dissociation process, and perform the moment calculations to describe the dynamics of such networks. For nondirectional networks with finite numbers of nodes and connections, the moments are the total number of nodes, the total number of connections, and the degree (the average number of connections per node), represented by the average moment. Size independent rate coefficients yield an exponential network describing the network without preferential attachment, and size dependent rate coefficients produce a power law network with preferential attachment. The model quantitatively describes accelerating network growth data for a supercomputer (Earth Simulator), for regulatory gene networks, and for the Internet.     

Statistical Property and Model for the Inter-Event Time of Terrorism Attacks

The inter-event time of terrorism attack events is investigated by empirical data and model analysis. Empirical evidence shows that it follows a scale-free property. In order to understand the dynamic mechanism of such a statistical feature, an opinion dynamic model with a memory effect is proposed on a two-dimensional lattice network. The model mainly highlights the role of individual social conformity and self-affirmation psychology. An attack event occurs when the order parameter indicating the strength of public opposition opinion is smaller than a critical value. Ultimately, the model can reproduce the same statistical property as the empirical data and gives a good understanding for the possible dynamic mechanism of terrorism attacks.     

The network of commuters in London

We have studied the directed and weighted network in which the wards of London are vertices and two vertices are connected whenever there is at least one person commuting to work from one ward to another. Remarkably the in-strength and in-degree distribution tail is a power law with exponent around −2, while the out-strength and out-degree distribution tail is exponential. We propose a simple square lattice model to explain the observed empirical behaviour.     

Network traffic behaviour near phase transition point

We explore packet traffic dynamics in a data network model near phase transition point from free flow to congestion. The model of data network is an abstraction of the Network Layer of the OSI (Open Systems Interconnect) Reference Model of packet switching networks. The Network Layer is responsible for routing packets across the network from their sources to their destinations and for control of congestion in data networks. Using the model we investigate spatio-temporal packets traffic dynamics near the phase transition point for various network connection topologies, and static and adaptive routing algorithms. We present selected simulation results and analyze them.     

Kauffman networks with threshold functions

We investigate Threshold Random Boolean Networks with K = 2 inputs per node, which are equivalent to Kauffman networks, with only part of the canalyzing functions as update functions. According to the simplest consideration these networks should be critical but it turns out that they show a rich variety of behaviors, including periodic and chaotic oscillations. The analytical results are supported by computer simulations.     

Statistical dynamics of religion evolutions

A religion affiliation can be considered as a “degree of freedom” of an agent on the human genre network. A brief review is given on the state of the art in data analysis and modelization of religious “questions” in order to suggest and if possible initiate further research, after using a “statistical physics filter”. We present a discussion of the evolution of 18 so-called religions, as measured through their number of adherents between 1900 and 2000. Some emphasis is made on a few cases presenting a minimum or a maximum in the investigated time range—thereby suggesting a competitive ingredient to be considered, besides the well accepted “at birth” attachment effect. The importance of the “external field” is still stressed through an Avrami late stage crystal growth-like parameter. The observed features and some intuitive interpretations point to opinion based models with vector, rather than scalar, like agents.     

Emergence of scale-free behavior in networks from limited-horizon linking and cost trade-offs

We study network growth from a fixed set of initially isolated nodes placed at random on the surface of a sphere. The growth mechanism we use adds edges to the network depending on strictly local gain and cost criteria. Only nodes that are not too far apart on the sphere may be considered for being joined by an edge. Given two such nodes, the joining occurs only if the gain of doing it surpasses the cost. Our model is based on a multiplicative parameter λ that regulates, in a function of node degrees, the maximum geodesic distance that is allowed between nodes for them to be considered for joining. For n nodes distributed uniformly on the sphere, and for within limits that depend on cost-related parameters, we have found that our growth mechanism gives rise to power-law distributions of node degree that are invariant for constant . We also study connectivity- and distance-related properties of the networks.     

Community structure in Congressional cosponsorship networks

We study the United States Congress by constructing networks between Members of Congress based on the legislation that they cosponsor. Using the concept of modularity, we identify the community structure of Congressmen, who are connected via sponsorship/cosponsorship of the same legislation. This analysis yields an explicit and conceptually clear measure of political polarization, demonstrating a sharp increase in partisan polarization which preceded and then culminated in the 104th Congress (1995-1996), when Republicans took control of both chambers of Congress. Although polarization has since waned in the U.S. Senate, it remains at historically high levels in the House of Representatives.     

The emergence of coordination in public good games

In physical models it is well understood that the aggregate behaviour of a system is not in one to one correspondence with the behaviour of the average individual element of that system. Yet, in many economic models the behaviour of aggregates is thought of as corresponding to that of an individual. A typical example is that of public goods experiments. A systematic feature of such experiments is that, with repetition, people contribute less to public goods. A typical explanation is that people “learn to play Nash” or something approaching it. To justify such an explanation, an individual learning model is tested on average or aggregate data. In this paper we will examine this idea by analysing average and individual behaviour in a series of public goods experiments. We analyse data from a series of games of contributions to public goods and as is usual, we test a learning model on the average data. We then look at individual data, examine the changes that this produces and see if some general model such as the EWA (Expected Weighted Attraction) with varying parameters can account for individual behaviour. We find that once we disaggregate data such models have poor explanatory power. Groups do not learn as supposed, their behaviour differs markedly from one group to another, and the behaviour of the individuals who make up the groups also varies within groups. The decline in aggregate contributions cannot be explained by resorting to a uniform model of individual behaviour. However, the Nash equilibrium of such a game is a total payment for all the individuals and there is some convergence of the group in this respect. Yet the individual contributions do not converge. How the individuals “self-organsise” to coordinate, even in this limited way remains to be explained.     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Spectral methods and cluster structure in correlation-based networks

We investigate how in complex systems the eigenpairs of the matrices derived from the correlations of multichannel observations reflect the cluster structure of the underlying networks. For this we use daily return data from the NYSE and focus specifically on the spectral properties of weight W_ij=|C|_ij−δ_ij and diffusion matrices D_ij=W_ij/s_j−δ_ij, where C_ij is the correlation matrix and s_i=∑_jW_ij the strength of node j. The eigenvalues (and corresponding eigenvectors) of the weight matrix are ranked in descending order. As in the earlier observations, the first eigenvector stands for a measure of the market correlations. Its components are, to first approximation, equal to the strengths of the nodes and there is a second order, roughly linear, correction. The high ranking eigenvectors, excluding the highest ranking one, are usually assigned to market sectors and industrial branches. Our study shows that both for weight and diffusion matrices the eigenpair analysis is not capable of easily deducing the cluster structure of the network without a priori knowledge. In addition we have studied the clustering of stocks using the asset graph approach with and without spectrum based noise filtering. It turns out that asset graphs are quite insensitive to noise and there is no sharp percolation transition as a function of the ratio of bonds included, thus no natural threshold value for that ratio seems to exist. We suggest that these observations can be of use for other correlation based networks as well.     

Dynamics-driven evolution to structural heterogeneity in complex networks

The mutual influence of dynamics and structure is a central issue in complex systems. In this paper we study by simulation slow evolution of network under the feedback of a local-majority-rule opinion process. If performance-enhancing local mutations have higher chances of getting integrated into its structure, the system can evolve into a highly heterogeneous small-world with a global hub (whose connectivity is proportional to the network size), strong local connection correlations and power-law-like degree distribution. Networks with better dynamical performance are achieved if structural evolution occurs much slower than the network dynamics. Structural heterogeneity of many biological and social dynamical systems may also be driven by various dynamics-structure coupling mechanisms.     

Dynamical evolution of clustering in complex network of earthquakes

The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe, N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamical property of the earthquake network constructed in California and report the discovery that the values of the clustering coefficient remain stationary before main shocks, suddenly jump up at the main shocks, and then slowly decay following a power law to become stationary again. Thus, the dynamical network approach characterizes main shocks in a peculiar manner.