Effects of Some Topological Ingredients on the Evolutionary Ultimatum Game

This study aims at figuring out the crucial topological ingredients which affect the outcomes of the ultimatum game located on different networks,encompassing the regular network,the random network,the small world network,and the scale-free network.With the aid of random interchanging algorithm,we investigate the relations between the outcomes of the ultimatum game and some topological ingredients,including the average range,the clustering coefficient and the heterogeneity,and so forth.It is found that for the regular,random and small-work networks,the average range and the clustering coefficient have evident impacts on the ultimatum game,while for the scale-free network the original degree heterogeneity and the underlying rich-club characterizations are the mainly important topologica ingredients that influence the outcomes of ultimatum game substantially.     

Synchronization in evolving snowdrift game model

The interaction between the evolution of the game and the underlying network structure with evolving snowdrift game model is investigated. The constructed network follows a power-law degree distribution typically showing scale-free feature. The topological features of average path length, clustering coefficient, degree-degree correlations and the dynamical feature of synchronizability are studied. The synchronizability of the constructed networks changes by the interaction. It will converge to a certain value when sufficient new nodes are added. It is found that initial payoffs of nodes greatly affect the synchronizability. When initial payoffs for players are equal, low common initial payoffs may lead to more heterogeneity of the network and good synchronizability. When initial payoffs follow certain distributions, better synchronizability is obtained compared to equal initial payoff. The result is also true for phase synchronization of nonidentical oscillators.     

Evolutionary minority game on complex networks

In this work we investigate the dynamics of networked evolutionary minority game (NEMG) wherein each agent is allowed to evolve its strategy according to the information obtained from its neighbors in the network. We investigate four kinds of networks, including star network, regular network, random network and scale-free network. Simulation results indicate that the dynamics of the system depends crucially on the structure of the underlying network. The strategy distribution in a star network is sensitive to the precise value of the mutation magnitude L, in contrast to the strategy distribution in regular, random and scale-free networks, which is easily affected by the value of the prize-to-fine ratio R. Under a simple evolutionary scheme, the networked system with suitable parameters evolves to a high level of global coordination among its agents. In particular, the performance of the system is correlated to the clustering property of the network, where larger clustering coefficient leads to better performance.     

Hierarchical Characterization of Complex Networks

While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be obtained by considering further neighborhoods. The current work considers the concept of virtual hierarchies established around each node and the respectively defined hierarchical node degree and clustering coefficient (introduced in cond-mat/0408076), complemented by new hierarchical measurements, in order to obtain a powerful set of topological features of complex networks. The interpretation of such measurements is discussed, including an analytical study of the hierarchical node degree for random networks, and the potential of the suggested measurements for the characterization of complex networks is illustrated with respect to simulations of random, scale-free and regular network models as well as real data (airports, proteins and word associations). The enhanced characterization of the connectivity provided by the set of hierarchical measurements also allows the use of agglomerative clustering methods in order to obtain taxonomies of relationships between nodes in a network, a possibility which is also illustrated in the current article.     

Effects of the planarity and heterogeneity of networks on evolutionary two-player games

We study the effects of the planarity and heterogeneity of networks on evolutionary two-player symmetric games by considering four different kinds of networks, including two types of heterogeneous networks: the weighted planar stochastic lattice(a planar scale-free network) and the random uncorrelated scale-free network with the same degree distribution as the weighted planar stochastic lattice; and two types of homogeneous networks: the hexagonal lattice and the random regular network with the same degree k_0= 6 as the hexagonal lattice. Using extensive computer simulations, we found that both the planarity and heterogeneity of the network have a significant influence on the evolution of cooperation, either promotion or inhibition, depending not only on the specific kind of game(the Harmony, Snowdrift, Stag Hunt or Prisoner's Dilemma games), but also on the update rule(the Fermi, replicator or unconditional imitation rules).     

Evolutionary ultimatum game on complex networks under incomplete information

This paper studies the evolutionary ultimatum game on networks when agents have incomplete information about the strategies of their neighborhood agents. Our model assumes that agents may initially display low fairness behavior, and therefore, may have to learn and develop their own strategies in this unknown environment. The Genetic Algorithm Learning Classifier System (GALCS) is used in the model as the agent strategy learning rule. Aside from the Watts-Strogatz (WS) small-world network and its variations, the present paper also extends the spatial ultimatum game to the Barabási-Albert (BA) scale-free network. Simulation results show that the fairness level achieved is lower than in situations where agents have complete information about other agents’ strategies. The research results display that fairness behavior will always emerge regardless of the distribution of the initial strategies. If the strategies are randomly distributed on the network, then the long-term agent fairness levels achieved are very close given unchanged learning parameters. Neighborhood size also has little effect on the fairness level attained. The simulation results also imply that WS small-world and BA scale-free networks have different effects on the spatial ultimatum game. In ultimatum game on networks with incomplete information, the WS small-world network and its variations favor the emergence of fairness behavior slightly more than the BA network where agents are heterogeneously structured.     

一类高聚类系数的加权无标度网络及其同步能力分析

针对真实世界中大规模网络都具有明显聚类效应的特点, 提出一类具有高聚类系数的加权无标度网络演化模型, 该模型同时考虑了优先连接、三角结构、随机连接和社团结构等四种演化机制. 在模型演化规则中, 以概率p增加单个节点, 以概率1–p增加一个社团. 与以往研究的不同在于新边的建立, 以概率φ在旧节点之间进行三角连接, 以概率1–φ进行随机连接. 仿真分析表明, 所提出的网络度、强度和权值分布都是服从幂律分布的形式, 且具有高聚类系数的特性, 聚类系数的提高与社团结构和随机连接机制有直接的关系. 最后通过数值仿真分析了网络演化机制对同步动态特性的影响, 数值仿真结果表明, 网络的平均聚类系数越小, 网络的同步能力越强. 关键词： 无标度网络 加权网络 聚类系数 同步能力     

基于Web 2.0的边与节点同时增长网络模型

模拟了Web2.0网络的发展过程并研究其拓扑结构,分析某门户网站实际博客数据的度分布、节点度时间变化,发现与先前的无标度网络模型有所差别.根据真实网络的生长特点,提出了边与节点同时增长的网络模型,包括随机连接及近邻互联的网络构造规则.仿真研究表明,模拟的网络更接近实际,在没有优先连接过程时,模型能得到幂率的度分布;并且网络有更大的聚类系数以及正的度相关性。     

Spatial Games Based on Pursuing the Highest Average Payoff

We propose a strategy updating mechanism based on pursuing the highest average payoff to investigate the prisoner＇s dilemma game and the snowdrift game. We apply the new rule to investigate cooperative behaviours on regular, small-world, scale-free networks, and find spatial structure can maintain cooperation for the prisoner＇s dilemma game. fn the snowdrift game, spatial structure can inhibit or promote cooperative behaviour which depends on payoff parameter. We further study cooperative behaviour on scale-free network in detail. Interestingly, non-monotonous behaviours observed on scale-free network with middle-degree individuals have the lowest cooperation level. We also find that large-degree individuals change their strategies more frequently for both games.     

Properties of wealth distribution in multi-agent systems of a complex network

We present a simple model for examining the wealth distribution with agents playing evolutionary games (the Prisoners’ Dilemma and the Snowdrift Game) on complex networks. Pareto’s power law distribution of wealth (from 1897) is reproduced on a scale-free network, and the Gibbs or log-normal distribution for a low income population is reproduced on a random graph. The Pareto exponents of a scale-free network are in agreement with empirical observations. The Gini coefficient of an ER random graph shows a sudden increment with game parameters. We suggest that the social network of a high income group is scale-free, whereas it is more like a random graph for a low income group.     

Deterministic weighted scale-free small-world networks

We propose a deterministic weighted scale-free small-world model for considering pseudofractal web with the co-evolution of topology and weight. Considering the fluctuations in traffic flow constitute a main reason for congestion of packet delivery and poor performance of communication networks, we suggest a recursive algorithm to generate the network, which restricts the traffic fluctuations on it effectively during the evolutionary process. We provide a relatively complete view of topological structure and weight dynamics characteristics of the networks such as weight and strength distribution, degree correlations, average clustering coefficient and degree-cluster correlations as well as the diameter.     

Deterministic scale-free small-world networks of arbitrary order

In many real-life networks, both the scale-free distribution of degree and small-world behavior are important features. There are many random or deterministic models of networks to simulate these features separately. However, there are few models that combine the scale-free effect and small-world behavior, especially in terms of deterministic versions. What is more, all the existing deterministic algorithms running in the iterative mode generate networks with only several discrete numbers of nodes. This contradicts the purpose of creating a deterministic network model on which we can simulate some dynamical processes as widely as possible. According to these facts, this paper proposes a deterministic network generation algorithm, which can not only generate deterministic networks following a scale-free distribution of degree and small-world behavior, but also produce networks with arbitrary number of nodes. Our scheme is based on a complete binary tree, and each newly generated leaf node is further linked to its full brother and one of its direct ancestors. Analytical computation and simulation results show that the average degree of such a proposed network is less than 5, the average clustering coefficient is high (larger than 0.5, even for a network of size 2 million) and the average shortest path length increases much more slowly than logarithmic growth for the majority of small-world network models.     

Synchronizability of Highly Clustered Scale-Free Networks

We consider the effect of clustering coefficient on the synchronizability of coupled oscillators located on scale-free networks. The analytic result for the value of clustering coefficient aiming at a highly clustered scale-free network model, the Holme-Kim model is obtained, and the relationship between network synchronizability and clustering coefficient is reported. The simulation results strongly suggest that the more clustered the network, the poorer the synchronizability.     

Cascading Failures of Complex Networks Based on Two-Step Degree

We propose a new concept, two-step degree. Defining it as the capacity of a node of complex networks, we establish a novel capacity-load model of cascading failures of complex networks where the capacity of nodes decreases during the process of cascading failures. For scale-free networks, we find that the average two-step degree increases with the increase of the heterogeneity of the degree distribution, showing that the average two- step degree can be used for measuring the heterogeneity of the degree distribution of complex networks. In addition, under the condition that the average degree of a node is given, we can design a scale-free network with the optimal robustness to random failures by maximizing the average two-step degree.     

The coevolutionary ultimatum game on different network topologies

In this paper, a model of ultimatum game is discussed from the coevolutionary perspective, where strategy dynamics and structure dynamics coexist. The interplay between structure dynamics and strategy dynamics leads to overwhelmingly interesting evolved topology and fairness behaviors. It is found that fair division emerges for specific ratios of structure updating probability to strategy updating probability. Furthermore, it is shown that the initial structures have no essentially different effect on the coevolutionary results. In particular, the results for strategy are almost similar whenever the initial structure is set to be the nearest-neighbor coupled network, the ER random network or the scale-free network. Besides, the effects of other spatial factors are also investigated, e.g. the population size has a positive influence on the offer, while the average degree has a negative effect. In addition, one extrinsic factor, the background payoff, is also of great importance in promoting fair divisions. Apart from above, we study the properties of the evolved networks, which have the small-world effect and positive assortative behaviors.     

Boolean game on scale-free networks

Inspired by the local minority game, we propose a network Boolean game and investigate its dynamical properties on scale-free networks. The system can self-organize to a stable state with better performance than the random choice game, although only the local information is available to each agent. By introducing the heterogeneity of local interactions, we find that the system will achieve the best performance when each agent's interaction frequency is linearly correlated with its information capacity. Generally, the agents with more information gain more than those with less information, while in the optimal case, each agent almost has the same average profit. In addition, we investigate the role of irrational factor and find an interesting symmetrical behavior.     

Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices

In this paper, firstly, we study analytically the topological features of a family of hierarchical lattices (HLs) from the view point of complex networks. We derive some basic properties of HLs controlled by a parameter q: scale-free degree distribution with exponent γ=2+ln 2/(ln q), null clustering coefficient, power-law behavior of grid coefficient, exponential growth of average path length (non-small-world), fractal scaling with dimension d_B=ln (2q)/(ln 2), and disassortativity. Our results show that scale-free networks are not always small-world, and support the conjecture that self-similar scale-free networks are not assortative. Secondly, we define a deterministic family of graphs called small-world hierarchical lattices (SWHLs). Our construction preserves the structure of hierarchical lattices, including its degree distribution, fractal architecture, clustering coefficient, while the small-world phenomenon arises. Finally, the dynamical processes of intentional attacks and collective synchronization are studied and the comparisons between HLs and Barabási-Albert (BA) networks as well as SWHLs are shown. We find that the self-similar property of HLs and SWHLs significantly increases the robustness of such networks against targeted damage on hubs, as compared to the very vulnerable non fractal BA networks, and that HLs have poorer synchronizability than their counterparts SWHLs and BA networks. We show that degree distribution of scale-free networks does not suffice to characterize their synchronizability, and that networks with smaller average path length are not always easier to synchronize.     

Synchronizability of dynamical scale-free networks subject to random errors

In this paper the robustness of network synchronizability against random deletion of nodes, i.e. errors, in dynamical scale-free networks was studied. To this end, two measures of network synchronizability, namely, the eigenratio of the Laplacian and the order parameter quantifying the degree of phase synchrony were adopted, and the synchronizability robustness on preferential attachment scale-free graphs was investigated. The findings revealed that as the network size decreases, the robustness of its synchronizability against random removal of nodes declines, i.e. the more the number of randomly removed nodes from the network, the worse its synchronizability. We also showed that this dependence of the synchronizability on the network size is different with that in the growing scale-free networks. The profile of a number of network properties such as clustering coefficient, efficiency, assortativity, and eccentricity, as a function of the network size was investigated in these two cases, growing scale-free networks and those with randomly removed nodes. The results showed that these processes are also different in terms of these metrics.     

Impact of degree heterogeneity on the behavior of trapping in Koch networks

Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent γ of power-law degree distribution P(k) ～ k(-γ), which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent γ in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number N, which is obviously independent of γ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where γ influences qualitatively the MFPT of trapping problem.     

Cross over of recurrence networks to random graphs and random geometric graphs

Recurrence networks are complex networks constructed from the time series of chaotic dynamical systems where the connection between two nodes is limited by the recurrence threshold. This condition makes the topology of every recurrence network unique with the degree distribution determined by the probability density variations of the representative attractor from which it is constructed. Here we numerically investigate the properties of recurrence networks from standard low-dimensional chaotic attractors using some basic network measures and show how the recurrence networks are different from random and scale-free networks. In particular, we show that all recurrence networks can cross over to random geometric graphs by adding sufficient amount of noise to the time series and into the classical random graphs by increasing the range of interaction to the system size. We also highlight the effectiveness of a combined plot of characteristic path length and clustering coefficient in capturing the small changes in the network characteristics.