Epidemic spreading in lattice-embedded scale-free networks

We study geographical effects on the spread of diseases in lattice-embedded scale-free networks. The geographical structure is represented by the connecting probability of two nodes that is related to the Euclidean distance between them in the lattice. By studying the standard susceptible-infected model, we found that the geographical structure has great influences on the temporal behavior of epidemic outbreaks and the propagation in the underlying network: the more geographically constrained the network is, the more smoothly the epidemic spreads, which is different from the clearly hierarchical dynamics that the infection pervades the networks in a progressive cascade across smaller-degree classes in Barabási–Albert scale-free networks.     

Emergence of complex networks from diffusion on fractal lattices. A special case of the Sierpinski gasket and tetrahedron

A new approach to the assemblage of complex networks displaying the scale-free architecture is proposed. While the growth and the preferential attachment of incoming nodes assure an emergence of such networks according to the Barabási–Albert model, it is argued here that the preferential linking condition needs not to be a principal rule. To assert this statement a simple computer model based on random walks on fractal lattices is introduced. It is shown that the model successfully reproduces the degree distributions, the ultra-small-worldness and the high clustering arising from the topology of scale-free networks.     

Effective usage of shortest paths promotes transportation efficiency on scale-free networks

With rapid economic and social development, the problem of traffic congestion is getting more and more serious. Accordingly, network traffic models have attracted extensive attention. In this paper, we introduce a shortest-remaining-path-first queuing strategy into a network traffic model on Barabási–Albert scale-free networks under efficient routing protocol, where one packet’s delivery priority is related to its current distance to the destination. Compared with the traditional first-in-first-out queuing strategy, although the network capacity has no evident changes, some other indexes reflecting transportation efficiency are significantly improved in the congestion state. Extensive simulation results and discussions are carried out to explain the phenomena. Our work may be helpful for the designing of optimal networked-traffic systems.     

Maximum entropy networks are more controllable than preferential attachment networks

A maximum entropy (ME) method to generate typical scale-free networks has been recently introduced. We investigate the controllability of ME networks and Barabási–Albert preferential attachment networks. Our experimental results show that ME networks are significantly more easily controlled than BA networks of the same size and the same degree distribution. Moreover, the control profiles are used to provide insight into control properties of both classes of network. We identify and classify the driver nodes and analyze the connectivity of their neighbors. We find that driver nodes in ME networks have fewer mutual neighbors and that their neighbors have lower average degree. We conclude that the properties of the neighbors of driver node sensitively affect the network controllability. Hence, subtle and important structural differences exist between BA networks and typical scale-free networks of the same degree distribution.     

Influence of different initial distributions on robust cooperation in scale-free networks: A comparative study

We study the evolutionary Prisoner's dilemma game on scale-free networks, focusing on the influence of different initial distributions for cooperators and defectors on the evolution of cooperation. To address this issue, we consider three types of initial distributions for defectors: uniform distribution at random, occupying the most connected nodes, and occupying the lowest-degree nodes, respectively. It is shown that initial configurations for defectors can crucially influence the cooperation level and the evolution speed of cooperation. Interestingly, the situation where defectors initially occupy the lowest-degree vertices can exhibit the most robust cooperation, compared with two other distributions. That is, the cooperation level is least affected by the initial percentage of defectors. Moreover, in this situation, the whole system evolves fastest to the prevalent cooperation. Besides, we obtain the critical values of initial frequency of defectors above which the extinction of cooperators occurs for the respective initial distributions. Our results might be helpful in explaining the maintenance of high cooperation in scale-free networks.     

Other-regarding preference and the evolutionary prisoner’s dilemma on complex networks

Prevailing models of the evolutionary prisoner’s game on networks always assume that agents are pursuing their own profit maximization. But the results from experimental games show that many agents have other-regarding preference. In this paper, we study the emergence of cooperation from the prisoner’s dilemma game on complex networks while some agents exhibit other-regarding preference such as inequality aversion, envious and guilty emotions. Contrary to common ideas, the simulation results show that the existence of inequality aversion agents does not promote cooperation emergence on a BA (Barabási and Albert) scale-free network in most situations. If the defection attraction is big and agents exhibit strong preference for inequality aversion, the frequency of cooperators will be lower than in situations where no inequality aversion agents exist. In some cases, the existence of the inequality agents will even induce the frequency of cooperators to zero, a feature which is not observed in previous research on the prisoner’s dilemma game when the underlying interaction topology is a BA scale-free network. This means that if an agent cares about equality too much, it will be difficult for cooperation to emerge and the frequency of cooperators will be low on BA networks. The research on the effect of envy or guilty emotions on the emergence of cooperation in the prisoner’s dilemma game on BA networks obtains similar results, though some differences exist. However, simulation results on a WS (Watts and Strogatz) small-world network display another scenario. If agents care about the inequality of agents very much, the WS network favors cooperation emergence in the prisoners’ dilemma game when other-regarding agents exist. If the agent weight on other-regarding is lowered, the cooperation frequencies emerging on a WS network are not much different from those in situations without other-regarding agents, although the frequency of cooperators is lower than those of the situation without other-regarding preference agents sometimes. All the simulation results imply that inequality aversion and its variations can have important effects on cooperation emergence in the prisoner’s dilemma game, and different network topologies have different effects on cooperation emergence in the prisoner’s dilemma game played on complex networks.     

Research on the search ability of Brownian particles on networks with an adaptive mechanism

In this paper, we focus on the search ability of Brownian particles with an adaptive mechanism. In the adaptive mechanism, nodes are allowed to be able to change their own accepting probability according to their congestion states. Two searching-traffic models, the static one in which nodes have fixed accepting probability to the incoming particles and the adaptive one in which nodes have adaptive accepting probability to the incoming particles are presented for testing the adaptive mechanism. Instead of number of hops, we use the traveling time, which includes not only the number of hops for a particle to jump from the source node to the destination but also the time that the particle stays in the queues of nodes, to evaluate the search ability of Brownian particles. We apply two models to different networks. The experiment results show that the adaptive mechanism can decrease the network congestion and the traveling time of the first arriving particle. Furthermore, we investigate the influence of network topologies on the congestion of networks by addressing several main properties: degree distribution, average path length, and clustering coefficient. We show the reason why random topologies are more able to deal with congested traffic states than others. We also propose an absorption strategy to deal with the additional Brownian particles in networks. The experiment results on Barabási–Albert (BA) scale-free networks show that the absorption strategy can increase the probability of a successful search and decrease the average per-node particles overhead for our models.     

Efficient routing on small complex networks without buffers

In this paper, we are exploring strategies for the reduction of the congestion in the complex networks. The nodes without buffers are considered, so, if the congestion occurs, the information packets will be dropped. The focus is on the efficient routing. The routing strategies are compared using two generic models, i.e., Barabàsi–Albert scale-free network and scale-free network on lattice, and the academic router networks of the Netherlands and France. We propose a dynamic deflection routing algorithm which automatically extends path of the packet before it arrives at congested node. The simulation results indicate that the dynamic routing strategy can further reduce the number of dropped packets in a combination with the efficient path routing proposed by Yan et al. [5].     

Ranking the spreading influence in complex networks

Identifying the node spreading influence in networks is an important task to optimally use the network structure and ensure the more efficient spreading in information. In this paper, by taking into account the shortest distance between a target node and the node set with the highest k$k$-core value, we present an improved method to generate the ranking list to evaluate the node spreading influence. Comparing with the epidemic process results for four real networks and the Barabási–Albert network, the parameterless method could identify the node spreading influence more accurately than the ones generated by the degree k$k$, closeness centrality, k$k$-shell and mixed degree decomposition methods. This work would be helpful for deeply understanding the node importance of a network.     

Time evolution of the degree distribution of model A of random attachment growing networks

For random growing networks, Barabás and Albert proposed a kind of model in Barabás et al. [Physica A 272 (1999) 173], i.e. model A. In this paper, for model A, we give the differential format of master equation of degree distribution and obtain its analytical solution. The obtained result P(k, t) is the time evolution of degree distribution. P(k, t) is composed of two terms. At given finite time, one term decays exponentially, the other reflects size effect. At infinite time, the degree distribution is the same as that of Barabás and Albert. In this paper, we also discuss the normalization of degree distribution P(k, t) in detail.     

Evolutionary prisoner's dilemma game on Barabási–Albert scale-free networks

An evolutionary prisoner's dilemma game (PDG) with players located on Barabási–Albert scale-free networks (BASFN) with average connectivity 8$8$ is studied in detail. The players are pure strategists and can adopt two strategies: either defect or cooperate. Several alternative update rules determining the evolution of each player's strategy are considered. Using Monte Carlo (MC) simulations we implemented both synchronous and asynchronous update dynamics to calculate the average density of cooperators _ρC${\rho }_C$ as a function of the temptation-to-defect b$b$ in the equilibrium state. For the sake of comparison, evolution of cooperative behavior on random regular graphs (RRG) and regular lattices (RL) with the same total size and average connectivity as BASFN are also investigated. We find the overall result that cooperation is sometimes inhibited and sometimes enhanced on the BASFN, with respect to the cases on the RRG and RL. The differences depend on the detailed evaluation function of the players’ success (average payoffs of the players in the present case), the different update rules that determine a player's future strategy, the synchronous and asynchronous events of strategy-updating, and also on the temptation to defect.     

The roles of heterogeneous investment mechanism in the public goods game on scale-free networks

The public goods game is an important theoretical model for investigating the emergence of cooperation in the multi-player social dilemma. It has been proven that scale-free networks can significantly promote cooperation, but fail to sustain cooperation when the player obtains the normalized payoff. In this paper, we introduce heterogeneous investment mechanism into the public goods game on scale-free networks, and study the evolution of cooperation in both cases of accumulated and normalized payoff. Our research reveals that the heterogeneous investment mechanism can obviously facilitate cooperation as the adjusted parameter α increases. The increase of α allows cooperators to emerge under lower values of r. In the case of accumulated payoff, cooperators always firmly occupy the hubs, and the population keeps high cooperation level. In the case of normalized payoff, the increase of α changes the situation that the hubs are easily invaded by defectors, and inhibits the spread of defectors.     

Prisoner's Dilemma Game on Clustered Scale-Free Networks under Different Initial Distributions

The evolutionary prisoner＇s dilemma game is investigated under different initial distributions for cooperators and defectors on scale-free networks with a tunable clustering coefficient. It is found that, on the one hand, cooperation can be enhanced with the increasing clustering coefficient when only the most connected nodes are occupied by cooperators initially. On the other hand, if cooperators just occupy the lowest-degree nodes at the beginning, then the higher the value of the clustering coefficient, the more unfavorable the environment for cooperators to survive for the increment of temptation to defect. Thereafter, we analytically argue these nontrivial phenomena by calculating the cooperation probability of the nodes with different degrees in the steady state, and obtain the critical values of initial frequency of cooperators below which cooperators would vanish finally for the two initial distributions.     

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.     

Cooperation in N-person evolutionary snowdrift game in scale-free Barabási-Albert networks

Cooperation in the N-person evolutionary snowdrift game (NESG) is studied in scale-free Barabási-Albert (BA) networks. Due to the inhomogeneity of the network, two versions of NESG are proposed and studied. In a model where the size of the competing group varies from agent to agent, the fraction of cooperators drops as a function of the payoff parameter. The networking effect is studied via the fraction of cooperative agents for nodes with a particular degree. For small payoff parameters, it is found that the small-k agents are dominantly cooperators, while large-k agents are of non-cooperators. Studying the spatial correlation reveals that cooperative agents will avoid to be nearest neighbors and the correlation disappears beyond the next-nearest neighbors. The behavior can be explained in terms of the networking effect and payoffs. In another model with a fixed size of competing groups, the fraction of cooperators could show a non-monotonic behavior in the regime of small payoff parameters. This non-trivial behavior is found to be a combined effect of the many agents with the smallest degree in the BA network and the increasing fraction of cooperators among these agents with the payoff for small payoffs.     

Opinion evolution on a BA scaling network

In this paper, the dynamics of opinion formation is investigated based on a BA (Barabási–Albert) scale-free network, using a majority–minority rule governed by parameter q$q$. As the value of q$q$ is smoothly varied, a phase transition occurs between an ordered phase and a disordered one. By performing extensive Monte Carlo simulations, we show that the phase transition is dependent on the system size, as well as on m$m$, the number of edges added at each time step during the growth of the BA scaling network. Additionally, some theoretical analysis is given based on mean-field theory, by neglecting fluctuations and correlations. It is observed that the theoretical results coincide with results from simulations, especially for very large m$m$.     

Individual’s strategy characterized by local topology conditions in prisoner’s dilemma on scale-free networks

We introduce a “gradient” to find out the defectors, and further a “topology potential” to characterize the individual’s strategy preference in the prisoner’s dilemma on scale-free networks. It is shown that the cooperators typically locate on the nodes with high topology potential and the defectors are mainly found on the nodes with small topology potential. A critical topology potential is found for the nodes where cooperators are nip and tuck with defectors. So the information of node’s degree, gradient and topology potential together can predict individual’s strategy decision in the prisoner’s dilemma on the complex networks.     

Evolution of cooperation in lattice population with adaptive interaction intensity

We study the evolutionary Prisoner’s Dilemma game among individuals endowed with adaptively interaction intensity. Individuals adjust their interaction intensity according to the rules “payoff increase-high intensity, payoff decrease-low intensity”: if an individual’s payoff increases compared with that in the previous generation, he raises his interaction intensity; otherwise, he reduces the probability of interaction. We find that if individuals can adjust their interaction intensity with a proper scale, cooperation can be promoted. Interestingly, individuals with low interaction intensity usually hold the boundary of cooperator cluster. Such spatial distribution can alleviate the exploitation from defectors to cooperators since the interaction between cooperators and defectors is weakened. We hope our work can yield some insight into investigation of the evolution of cooperation in structured population.     

Effect of Growing Size of Interaction Neighbors on the Evolution of Cooperation in Spatial Snowdrift Game

In this paper,we study the influence of the size of interaction neighbors(k) on the evolution of cooperation in the spatial snowdrift game.At first,we consider the effects of noise K and cost-to-benefit ratio r,the simulation results indicate that the evolution of cooperation depends on the combined action of noise and cost-to-benefit ratio.For a lower r,the cooperators are multitudinous and the cooperation frequency ultimately increases to 1 as the increase of noise.However,for a higher r,the defectors account for the majority of the game and dominate the game if the noise is large enough.Then we mainly investigate how k influences the evolution of cooperation by varying the noise in detail.We find that the frequency of cooperators is closely related to the size of neighborhood and cost-to-benefit ratio r.In the case of lower r,the augmentation of k plays no positive role in promoting the cooperation as compared with that of k = 4,while for higher r the cooperation is improved for a growing size of neighborhood.At last,based on the above discussions,we explore the cluster-forming mechanism among the cooperators.The current results are beneficial to further understand the evolution of cooperation in many natural,social and biological systems.     

Evolution of ethnocentrism on undirected and directed Barabási-Albert networks

Using Monte Carlo simulations, we study the evolution of contingent cooperation and ethnocentrism in the one-shot game. Interactions and reproduction among computational agents are simulated on undirected and directed Barabási-Albert (BA) networks. We first replicate the Hammond-Axelrod model of in-group favoritism on a square lattice and then generalize this model on undirected and directed BA networks for both asexual and sexual reproduction cases. Our simulations demonstrate that irrespective of the mode of reproduction, the ethnocentric strategy becomes common even though cooperation is individually costly and mechanisms such as reciprocity or conformity are absent. Moreover, our results indicate that the spread of favoritism towards similar others highly depends on the network topology and the associated heterogeneity of the studied population.