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.     

Evolutionary Prisoner＇s Dilemma on a Weighted Scale-Free Network

We investigate the evolutionary prisoner＇s dilemma on a weighted scale-free network where the diversity of teaching ability is introduced to the network in the form of weight. Though the diversity of teaching ability is not a sufficient condition for the enhancement of cooperation, we find that the degree-dependent teaching ability plays an active role in the evolution of cooperation. A new phenomenon is found when the degree-dependent teaching ability is used： the distribution of the cooperator frequency displays a two-peak structure for a certain parameter range. We also investigate the effects of the degree-degree correlation of the network on the evolution of cooperation in the presence of the diversity of the teaching ability.     

Effects of Inertia on Evolutionary Prisoner's Dilemma Game

Considering the inertia of individuals in real life, we propose a modified Fermi updating rule, where the inertia of players is introduced into evolutionary prisoner's dilemma game (PDG) on square lattices. We mainly focus on how the inertia affects the cooperative behavior of the system. Interestingly, we find that the cooperation level has a nonmonotonic dependence on the inertia: with small inertia, cooperators will soon be invaded by defectors; with large inertia, players are unwilling to change their strategies and the cooperation level remains the same as the initial state; while a moderate inertia can induce the highest cooperation level. Moreover, effects of environmental noise and individual inertia are studied. Our work may be helpful in understanding the emergence and persistence of cooperation in nature and society.     

Effects of Inertia on Evolutionary Prisoner's Dilemma Game

Considering the inertia of individuals in real life,we propose a modified Fermi updating rule,where the inertia of players is introduced into evolutionary prisoner's dilemma game(PDG) on square lattices.We mainly focus on how the inertia affects the cooperative behavior of the system.Interestingly,we find that the cooperation level has a nonmonotonic dependence on the inertia:with small inertia,cooperators will soon be invaded by defectors;with large inertia,players are unwilling to change their strategies and the cooperation level remains the same as the initial state;while a moderate inertia can induce the highest cooperation level.Moreover,effects of environmental noise and individual inertia are studied.Our work may be helpful in understanding the emergence and persistence of cooperation in nature and society.     

Evolutionary Prisoner's Dilemma Game Based on Pursuing Higher Average Payoff

We investigate the prisoner＇s dilemma game based on a new rule： players will change their current strategies to opposite strategies with some probability if their neighbours＇ average payoffs are higher than theirs. Compared with the cases on regular lattices （RL） and Newman-Watts small world network （NW）, cooperation can be best enhanced on the scale-free Barabasi-Albert network （BA）. It is found that cooperators are dispersive on RL network, which is different from previously reported results that cooperators will form large clusters to resist the invasion of defectors. Cooperative behaviours on the BA network are discussed in detail. It is found that large-degree individuals have lower cooperation level and gain higher average payoffs than that of small-degree individuals. In addition, we find that small-degree individuals more frequently change strategies than do large- degree individuals.     

Opinion Spreading with Mobility on Scale-Free Networks

A continuum opinion dynamic model is presented based on two rules. The first one considers the mobilities of the individuals, the second one supposes that the individuals update their opinions independently. The results of the model indicate that the bounded confidence εc, separating consensus and incoherent states, of a scale-free network is much smaller than the one of a lattice. If the system can reach the consensus state, the sum of all individuals＇ opinion change Oc（t） quickly decreases in an exponential form, while if it reaches the incoherent state finally, Oc（t） decreases slowly and has the punctuated equilibrium characteristic.     

Evolutionary Prisoner＇s Dilemma Game Based on Division of Work

We propose a new two-type-player prisoner＇s dilemma game based on the division of work on a square lattice, in which a fraction of the population μ are assigned type A and the rest B. In a one-shot two-player game, we let both of their original payoffs be scaled by a same multiplicative factor α 〉 1, if two neighboring players are of different types; however we leave the payoffs unchanged if they are of the same type. Then we show that combined with the two-type setup, the square lattice can assist to induce different social ranks according to players＇ abilities to collect payoffs. Simulation results show that the density of cooperation is significantly promoted for a wide range of the temptation to defection parameters and that there are optimal values for both α and μ leading to the maximal cooperation level. We reach these results by analyzing the distribution of the players in the social ranks and we also show some typical snapshots of the system.     

Prisoner's Dilemma Game on Two Types of Positively Correlated Networks

We study the effects of degree correlations on the evolution of cooperation in the prisoner＇s dilemma game with individuals located on two types of positively correlated networks. It is shown that the positive degree correlation can either promote or inhibit the emergence of cooperation depending on network configurations. Furthermore, we investigate the probability to cooperate as a function of connectivity degree, and find that high-degree individuals generally have a higher tendency to cooperate. Finally, it is found that small-degree individuals usually change their strategy more frequently, and such change is shown to be unfavourable to cooperation for both kinds of networks.     

Prisoner’s Dilemma Game with Cooperation-Defection Dominance Strategies on Correlational Multilayer Networks

As multilayer networks are widely applied in modern society, numerous studies have shown the impact of a multilayer network structure and the network nature on the proportion of cooperators in the network. In this paper, we use Barabási–Albert scale-free networks (BA) and Watts and Strogatz networks (WS) to build a multilayer network structure, and we propose a new strategy-updating rule called “cooperation-defection dominance”, which can be likened to dominant and recessive traits in biogenetics. With the newly constructed multilayer network structure and the strategy-updating rules, based on the simulation results, we find that in the BA-BA network, the cooperation dominance strategy can make the networks with different rs show a cooperative trend, while the defection dominance strategy only has an obvious effect on the network cooperation with a larger r. When the BA network is connected to the WS network, we find that the effect of strategy on the proportion of cooperators in the network decreases, and the main influencing factor is the structure of the network. In the three-layer network, the cooperation dominance strategy has a greater impact on the BA network, and the proportion of the cooperators is enhanced more than under the natural evolution strategy, but the promotion effect is still smaller than that of the two-layer BA network because of the WS network. Under the defection dominance strategy, the WS layer appears different from the first two strategies, and we conclude through simulation that when the payoff parameter is at the middle level, its cooperator proportion will be suppressed, and we deduce that the proportion of cooperators and defectors, as well as the payoff, play an important role.     

A Two-Stage Contact Process on Scale-Free Networks

We study a two-stage contact process on scale-free networks as a model for the spread of epidemics. We show that any virus starting from a single vertex with arbitrarily small infection rates can last for a super-polynomial time with positive probability if the power law exponent α>2. This is in sharp contrast with the mean-field analysis. The estimation of the metastable density is also provided.     

Consensus of Multi-Agent Systems with Prestissimo Scale-Free Networks

In this paper, the relations of the network topology and the moving consensus of multi-agent systems are studied. A consensus-prestissimo scale-free network model with the static preferential-consensus attachment is presented on the rewired link of the regular network. The effects of the static preferential-consensus BA network on the algebraic connectivity of the topology graph are compared with the regular network. The robustness gain to delay is analyzed for variable network topology with the same scale. The time to reach the consensus is studied for the dynamic network with and without communication delays. By applying the computer simulations, it is validated that the speed of the convergence of multi-agent systems can be greatly improved in the preferential-consensus BA network model with different configuration.     

Robustness of Cooperation on Highly Clustered Scale-Free Networks

We study the effect of mutation on the evolutionary prisoner＇s dilemma in highly clustered scale-free networks. It is found that cooperation is more sensitive and vulnerable to strategy mutation in more highly clustered networks. For small mutation rates, high clustering coefficient promotes cooperation. For medium mutation rates, high clustering coefficient inhibits the emergence of cooperation. For large mutation rates, cooperation is insensitive to clustering property. We provide explanations for the effects of clustering on cooperation with varied mutation rates.     

Sandpile Dynamics Driven by Degree on Scale-Free Networks

We introduce a sandpile model driven by degree on scale-free networks, where the perturbation is triggered at nodes with the same degree. We numerically investigate the avalanche behaviour of sandpile driven by different degrees on scale-free networks. It is observed that the avalanche area has the same behaviour with avalanche size. When the sandpile is driven at nodes with the minimal degree, the avalanches of our model behave similarly to those of the original Bak-Tang-Wiesenfeld （BTW） model on scale-free networks. As the degree of driven nodes increases from the minimal value to the maximal value, the avalanche distribution gradually changes from a clean power law, then a mixture of Poissonian and power laws, finally to a Poisson-like distribution. The average avalanche area is found to increase with the degree of driven nodes so that perturbation triggered on higher-degree nodes will result in broader spreading of avalanche propagation.     

Relaxation Dynamics in Condensation on Weighted Scale-Free Networks

Most of the realistic networks are weighted scale-free networks. How this structure influences the condensation on it is a challenging problem. Recently, we make a first step to discuss its condensation [Phys. Rev. E 74 （2006） 036101] and here we focus on its evolutionary process of phase transition. In order to show how the weighted transport influences the dynamical properties, we study the relaxation dynamics in a zero range process on weighted scale-free networks. We find that there is a hierarchical relaxation dynamics in the evolution and there is a scaling relation between the relaxation time and the jumping exponent. The relaxation dynamics can be illustrated by a mean-field equation. The theoretical predictions are confirmed by our numerical simulations.     

Cooperative Dynamics in Lattice-Embedded Scale-Free Networks

We investigate cooperative behaviors of lattice-embedded scale-free networking agents in the prisoner＇s dilemma game model by employing two initial strategy distribution mechanisms, which are specific distribution to the most connected sites （hubs） and random distribution. Our study indicates that the game dynamics crucially depends on the underlying spatial network structure with different strategy distribution mechanism. The cooperators＇ specific distribution contributes to an enhanced level of cooperation in the system compared with random one, and cooperation is robust to cooperators＇ specific distribution but fragile to defectors＇ specific distribution. Especially, unlike the specific case, increasing heterogeneity of network does not always favor the emergence of cooperation under random mechanism. Furthermore, we study the geographical effects and find that the graphically constrained network structure tends to improve the evolution of cooperation in random case and in specific one for a large temptation to defect.     

Improvement of Synchronizability of Scale-Free Networks

We investigate the factors that affect synchronizability of coupled oscillators on scale-free networks. Using the memory Tabu search （MTS） algorithm, we improve the eigen-ratio Q of a coupling matrix by edge intercrossing. The numerical results show that the synchronizatlon-improved scale-free networks should have distinctive both small average distance and larger clustering coefficient, which are consistent with some real-world networks. Moreover, the synchronizability-improved networks demonstrate the disassortative coefficient.     

Trajectory Control of Scale-Free Dynamical Networks with Exogenous Disturbances

In this paper, the trajectory control of multi-agent dynamical systems with exogenous disturbances is studied. Suppose multiple agents composing of a scale-free network topology, the performance of rejecting disturbances for the low degree node and high degree node is analyzed. Firstly, the consensus of multi-agent systems without disturbances is studied by designing a pinning control strategy on a part of agents, where this pinning control can bring multiple agents' states to an expected consensus track. Then, theinfluence of the disturbances is considered by developing disturbance observers, and disturbance observers based control (DOBC) are developed for disturbances generated by an exogenous system to estimate the disturbances. Asymptotical consensus of the multi-agent systems with disturbances under the composite controller can be achieved for scale-free network topology. Finally, by analyzing examples of multi-agent systems with scale-free network topology and exogenous disturbances, the verities of the results are proved. Under the DOBC with the designed parameters, the trajectory convergence of multi-agent systems is researched by pinning two class of the nodes. We have found that it has more stronger robustness to exogenous disturbances for the high degree node pinned than that of the lowdegree node pinned.