We have investigated the influence of the average degree （k） of network on the location of an order-disorder transition in opinion dynamics. For this purpose, a variant of majority rule （VMR） model is applied to Watts-Strogatz （WS） small-world networks and Barabasi-Albert （BA） scale-free networks which may describe some non-trivial properties of social systems. Using Monte Carlo simulations, we find that the order-disorder transition point of the VMR model is greatly affected by the average degree （k） of the networks; a larger value of （k） results in a more ordered state of the system. Comparing WS networks with BA networks, we find WS networks have better orderliness than BA networks when the average degree （k） is small. With the increase of （k）, BA networks have a more ordered state. By implementing finite-size scaling analysis, we also obtain critical exponents β/v, γ/u and 1/v for several values of average degree （k）. Our results may be helpful to understand structural effects on order-disorder phase transition in the context of the majority rule model.
We investigate an evolutionary snowdrift game on a square N ： L × L lattice with periodic boundary conditions, where a population of no （no ≤ N） players located on the sites of this lattice can either cooperate with or defect from their nearest neighbours. After each generation, every player moves with a certain probability p to one of the player＇s nearest empty sites. It is shown that, when p = 0, the cooperative behaviour can be enhanced in disordered structures. When p 〉 0, the effect of mobility on cooperation remarkably depends on the payoff parameter r and the density of individuals ρ （ρ = no/N）. Compared with the results of p = 0, for small r, the persistence of cooperation is enhanced at not too small values of p; whereas for large r, the introduction of mobility inhibits the emergence of cooperation at any p 〈 1; for the intermediate value of r, the cooperative behaviour is sometimes enhanced and sometimes inhibited, depending on the values of p and p. In particular, the cooperator density can reach its maximum when the values of p and p reach their respective optimal values. In addition, two absorbing states of all cooperators and all defectors can emerge respectively for small and large r in the case of p 〉 0.
By using the well-known Ikeda model as the node dynamics, this paper studies synchronization of time-delay systems on small-world networks where the connections between units involve time delays. It shows that, in contrast with the undelayed case, networks with delays can actually synchronize more easily. Specifically, for randomly distributed delays, time-delayed mutual coupling suppresses the chaotic behaviour by stabilizing a fixed point that is unstable for the uncoupled dynamical system.
The collective synchronization of a system of coupled logistic maps on random community networks is investigated. It is found that the synchronizability of the community network is affected by two factors when the size of the network and the number of connections are fixed. One is the number of communities denoted by the parameter rn, and the other is the ratio σ of the connection probability p of each pair of nodes within each community to the connection probability q of each pair of nodes among different communities. Theoretical analysis and numerical results indicate that larger rn and smaller σ are the key to the enhancement of network synchronizability. We also testify synchronous properties of the system by analysing the largest Lyapunov exponents of the system.
An evolutionary prisoner＇s dilemma game is investigated on two-layered complex networks respectively representing interaction and learning networks in one and two dimensions. A parameter q is introduced to denote the correlation degree between the two-layered networks. Using Monte Carlo simulations we studied the effects of the correlation degree on cooperative behaviour and found that the cooperator density nontrivially changes with q for different payoff parameter values depending on the detailed strategy updating and network dimension. An explanation for the obtained results is provided.
