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In this paper, the relations of the scale-free network topology and the moving consensus of multi-agent systems are studied. A synchronization-preferential BA (SPBA) network with the combinatorial preferential attachment is presented for the prestissimo consensus. The effects of the SPBA network on the algebraic connectivity of the topology graph are compared with the standard BA network. The robustness gain to delay is analyzed for variable network topologies with the same size. The characteristics of dramatic increase for the algebraic connectivity and slightly increase for the robustness gain to delay are unfolded in SPBA network. By comparing the convergence time of the different network structures, the consensuses are studied for multiagent systems with and without communication delays.

We have introduced a vector Markov chain of the vertex number with degree k in network evolving process as a framework of theoretical analysis and proved the stability of the BA-1 model and the LCD-1 model. In this paper, we use the vertex-number-evolving Markov chain to prove rigorously the existence of the steady-state degree distribution P(k) for a special case of the initial attraction model allowing multiple edges. The application of our approach to the LCD-m model, the result shows that it is more simpler than Bollobas’ method.

The voluntary participation mechanism has been demonstrated as a natural extension to promote the cooperative behavior in evolutionary games. Apart from the cooperator strategy and defector strategy in the original Prisoner’s Dilemma Game (PDG), players can choose the additional loner strategy to refuse to participate and get some small but fixed income Q. We allow for voluntary participation in the evolutionary PDG on regular lattices and scale-free networks, and focus on the strategy density and the evolution behavior of the system. Simulation shows that the system behavior is sensitive to the population structure. For lattices, the densities of cooperator, defector and loners exhibit non-trivial behaviors with the increment of Q and temptation parameter b except for the density of loners as a function of Q. For scale-free networks, the densities of cooperator and defectors monotonously decrease with the increment of Q, and the densities of defectors and loners monotonously increase with the increment of b. Moreover, this extension can lead to “rock-scissors-paper”-like cyclic dominance of the three types of players on regular lattices. But for scale-free networks, the loners cannot coexist with cooperators or defectors.

In this paper, we introduced the geographical effect into susceptible-infected-susceptible (SIS) model on a growing network proposed by Xie et al. [Phys. Rev. E 75 (2007) 036106] to investigate the effect of geographical distance on epidemic spreading. In our SIS model, the susceptible individual i can be infected by infectious neighbors with probability θ i : θ i = 1 − ∏ j ∈ N ( i ) ( 1 − λ j ) , λ j = min ( 1 , λ 0 × ( d i j / min k ∈ N ( i ) ( d i k ) ) − β ) , where λ 0 is the average transmission rate of the virus, j is one of the infected neighbors ( N ( i ) ) of susceptible individual i , λ j is transmission rate of individual j , d i j is Euclidean distance between individual i and j , and β is the tunable parameter. Simulation results show that the steady density of infected individuals monotonously decreases with the increment of β and the epidemic threshold emerges in the scale-free networks when the effect of geographical distance is taken into account. Moreover, when the network is star-like, the density of the infected individuals shows large amplitude oscillations.

We study the role of social diversity in the evolutionary game and opinion dynamics on scale-free networks. Each individual is assigned a weight that is proportional to the power of its degree, where the exponent α is an adjustable parameter that controls the level of diversity among individuals in the network. It is found that there exists an optimal value of α , leading to the highest cooperation level or the fastest consensus. We give some qualitative explanations for the emergence of optimal α .

We propose a new mechanism leading to scale-free networks which is based on the presence of an intrinsic character of a vertex called fitness. In our model, at each vertex i a fitness x i , drawn from a given probability distribution function f ( x ) , is assigned. During network evolution, with rate p we add a vertex j and connect to an existing vertex i of selected preferentially to a linking probability function g ( x i , x j ) which depends on the fitnesses of the two vertices involved and, with rate 1 − p we create an edge between two already existed vertices i and j, with a probability also preferential to the connection function g ( x i , x j ) . For the proper choice of g, the resulting networks have power-law distributions of connectivity and small-world properties, irrespective of the fitness distribution of vertices.

Although the scale invariance is a main feature of growing networks, evidence from a few modelled networks show that the finite-size effect of network at a fixed time t cannot be ignored. We propose a concept of time-dependent scale-free networks, and prove the criteria for the stability and scale-free of growing networks by the degree-growing Markov chain. Our results show the importance of rigorous theoretical analysis for the network science.