Abstract
We present a stochastic dynamics model of coupled evolution for the binary states of nodes and links in a complex network. In the context of opinion formation node states represent two possible opinions and link states represent positive or negative relationships. Dynamics proceeds via node and link state update towards pairwise satisfactory relations in which nodes in the same state are connected by positive links or nodes in different states are connected by negative links. By a mean-field rate equations analysis and Monte Carlo simulations in random networks we find an absorbing phase transition from a dynamically active phase to an absorbing phase. The transition occurs for a critical value of the relative time scale for node and link state updates. In the absorbing phase the order parameter, measuring global order, approaches exponentially the final frozen configuration. Finite-size effects are such that in the absorbing phase the final configuration is reached in a characteristic time that scales logarithmically with system size, while in the active phase, finite-size fluctuations take the system to a frozen configuration in a characteristic time that grows exponentially with system size. There is also a class of finite-size topological transition associated with group splitting in the network of these final frozen configurations.
Highlights
The linear terms are obtained by the variation of densities due to the direct update of nodes and links in the real time steps
As a way of example, we derive in detail the first term of the first equation: In any update step, with probability ρa a b and changes pair a is randomly the global density chosen
To show how we obtain the nonlinear terms, we derive the first non linear term of the fifth equation. This term is the implication of a node update from the pair connection a to f
Summary
Coupled evolution of node and links property in the imitating process. Let us consider a connected network defined as a set of nodes and links. For an Erdős-Rényi network, Eqs (1–3) imply that the density of white links in the steady state should be xst = 1/2 in the active phase, while it depends on the initial condition in the frozen phase. We mean that the nodes organize in several groups, defining a group as a set of nodes holding the same opinion and connected by friendly links among themselves and by unfriendly links to the members of other groups This group splitting structure appears both when the absorbing configuration has been reached from a finite-size fluctuation of an active phase, p < pc (μ), see Fig. 9(c,d), or when it corresponds to the frozen phase in parameter spacTeh,epn>umpcb(eμr)o, fseceomFipg.o1n0e(nat,sbi)s. In 12 we show that fgiant (G) can be well represented by a Gaussian distribution
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