Fragmentation transitions in a coevolving nonlinear voter model

Abstract

We study a coevolving nonlinear voter model describing the coupled evolution of the states of the nodes and the network topology. Nonlinearity of the interaction is measured by a parameter q. The network topology changes by rewiring links at a rate p. By analytical and numerical analysis we obtain a phase diagram in p,q parameter space with three different phases: Dynamically active coexistence phase in a single component network, absorbing consensus phase in a single component network, and absorbing phase in a fragmented network. For finite systems the active phase has a lifetime that grows exponentially with system size, at variance with the similar phase for the linear voter model that has a lifetime proportional to system size. We find three transition lines that meet at the point of the fragmentation transition of the linear voter model. A first transition line corresponds to a continuous absorbing transition between the active and fragmented phases. The other two transition lines are discontinuous transitions fundamentally different from the transition of the linear voter model. One is a fragmentation transition between the consensus and fragmented phases, and the other is an absorbing transition in a single component network between the active and consensus phases.

Figure 1

Schematic illustration of update rule of the coevolving nonlinear voter model. Each node is in either up (red circle) or down (blue rounded square) state. Solid and dashed lines indicate respectively inert and active links. At each step, we randomly choose a node i. And we choose one of its neighbors j connected by an active link with a probability ((a _i)/(k _i))^q. Then, we rewire an active link with a probability p and copy the state of the neighbor with a probability 1 − p.

Figure 2

Flow diagram on the plane (m, ρ) for q = 0.5 and 2 below and above p _c = 0.875 (q = 0.5) and 0.8 (q = 2) obtained from pair approximation. The filled (open) circles denote the stable (unstable) fixed points. The point at m = 0 is stable for q = 0.5 but is unstable for q = 2.

Figure 3

The size S of giant component, magnetization |m|, and the density ρ of active link for the network at the steady state for (a) q = 0.5, (b) q = 1, and (c) q = 2 on random regular networks with 〈k〉 = 8, N = 10³, and initial condition m = 0 averaged over 10⁴ realizations. The right panel represents an typical trajectory to steady state below and above p _c on (m, ρ) space.

Figure 4

Rescaled magnetization μ and time to steady state state τ for (a,b) q = 0.5, (c,d) q = 1, and (e,f) q = 2 with different network size N and 〈k〉 = 8, averaged over 10⁵ runs. Figures (a) q = 0.5 and (c) q = 1 show scaling of magnetization in a form μ = N ^β/ν f(N ^1/ν(p − p _c)) with p _c ≈ 0.83 and 0.68, respectively. When q = 0.5, (b) τ exponentially grows with respect to N below p _c. For q = 2, (e) the transition of magnetization is getting sharp as N increases, with (f) non-diverging τ at the transition.

Figure 5

(a) Phase diagram with respect to p and q shows consensus, coexistence, and fragmented phases, obtained numerically on degree regular networks with 〈k〉 = 8, N = 10⁴ and initial condition m = 0, averaged over 10³ realizations. Examples of network configuration at the steady-state of the coevolution model are also shown with N = 200 and (p, q) = (0.2, 0.5) for coexistence, (0.2, 2) for consensus, and (0.8,0.5) for fragmentation. Size of giant component, magnetization, and density of active links at (b) p = 0.55 and (c) p = 0.75 are also shown.