Generic Absorbing Transition in Coevolution Dynamics

We study a coevolution voter model on a complex network. A mean-field approximation reveals an absorbing transition from an active to a frozen phase at a critical value $p_c=\frac{\mu -2}{\mu -1}$ that only depends on the average degree $\mu$ of the network. In finite-size systems, the active and frozen phases correspond to a connected and a fragmented network, respectively. The transition can be seen as the sudden change in the trajectory of an equivalent random walk at the critical point, resulting in an approach to the final frozen state whose time scale diverges as $\tau \sim |p_c-p{|}^{-1}$ near $p_c$.

Figure 1

Update events and the associated changes in the density of activelinks $\rho$ andthe link magnetization $m={\rho }_{++}-{\rho }_{--}$ when two neighbors $i$ and $j$ withstates $S_i=s$ and $S_j=-s$ are chosen ($s=\pm 1$).

Figure 2

Stationary density of active links ${\rho }_s$ and the two types of inert links ${\rho }_{++}$ and ${\rho }_{--}$ vs the rewiring probability $p$ as describedby the mean-field theory for a network with average degree $\mu =4$. The critical point $p_c$ separates an active from a frozen phase.

Figure 3

(a) Average relative size of the largest network component $S$ and(b) absolute value of the link magnetization $m$ vs $p$ in thefinal frozen state. (c) Average convergence time $\tau$ persystem size $N$ vs $p$. Inset:scaling of $\tau$ for $p\gtrsim p_c\simeq 0.38$, indicating that $\tau \sim \frac{N}{z}\ln (z)$ with $z=\mu (p-p_c)N$. The solid line has slope $-1$. (d) Stationary density of active links in survivingruns ${\rho }^{\mathrm{surv}}$. Inset: average time evolution of $\rho$.The averages are over ${10}^4$ realizations of networks with $\mu =4$ and sizes $N=250$ (circles), 1000 (squares), and 4000 (diamonds).

Figure 4

Typical trajectories of the random walk for a network of size $N={10}^4$ and $\mu =4$. The upper ($p=0$) and lower ($p=0.35$) parabolas are the trajectories for rewiring rates belowthe transition point $p\simeq 0.38$, while the quasivertical line is for $p=0.4$. Insets: Time evolution of the density of active links $\rho$ (right)and the ratio between $\rho$ andthe product ${\sigma }_{+}{\sigma }_{-}$ (left)in a single realization for different values of $p$.