Time-scale competition leading to fragmentation and recombination transitions in the coevolution of network and states

We study the coevolution of network structure and node states in a model of multiple state interacting agents. The system displays two transitions, network recombination and fragmentation, governed by time scales that emerge from the dynamics. The recombination transition separates a frozen configuration, composed by disconnected network components whose agents share the same state, from an active configuration, with a fraction of links that are continuously being rewired. The nature of this transition is explained analytically as the maximum of a characteristic time. The fragmentation transition, that appears between two absorbing frozen phases, is an anomalous order-disorder transition, governed by a crossover between the time scales that control the structure and state dynamics.

Figure 1

(Color online) Average relative size of the largest network component (circles) and largest domain (solid line) in the stationary configuration vs $q$, for $N=2500$, averaged over 400 realizations. The vertical lines at $q_c=85$ and $q^{*}=1875$ indicate the transition points between the different phases. Inset: Fluctuations are maximum at the critical point $q_c$.

Figure 2

Size distribution of network components for $N=2500$ and values of $q$ (a), (b) below, (c) at, and (d) above the transition point $q_c\simeq 85$. The dashed line represents a power law with exponent $-1.3\pm 0.02$.

Figure 3

(Color online) Average relative size of the largest network component $S∕N$ vs $q$ for system sizes $N=2500$, 6400, 10 000, and 14 400 (left to right), averaged over 400 realizations. Inset: Finite size scaling. The data collapse below the scaled transition point $Q_c=q_c{N}^{-\alpha }$, with $\alpha =0.82$.

Figure 4

Time evolution of the density of domains $n_d$ (solid line) and network components $n_c$ (dashed line), for $N=2500$ and values of $q$ (a) below, (b) at, and (c) above the transition point $q_{\tau }=194\pm 10$. Each inset is a zoom of the region that shows the approach to the final configuration.

Figure 5

(Color online) Convergence times ${\tau }_c$ (circles), ${\tau }_d$ (squares), and $\tau$ (solid line) vs $q$ for $N=2500$ and averaged over 500 configurations. The result from Eq. (5) (dashed line) is compared with $\tau$ and ${\tau }_c$ for $q>q_{\tau }=194\pm 10$. Inset: Relative difference between the transition points $q_{\tau }$ and $q_c$ vs $1∕N$ in log-log scale. The dashed line has slope $0.30\pm 0.01$.