Analytic description of adaptive network topologies in a steady state

In many complex systems, states and interaction structure coevolve towards a dynamic equilibrium. For the adaptive contact process, we obtain approximate expressions for the degree distributions that characterize the interaction network in such active steady states. These distributions are shown to agree quantitatively with simulations except when rewiring is much faster than state update and used to predict and to explain general properties of steady-state topologies. The method generalizes easily to other coevolutionary dynamics.

Figure 1

(a) Determining self-consistent $\langle k_{\mathrm{S},\mathrm{I}}\rangle$ for $a=2$. Solid lines bound the region given by Eq. (12), the dashed line is the actual solution to normalization constraints on Eqs. (4). Imposing $\langle k\rangle =10$ further restricts allowed $\langle k_{\mathrm{S},\mathrm{I}}\rangle$ to the shaded area; feeding in $\left[I\right]=0.89$ as given by the PA for $p=r=1$ yields final constraint (dash-dotted line). Solution to full constraint problem (red circle) is compared to PA prediction (blue square) and simulations (green triangle). [(b) and (c)] Regions (P) in $\langle k\rangle ,a$ plane where $P_{\mathrm{S}}\left(k\right)$ (b) and $P_{\mathrm{I}}\left(k\right)$ (c) peak away from $k=0$ for all choices of $r$, compared to boundary of the largest possible (i.e., for $r=0$) simple active phase in PA (red dashed line).

Figure 2

Stationary statistics of $S$ nodes (blue circles and solid lines) and $I$ nodes (red squares and dashed lines) for $\langle k\rangle =10$, $r=p=1$, and $\left[I\right]=0.89$. Main plots: $P_{\mathrm{S},\mathrm{I}}\left(k\right)$ in simulations (symbols) and model (lines), compared to Poissonian distributions of same $\langle k\rangle$ (black dotted line). Plot insets: $f_{\mathrm{S},\mathrm{I}}$ in simulations; $k$ dependency (symbols) and mean-field approximation (lines). Text insets: $\langle k_{\mathrm{S},\mathrm{I}}\rangle$ (${\sigma }_{\mathrm{S},\mathrm{I}}^2$) in simulations. (a) Static Barabási-Albert graph. (b) $a=0.01$. (c) $a=1$; additionally $P\left(k\right)$ for $r=0.2$ (black crosses, $\left[I\right]=0.98$) and $r=5$ (black pluses, $\left[I\right]=0.44$). All distributions are Poissonian and coincide. (d) $a=3$.