Figure 1

Dynamical phase transition in the scaling behavior of the mean time to reach consensus $T_C$ as a function of network size $N$. At the critical value $\rho =1/2$ the growth of $T_C$ is linear, while being exponential for $\rho$ above the critical point, and logarithmic below. In each case circles give the average over 100 simulation runs with $k=10$ and $\varphi =0.1$, while the solid lines show the result of the theory developed in the main text—see Eq. (7).Reuse & Permissions

Figure 2

(a) Dependence of consensus time $T_C$ on the rewiring rate $\varphi$, when $\varepsilon =0$ and $k=10$. Orange circles show the average of 100 samples, the solid black line is the theoretical prediction (8), while the dashed line indicates the point of the fragmentation transition, as derived in Ref. [8]. (b) Comparison between the slow manifold (4) and typical simulation trajectories for $\varphi =0.5$ (red) and $\varphi =0.7$ (blue). The discrepancy between simulations and theory in the case $\varphi =0.7$ illustrates the breakdown of the pair-level approximation close to the fragmentation transition.Reuse & Permissions

Figure 3

Large $N$ scaling for $\rho$ either side of the critical value ${\rho }_c=1/2$. On both plots circles show the average consensus time for 100 simulation runs with $k=10$ and $\varphi =0.1$. On the left the black line is given by Eq. (9), while on the right the slope is given by Eq. (11), whereas the intercept has been chosen manually for comparison.Reuse & Permissions

Figure 4

Consensus time in other model specifications. (a) For link update rules with moderate $k$, consensus time does not depend strongly on the rewiring rule. Theoretical predictions of Eq. (8) (solid line) and the large $k$ limit (dashed line) are compared to simulation results averaged over 1000 networks for rewire to same (orange circles) and rewire to random (green squares). (b) Linear scaling for node direct updates in dense networks with $k=N/10$. The theoretical prediction of $T_C\sim N$ (solid line) is compared to simulation results averaged over 100 samples (purple circles). Parameters: $\varphi =0.5$, $N=1000$.Reuse & Permissions