Onset of synchronization in large networks of coupled oscillators

We study the transition from incoherence to coherence in large networks of coupled phase oscillators. We present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength. We find that, under appropriate conditions, the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix. We show how, with an additional assumption, a mean-field approximation recently proposed is recovered from our results. We test our theory with numerical simulations and find that it describes the transition when our assumptions are satisfied. We find that our theory describes the transition well in situations in which the mean-field approximation fails. We study the finite-size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase.

Figure 1

Different approximations and the assumptions leading to them. See text for details.

Figure 2

Largest eigenvalue $\lambda$ (diamonds) and $⟨d^2⟩∕⟨d⟩$ (stars) as a function of $\gamma$ for $N=5000$ and $d_0=20$.

Figure 3

Order parameter $r^2$ obtained from numerical solution of Eq. (3) (triangles), time-averaged theory (solid line), mean-field theory (long-dashed line), and perturbation theory (short-dashed line) as a function of $k∕k_c$ for network (i), with the degree of the nodes uniformly distributed in {50,…,149}. All curves are obtained using the same single random network realization.

Figure 4

Order parameter $r^2$ obtained from numerical solution of Eq. (3) (triangles), time-averaged theory (solid line), frequency distribution approximation (stars), and mean-field theory (long-dashed line) as a function of $k∕k_c$ for degree distributions given by $p\left(d\right)\propto d^{-\gamma }$ if $50⩽d⩽2000$ and $p\left(d\right)=0$ otherwise, with (a) $\gamma =2$, (b) $\gamma =2.5$, (c) $\gamma =3$, and (d) $\gamma =4$. All curves in each figure are obtained using the same single random network realization.

Figure 5

Order parameter $r^2$ obtained from numerical solution of Eq. (3) (solid lines) and growth rate $\mathrm{Re}\left(s\right)$ (dashed lines) for a network with the degree of all nodes $d=20$, $d=50$, and $d=100$ as a function of $k∕k_c$. The arrows indicate which network corresponds to the given curve.