Noise-induced synchronization in small world networks of phase oscillators

A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

Figure 1

Order parameter ($r$) versus time (on a logarithmic scale) for ER [dark-gray (fuchsia) dashed curve], Barabási-Albert [light-gray (green) dashed curve], and WS networks for five initial conditions (solid curves for, from top to bottom, WS-a to WS-e). $N=1000$ and $\langle k\rangle =10$.

Figure 2

Steady-state phase configurations in two helical patterns of a regular network and corresponding steady states of WS networks with $N=1000$ and $\langle k\rangle =10$. WS-b, $\lambda =1000$; WS-e, $\lambda =50$. $\lambda$ is the wavelength of helical states in the regular network.

Figure 3

Density plot of correlation matrix elements ($D_{ij}$) for four helical states of a regular network and corresponding stationary states in a WS network with $N=1000$ and $\langle k\rangle =10$. (b) $\lambda =1000$, (c) $\lambda =250$, (d) $\lambda =100$, and (e) $\lambda =50$. $\lambda$ is the wavelength of helical states in the regular network.

Figure 4

Stationary order parameter versus reduced noise intensity for the four network types. Left: Regular, ER, SF, and fully synchronized states of WS. Right: Four phase-locked states of WS corresponding to states represented in Fig. 3. The number of nodes and mean degree for the three networks are $N=1000$ and $\langle k\rangle =10$.

Figure 5

Density plot of correlation matrix elements ($D_{ij}$) for a steady state with four point defects on an SW network and different noise intensities: (a) $g=0$, (b) $g=2$, (c) $g=3$, (d) $g=4$, (e) $g=5$, (f) $g=6$, (g) $g=7$, and (h) $g=8$. The number of nodes and mean degree for the three networks are $N=1000$ and $\langle k\rangle =10$.

Figure 6

Probability distribution function of correlation matrix elements ($D_{ij}$) for different noise intensities: (a) $g=0$, (b) $g=2$, (c) $g=3$, (d) $g=4$, (e) $g=5$, (f) $g=6$, (g) $g=7$, and (h) $g=8$. The number of nodes and mean degree for the three networks are $N=1000$ and $\langle k\rangle =10$.

Figure 7

Left: Complexity of small-world network versus noise intensity for $N=1000$ and $\langle k\rangle =10$. These plots correspond to initial conditions leading to the helical state in parent regular networks with wavelength $\lambda =1000$, 250, 100, and 10. Right: Complexity of ER and SF networks versus noise intensity for $N=1000$ and $\langle k\rangle =10$.

Figure 8

Phase diagram of the noisy Kuramoto model for an SW network with $N=1000$ and $\langle k\rangle =10$, in g-p space. $g_s$ [light-gray (green) line] and $g_c$ [dark-gray (red) line] denote the noise strengths at the onsets of stochastic synchronization and desynchronization, respectively.