Synchronization in symmetric bipolar population networks

We analyze populations of Kuramoto oscillators with a particular distribution of natural frequencies. Inspired by networks where there are two groups of nodes with opposite behaviors, as for instance, in power-grids where energy is either generated or consumed at different locations, we assume that the frequencies can take only two different values. Correlations between the value of the frequency of a given node and its topological localization are considered in both regular and random topologies. Synchronization is enhanced when nodes are surrounded by nodes of the opposite frequency. The theoretical result presented in this paper is an analytical estimation for the minimum value of the coupling strength between oscillators that guarantees the achievement of a globally synchronized state. This analytical estimation, which is in a very good agreement with numerical simulations, provides a better understanding of the effect of topological localization of natural frequencies on synchronization dynamics.

Figure 1

Illustration of the swap operation between nodes $i$ and $j$ in the reshuffling procedure. To represent visually the bimodal distribution of frequencies, we assign colors to nodes according to their nominal frequencies ${\omega }_i$. Column (a) corresponds to the iteration step $t$; two nodes $i$ and $j$ are chosen at random and we compute their frequency similarity $S_i$ and $S_j$. Column (b) represents the iteration step $t+1$, where the frequencies of the nodes have been interchanged since the overall network similarity increases.

Figure 2

(Color online) Order parameter $r$ [see Eq. (2)] as a function of the coupling strength $\sigma$ for random allocations of natural frequencies on two types of networks. Circles and squares correspond, respectively, to a RR network with similarity $S_{\mathcal{G}}=0.44$ and a Barabasi-Albert SF network with similarity $S_{\mathcal{G}}=0.624$ (number of nodes $N=500$ in both cases). Each point is an average over 30 independent simulations with initially uniformly distributed random phases and the error bars show the standard deviation.

Figure 3

(Color online) Synchronization (averaged stationary value of the proposed order parameter $r_{\omega }$) as a function of the coupling strength $\sigma$ for RR and SF networks ($N=500$ and $\overline{k}=3$) and different network similarities. (a) Original network similarity $\left(S_{RR}=0.467,S_{SF}=0.475\right)$. (b) Low network similarity $\left(S_{RR}=0.099,S_{SF}=0.144\right)$. (c) High network similarity $\left(S_{RR}=0.904,S_{SF}=0.859\right)$. Each point corresponds to an average over 30 independent simulations with uniformly distributed random phases as initial conditions and the error bars stand for the standard deviation. The vertical dashed lines correspond to the approximate analytical solutions for the critical value as calculated in Sec. 5.

Figure 4

Distributions of natural frequencies with low (left) and high (right) network similarity on two regular lattices with node degrees 2 (top) and 4 (bottom). White and black colors correspond, respectively, to $+1$ and $-1$ natural frequencies. See main text for details.

Figure 5

(Color online) Synchronization in one-dimensional regular networks $\left(N=500\right)$ for different degrees and network similarities. Labels correspond to those of Fig. 4. Averaged frequency dispersion $\left(⟨r_{\omega }⟩\right)$ as a function of the coupling $\sigma$. Vertical dashed lines indicate our analytical estimate for the minimum coupling strength needed for synchronization ${\sigma }_c$ [0.5 in cases (a) and (c), 125 in case (b) and 41.6 in case (d)].