Optimal Synchronization of Complex Networks

We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators’ frequencies and that can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure.

Figure 1

Constrained frequency allocation: (a) $r$ vs $K$ for optimal allocation (blue circles) compared with random allocations drawn from normal (red triangles), uniform (green pluses), and Laplace (orange squares) distributions. (b) $r$ vs $K$ for a prechosen set of normally distributed frequencies with random (red triangles), first-order (black crosses), and near-optimal (blue circles) allocations. The near-optimal allocation was obtained from $10^6$ proposed frequency exchanges. Networks are scale free (SF) with $N=1000$, $\gamma =3$, and $d_0=2$.

Figure 2

Optimal network design: (a)–(b) $r$ vs $K$ for initial (red crosses) and rewired (blue circles) networks with normal and power-law distributed frequencies. (c)–(d) Degree distributions of the initial (red crosses) and rewired (blue circles) networks. (e)–(f) Illustrations for networks of $N=40$ and $N=36$ nodes, respectively after rewiring.

Figure 3

Eigenvector alignments: (a) $r$ vs $K$ for frequency alignments $\mathit{\omega }\propto {\mathit{v}}^{100},{\mathit{v}}^{200},\dots ,{\mathit{v}}^{1000}$ (red to blue). (b) $r$ vs $i$ for $\mathit{\omega }\propto {\mathit{v}}^i$ with fixed $K=0.08$ (blue circles), 0.25 (red crosses), and 1 (green triangles). Each point is averaged over 50 realizations of SF networks with $N=1000,$ $\gamma =3$ and $d_0=2$.

Figure 4

Correlations in optimized networks: (a),(b) Frequency magnitude $|{\omega }_i|$ vs degree $d_i$ and (c),(d) average neighbor frequency $⟨{\omega }_i⟩$ vs frequency ${\omega }_i$ for networks with optimally chosen frequencies (green triangles), prechosen frequencies (blue circles), and a rewired network (red crosses). Networks are SF with $N=1000$, $\gamma =3$, and $d_0=3$. Frequencies for the arranged and rewired cases are normally distributed.