Network synchronization, diffusion, and the paradox of heterogeneity

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Figure 1

Eigenratio $R$ as function of $\beta$: (a) random SFN’s with $\gamma =3$ (●), $\gamma =5$ (∎), and $\gamma =\infty$ (solid line), for $k_{\mathit{min}}=10$; (b) networks with expected scale-free sequence for $\gamma =3$ and ${\tilde{k}}_{\mathit{min}}=10$; (c) growing SFN’s for $\gamma =3$ and $m=10$; (d) SWN’s with $M=256$ (●) and $M=512$ (∎), for $\kappa =1$. Each curve is the result of an average over 50 realizations for $N=1024$.

Figure 2

Eigenratio $R$ as a function of the scaling exponent $\gamma$: (a) random SFN’s, (b) networks with expected scale-free sequence, and (c) growing SFN’s, for $\beta =1$ (●) and $\beta =0$ (엯). The other curves are the approximations of the eigenratio in Eqs. (20) (dotted lines) and (21) (solid lines), and the eigenratio for $\beta =1$ (dashed lines) and $\beta =0$ (dot-dashed lines) at $\gamma =\infty$. The ◇ symbols correspond to random homogeneous networks with the same mean degree of the corresponding SFN’s (the degrees are indicated in the figure). The other network parameters are the same as in Fig. 1.

Figure 3

Eigenratio $R$ as a function of the number of oscillators for $\gamma =3$ and the SFN models in Figs. 2a, 2b, 2c, respectively. Dotted lines are guides for the eyes. The legend and other parameters are the same as in Fig. 2.

Figure 4

Normalized cost $C∕N{\alpha }_1$ as a function of the scaling exponent $\gamma$ for random SFN’s with $\beta =1$ (●) and $\beta =0$ (엯), and for random homogeneous networks with the same mean degree (◇). The dashed line corresponds to $\gamma =\infty$. The other parameters are the same as in Fig. 1.

Figure 5

Normalized cost $C∕N{\alpha }_1$ as a function of $\beta$ for random SFN’s with scaling exponent $\gamma =3$. The other parameters are the same as in Fig. 1.

Figure 6

Synchronization region $({\sigma }_{\mathit{min}}^{*},{\sigma }_{\mathit{max}}^{*})$ as a function of $\gamma >2.8$ in random SFN’s of logistic maps, for $\beta =1$ (●) and $\beta =0$ (엯). The bifurcation parameter is (a) $a=3.58$ and (b) $a=4.0$. Averages are taken over 20 realizations of the networks. The other parameters are the same as in Fig. 1.