Assortative and modular networks are shaped by adaptive synchronization processes

Modular organization and degree-degree correlations are ubiquitous in the connectivity structure of biological, technological, and social interacting systems. So far most studies have concentrated on unveiling both features in real world networks, but a model that succeeds in generating them simultaneously is needed. We consider a network of interacting phase oscillators, and an adaptation mechanism for the coupling that promotes the connection strengths between those elements that are dynamically correlated. We show that, under these circumstances, the dynamical organization of the oscillators shapes the topology of the graph in such a way that modularity and assortativity features emerge spontaneously and simultaneously. In turn, we prove that such an emergent structure is associated with an asymptotic arrangement of the collective dynamical state of the network into cluster synchronization.

Figure 1

Top: (a) Global synchronization indicator $R^G$ (see text for definition), (b) local synchronization indicator, from Eq. (5), and (c) modularity index $\mathcal{M}$, from Eq. (6), vs $p_c$. Each curve corresponds to a different value of the coupling strength $\sigma$, represented with the symbols and colors shown in the legend of (c). Bottom: (d) Transition to synchronization in the model, using the coupling strength $\sigma$ as the order parameter, for $p_c=0.25$, and typical network configurations obtained for (e) $p_c=0.7$, (f) $p_c=0.8$, and (g) $p_c=0.9$ (in all cases $\sigma =0.2$). The three specific values of the correlation threshold are marked with red upward pointing arrows in all the three top panels.

Figure 2

Assortativity coefficient $r$ [as defined in Eq. (7)] vs $p_c$, for different values of $\sigma$ (same stipulations for colors and symbols of the curves as in the caption of Fig. 1). The plot in the inset depicts the slope of the best linear fit of the function $k_{nn}\left(k\right)$ vs $p_c$. Notice that both plots clearly indicate a transition to an assortative configuration for $p_c>0.6$, i.e., in correspondence with the transition to increasingly pronounced modular networks highlighted in Fig. 1c.

Figure 3

Frequency of the oscillating links in the network ${\Omega }^{\text{expt}}$ vs the theoretically predicted frequency ${\Omega }^{\text{th}}$ (see text for the definition of both quantities). When the Pearson's linear correlation coefficient is calculated, a $p$ value of $3\times {10}^{-10}$ is found, indicating that the trend is significative. The red solid line is the best fit of our data, while the blue dashed line is the theoretical prediction from Ref. [16].