Effects of degree correlations on the explosive synchronization of scale-free networks

Abstract

We study the organization of finite-size, large ensembles of phase oscillators networking via scale-free topologies in the presence of a positive correlation between the oscillators' natural frequencies and the network's degrees. Under those circumstances, abrupt transitions to synchronization are known to occur in growing scale-free networks, while the transition has a completely different nature for static random configurations preserving the same structure-dynamics correlation. We show that the further presence of degree-degree correlations in the network structure has important consequences on the nature of the phase transition characterizing the passage from the phase-incoherent to the phase-coherent network state. While high levels of positive and negative mixing consistently induce a second-order phase transition, moderate values of assortative mixing, such as those ubiquitously characterizing social networks in the real world, greatly enhance the irreversible nature of explosive synchronization in scale-free networks. The latter effect corresponds to a maximization of the area and of the width of the hysteretic loop that differentiates the forward and backward transitions to synchronization.

The Kuramoto model in complex networks

“…Surprisingly, networks statically constructed with the very same size N , average degree k and exponent γ through the CM do not present the same dependency on A as observed in Fig. 31(a) [142]. In this case, the hysteresis is maximized for slightly negative values of A, besides having significant lower values (Fig.…”

“…3.2, the critical coupling λ c was observed to be positively correlated with the network assortativity [141]. The irreversibility of the phase transition was further investigated in [142]. The hysteric behavior of the order parameter is maximized for positive values of the assortativity coefficient A in growing SF networks, as shown in Fig.…”

“…Curiously, the thorough analysis of the influence of assortative mixing in the dynamics of Kuramoto oscillators has been only addressed very recently and mostly in the context of correlation between natural frequencies and degrees [137][138][139][140][141][142] (See Sec. 5).…”

“…31(b)). This interesting phenomenon was explained in [142] by studying the average betweenness centrality among the highest connected hubs as a function of the degree-mixing. It was found that hubs belonging to networks constructed through growing processes exhibit significant higher betweenness centrality than those in networks generated by the CM.…”

“…In other words, in growing SF networks, the hubs segregate more efficiently the low-degree nodes, whereas in the CM the highly connected nodes are more homogeneously distributed throughout the network. Thus, since explosive synchronization is induced by the large frequency gaps between neighbors, frustrating in this way the paths to synchronization [142,268,273], growing SF networks are more vulnerable to have paths frustrated by such gaps, explaining the higher irreversibility in the transition than in the CM. Furthermore, as the networks are rewired towards extreme values of positive or negative assortativity A, other topological properties, such as average shortest path length, clustering coefficient and even modular structure [106], are dramatically changed up to a point that the frequency gap is not sufficient to yield an abrupt formation of a synchronous giant component [142,268].…”