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

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.

Figure 1

Comparative results on ES between SF networks belonging to two different ensembles: preferential attachment (PA) and configuration model (CM). Top row: Forward (solid lines) and backward (dashed lines) synchronization curves for PA (left panel) and CM (right panel) SF networks with exactly the same degree distribution. The area of the hysteresis is depicted as a blue shaded area in the PA-SF network while the synchronization jumps are marked in both cases. Bottom row: Probability density functions of the area of hysteresis (left panel) and of the synchronization jumps (right panel) for 400 PA (CM) network realizations. In all cases, $N={10}^3,\langle k\rangle =6,\gamma =2.4$, and natural frequencies ${\omega }_i=k_i$. The CM network realizations are constructed using the degree sequences of the PA networks.

Figure 2

ES as a function of the degree mixing. Top row: Area of the hysteretic region delimited by the forward and backward synchronization curves vs the Pearson correlation coefficient $r$ for PA (left panel) and CM (right panel) networks with different values (reported in the legend) of the exponent $\gamma$ of the degree distribution $P\left(k\right)\sim k^{-\gamma }$. Each point is an average over ten different simulations. Bottom row: Forward (solid lines) and backward (dashed lines) synchronization curves for PA networks $\left(\gamma =2.4\right)$ displaying different levels of assortative (left panel) and disassortative (right panel) mixing. Curves are coded accordingly to the specific value of the parameter $r$ (reported in the legend of each panel). In all cases, networks are SF with $N=5\times {10}^3,\langle k\rangle =6$, and ${\omega }_i=k_i$.

Figure 3

Area of hysteresis as a function of $r$ for PA networks with different amounts of heterogeneity. In the left panel, the different curves correspond to different values of the mean degree $\langle k\rangle$ (reported in the legend), while in the right panel $\langle k\rangle =6$ and the network heterogeneity is varied by means of increasing the parameter $\alpha$ (see the legend for the color and symbol code of the different reported curves), from pure SF $\left(\alpha =0\right)$ to $\alpha =0.3$ $(\alpha =1$ corresponds to a pure ER network). In all cases, $N={10}^3,\gamma =2.4$, and ${\omega }_i=k_i$, and each point is an average of ten simulations.

Figure 4

Behavior of the critical parameters characterizing the synchronization transition. Critical coupling strengths (left panel) and synchronization jumps $\Delta S_{max}$ of the order parameter (right panel) at the synchronization transitions during the forward (solid symbols) and backward (hollow symbols) continuations for growing PA $(\circ )$ and static CM $(△)$ SF networks as a function of the degree mixing $r$. In all cases, $N=5\times {10}^3,\langle k\rangle =6$, and $\gamma =2.4$. The vertical dashed line marks $r=0$. The inset of the left panel reports the corresponding width of the hysteresis curves, calculated as $\Delta {\sigma }_c=|{\sigma }_c^{fw}-{\sigma }_c^{bk}|$ for PA $(\circ )$ and CM $(△)$ SF networks, with “fw” and “bk” indicating “forward” and “backward,” respectively.

Figure 5

Dependence of the network structural properties on the degree mixing. Betweenness centrality of the first three higher-degree hubs (left panel) and mean network distance of the hub of the network (right panel) as a function of the Pearson correlation coefficient $r$ for PA (red and solid symbols) and CM (blue and hollow symbols) SF networks with different values of the exponent $\gamma$ of the degree distribution $P\left(k\right)\sim k^{-\gamma }$ as indicated in the legend. Each point is the average of ten network realizations with $N=5\times {10}^3$ and $\langle k\rangle =6$.