Effective centrality and explosive synchronization in complex networks

Synchronization of networked oscillators is known to depend fundamentally on the interplay between the dynamics of the graph's units and the microscopic arrangement of the network's structure. We here propose an effective network whose topological properties reflect the interplay between the topology and dynamics of the original network. On that basis, we are able to introduce the effective centrality, a measure that quantifies the role and importance of each network's node in the synchronization process. In particular, in the context of explosive synchronization, we use such a measure to assess the propensity of a graph to sustain an irreversible transition to synchronization. We furthermore discuss a strategy to induce the explosive behavior in a generic network, by acting only upon a fraction of its nodes.

Figure 1

Synchronization transition curves (black circles) compared with the modified adjacency matrix $A_{ij}\left(1+\delta \mathrm{\Delta }{\omega }_{ij}\right)$ in (a) ER, $\langle k\rangle =30$ and (b) SF $\langle k\rangle =12$ networks for positive and negative $\delta$ values. Red triangles ($\delta >0$) and blue squares ($\delta <0$) show how a local perturbation enhances/frustrates the synchronization more efficiently than a global perturbation over all links [dashed lines, where $\delta \mathrm{\Delta }{\omega }_{ij}\to \delta \langle \mathrm{\Delta }\omega \rangle$, being $\langle \mathrm{\Delta }\omega \rangle$ the average over nonzero values of the detuning matrix].

Figure 2

Comparison between synchronization centrality ${\mathrm{\Lambda }}_i^C$ (black squares) and topological centrality ${\mathrm{\Lambda }}_i^A$ (red dots). (a) ER networks, $\langle k\rangle =50, {\mathrm{\Lambda }}_i^C$, and ${\mathrm{\Lambda }}_i^A$ are reported vs the nodes' natural frequencies ${\omega }_i$; (b) SF networks, $\langle k\rangle =12, {\mathrm{\Lambda }}_i^C$, and ${\mathrm{\Lambda }}_i^A$ are plotted vs the node degrees $k_i$. All data refer to ensemble averages over 100 different network realizations.

Figure 3

Average percentage of coincidence between the third of the nodes with the highest (red triangles) and lowest (blue circles) dynamical ${\mathrm{\Lambda }}_i^S$ and effective ${\mathrm{\Lambda }}_i^C$ centralities (solid symbols) and the corresponding percentage between dynamical ${\mathrm{\Lambda }}_i^S$ and topological ${\mathrm{\Lambda }}_i^A$ centralities (empty symbols). Calculations are performed on ER networks with $\langle k\rangle =50$, and refer to 10 realizations of the network's topology and frequency distribution (see text for details).

Figure 4

Effective centrality of explosive networks (${\mathrm{\Lambda }}_i^{\tilde{C}}$, red clouds) versus that of non-explosive networks (${\mathrm{\Lambda }}_i^C$, cyan clouds) for ER (left panels) and SF (right panels) topologies (see text for definitions). (a) and (b) account for the weighted method of Ref. [21] with a uniform natural frequency distribution in $[-0.5,0.5]$. (c) and (d) correspond to the weighted method of Ref. [31] with the same uniform frequency distribution (see text for the methods' description). (e) is the same as in (d) but natural frequencies are here chosen from the positive values of a Gaussian distribution, in a way that ES is no longer induced. Finally, (f) corresponds to a SF network where the degree-frequency correlation method from Ref. [15] is applied and the same uniform frequency distribution as for (a)–(d) is used. The inset in (f) is a zoom centered on the highest-degree nodes. All data refer to averages over 100 realizations.

Figure 5

Maximum jump size in the synchronization curve $\sigma \left(r\right)$ vs fraction of perturbed nodes chosen along several rankings: effective (back squares), distance (red dots), topology (magenta diamonds), and random (blue asterisks). (a) ER networks $\langle k\rangle =30$, weighting method from Ref. [21]. (b) SF networks $\langle k\rangle =6$ imposing ${\omega }_i=k_i$ for the selected nodes as in Ref. [15]. In all cases, data refer to averages over 50 realizations.