# American Institute of Mathematical Sciences

July  2018, 23(5): 1931-1944. doi: 10.3934/dcdsb.2018189

## Explosive synchronization in mono and multilayer networks

 1 Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain 2 Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Madrid, Spain 3 CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy

* Corresponding author: I. Leyva

Received  April 2017 Revised  August 2017 Published  July 2018 Early access  May 2018

Fund Project: Work partly supported by the Spanish Ministry of Economy under project FIS2013-41057-P and by GARECOM, Group of Research Excelence URJC-Banco de Santander. Authors acknowledge the computational resources and assistance provided by CRESCO, the supercomputing center of ENEA in Portici, Italy.

Explosive synchronization, an abrupt transition to a collective coherent state, has been the focus of an extensive research since its first observation in scale-free networks with degree-frequency correlations. In this work, we report several scenarios where a first-order transition to synchronization occurs driven by the presence of a dependence between dynamics and network structure. Therefore, different mechanisms are shown to be able to prevent the formation of a giant synchronization cluster for sufficient large values of the coupling constant in both mono and multilayer networks. Using the Kuramoto model as a reference, we show how for an arbitrary network topology and frequency distribution, a very general weighting procedure acting on the weight of the links delays the synchronization transition forming independent synchronization clusters which suddenly merge above a critical threshold of the coupling constant. A completely different scenario in adaptive and multilayer networks is introduced which gives rise to the emergence of an explosive synchronization when a feedback between the dynamics and structure is operating by means of dependence links weighted through the order parameter.

Citation: Inmaculada Leyva, Irene Sendiña-Nadal, Stefano Boccaletti. Explosive synchronization in mono and multilayer networks. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1931-1944. doi: 10.3934/dcdsb.2018189
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(a) Phase synchronization level $r$ vs the coupling strength $\lambda$, for different values of the frequency gap $\gamma$ (see. Eq. (2)) at $\langle k\rangle = 40$ with network size N = 500. Frequency distribution $g(\omega)$ is chosen to be homogeneous in the [0, 1] interval. (b) Same as in (a), but for different values of the average degree $\langle k\rangle$ at $\gamma = 0.4$. In both panels, the legends report the color and symbol codes for the different plotted curves. In (c) and (d), the degree $k_i$ that each node achieves after the network growth is completed (upper plots) and the average of the natural frequencies $\langle \omega_j\rangle$ of the neighboring nodes ($j\in {\it N}(i)$, bottom plots) are reported vs. the node's natural frequency $\omega_i$, for $\langle k \rangle = 100$ and frequency gaps $\gamma = 0.0$ (c), and $\gamma = 0.4$ (d). The red solid line in (d) is a sketch of the theoretical prediction $f(\omega)$. Panels (e) and (f) show $r$ (color coded according to the color bar) in the parameter space ($\lambda, \gamma$) for (e) $\langle k \rangle = 20$ and (f) $\langle k \rangle = 60$. The horizontal dashed lines mark the separation between the region of the parameter space where a second-order transition occurs (below the line) and that in which the transition is instead of the first order type (above the line). The yellow striped area delimits the hysteresis region
(Top row) $r$ vs. $\lambda$. Synchronization transition schemes for different $g(\omega)$ or network construction rules, for system size N = 500. (Bottom row) The final node degree $k_i$ as a function of $\omega_i$. (a)-(b) Rayleigh distribution for $\gamma = 0.3$. In (b), the red solid line depicts the theoretical prediction $f(\omega)$, obtained with the same method of the red solid line in panel (d) of Fig. 1; (c)-(d) uniform frequency distribution, but network constructed accordingly to a local mean field condition (see text) for $\gamma = 0$, and $\gamma = 0.4$ (see legend for color code). In panel (d) $\gamma = 0.4$. In all cases, $\langle k\rangle = 60$
(a) Synchronization schemes ($r$ vs. $\lambda /\langle k\rangle$) for ER networks of size $N = 500$, $\langle k\rangle$ = 30, for the un-weighted case (dark blue squares), and weighted cases for several frequency distributions within the range $[0, 1]$. (b) Node strengths $s_i$ (see text for definition) vs. natural frequencies $\omega_i$, for the un-weighted (dark blue squares) and weighted (light blue circles) networks reported in (a). The solid line corresponds to the analytical prediction
(a) $I$ as a function of $\mu = \lambda r$ (solid curve) as given by Eq. (16). The dashed lines are those straight lines whose intersection with $I$ marks the backward ($\lambda_1^c$) and forward ($\lambda_2^c$) critical points of ES for a fully connected network and for a uniform frequency distribution. (b) The corresponding synchronization order parameter $r$ as a function of the coupling strength. Solid (dashed) curves correspond to the stable (unstable) solution. Dotted vertical lines mark the region of hysteresis defined by $\lambda_1^c$ and $\lambda_2^c$ in (a)
Plots of the matrix $r_{ij}$ (Eq. (17)) for fully connected (a)-(d) and ER (e-h) networks. Oscillators are labeled in ascending order of the frequency $\omega_i$. The critical coupling strengths where the ES takes place are $\lambda_c$ = 2.13 in $\lambda_c = 2.17$ respectively
Comparison between synchronization centrality $\Lambda^C_i$ (black squares) and topological centrality $\Lambda^A_i$ (red dots). (a) ER networks, $\langle k \rangle$ = 50, $\Lambda^C_i$ and $\Lambda^A_i$ are reported vs. the nodes' natural frequencies $\omega_i$; (b) SF networks, $\langle k \rangle$ = 12, $\Lambda^C_i$ and $\Lambda^A_i$ are plotted vs. the node degrees $k_i$. All data refer to ensemble averages over 100 different network realizations for N = 500 nodes
Forward (black line with squares) and backward (red line with circles) synchronization transitions for a single network with N = 1000 and f = 1. The insets report the dependence on $f$ of the average transition points $\lambda_F$ and $\lambda_B$ for ten realizations
Analytical solutions of Eq. (23) for the order parameter $r$ vs. $\lambda$ for the ER network with $f$ = 1
Synchronization parameters $r^{1}$ and $r^{2}$ vs. $\lambda$ for two inter-dependent networks with $N = 10^3$ and $f = 1$. Squares and circles (triangles and stars) denote the forward (backward) transitions, and the insets report the average width $\langle d\rangle$ of the hysteresis loop as a function of $f$. Here layer 1 is a ER network with $\langle k\rangle = 12$. From (a) to (b), layer 2 is ER network with $\langle k\rangle = 12$ and $\langle k\rangle = 6$, and $g(w_i^2)$ is a random homogeneous distribution in the range $[-1, 1]$. From (c) to (d), layer 2 is ER network and SF network with $\langle k\rangle = 12$, and $g(w_i^2)$ is Lorentzian distribution and random homogeneous fashion
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