Basin of Attraction Determines Hysteresis in Explosive Synchronization

Abstract

Spontaneous explosive emergent behavior takes place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. A central feature of such explosive transitions is a hysteretic behavior at the transition to synchronization. We unravel the underlying mechanisms and show that the dynamical origin of the hysteresis is a change of basin of attraction of the synchronization state. Our findings hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, reveal how microscopic network parameters such as node degree and frequency affect the global network properties and can be used for network design and control.

The Kuramoto model in complex networks

“…This gives also insights into the dynamics of SF networks with correlation between frequencies and degrees. In their topology hubs can locally form star-like structures leading to a frequency mismatch between oscillators and, consequently, yielding a first-order transition [249,254]. Zou et al [254] also addressed the relation between the dynamics of star and SF networks using a different approach, namely by analyzing the basin of attraction of the synchronized state.…”

“…The authors defined first the state space of Eq. 190 and 191 as a K + 1 dimensional torus T K+1 [254] , where K is the number of peripheral nodes in the star networks, as previously defined. By setting ω H = Kω and ω j = ω (j = 1, ..., K) in Eqs.…”

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

“…Many works addressed the latter kind of question recently, see e.g. references [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. For interesting recent works that highlight the special interplay between dynamics and network structure in neuronal systems, we refer to [27,28].…”

Disorder induces explosive synchronization

“…[16]. Subsequently, significant attention has been paid to the further exploration of degree-frequency correlations [17][18][19][20] and in particular explosive synchronization [21][22][23][24][25][26][27][28][29][30][31][32]. While this research has augmented our understanding of explosive synchronization and its relationship with dynamical and structural correlations, in each case strong conditions are necessarily imposed on either the heterogeneity of the network, its link weights, or its initial construction to engineer first-order phase transitions.…”