Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization

Abstract

Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in complex networks' structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the network's structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks' dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and debates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.

Heterogeneous behavioral adoption in multiplex networks

“…Moreover, Boccaletti et al discovered that instead of classic result of the second-order type, i.e. continuous and reversible, in complex networks' structure and dynamics, the synchronizing process rather reminds first-order (discontinuous and irreversible) transitions [19]. For evolutionary games in multiplex network, interactions between layers influence evolution of cooperation [20][21][22].…”

Delay regulated explosive synchronization in multiplex networks

“…In recent years, a novel synchronization process exhibiting hypersensitivity or explosiveness in the natural physical and biological systems, called explosive synchronization (ES), has drawn much attention in the scientific community [1][2][3][4][5][6][7][8]. The epileptic seizures in the brain [9], chronic pain in the Fibromyalgia brain [10], the cascading failure of power grids [11], and the jamming of the internet [12] have shown ES transitions with small initial perturbations.…”

Synchronization clusters emerge as the result of a global coupling among classical phase oscillators

“…Recently, a special phase coherence (called the Bellerophon state) has been characterized in globally coupled nonidentical phase oscillators. Such a state is a quantized, time-dependent, clustered state which emerges close to a first-order-like transition to synchronization [2][3][4][5]. There are two main backgrounds of this work: the first is explosive synchronization, i.e.…”

“…Later, Zhang et al proposed a frequencyweighted Kuramoto model, which can exhibit first-order-like synchronization transition on generic networkʼs topologies for typical frequency distributions [8][9][10][11][12]. Furthermore, it was revealed that the mechanism at the basis of such abrupt synchronization transition is similar to that underlying explosive percolation [2], where the formation of a giant component is controlled by a suppressive rule [10]. Finally, first-order-like synchronization transition was also found in adaptive and multi-layer networks [11].…”