Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

Abstract

Collective behavior in large ensembles of dynamical units with non-pairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.PACS numbers: 05.45.Xt, 89.75.Hc Research into the macroscopic dynamics of large ensembles of coupled oscillators have extended our understanding of natural and engineered systems ranging from cell cycles to power grids [1][2][3][4][5]. However, with few exceptions (including [6-8]), little attention has been paid to the synchronization dynamics of coupled oscillator systems where interactions are not pair-wise, but rather n-way, with n ≥ 3. Such interactions are called "simplicial", where an n-simplex represents an interaction between n + 1 units, so 2-simplices describe three-way interactions, etc [9]. Recent advances suggest that simplicial interactions may be vital in general oscillator systems [10-12] and may play an important role in brain dynamics [13][14][15] and other complex systems phenomena such as, the dynamics of collaborations [16] or social contagion [17]. In particular, interactions in 2-simplices (named holes or cavities) are important because they can describe correlations in neuronal spiking activity (that can be mapped to phase oscillators [18]) in the brain [19] providing a missing link between structure and function. In fact, coupled oscillator systems that display clustering and multi-branch entrainment have been shown to be useful models for memory and information storage [20][21][22][23][24][25][26]. Despite these findings, the general collective dynamics of coupled oscillator simplices and their utility in storing information are poorly understood.In this work we study large coupled oscillator simplicial complexes, considering the impact of 2-simplices, i.e., threeway interactions, on collective behavior. Specifically, we consider the 2-and 1-simplex multilayer system given bẏThe dynamics in the θ-layer are the natural generalization of the classical Kuramoto model [27] with 2-simplex interactions (namely, coupling is sinusoidal and diffusive), where θ i represents the phase of oscillator i with i = 1, . . . , N , ω i is its natural frequency which is assumed to be drawn from the distribution g(ω), an...

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The spectral dimension of simplicial complexes: a renormalization group theory

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Abrupt phase transition of epidemic spreading in simplicial complexes

“…Recently, the network science community has turned its attention to network geometry [6-9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems.…”

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