Abstract
Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Recent work in physics and neuroscience highlights the importance of higher-order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior. Here we show that higher-order interactions between coupled phase oscillators, encoded microscopically in a simplicial complex, give rise to added nonlinearity in the macroscopic system dynamics that induces abrupt synchronization transitions via hysteresis and bistability of synchronized and incoherent states. Moreover, these higher-order interactions can stabilize strongly synchronized states even when the pairwise coupling is repulsive. These findings reveal a self-organized phenomenon that may be responsible for the rapid switching to synchronization in many biological and other systems that exhibit synchronization without the need of particular correlation mechanisms between the oscillators and the topological structure.
Highlights
Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems
Recent work in physics and neuroscience have highlighted the importance of higher order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior[13,14,15,16,17,18,19,20], prompting the network science community to turn its attention to higher order structures to better represent the kinds of interactions that one can find beyond typical pairwise interactions[21,22,23]
While simplicial complexes have been proven to be very useful for analysis and computation in high dimensional data sets, e.g., using persistent homologies[17], little is understood about their role in shaping dynamical processes, save for a handful of examples[25,26,27,28,29]
Summary
Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Our understanding of both natural and man-made systems has significantly improved by studying how network structures and dynamical processes combine to shape overall system behaviors This interplay gives rise to nonlinear phenomena like switch-like abrupt transitions to synchronization[8,9,10] and cluster states[11,12]. Recent work in physics and neuroscience have highlighted the importance of higher order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior[13,14,15,16,17,18,19,20], prompting the network science community to turn its attention to higher order structures to better represent the kinds of interactions that one can find beyond typical pairwise interactions[21,22,23] These higher order interactions are often encoded in simplicial complexes[24] 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. The collective dynamics of sources and loads in large-scale power grids provides another important application where abrupt synchronization transitions play an important role[35]
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