Dynamical systems on Hypergraphs

Abstract

Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider a general class of dynamical systems anchored on hypergraphs. Hyperedges of arbitrary size ideally encircle individual units so as to account for multiple, simultaneous interactions. These latter are mediated by a combinatorial Laplacian, that is here introduced and characterised. The formalism of the Master Stability Function is adapted to the present setting. Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs. The response to externally imposed perturbations bears the imprint of the higher order nature of the interactions. I. INTRODUCTION.Network science [1,2] has proved successful in describing many real-world systems [3-5], which, despite inherent differences, share common structural features. Even more interestingly, dynamical processes and hosting networks are indissolubly entangled with the ensuing patterns that reflect in fact the complex topology of the supports to which they are anchored [6][7][8].Networks constitute abstract frameworks, where pairwise interactions among generic agents, represented by nodes, are schematised by edges. Stated simply, two agents are connected if they interact. Hence by their very first definition, networks encode for binary relationships among units. This descriptive framework is sufficiently accurate in many cases of interest, although several examples exist of systems for which it holds true just as a first order approximation [9,10]. The relevance of high-orders structures has been indeed emphasised in the context of functional brain networks [11,12], in applications to protein interaction networks [13], to the study of ecological communities [14] and co-authorship networks [15,16].Starting from this observation, higher-order models have been developed so as to capture the many body interactions among interacting units. The most notable examples are simplicial complexes [17][18][19] and hypergraphs [20][21][22], non trivial mathematical generalisations of ordinary networks that are currently attracting a lot of interest. The concept of simplicial complexes has been for instance invoked to address problems in epidemic spreading [23,24] or synchronisation phenomena [25,26]. Our work is positioned in the framework of hypergraphs, a domain of investigation which is still in its infancy. In this respect, we mention applications to social contagion model [27,28], to the modelling of random walks [16] and to the study of synchronisation [29,30] and diffusion [28].Hypergraphs constitute indeed a very flexible paradigm: an arbitrary number of agents are allowed to interact, thus extending beyond the limit of binary interactions of conventional network models. On the other hand, hypergraphs define a leap forward as compared to simplicial com...

Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions

“…However, the associated tensors describing those higher-order structures are more involved than matrices, especially when combined with the analysis of dynamical processes [37][38][39][40][41][42]. There have been several efforts to generalize the master stability function (MSF) formalism [17] to these settings, for which different variants of an aggregated Laplacian have been proposed [43][44][45][46]. The aggregated Laplacian captures interactions of all orders in a single matrix, whose spectral decomposition allows the stability analysis to be decoupled into structural and dynamical components, just like the usual MSF for pairwise interactions.…”