Phase transitions and stability of dynamical processes on hypergraphs

Abstract

Hypergraphs naturally represent higher-order interactions, which persistently appear in social interactions, neural networks, and other natural systems. Although their importance is well recognized, a theoretical framework to describe general dynamical processes on hypergraphs is not available yet. In this paper, we derive expressions for the stability of dynamical systems defined on an arbitrary hypergraph. The framework allows us to reveal that, near the fixed point, the relevant structure is a weighted graph-projection of the hypergraph and that it is possible to identify the role of each structural order for a given process. We analytically solve two dynamics of general interest, namely, social contagion and diffusion processes, and show that the stability conditions can be decoupled in structural and dynamical components. Our results show that in social contagion process, only pairwise interactions play a role in the stability of the absorbing state, while for the diffusion dynamics, the order of the interactions plays a differential role. Our work provides a general framework for further exploration of dynamical processes on hypergraphs.

Dynamical systems on hypergraphs

“…We hereby focus on hypergraphs [8,17,23], versatile tools with a broad potential that is still being fully elucidated. Hypergraphs have been applied to different fields from social contagion model [15,20], to the modelling of random walks [11], from the study of synchronisation [28,35,12] and diffusion [20], to non-linear consensus [37], via the emergence of Turing patterns [12]. It is also worth mentioning an alternative approach to high-order interactions which exploits the notion of simplicial complexes [16,14,41].…”

Full reconstruction of simplicial complexes from binary time-series data

“…High-order interactions are nonetheless ubiquitous in complex networks and its importance has been gradually recognized with an accumulating interest, eventually generating an explosive growth of research recently [18][19][20][21][22][23][24]. The structure of networks with high-order connections, also known as simplicial complexes, are represented by tensors of high orders.…”

“…For example, the drive-response and adaptive synchronization methods use data from continuous-time nonlinear coupled systems [2,3,5,11], while the maximum likelihood estimation method is suitable for data from discrete-time dynamics [13,14,16]. In this paper, motivated by the fact that high-order networks have become a stateof-the-art subfield of research in network science [18][19][20][21][22][23][24], we develop a reconstruction framework for finding from time-series data network topology with high-order interactions.…”

“…A family of random walks on hypergraphs with a parameter controlling the bias of the dynamics towards hyperedges of small or large size was constructed and the impacts of walk strategy and walk time on community detection were elucidated [31]. The stability conditions of the general dynamical processes on hypergraphs were found [18], and a social contagion model on hypergraphs was constructed which presents dynamical phenomena such as first-and second-order transitions, bistability and hysteresis [32]. A simplicial model of social contagion was proposed and it was demonstrated that the reinforce- ment mechanisms in 2-simplex can lead to a discontinuous phase transition [33].…”

Coupled hypergraph maps and chaotic cluster synchronization

“…Historically, the need to model higher-order interactions was observed very early within ecology [10,11], where the interaction between more than two species is very common in predator-prey systems due to intertwining of various competitive, mutualistic, or parasitic effects. Hence, higher-order interactions have recently become a common physical mod-elling principle and we refer to [12] for a detailed survey of the area, which displays a high level of recent activity [13][14][15][16][17] in the context of dynamics. In this work, we are interested in the effect that higher-order coupling between nodes can have on the interactions between coupled maps.…”