Uncovering higher-order interactions from memory in observed dynamics

Abstract

Many complex systems involve group interactions, which can readily be represented by higher-order structures such as hypergraphs. However, when observing a system's dynamical evolution, establishing whether that evolution actually results from higher-order interactions, or whether pairwise interactions suffice to explain the data, remains a challenge. Even in the simple case of a temporal sequence of nodes visited during a diffusion process, such a task is intricate. Indeed, group interactions leave their traces as memory on the observed nodes, while path dependence in a purely pairwise process can mimic that same footprint without any group structure. To resolve this, we propose an inference pipeline based on a hidden Markov model (HMM) that tests, directly from an observed sequence of nodes carrying no group labels, whether higher-order interactions are required to describe the dynamics. The HMM latent states represent higher-order interactions, seen as hyperedges of an unobserved hypergraph, so that their detection becomes a model selection problem between the HMM class and a class of null models without higher-order interactions. We first validate this approach on synthetic node sequences generated from random walks on graphs and hypergraphs, showing that it recovers both the number and the composition of the underlying hyperedges, while correctly rejecting them when the generating process is pairwise. As a second step, we consider a validation based on empirical data, namely empirical trajectories of ball possession in football matches. This application shows that the pipeline goes beyond providing statistical evidence of the presence of higher-order interactions, and yields interpretable underlying structures: the recovered hyperedges map directly onto recognizable tactical team formations.