What Are Higher-Order Networks?

Abstract

Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higherorder networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.

Pathways to discontinuous transitions in interacting contagion dynamics

“…This complex contagion reflects that transmission in social dynamics not only depends on the pairwise interaction between the transmitter and the receptor of the information but also on the context of the latter [32,33]. Mathematically, complex contagions are modeled through synergistic transmission rates [34], or following higher-order approaches that explicitly incorporate the influence of group structures in the transmission process [35][36][37]. These features prompt the emergence of discontinuous transitions in epidemic dynamics when being coupled with social dynamics, even in the absence of the aforementioned bi-directional enhancement.…”

Encapsulation structure and dynamics in hypergraphs

“…Driven by the availability of datasets with richer connectivity information in recent years, different frameworks have emerged to enrich the network representation, leading to different types of higher-order networks [29]. One branch of this research has extended pairwise graph-based models to multiway interaction frameworks, most notably as hypergraphs or simplicial complexes, to account for group interactions among arbitrary numbers of nodes [3,7,53].…”

Chaos and Chaos Control in Network Dynamical Systems

“…Due to the symmetry properties of G (4) , the right hand side is invariant under exchanging x i and x j for i, j ̸ = k. As we established before, this cannot realize the cubic Guckenheimer-Holmes system (2.1). □ 4.6.…”

“…, N } is given by x k ∈ R d and evolves according to the network interactions. If the interactions take place between pairs of nodes, then a graph G = (V, E) is the traditional combinatorial object that captures the interaction structure, where each unit corresponds to a vertex v ∈ V and the (additive) pairwise interactions take place along edges e ∈ E. However, recent work has highlighted the importance of "higher-order" nonadditive interactions between three or more units [3,4]: In analogy to dynamical systems on graphs, nonadditive interactions have been associated with hyperedges e ∈ E in a hypergraph H = (V, E). While numerous definitions of network dynamical systems on hypergraphs (whether undirected, directed, weighted, etc.)…”

“…In particular, we assume that the formal series expansions (2.3) contain only terms that are simultaneously of odd degrees in the internal variable and of even degrees in the input variables. Note that for G (4) we have y j = z for some j, as there are only three different variables and no variable can be a coupling variable twice.…”