A statistical perspective on higher-order interactions modeling

Abstract

Modeling higher-order interactions (HOI) has emerged as a crucial challenge in complex systems analysis, as many phenomena cannot be fully captured by pairwise relationships alone. Hypergraphs, which generalize graphs by allowing interactions among more than two entities, provide a powerful framework for representing such intricate dependencies. Adopting a statistical and probabilistic perspective on hypergraph modeling, we propose a guided tour through this emerging research area. We begin by illustrating the ubiquity of HOI in real-world systems, where interactions often involve groups of entities rather than isolated pairs. We then introduce the foundational concepts and notations of hypergraphs, discussing their descriptive statistics, graph-based representations, and the challenges associated with their complexity. We further explore a variety of statistical models for hypergraphs and address the critical task of node clustering. We conclude by outlining some open challenges in the field.

Figures (3)

• Figure 1: A hypergraph with 7 nodes and 3 hyperedges: $e_1=\{1,2,3,4\}$, $e_2=\{5,6,7\}$ and $e_3=\{3,6\}$.
• Figure 2: Graph representations of the hypergraph from Fig. \ref{['fig:toy_hypergraph']}. (a) Clique graph; (b) Line graph; (c) Bipartite graph.
• Figure 3: (a) A bipartite graph $\mathcal{G}$; (b) Projection of $\mathcal{G}$ into the space of multisets hypergraphs with self-loops, choosing the top nodes of $\mathcal{G}$ as the new set of nodes. Hyperedges are $\{1,2\}, \{1\}, \{1,2,3\}$ and $\{1,2\}$. The applications from (a) to (b) are invertible bijections, one being the inverse of the other; (c) Projection of $\mathcal{G}$ on the simple hypergraphs subspace: the multiplicity of hyperedge $\{1,2\}$ and the self-loop $\{1\}$ have been removed. (d) Embedding of the simple hypergraph from (c) in the bipartite graphs space. Note that (a) and (d) are not the same bipartite graph.