Modeling temporal hypergraphs

TL;DR

This paper introduces Relational Hyperevent Models (RHEM) as a flexible, Cox-like framework for temporal hypergraphs, enabling tailored null distributions that condition on selected higher-order temporal statistics. By modeling the next hyperevent probability with a log-linear form and matching expected statistics to observed values, RHEM facilitate tests of patterns such as preferential attachment, subset repetition, triadic closure, assortativity, and homophily, while generalizing to directed, multilayer, typed, and weighted hyperedges. The Les Misérables actor network serves as a running illustration, demonstrating how RHEM quantify over- or under-representation of higher-order configurations beyond node degrees and dyads, and how chapter-level probabilities can be interpreted relative to potential alternatives. The framework is extensible, supported by case-control estimation, and applicable across diverse domains; the authors also point to software tooling (eventnet) for computing temporal hyperedge statistics and fitting models.

Abstract

Networks representing social, biological, technological or other systems are often characterized by higher-order interaction involving any number of nodes. Temporal hypergraphs are given by ordered sequences of hyperedges representing sets of nodes interacting at given points in time. In this paper we discuss how a recently proposed model family for time-stamped hyperedges - relational hyperevent models (RHEM) - can be employed to define tailored null distributions for temporal hypergraphs and to test and control for complex dependencies in hypergraph dynamics. RHEM can be specified with a given vector of temporal hyperedge statistics - functions that quantify the structural position of hyperedges in the history of previous hyperedges - and equate expected values of these statistics with their empirically observed values. This allows, for instance, to analyze the overrepresentation or underrepresentation of temporal hyperedge configurations in a model that reproduces the observed distributions of possibly complex sub-configurations, including but going beyond node degrees. Concrete examples include, but are not limited to, preferential attachment, repetition of subsets of any given size, triadic closure, homophily, and degree assortativity for subsets of any order.

Figures (4)

• Figure 1: Number of chapters with a given number of actors (i. e., given the hyperedge size).
• Figure 2: (Adapted from lerner2022dynamic.) Illustrating subset repetition based on four prior events $e_1,\dots,e_4$ on the hyperedges $I=\{A,B,C\}$ and $I'=\{D,E,F\}$ at time $t>t_4$. For $I$ it is $subrep^{(1)}(I',E_{<t})=3$, $subrep^{(2)}(I',E_{<t})=3$, and $subrep^{(3)}(I',E_{<t})=1$. For $I'$ it is $subrep^{(1)}(I',E_{<t})=6$, $subrep^{(2)}(I',E_{<t})=3$, and $subrep^{(3)}(I',E_{<t})=1$. Moreover, Event $e_4$ is an illustration of triadic closure (defined further below), as the event connecting Nodes $E$ and $F$, closes a two-path $E$--$D$--$F$, which has been established in two previous events, $e_2$ and $e_3$.
• Figure 3: Chapter probabilities estimated by Model 3, in Table \ref{['tab:models_subrep']}.
• Figure 4: Probability that the chapter's probability estimated by Model 3, in Table \ref{['tab:models_subrep']} is larger than the estimated probability of a sampled alternative "non-event" hyperedge of the same size.