Understanding High-Order Network Structure using Permissible Walks on Attributed Hypergraphs

TL;DR

This work develops the permissible walk graph, a generalization of the $s$-line graph for attributed hypergraphs, by constructing an attributed $s$-line graph $\mathcal{L}_s=(E,K,\tau,\zeta)$ from $H=(V,E,\phi,\epsilon,\gamma)$ and then restricting edges via a predicate $q:A\times A\rightarrow\{0,1\}$ to form the permissible walk graph $P_q=(E,Q,\tau,\zeta|_Q)$. The methodology preserves rich attribute information (e.g., temporal, categorical) and enables directionality and flow analysis on high-order interactions, demonstrated on a temporally attributed Reddit dataset to reveal intra- and inter-subreddit dynamics that are not captured by conventional $s$-line graphs. A toy example shows edge marginalization and the construction of multiple attribute-aware permissible walks, while the Reddit application illustrates the framework’s ability to quantify information flow through a class interaction matrix and to reveal temporal evolution across $s$-levels. The framework opens avenues for clustering and graph neural network applications in attributed higher-order networks and provides a basis for exploring marginalization across diverse attribute types.

Abstract

Hypergraphs have been a recent focus of study in mathematical data science as a tool to understand complex networks with high-order connections. One question of particular relevance is how to leverage information carried in hypergraph attributions when doing walk-based techniques. In this work, we focus on a new generalization of a walk in a network that recovers previous approaches and allows for a description of permissible walks in hypergraphs. Permissible walk graphs are constructed by intersecting the attributed $s$-line graph of a hypergraph with a relation respecting graph. The attribution of the hypergraph's line graph commonly carries over information from categorical and temporal attributions of the original hypergraph. To demonstrate this approach on a temporally attributed example, we apply our framework to a Reddit data set composed of hyperedges as threads and authors as nodes where post times are tracked.

Figures (8)

• Figure 1: Comparison between the interaction matrix of the $s$-line graph for $s=10$ in (b) and permissible walk graph in (c) from the attributed hypergraph associated to the China_Flu and COVID19 subreddits of Reddit.
• Figure 1: The pipeline for generating the permissible walk graph starting with an attributed hypergraph, which is used to construct the attributed $s$-line graph using an attribute transferring function. The permissible walk graph is then constructed as an attribute respecting subgraph of the $s$-line graph.
• Figure 1: Attributed hypergraph of meetings with meeting time intervals as edge attributes and topics as incidence attributes.
• Figure 1: Trace $T(t)$ (interval count at time $t$) of the collection of temporal intervals $I_j$ associated to each thread for COVID-19 related subreddits from the PAPERCRANE Papercrane2022 dataset.
• Figure 2: $s$-line graphs for hypergraph in Fig. \ref{['fig:pipeline_permissible_walks']} for $s \in \{0,1,2,3\}$ with undirected edges.

Theorems & Definitions (7)

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