Inference and Visualization of Community Structure in Attributed Hypergraphs Using Mixed-Membership Stochastic Block Models

TL;DR

This study proposes a framework, HyperNEO, that combines mixed-membership stochastic block models for hypergraphs with dimensionality reduction methods and generates a node layout that largely preserves the community memberships of nodes.

Abstract

Hypergraphs represent complex systems involving interactions among more than two entities and allow the investigation of higher-order structure and dynamics in complex systems. Node attribute data, which often accompanies network data, can enhance the inference of community structure in complex systems. While mixed-membership stochastic block models have been employed to infer community structure in hypergraphs, they complicate the visualization and interpretation of inferred community structure by assuming that nodes may possess soft community memberships. In this study, we propose a framework, HyperNEO, that combines mixed-membership stochastic block models for hypergraphs with dimensionality reduction methods. Our approach generates a node layout that largely preserves the community memberships of nodes. We evaluate our framework on both synthetic and empirical hypergraphs with node attributes. We expect our framework will broaden the investigation and understanding of higher-order community structure in complex systems.

Figures (6)

• Figure 1: Overview of the present framework. $N$: number of nodes. $K$: number of communities.
• Figure 2: Node layouts in synthetic hypergraphs with or without community structures. We set $(N, k, b) = (500, 3, 1)$ for generating all synthetic hypergraphs via the HSBM. We set $a=1$ in (a), (c), (e), and (g); we set $a = 10$ in (b), (d), (f), and (h); we show the results for t-SNE in (a) and (b), those for UMAP in (c) and (d), those for TriMap in (e) and (f), and those for PaCMAP in (g) and (h).
• Figure 3: Node layouts in synthetic hypergraphs with soft or hard community memberships. We set $(N, D, w_{\text{in}}) = (500, 3, 0.1)$ for generating all synthetic hypergraphs via the Hy-MMSBM. We set $\mu=1.0$ in (a), (c), (e), and (g); we set $\mu=0.8$ in (b), (d), (f), and (h); we show the results for t-SNE in (a) and (b), those for UMAP in (c) and (d), those for TriMap in (e) and (f), and those for PaCMAP in (g) and (h). We refer to the set of nodes that have the community membership $[\mu, 1-\mu]$ as 'the first set of nodes' and the remaining nodes as 'the second set of nodes'.
• Figure 4: Node layouts in synthetic hypergraphs with node attributes independent of or strongly associated with community structures. We set $(N, k, a, b) = (500, 3, 4, 1)$ for generating all synthetic hypergraphs via the HSBM. We set $r=0.5$ in (a), (c), (e), and (g); we set $r=0.9$ in (b), (d), (f), and (h); we show the results for t-SNE in (a) and (b), those for UMAP in (c) and (d), those for TriMap in (e) and (f), and those for PaCMAP in (g) and (h).
• Figure 5: Inference and visualization of community structure in the high-school hypergraph. (a) and (b): Inferred community membership matrices of (a) Hy-MMSBM and (b) HyCoSBM. (c): Inferred matrix $\bm{\beta}$ of HyCoSBM. (d) and (e): Node layouts generated using (d) Hy-MMSBM and (e) HyCoSBM. In panels (a) and (b), for visualization purposes, we arranged the $N$ row indices according to the attributes of the nodes, and we arranged the $K$ column indices arbitrarily.