Geometric characterisation of structural and regular equivalences in undirected (hyper)graphs

TL;DR

The paper addresses the problem of characterising and computing structural and regular equivalence in undirected graphs and hypergraphs by bridging network similarity with geometry via Ollivier–Ricci curvature. It develops a curvature-based characterisation of structural equivalence for graphs and hypergraphs and introduces neighborhood-graph constructions $G_n$, $G_n^{NB}$, and $G_n^{walk}$ to derive regular partitions. It further extends to undirected hypergraphs by defining weak/strong equivalences and two curvature-based random walks (EN-ORC and EE-ORC), establishing a hierarchy of equivalences and showing how connected components of $G_n$ or $H_n$ yield regular partitions under suitable conditions. The work provides a new computational route to regular partitions in hypergraphs and links curvature measures with classical notions of vertex similarity, with potential applications in representation learning and network analysis.

Abstract

Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected hypergraphs and provide a characterisation of structural and regular equivalences of undirected graphs and hypergraphs through neighbourhood graphs and Ollivier-Ricci curvature. Our characterisation sheds new light on these similarity notions opening a new avenue for their exploration. These characterisations also enable the construction of a possibly wide family of regular partitions, thereby offering a new route to a task that has so far been computationally challenging.

Figures (8)

• Figure 1: An illustration of the construction in the argument of the proof of Lemma \ref{['lem:G2']} for $G$ non-bipartite.
• Figure 2: Four different types of graphs that arise as $n$-neighbourhood graphs of $G$.
• Figure 3: (Left) a graph $G$ and (right) its $2$-neighborhood graph $G_2$. While all connected components of $G_2$ are complete, the connected component on the nodes $a,b,c$ does not induce a structural class of $G$.
• Figure 4: A graph $G$ (left), and the partition of its set of nodes into two blocks induced by the two connected components of $G_4$, where we color vertices in the same block by the same color (right). The two connected components of $G_4$ do not induce regular classes on $G$: the nodes $x$ and $y$ belong to the same class, however, while $y$ is connected through an edge to an orange node, there is no edge from $x$ to an orange node.
• Figure 5: An example of a graph $G$ with vertices of degree $1$ and in which the components of $G_3$ give regular partitions.

Theorems & Definitions (50)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10: Ollivier Ricci Curvature on Graphs