Finding Influential Cores via Normalized Ricci Flows in Directed and Undirected Hypergraphs with Applications

TL;DR

This work introduces a curvature-guided discrete diffusion framework to identify influential cores in both directed and undirected hypergraphs, leveraging hypergraph Ricci flow with topological surgery to reveal cohesive, central substructures. Core quality is assessed via connectivity, size, cohesiveness, centrality, and statistical significance, with directed and undirected cases treated via tailored P_left/P_right calculations and centrality metrics using Earth Mover's Distance. The authors demonstrate the method on seven metabolic directed hypergraphs and two co-authorship undirected hypergraphs, report rapid initial convergence of the flows, and provide interpretive analyses of the resulting cores. A theoretical result shows that a previously proposed normalized Ricci-flow update can produce negative edge weights in graphs, underscoring the practical advantage of their sigmoid-based normalization, and the work highlights potential biological and scholarly insights gained from core perturbations and collaborations.

Abstract

Many biological and social systems are naturally represented as edge-weighted directed or undirected hypergraphs since they exhibit group interactions involving three or more system units as opposed to pairwise interactions that can be incorporated in graph-theoretic representations. However, finding influential cores in hypergraphs is still not as extensively studied as their graph-theoretic counter-parts. To this end, we develop and implement a hypergraph-curvature guided discrete time diffusion process with suitable topological surgeries and edge-weight re-normalization procedures for both undirected and directed weighted hypergraphs to find influential cores. We successfully apply our framework for directed hypergraphs to seven metabolic hypergraphs and our framework for undirected hypergraphs to two social (co-authorship) hypergraphs to find influential cores, thereby demonstrating the practical feasibility of our approach. In addition, we prove a theorem showing that a certain edge weight re-normalization procedure in a prior research work for Ricci flows for edge-weighted graphs has the undesirable outcome of modifying the edge-weights to negative numbers, thereby rendering the procedure impossible to use. To the best of our knowledge, this seems to be one of the first articles that formulates algorithmic approaches for finding core(s) of (weighted or unweighted) directed hypergraphs.

Figures (7)

• Figure 1: $\mathsf{LP}$-formulation for Emd on hypergraph $H=(V,E,w)$ corresponding to distributions ${\mathbb P}_{\mathrm{left}}$ and ${\mathbb P}_{\mathrm{right}}$. Comments are enclosed by (* and *).
• Figure 2: An illustration of the calculations of ${\mathbb P}_{\mathrm{left}}$ and ${\mathbb P}_{\mathrm{right}}$ for an undirected hypergraph, as outlined in Section \ref{['sec-ricci-undir']}, where the node set is $\{ s_1, s_2, s_3, s_4, s_5, s_6, s_7, s_8 \}$ and the four hyperedges are $\{s_1,s_2,s_3,s_4\}$, $\{s_1,s_5,s_7\}$, $\{s_5,s_6\}$ and $\{s_2,s_8\}$.
• Figure 3: A visual illustration of the intuition behind our approach of finding cores by using Ricci flows with topological surgery as discussed in Section \ref{['sec-flow-surg']}. The notation "$\approx 0"$ refers to a function $f(n)$ such that $\lim_{n\to\infty}f(n)=0$. The nodes are colored blue and red for visual clarity: red nodes are involved in cliques of hyperedges of two nodes (i.e., cliques of edges) and all the blue nodes together with an equal number of red nodes appear in a single hyperedge of $2n$ nodes. The cliques are enclosed by dotted black bounding boxes for visual clarity (the cliques do not correspond to hyperedges). The second figure from top indicates the hypergraph after one iteration of Ricci flow but before the weight renormalization. The thicknesses of the red edges are reduced to indicate the decrease of their weights from approximately $1$ to approximately $0$ and the thickness of the black hyperedge is increased to indicate an increase of its weight. The third figure from top shows that the black hyperedge of $2n$ nodes gets deleted as a result of weight renormalization and topological surgery, thus giving us the $n$ cores corresponding to the $n$ cliques.
• Figure 4: A visual illustration of the hypergraph-theoretic representation (Section \ref{['sec-data-metabolic']}) vs. two common graph-theoretic representations of biochemical reactions. If the reaction times are known accurately then they can be used as the weights of the corresponding hyperedges.
• Figure 5: A visual illustration of the hypergraph-theoretic representation (Section \ref{['sec-coauth-data']}) vs. graph-theoretic representation of co-authorships.

Theorems & Definitions (2)

• Theorem 1
• proof