[Paper Review] Coarse Ricci curvature of hypergraphs and its generalization

Masahiro Ikeda, Yu Kitabeppu+2 more|arXiv (Cornell University)|Feb 1, 2021

Geometric Analysis and Curvature Flows35 references4 citations

TL;DR

This paper introduces a novel notion of coarse Ricci curvature for hypergraphs using a nonlinear submodular hypergraph Laplacian, generalizing Lin-Lu-Yau's graph curvature. It establishes key geometric results—gradient estimates, lower eigenvalue bounds, and Bonnet-Myers diameter bounds—demonstrating that nonlinearity in the Laplacian does not obstruct curvature-based analysis, extending Riemannian curvature concepts to hypergraphs.

ABSTRACT

In the present paper, we introduce a concept of Ricci curvature on hypergraphs for a nonlinear Laplacian. We prove that our definition of the Ricci curvature is a generalization of Lin-Lu-Yau coarse Ricci curvature for graphs to hypergraphs. We also show a lower bound of nonzero eigenvalues of Laplacian, gradient estimate of heat flow, and diameter bound of Bonnet-Myers type for our curvature notion. This research leads to understanding how nonlinearity of Laplacian causes complexity of curvatures.

Motivation & Objective

• To define a notion of coarse Ricci curvature on hypergraphs that generalizes Lin-Lu-Yau’s graph curvature.
• To investigate how nonlinearity in the Laplacian affects curvature and geometric properties in discrete spaces.
• To establish functional inequalities and geometric bounds (e.g., diameter, eigenvalue) analogous to Riemannian geometry.
• To demonstrate that the submodular hypergraph Laplacian, though nonlinear and multivalued, supports a tractable curvature framework.

Proposed method

• Uses a submodular hypergraph Laplacian defined via subdifferentials of submodular set functions.
• Defines the normalized Laplacian as $\mathcal{L} = L \circ D^{-1}$ on a Hilbert space with inner product $\langle f,g \rangle = f^\top D^{-1}g$.
• Establishes that the normalized Laplacian is maximal monotone and the subdifferential of a convex functional $Q(\bar{g}) = \frac{1}{2} \sum_e \omega(e) f_e(\bar{g})^2$.
• Applies the heat flow $h_t$ and resolvent $J_\lambda$ via the maximal monotonicity of $\mathcal{L}$, enabling curvature analysis.
• Implements the curvature definition via contraction of $L^1$-Wasserstein distance, following Lin-Lu-Yau’s approach.
• Uses the normalized submodular transformation $F$ with $F_e(V) = 0$ to ensure $\mathcal{L}(\pi) = 0$, enabling measure-theoretic analysis.

Experimental results

Research questions

• RQ1 Can Lin-Lu-Yau’s coarse Ricci curvature for graphs be generalized to hypergraphs using a nonlinear Laplacian?
• RQ2 How does nonlinearity in the Laplacian affect the geometric and analytic properties of hypergraphs?
• RQ3 Does the submodular hypergraph Laplacian support a curvature notion that yields functional inequalities and diameter bounds similar to Riemannian geometry?
• RQ4 Can the curvature notion be defined without relying on clique expansion or random walks, given the lack of canonical random walks on hypergraphs?

Key findings

• The proposed curvature generalizes Lin-Lu-Yau’s coarse Ricci curvature to hypergraphs, preserving its foundational properties.
• A lower bound on nonzero eigenvalues of the Laplacian is established, linking curvature to spectral properties.
• A gradient estimate for the heat flow is proven: $|\nabla h_t f|^2(x) \leq e^{-2Kt} h_t |\nabla f|^2(x)$, valid for $t > 0$.
• A Bonnet-Myers type diameter bound is derived, showing that positive curvature implies finite diameter.
• The normalized Laplacian $\mathcal{L}$ is maximal monotone and the subdifferential of a convex functional, ensuring well-posedness of the heat flow and curvature definitions.
• The curvature framework is applicable to various hypergraph types, including directed hypergraphs, via the Lovász extension of cut functions.