Generalizing the Hypergraph Laplacian via a Diffusion Process with Mediators

Abstract

In a recent breakthrough STOC 2015 paper, a continuous diffusion process was considered on hypergraphs (which has been refined in a recent JACM 2018 paper) to define a Laplacian operator, whose spectral properties satisfy the celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion process is that each hyperedge directs flow only from vertices with the maximum density to those with the minimum density, while ignoring vertices having strict in-beween densities.In this work, we consider a generalized diffusion process, in which vertices in a hyperedge can act as mediators to receive flow from vertices with maximum density and deliver flow to those with minimum density. We show that the resulting Laplacian operator still has a second eigenvalue satsifying the Cheeger's inequality.Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance, and provides a powerful tool to enhance the development of spectral hypergraph theory. Moreover, since every vertex can participate in the new diffusion model at every instant, this can potentially have wider practical applications.Spectral graph theory, and specifically, the well-known Cheeger's inequality give a relationship between the edge expansion properties of a graph and the eigenvalues of some appropriately defined matrix [1,2]. Loosely speaking, for a given graph, its edge expansion or conductance gives a lower bound on the ratio of the number of edges leaving a subset S of vertices to the sum of vertex degrees in S. It is natural that graph conductance is studied in the context of graph partitioning or clustering [9,17,18], whose goal is to minimize the weight of edges crossing different clusters with respect to intra-cluster edges. The reader can refer to the standard references [5,8] for an introduction to spectral graph theory.Recent Generalization to Hypergraphs. In an edge-weighted hypergraph H = (V, E, w), an edge e ∈ E is a non-empty subset of V . The edges have positive weights indicated by w : E → R + . The weight of each vertex v ∈ V is its weighted degree w v := e∈E:v∈e w e . A subset S of vertices has weight w(S) := v∈S w v , and the edges it cuts is ∂S := {e ∈ E : e intersects both S and V \ S}.The conductance of S ⊆ V is defined as φ(S) := w(∂S) w(S) . The conductance of H is defined as:( 1.1) Until recently, it was an open problem to define a spectral model for hypergraphs. In a breakthrough STOC 2015 paper, Louis [13] considered a continuous diffusion process on hypergraphs (which has been refined in a recent JACM paper [3]), and defined an operator L w f := − df dt , where f ∈ R V is some appropriate vector associated with the diffusion process. As in classical spectral graph theory, L w has non-negative eigenvalues, and the all-ones vector 1 is an eigenvector with eigenvalue 0. Moreover, the operator L w has a second eigenvalue γ 2 , and the Cheeger's inequality can be recovered 1 for hypergraphs:

A nonlinear diffusion method for semi-supervised learning on hypergraphs

“…• MLP+Ω H This is the MLP baseline with a regularization term λΩ H added to the objective, where Ω H is the Laplacian energy associated to the hypergraph Laplacian based on mediators [8]. • HGNN This is the hypergraph neural network model of [12], which uses the clique-expansion Laplacian [42,1] for the hypergraph convolutional filter.…”

“…One simple case to define L as the Laplacian of the clique expansion graph of H [1,42], where the hypergraph is mapped to a graph on the same set of nodes by adding a clique among the nodes of each hyperedge. This is the approach used in HGNN [12], and other variants uses mediators instead of cliques in the hypergraph to graph reduction [8].…”

Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

“…One drawback of the clique expansion is that the resulting graph tends to be dense since a hyperedge is replaced by a number of edges that is quadratic in the size of the hyperedge. A similar idea is proposed in [111], but this convolutional neural network is based on a different hypergraph Laplacian shift (proposed in [112]), which only requires a linear number of edges for each hyperedge. This provides a more efficient training when compared with that of [110].…”

HyperGCN: A New Method of Training Graph Convolutional Networks on Hypergraphs

“…Another line of work uses mathematically appealing tensor methods Shashua et al [2006], Bulò and Pelillo [2009], Kolda and Bader [2009], but they are limited to uniform hypergraphs. Recent developments, however, work for arbitrary hypergraphs and fully exploit the hypergraph structure Hein et al [2013], Zhang et al [2017], Chan and Liang [2018], Li and Milenkovic [2018b], Chien et al [2019].…”

“…This hypergraph Laplacian ignores the hypernodes in K e := {k ∈ e : k = i e , k = j e } in the given epoch. Recently, it has been shown that a generalised hypergraph Laplacian in which the hypernodes in K e act as "mediators" Chan and Liang [2018] satisfies all the properties satisfied by the above Laplacian given by . The two Laplacians are pictorially compared in Figure 2.…”