Finding Cheeger cuts in hypergraphs via heat equation

Cheeger's inequality states that a tightly connected subset can be extracted from a graph G using an eigenvector of the normalized Laplacian associated with G. More specifically, we can compute a vertex subset in G with conductance O(ϕG), where ϕG is the minimum conductance of G. It has recently been shown that Cheeger's inequality can be extended to hypergraphs. However, as the normalized Laplacian of a hypergraph is no longer a matrix, we can only approximate its eigenvectors; this causes a loss in the conductance of the obtained subset. To address this problem, we here consider the heat equation on hypergraphs, which is a differential equation exploiting the normalized Laplacian. We show that the heat equation has a unique global solution and that we can extract a subset with conductance ϕG from the solution under a mild condition. An analogous result also holds for directed graphs.