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 subset with conductance O(√φG), where φG is the minimum conductance of a set in 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 to 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 solution and that we can extract a subset with conductance √φG from the solution. An analogous result also holds for directed graphs.