Finding Cheeger Cuts in Hypergraphs via Heat Equation
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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(√(phi_G)), where ϕ_G 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 √(phi_G) from the solution. An analogous result also holds for directed graphs.
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