Partitions of networks that are robust to vertex permutation dynamics

Abstract

Minimum disconnecting cuts of connected graphs provide fundamental information about the connectivity structure of the graph. Spectral methods are well-known as stable and efficient means of finding good solutions to the balanced minimum cut problem. In this paper we generalise the standard balanced bisection problem for static graphs to a new "dynamic balanced bisection problem", in which the bisecting cut should be minimal when the vertex-labelled graph is subjected to a general sequence of vertex permutations. We extend the standard spectral method for partitioning static graphs, based on eigenvectors of the Laplacian matrix of the graph, by constructing a new dynamic Laplacian matrix, with eigenvectors that generate good solutions to the dynamic minimum cut problem. We formulate and prove a dynamic Cheeger inequality for graphs, and demonstrate the effectiveness of the dynamic Laplacian matrix for both structured and unstructured graphs.