Locally computable approximations for spectral clustering and absorption times of random walks

We address the problem of determining a natural local neighbourhood or “cluster” associated to a given seed vertex in an undirected graph. We formulate the task in terms of absorption times of random walks from other vertices to the vertex of interest, and observe that these times are well approximated by the components of the principal eigenvector of the corresponding fundamental matrix of the graph’s adjacency matrix. We further present a locally computable gradient-descent method to estimate this Dirichlet-Fiedler vector, based on minimising the respective Rayleigh quotient. Experimental evaluation shows that the approximations behave well and yield well-defined local clusters.