Relevant Eigen - Subspace of a Graph : A Randomization Test

Determining the number of relevant dimensions in the eigen-space of a graph Laplacian matrix is a central issue in many spectral graph-mining applications. We tackle here the sub-problem of finding the “right" dimensionality of Laplacian matrices, especially those often encountered in the domains of social or biological graphs: the ones underlying large, sparse, unoriented and unweighted graphs with a power-law degree distribution. We present here the application of a randomization test to this problem. We validate our approach first on an artificial sparse and powerlaw type graph, with two intermingled clusters, then on two real-world social graphs (“Football-league”, “Mexican Politician Network”), where the actual, intrinsic dimensions appear to be 11 and 2 respectively ; we illustrate the optimality of the transformed dataspaces both visually, and numerically by means of a decision tree.