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This work introduces a novel nonparametric density index defined on graphs,the Sum-over-Forests (SoF) density index. It is based on a clear and intuitiveidea: high-density regions in a graph are characterized by the fact that theycontain a large amount of low-cost trees with high outdegrees while low-densityregions contain few ones. Therefore, a Boltzmann probability distribution onthe countable set of forests in the graph is defined so that large (high-cost)forests occur with a low probability while short (low-cost) forests occur witha high probability. Then, the SoF density index of a node is defined as theexpected outdegree of this node in a non-trivial tree of the forest, thusproviding a measure of density around that node. Following the matrix-foresttheorem, and a statistical physics framework, it is shown that the SoF densityindex can be easily computed in closed form through a simple matrix inversion.Experiments on artificial and real data sets show that the proposed indexperforms well on finding dense regions, for graphs of various origins.

The Sum-over-Forests density index: identifying dense regions in a graph