1The Sum-over-Forests density index: identifying dense regions in a graph

Abstract—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 intuitive idea: high-density regions in a graph are characterized by the fact that they contain a large amount of low-cost trees with high outdegrees while low-density regions contain few ones. Therefore, inspired by [1], a Boltzmann probability distribution on the 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 with a high probability. Then, the SoF density index of a node is defined as the expected outdegree of this node in a non-trivial tree of the forest, thus providing a measure of density around that node. Following the matrix-forest theorem [2], [3] and a statistical physics framework, it is shown that the SoF density index can be easily computed in closed form through a simple matrix inversion. Experiments on artificial and real data sets show that the proposed index performs well on finding dense regions, for graphs of various origins. Index Terms—Graph mining, density index, dense regions on graphs, matrix-forest theorem.