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The Sum-over-Forests Density Index: Identifying Dense Regions in a Graph

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6 Author(s)
Senelle, M. ; Louvain Sch. of Manage. (LSM), Univ. catholique de Louvain (UCL), Mons, Belgium ; Garcia-Diez, S. ; Mantrach, A. ; Shimbo, M.
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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 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, 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 on the set of forests, thus providing a measure of density around that node. Following the matrix-forest theorem 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 datasets show that the proposed index performs well on finding dense regions, for graphs of various origins.

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Pattern Analysis and Machine Intelligence, IEEE Transactions on  (Volume:36 ,  Issue: 6 )