Clustering, community partition and disjoint spanning trees

Abstract

Clustering method is one of the most important tools in statistics. In a graph theory model, clustering is the process of finding all dense subgraphs. A mathematically well-defined measure for graph density is introduced in this article as follows. Let G = ( V , E ) be a graph (or multi-graph) and H be a subgraph of G . The dynamic density of H is the greatest integer k such that min ∀ P {| E ( H / P )|/| V ( H / P )| − 1} > k where the minimum is taken over all possible partitions P of the vertex set of H , and H / P is the graph obtained from H by contracting each part of P into a single vertex. A subgraph H of G is a level- k community if H is a maximal subgraph of G with dynamic density at least k . An algorithm is designed in this paper to detect all level- h communities of an input multi-graph G . The worst-case complexity of this algorithm is upper bounded by O (| V ( G )| 2 h 2 ). This new method is one of few available clustering methods that are mathematically well-defined, supported by rigorous mathematical proof and able to achieve the optimization goal with polynomial complexity. As a byproduct, this algorithm also can be applied for finding edge-disjoint spanning trees of a multi-graph. The worst-case complexity is lower than all known algorithms for multi-graphs.