Mining the Largest Dense Vertexlet in a Weighted Scale-free Graph

An important problem of knowledge discovery that has recently evolved in various reallife networks is identifying the largest set of vertices that are functionally associated. The topology of many real-life networks shows scale-freeness, where the vertices of the underlying graph follow a power-law degree distribution. Moreover, the graphs corresponding to most of the real-life networks are weighted in nature. In this article, the problem of finding the largest group or association of vertices that are dense (denoted as dense vertexlet) in a weighted scale-free graph is addressed. Density quantifies the degree of similarity within a group of vertices in a graph. The density of a vertexlet is defined in a novel way that ensures significant participation of all the vertices within the vertexlet. It is established that the problem is NP-complete in nature. An upper bound on the order of the largest dense vertexlet of a weighted graph, with respect to certain density threshold value, is also derived. Finally, an O(n$^2$ log n) (n denotes the number of vertices in the graph) heuristic graph mining algorithm that produces an approximate solution for the problem is presented.