Mining the Largest Dense N - vertexlet in a Fuzzy Scale - free Graph

The topology of many real-life networks, e.g., the world wide web, social networks, ecological networks, genetic networks, comply with scale-free models. In scale-free models, the vertices of the underlying graph follow a power-law degree distribution. Observably, the graphs corresponding to most of the real-life networks are fuzzy in nature. An important problem of knowledge engineering that has evolved in various real-life networks is to identify the largest group of similar vertices in such networks that are functionally associated. Here, the problem of finding the largest group or association of vertices that are dense (denoted as dense N-vertexlet) in a fuzzy scale-free graph is addressed. Density quantifies the degree of similarity within a group of vertices in a graph. The density of an N-vertexlet is defined in a novel way that ensures significant participation of all the vertices in the N-vertexlet. First, it is established that the problem is NP-complete in nature. An upper bound on the size of the largest dense N-vertexlet in a fuzzy graph, with respect to certain density threshold value, is then derived. Finally, an O(n2 logn), n being the number of vertices in the graph, heuristic graph mining algorithm that produces an approximate solution for the problem is presented. The efficacy of the method is shown by applying it on some artificial and biological networks.