Isolated Vertices in Random Intersection Graphs

Abstract

For the structure analysis of non-metric data, it is natural to classify objects according to the properties they possess. An effective model to analyze the structure of similarities between objects is the random intersection graph generated by the random bipartite graph with bipartition $(\mathcal{V},\mathcal{W})$, where $\mathcal{V}$ is a set of objects, $\mathcal{W}$ is a set of properties, and according to some random procedure, edges join objects with their properties. In the related random intersection graph two vertices are joined by an edge if and only if they represent objects sharing at least s properties. In this paper we study the number of isolated vertices and its convergence to Poisson distribution. We generalize previous results obtained for special cases of the random model and for s = 1, only. Our approach leads us also to some interesting results on dependencies between the appearances of edges in the random intersection graph.