Testing for Characteristics of Attribute Linked Infinite Networks based on Small Samples

Abstract: The objective of this paper is to study the characteristics (geometric and otherwise) of very large attribute based undirected networks. Real-world networks are often very large and fast evolving. Their analysis and understanding present a great challenge. An Attribute based network is a graph in which the edges depend on certain properties of the vertices on which they are incident. In context of a social network, the existence of links between two individuals may depend on certain attributes of the two of them. We use the Lovasz type sampling strategy of observing a certain random process on a graph locally , i.e., in the neighborhood of a node, and deriving information about global properties of the graph. The corresponding adjacency matrix is our primary object of interest. We study the efficiency of recently proposed sampling strategies, modified to our set up, to estimate the degree distribution, centrality measures, planarity etc. The limiting distributions are derived using recently developed probabilistic techniques for random matrices and hence we devise relevant test statistics and confidence intervals for different parameters / hypotheses of interest. We hope that our work will be useful for social and computer scientists for designing sampling strategies and computational algorithms appropriate to their respective domains of inquiry. Extensive simulations studies are done to empirically verify the probabilistic statements made in the paper.