On the Growth of Large Independent Sets in Scale-Free Networks

Abstract

Independent sets are sets of mutually non-adjacent vertices of a network. For example, independent sets in social networks represent groups of people, who do not know anybody else within the group. In this paper, we investigate the growth of large independent sets in the famous Barabási-Albert (BA) model of scale-free complex networks. We formulate recurrent relations describing the cardinality of typical large independent sets and show that this cardinality seems to scale linearly with network size. This holds not only for the original BA model, where a new vertex brings a constant number of edges to the network, but also when the number of incoming edges is just bounded from above by a constant. Our finding is of a fundamental importance for community detection problems, since vertices of an independent set are naturally unlikely to belong to the same community. In other words, the number of communities in scale-free networks seems to be bounded from below by a linear function of network size.Keywordsindependent setscomplex networksscale-free networks