MATHEMATICAL EVOLUTION IN DISCRETE NETWORKS

This paper provides a mathematical explanation for the phenomenon of \triadic closure" so often seen in social networks. It appears to be a natural con- sequence when network change is constrained to be continuous. The concept of chordless cycles in the network's \irreducible spine" is used in the analysis of the network's dynamic behavior. A surprising result is that as networks undergo random, but continuous, pertur- bations they tend to become more structured and less chaotic. In this paper we explore the behavior of networks as they change under a sequence of simple, \continuous" transformations. We will start with a random network and apply a sequence of transformations which randomly adds edges (or connections) to it and randomly deletes edges. Periodically we observe the resulting network. The structure, or topology, of these show considerable similarity to that appearing in social networks. The rather surprising conclusion will be that \random change, provided it is continuous, leads to a less chaotic and more regular structure". This provides a purely mathematical basis for the presence of \triadic closure" in social networks.