Power series models of self-similarity in social networks

The evolution of a social network is associated with replicating self-similarity at many levels, the nature of interconnections can serve as a measure of the optimality of its organization. Closeness to self-similarity in the interconnections is proposed as a measure of the optimality of the organization. Two power series models are proposed to represent self-similarity and they are compared to the Zipf and Benford distributions. In contrast with the Zipf distribution where the middle term is the harmonic mean of the adjoining terms, our distribution considers the middle term to be the geometric mean. In one of the power series models, the scaling factor at one level is shown to be the golden ratio. A model for evolution of networks by oscillations between two different self-similarity measures is described.