Subgraphs in preferential attachment models

TL;DR

This paper analyzes how subgraph counts scale in preferential attachment models with power-law degree distributions, providing a framework to predict the expected number of various subgraphs based on the network size.

Contribution

It introduces a novel optimization approach and leverages Pólya urn representations to determine subgraph count scaling in preferential attachment networks.

Findings

Expected subgraph counts scale as a power of network size

Provides explicit formulas for different subgraph types

Framework applicable to various preferential attachment models

Abstract

We consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau>2$. For all subgraphs $H$, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a P\'olya urn model.