On the Degree Distribution of Pólya Urn Graph Processes

This paper presents a tighter bound on the degree distribution of arbitrary P\'{o}lya urn graph processes, proving that the proportion of vertices with degree $d$ obeys a power-law distribution $P(d) \propto d^{-\gamma}$ for $d \leq n^{\frac{1}{6}-\epsilon}$ for any $\epsilon > 0$, where $n$ represents the number of vertices in the network. Previous work by Bollob\'{a}s et al. formalized the well-known preferential attachment model of Barab\'{a}si and Albert, and showed that the power-law distribution held for $d \leq n^{\frac{1}{15}}$ with $\gamma = 3$. Our revised bound represents a significant improvement over existing models of degree distribution in scale-free networks, where its tightness is restricted by the Azuma-Hoeffding concentration inequality for martingales. We achieve this tighter bound through a careful analysis of the first set of vertices in the network generation process, and show that the newly acquired is at the edge of exhausting Bollob\'as model in the sense that the degree expectation breaks down for other powers.