On the Degree Distribution of P\'{o}lya Urn Graph Processes

TL;DR

This paper improves the bounds on the degree distribution in Pólya urn graph processes, demonstrating a broader power-law behavior for degrees up to a larger threshold than previous models, using advanced probabilistic analysis.

Contribution

It provides a significantly tighter bound on the degree distribution in Pólya urn graph processes, extending the known power-law range beyond previous limits.

Findings

Degree distribution follows a power-law for degrees up to n^{1/6 - ε}.

The bound improves upon the previous limit of n^{1/15}.

Analysis of early vertices is key to achieving the tighter bound.

Abstract

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.