Competition-Induced Preferential Attachment

TL;DR

This paper explains how preferential attachment can emerge from competition-driven local optimization processes, leading to a power law degree distribution with an upper cutoff in the resulting network models.

Contribution

It introduces a family of geometric growth models based on local tradeoff optimization that produce preferential attachment with a cutoff, connecting it to classical models like Barabási-Albert.

Findings

Degree distribution follows a power law up to a cutoff

The model generalizes to include parameters for cutoff and fertility

The process includes known models as special cases

Abstract

Models based on preferential attachment have had much success in reproducing the power law degree distributions which seem ubiquitous in both natural and engineered systems. Here, rather than assuming preferential attachment, we give an explanation of how it can arise from a more basic underlying mechanism of competition between opposing forces. We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics. This family admits an equivalent description as a graph process with no reference to the underlying geometry. Moreover, the resulting graph process is shown to be preferential attachment with an upper cutoff. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also introduce and rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a ``fertility'' parameter. Limiting cases of this process include the standard Barabasi-Albert preferential attachment model and the uniform attachment model. In the general case, we prove that the process has a power law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters.