Rank-based attachment leads to power law graphs

TL;DR

This paper analyzes how rank-based attachment models generate graphs with power law degree distributions, explaining the conditions under which such distributions emerge and their relation to the ranking scheme used.

Contribution

It provides a rigorous analysis of rank-based attachment models, identifying conditions for power law degree distributions and deriving the exponent based on the ranking scheme.

Findings

Power law degree distribution with exponent 1+1/a occurs when vertices are ranked by degree, age, or fitness.

A sharp threshold for power law behavior is identified based on initial rank bias.

Power law distributions can be explained by various ranking schemes, linking the exponent to attachment strength.

Abstract

We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power -a, for some a in (0,1). Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent 1+1/a whenever vertices are ranked according to their degree, their age, or a randomly chosen fitness value. We also investigate the case where the ranking is based on the initial rank of each vertex; the rank of existing vertices only changes to accommodate the new vertex. Here, we obtain a sharp threshold for power law behaviour. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, we obtain a power law degree distribution with exponent 1+1/a. This indicates that the power law degree distribution often observed in nature can be explained by a rank-based attachment scheme, based on a ranking scheme that can be derived from a number of different factors; the exponent of the power law can be seen as a measure of the strength of the attachment.