Rankbased attachment leads to power law graphs
Abstract
We investigate the degree distribution resulting from graph generation models based on rankbased attachment. In rankbased 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 rankbased 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 rankbased 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2009
 arXiv:
 arXiv:0908.3436
 Bibcode:
 2009arXiv0908.3436J
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 05C80;
 05C07
 EPrint:
 SIAM Journal of Discrete Math 24, 2010, pp. 420440