  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.
