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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 ―α, for some α ∈ (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/α 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 changes only to accommodate the new vertex. Here, we obtain a sharp threshold for power law behavior. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, do we obtain a power law degree distribution with exponent 1 + 1/α. 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.

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Janssen, Jeannette., &PraŁat, PaweŁ. (2010). Rank-Based Attachment Leads To Power Law Graphs. SIAM Journal on Discrete Mathematics, 24(2), 420-440.