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Rank-Based Attachment Leads to Power Law Graphs
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Rank-Based Attachment Leads to Power Law Graphs   (Citations: 3)
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We investigate the degree distribution resulting from graph genera- tion models based on rank-based attachment. In rank-based attachment, all ver- tices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power , for some 2 (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/ whenever vertices are ranked according to their degree, their age, or a randomly chosen fit- ness 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 ran- dom, we obtain a power law degree distribution with exponent 1 + 1/ . 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 dierent factors; the exponent of the power law can be seen as a measure of the strength of the attachment.
Journal: Siam Journal on Discrete Mathematics - SIAMDM , vol. 24, no. 2, pp. 420-440, 2010
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