Sudden emergence of a giant k-core in a random graph
Journal of Combinatorial Theory Series B
Proceedings of the 9th international World Wide Web conference on Computer networks : the international journal of computer and telecommunications netowrking
The degree sequence of a scale-free random graph process
Random Structures & Algorithms
Random evolution in massive graphs
Handbook of massive data sets
Random Structures & Algorithms
First to market is not everything: an analysis of preferential attachment with fitness
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Rank-based models of network structure and the discovery of content
WAW'11 Proceedings of the 8th international conference on Algorithms and models for the web graph
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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 $-\alpha$, for some $\alpha\in(0,1)$. Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent $1+1/\alpha$ 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 $1+1/\alpha$. 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.