Rank-Based Attachment Leads to Power Law Graphs

Abstract. 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 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/α. This indicates that the power law degree distribution oft...

• Type: Journal
• Year: 2010
• Where: SIAMDM
• Authors: Jeannette Janssen, Pawel Pralat
• Keywords: Degree Distribution, Power Law, Rank-based Attachment, SIAMDM 2010