NDLI: Improved Degree Bounds and Full Spectrum Power Laws in Preferential Attachment Networks

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Improved Degree Bounds and Full Spectrum Power Laws in Preferential Attachment Networks

Content Provider	ACM Digital Library
Author	Avin, Chen Nahum, Yinon Peleg, David Lotker, Zvi
Abstract	Consider a random preferential attachment model G(p) for network evolution that allows both node and edge arrivals. Starting with an arbitrary nonempty graph $G_{0},$ at each time step, there are two possible events: with probability p > 0 a new node arrives and a new edge is added between the new node and an existing node, and with probability 1 - p a new edge is added between two existing nodes. In both cases, the involved existing nodes are chosen at random according to preferential attachment, i.e., with probability proportional to their degree. G(p) is known to generate power law networks, i.e., the fraction of nodes with degree k is proportional to k-β. Here β=(4-p)/(2-p) is in the range (2,3]. Denoting the number of nodes of degree k at time t by $m_{k,t},$ we significantly improve some long-standing results. In particular, we show that $m_{k,t}$ is concentrated around its mean with a deviation of O(√t), which is independent of k. We also tightly bound the expectation $Em_{k,t}$ with an additive error of O(1/k), which is independent of t. These new bounds allow us to tightly estimate $m_{k,t}$ for a considerably larger k values than before. This, in turn, enables us to estimate other important quantities, e.g., the size of the k-rich club, namely, the set of all nodes with a degree at least k. Finally, we introduce a new generalized model, $G(p_{t},$ $r_{t},$ $q_{t}),$ which extends G(p) by allowing also time-varying probabilities for node and edge arrivals, as well as the formation of new components. We show that the extended model can produce power law networks with any exponent β in the range (1,∞). Furthermore, the concentration bounds established for $m_{k,t}$ in G(p) also apply in $G(p_{t},$ $r_{t},$ $q_{t}).$
Starting Page	45
Ending Page	53
Page Count	9
File Format	PDF
ISBN	9781450348874
DOI	10.1145/3097983.3098012
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-08-13
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Networks Preferential attachment Power law
Content Type	Text
Resource Type	Article

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