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Rich-Clubs in Preferential Attachment Networks

  • Yinon Nahum
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 813)

Abstract

Consider the general 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, either with probability \(p>0\) a new node arrives and connects to an existing node, or with probability \(1-p\) a new edge is added between two existing nodes. In both cases, the existing nodes are chosen at random with probability proportional to their degree. Letting the \(\delta \) - fraction rich club of the network be the smallest set of nodes which, collectively, hold a \(\delta \) fraction of the total degree in the network, we show that its size is concentrated around \(f_p\left( \delta \right) \cdot n_t\), where \(n_t\) is the number of nodes in the network, and \(f_p\) is a convex continuous piecewise-linear function. This answers the open question of whether or not the \(\delta \) - fraction rich club constitutes a constant fraction of the number of nodes in the network. We provide a full description of \(f_p\). Finally, we compare this with the size of the \(\delta \) - founders of the network defined as the smallest set of the first nodes to enter the network which, collectively, hold a \(\delta \) fraction of the total degree in the network.

Keywords

Preferential attachment Networks Rich club Founders 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael

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