Rich-Clubs in Preferential Attachment Networks
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.
KeywordsPreferential attachment Networks Rich club Founders
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