Highly dispersed networks generated by enhanced redirection

We analyze growing networks that are built by enhanced redirection. Nodes are sequentially added and each incoming node attaches to a randomly chosen 'target' node with probability 1 − r, or to the parent of the target node with probability r. When the redirection probability r is an increasing function of the degree of the parent node, with r → 1 as the parent degree diverges, networks grown via this enhanced redirection mechanism exhibit unusual properties, including (i) multiple macrohubs, i.e., nodes with degrees proportional to the number of network nodes N; (ii) non-extensivity of the degree distribution in which the number of nodes of degree k, Nk, scales as Nν−1/kν, with 1 < ν < 2; (iii) lack of self-averaging, with large fluctuations between individual network realizations. These features are robust and continue to hold when the incoming node has out-degree greater than 1 so that networks contain closed loops. The latter networks are strongly clustered; for the specific case of double attachment, the average local clustering coefficient is 〈Ci〉 = 4ln2 − 2 = 0.772 58… . (paper)