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---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, N_k, scales as N^{nu-1}/k^{nu}, with 1<nu<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 the double attachment, the average local clustering coefficient is <C_i>=4(ln2)-2=0.77258...