Complex scale-free networks with tunable power-law exponent and clustering

We introduce a network evolution process motivated by the network of citations in the scientific literature. In each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network. This set includes $m$ "ambassador" nodes and $l$ of each ambassador's descendants where $m$ and $l$ are random variables selected from any choice of distributions $p_{l}$ and $q_{m}$. The process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite. We show that the degree distributions of the networks generated after a large number of iterations are scale-free and derive an expression for the power-law exponent. In a particular case of the model where the number of ambassadors is always the constant $m$ and the number of selected descendants from each ambassador is the constant $l$, the power-law exponent is $(2l+1)/l$. For this example we derive expressions for the degree distribution and clustering coefficient in terms of $l$ and $m$. We conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale-free distributions that have been studied empirically.