Statistics of weighted treelike networks

We study the statistics of growing networks with a tree topology in which each link carries a weight (k(i)k(j))(theta), where k(i) and k(j) are the node degrees at the end points of link ij. Network growth is governed by preferential attachment in which a newly added node attaches to a node of degree k with rate A(k)=k+lambda. For general values of theta and lambda, we compute the total weight of a network as a function of the number of nodes N and the distribution of link weights. Generically, the total weight grows as N for lambda>theta-1 and superlinearly otherwise. The link weight distribution is predicted to have a power-law form that is modified by a logarithmic correction for the case lambda=0. We also determine the node strength, defined as the sum of the weights of the links that attach to the node, as function of k. Using known results for degree correlations, we deduce the scaling of the node strength on k and N.