We study the statistics of growing networks with a tree topology in which each link carries a weight ${\left(k_i{k}_j\right)}^{\theta }$, 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 $N$.

• Received 15 August 2004

DOI:https://doi.org/10.1103/PhysRevE.71.036124