Statistics of Weighted Networks

TL;DR

This paper analyzes weighted network growth models with preferential attachment, deriving how total network weight, link weight distribution, and node strength scale with network size and degree, revealing power-law behaviors and corrections.

Contribution

It provides analytical expressions for total weight, link weight distribution, and node strength in weighted networks with preferential attachment, including new scaling laws and corrections.

Findings

Total weight grows linearly with network size for certain parameters.

Link weight distribution follows a power law with logarithmic correction at specific conditions.

Node strength scales with degree and network size, revealing new scaling behaviors.

Abstract

We study the statistics of growing networks in which each link carries a weight (k_i k_j)^theta, where k_i and k_j are the node degrees at the endpoints 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 super-linearly 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.