Weighted scale-free network with self-organizing link weight dynamics

All crucial features of the recently observed real-world weighted networks are obtained in a model where the weight of a link is defined with a single non-linear parameter $\alpha$ as $w_{ij}=(s_is_j)^\alpha$, $s_i$ and $s_j$ are the strengths of two end nodes of the link and $\alpha$ is a continuously tunable positive parameter. In addition the definition of strength as $s_i= \Sigma_j w_{ij}$ results a self-organizing link weight dynamics leading to a self-consistent distribution of strengths and weights on the network. Using the Barab\'asi-Albert growth dynamics all exponents of the weighted networks which are continuously tunable with $\alpha$ are obtained. It is conjectured that the weight distribution should be similar in any scale-free network.