Transport and percolation theory in weighted networks

We study the distribution P(σ) of the equivalent conductance σ for Erd\"os-R\'enyi (ER) and scale-free (SF) weighted resistor networks with N nodes. Each link has conductance g≡e^-ax, where x is a random number taken from a uniform distribution between 0 and 1 and the parameter a represents the strength of the disorder. We provide an iterative fast algorithm to obtain P(σ) and compare it with the traditional algorithm of solving Kirchhoff equations. We find, both analytically and numerically, that P(σ) for ER networks exhibits two regimes: (i) A low conductance regime for σ<e^-apc, where pc=1∕⟨k⟩ is the critical percolation threshold of the network and $⟨k⟩$ is the average degree of the network. In this regime P(σ) is independent of N and follows the power law P(σ)~σ^-α, where α=1-⟨k⟩∕a. (ii) A high conductance regime for σ>e^-apc in which we find that P(σ) has strong N dependence and scales as P(σ)~f(σ,apc∕N1∕3). For SF networks with degree distribution P(k)~k^-λ, kₘᵢₙ≤k≤kₘₐₓ, we find numerically also two regimes, similar to those found for ER networks.