Spectra of random networks with arbitrary degrees

We derive a message-passing method for computing the spectra of locally treelike networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs with arbitrary node degrees---the so-called configuration model. We find that the latter approximation works well for all but the sparsest of networks. We also derive bounds on the position of the band edges...