Exact Rank Reduction of Network Models

Abstract

With the advent of the big data era, generative models of complex networks are becoming elusive from direct computational simulation. We present an exact, linear-algebraic reduction scheme of generative models of networks. By exploiting the bilinear structure of the matrix representation of the generative model, we separate its null eigenspace and reduce the exact description of the generative model to a smaller vector space. After reduction, we group generative models in universality classes according to their rank and metric signature and work out, in a computationally affordable way, their relevant properties (e.g., spectrum). The reduction also provides the environment for a simplified computation of their properties. The proposed scheme works for any generative model admitting a matrix representation and will be very useful in the study of dynamical processes on networks, as well as in the understanding of generative models to come, according to the provided classification.