Statistical Mechanics of Multi-Edge Networks

Statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones. There is, however, a subtle difference between networks where weights are continuos variables and those where they account for discrete, distinguishable events, which we call multi-edge networks. In this work we face this problem introducing multi-edge networks as graphs where multiple (distinguishable) connections between nodes are considered. We develop a statistical mechanics framework where it is possible to get information about the most relevant observables given a large spectrum of linear and nonlinear constraints including those depending both on the number of multi-edges per link and their binary projection. The latter case is particularly interesting as we show that binary projections can be understood from multi-edge processes. The implications of these results are important as many real agent based problems mapped onto graphs require of this treatment for a proper characterization of its collective behavior.