Maximum entropy modelling of sub-optimal transport

Many natural systems involve structures shaped by competing forces: efficiency, randomness, and incomplete information. To the best of our knowledge, there is currently no robust method to assess the presence of optimization processes in real networks. Here, we introduce a class of bipartite random graphs that bridges two foundational approaches: maximum entropy models and optimal transport theory. By tuning a single parameter, our model generates a continuous family of network configurations, ranging from fully random to cost-minimizing structures. This transition is governed by a variational principle analogous to free energy in statistical physics, where entropy and transport cost play competing roles. We analytically and numerically characterize how dense, entropic graphs evolve into sparse, efficient structures, revealing the most probable network configurations under partial optimization. Beyond clarifying the conceptual link between entropy-based and cost-based methods, our framework offers a generative model for systems where the structure emerges from random and constrained environments.