The concept of a flow network - a set of nodes connected by flow paths - encompasses many different disciplines, including electrical, pipe flow, transportation, chemical reaction, ecological, epidemiological, economic and human social networks. Over the past two years, we have developed a maximum entropy (MaxEnt) method to infer the stationary state of a flow network, subject to “observable” constraints on expectations of various parameters, “physical” constraints such as conservation (Kirchhoff's) laws and frictional properties, and “graphical” constraints due to uncertainty in the network structure itself. The method enables the probabilistic prediction of physical parameters and (if necessary) the graphical properties of the network, when there is insufficient information to obtain a closed-form solution. A number of analytical, semi-analytical and numerical tools have been developed for the handling of nonlinear constraints, and for extracting analytical and/or numerical solutions. The method is demonstrated by application to the analysis of (i) a 1123-node, 1140-pipe urban water distribution network; (ii) a 327-node urban electrical power network with distributed sources; and (iii) an urban road network.