Probabilistic Analysis of Flow Networks using the Maximum Entropy Method

Abstract

A maximum entropy (MaxEnt) framework is developed to predict mean flow rates in hydraulic pipe, electrical and transportation networks, without sufficient constraints to enable a closed-form solution. These analyses are each based on a continuous relative entropy defined on a reduced parameter set of flow rates or other network properties such as potential differences. This entropy is maximized subject to observable constraints, e.g. in pipe networks, these include a subset of flow rates and/or potential differences, the frictional properties of each pipe and physical constraints arising from Kirchhoff's first and second laws. The analysis of pipe flow networks is demonstrated by application to simple networks, and to a water distribution network in Canberra. A Bayesian inference method is also developed and compared with the MaxEnt method. Both methods of inference update the prior to the posterior probability density function by the inclusion of new information, in the form of data or constraints. Soft constraints, applied to the MaxEnt method through the prior, in addition to standard moment constraints. It is shown that when the prior is Gaussian, both Bayesian inference and the MaxEnt method with soft prior constraints give the same posterior means but their covariances are different. In addition, three new probabilistic formulations of the MaxEnt method for the modeling of transport networks are presented. In contrast, virtually all MaxEnt transport models developed since the pioneering work of Wilson (1967) adopt a 'parametric' formulation, based directly on trip counts or flow rates rather than probabilities. New probabilistic formulations are presented for the prediction of trip flow rates by the gravity model or proportional route assignment, and the prediction of trip and path flow rates using equilibrium route assignment. Quasi-Newton methods are often used to solve network problems. A MaxEnt derivation of many commonly used quasi-Newton rules is presented. This derivation interprets the elements of the Hessian as means of a multivariate probability distribution, while the variance is chosen to represent the uncertainty about the mean. This interpretation is more intuitive than previous maximum-entropy formulations of quasi-Newton rules, in which the Hessian elements are variances while the mean is fixed to zero.