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In this manuscript we review new ideas and first results on application ofthe Graphical Models approach, originated from Statistical Physics, InformationTheory, Computer Science and Machine Learning, to optimization problems ofnetwork flow type with additional constraints related to the physics of theflow. We illustrate the general concepts on a number of enabling examples frompower system and natural gas transmission (continental scale) and distribution(district scale) systems.

Graphical Models and Belief Propagation-hierarchy for Optimal Physics-Constrained Network Flows