@article{NivenAbelSchlegeletal.2019,
  author    = {Niven, Robert K. and Abel, Markus and Schlegel, Michael and Waldrip, Steven H.},
  title     = {Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications},
  series = {Entropy},
  volume    = {21},
  journal   = {Entropy},
  number    = {8},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e21080776},
  pages     = {776},
  year      = {2019},
  abstract  = {The concept of a "flow network"-a set of nodes and links which carries one or more flows-unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include "observable" constraints on various parameters, "physical" constraints such as conservation laws and frictional properties, and "graphical" constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks.},
  language  = {en}
}
@misc{WaldripNivenAbeletal.2017,
  author    = {Waldrip, Steven H. and Niven, Robert K. and Abel, Markus and Schlegel, Michael},
  title     = {Consistent maximum entropy representations of pipe flow networks},
  series = {AIP conference proceedings},
  volume    = {1853},
  journal   = {AIP conference proceedings},
  number    = {1},
  publisher = {American Institute of Physics},
  address   = {Melville},
  isbn      = {978-0-7354-1527-0},
  issn      = {0094-243X},
  doi       = {10.1063/1.4985365},
  year      = {2017},
  abstract  = {The maximum entropy method is used to predict flows on water distribution networks. This analysis extends the water distribution network formulation of Waldrip et al. (2016) Journal of Hydraulic Engineering (ASCE), by the use of a continuous relative entropy defined on a reduced parameter set. This reduction in the parameters that the entropy is defined over ensures consistency between different representations of the same network. The performance of the proposed reduced parameter method is demonstrated with a one-loop network case study.},
  language  = {en}
}
@misc{WaldripNivenAbeletal.2017,
  author    = {Waldrip, Steven H. and Niven, Robert K. and Abel, Markus and Schlegel, Michael},
  title     = {Maximum entropy analysis of transport networks},
  series = {AIP conference proceedings},
  volume    = {1853},
  journal   = {AIP conference proceedings},
  number    = {1},
  publisher = {American Institute of Physics},
  address   = {Melville},
  isbn      = {978-0-7354-1527-0},
  issn      = {0094-243X},
  doi       = {10.1063/1.4985364},
  pages     = {8},
  year      = {2017},
  abstract  = {The maximum entropy method is used to derive an alternative gravity model for a transport network. The proposed method builds on previous methods which assign the discrete value of a maximum entropy distribution to equal the traffic flow rate. The proposed method however, uses a distribution to represent each flow rate. The proposed method is shown to be able to handle uncertainty in a more elegant way and give similar results to traditional methods. It is able to incorporate more of the observed data through the entropy function, prior distribution and integration limits potentially allowing better inferences to be made.},
  language  = {en}
}