Abstract
This study examines MaxEnt methods for probabilistic inference of the state of flow networks, including pipe flow, electrical and transport networks, subject to physical laws and observed moments. While these typically assume networks of invariant graph structure, we here consider higher-level MaxEnt schemes, in which the network structure constitutes part of the uncertainty in the problem specification. In physics, most studies on the statistical mechanics of graphs invoke the Shannon entropy \(H_G^{Sh} = - \sum \nolimits _{\Omega _G} P(G) \ln P(G)\), where G is the graph and \(\Omega _G\) is the graph ensemble. We argue that these should adopt the relative entropy \(H_G = - \sum \nolimits _{\Omega _G} P(G) \ln {P(G)}/{Q(G)}\), where Q(G) is the graph prior associated with the graph macrostate G. By this method, the user is able to employ a simplified accounting over graph macrostates rather than need to count individual graphs. Using combinatorial methods, we here derive a variety of graph priors for different graph ensembles, using different macrostate partitioning schemes based on the node or edge counts. A variety of such priors are listed herein, for ensembles of undirected or directed graphs.
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Acknowledgements
This project acknowledges funding support from the Australian Research Council Discovery Projects Grant DP140104402, Go8/DAAD Australia-Germany Joint Research Cooperation Scheme RG123832 and the French Agence Nationale de la Recherche Chair of Excellence (TUCOROM) and the Institute Prime, Poitiers, France.
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Niven, R.K., Schlegel, M., Abel, M., Waldrip, S.H., Guimera, R. (2018). Maximum Entropy Analysis of Flow Networks with Structural Uncertainty (Graph Ensembles). In: Polpo, A., Stern, J., Louzada, F., Izbicki, R., Takada, H. (eds) Bayesian Inference and Maximum Entropy Methods in Science and Engineering. maxent 2017. Springer Proceedings in Mathematics & Statistics, vol 239. Springer, Cham. https://doi.org/10.1007/978-3-319-91143-4_25
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DOI: https://doi.org/10.1007/978-3-319-91143-4_25
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