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The TuttePotts connection in the presence of an external magnetic field
, 2011
"... The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we d ..."
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The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the Vpolynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The Vpolynomial generalizes Noble and Welsh’s Wpolynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the Vpolynomial, and hence a polynomial with deletioncontraction reduction and FortuinKasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models. This eprint is an extended version, including additional background information and
Model Reductions for Inference: Generality of Pairwise, Binary, and Planar Factor Graphs
, 2013
"... We offer a solution to the problem of efficiently translating algorithms between different types of discrete statistical model. We investigate the expressive power of three classes of model—those with binary variables, with pairwise factors, and with planar topology—as well as their four intersectio ..."
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We offer a solution to the problem of efficiently translating algorithms between different types of discrete statistical model. We investigate the expressive power of three classes of model—those with binary variables, with pairwise factors, and with planar topology—as well as their four intersections. We formalize a notion of “simple reduction ” for the problem of inferring marginal probabilities and consider whether it is possible to “simply reduce ” marginal inference from general discrete factor graphs to factor graphs in each of these seven subclasses. We characterize the reducibility of each class, showing in particular that the class of binary pairwise factor graphs is able to simply reduce only positive models. We also exhibit a continuous “spectral reduction” based on polynomial interpolation, which overcomes this limitation. Experiments assess the performance of standard approximate inference algorithms on the outputs of our reductions.
ON THE POTTS MODEL PARTITION FUNCTION IN AN EXTERNAL FIELD
, 2012
"... We study the partition function of the Potts model in an external (magnetic) field, and its connections with the zerofield Potts model partition function. Using a deletioncontraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zerofie ..."
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We study the partition function of the Potts model in an external (magnetic) field, and its connections with the zerofield Potts model partition function. Using a deletioncontraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zerofield partition function. We also show that Z can be written as a sum over the spanning trees, and the spanning forests, of a graph G. Our results extend to Z the wellknown spanning tree expansion for the zerofield partition function that arises though its connections with the Tutte polynomial.
Yusuke Watanabe y Kenji Fukumizu
"... graph polynomials satisfying deletioncontraction relations ..."
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