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Markov Bases of Binary Graph Models
Mike Develin1 and Seth Sullivant2
1American Institute of Mathematics, 360 Portage Ave., Palo Alto, CA 94306-2244, USA
develin@post.harvard.edu
2Department of Mathematics, University of California-Berkeley, Berkeley, CA 94720-3840, USA
seths@math.berkeley.edu
Annals of Combinatorics 7 (4) p.441-466 December, 2003
AMS Subject Classification: 05C99, 13P10, 62Q05
Abstract:
This paper is concerned with the topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We describe a degree four Markov basis for the model when the underlying graph is a cycle and generalize this result to the complete bipartite graph K2, n. We also give a combinatorial classification of degree two and three Markov basis moves as well as a Buchberger-free algorithm to compute moves of arbitrary given degree. Finally, we compute the algebraic degree of the model when the underlying graph is a forest.
Keywords: Markov bases, contingency tables, graphical models, hierarchical models, toric ideals

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