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A Spectral Approach to Analysing Belief Propagation for 3-Colouring

Published online by Cambridge University Press:  24 March 2009

Laboratory for Foundations of Computer Science, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK (e-mail:
Department of Statistics, University of California at Berkeley, 367 Evans Hall, Berkeley, CA 94720-3860, USA and Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel (e-mail:
Computer Science Division, University of California Berkeley, CA 94720, USA (e-mail:


Belief propagation (BP) is a message-passing algorithm that computes the exact marginal distributions at every vertex of a graphical model without cycles. While BP is designed to work correctly on trees, it is routinely applied to general graphical models that may contain cycles, in which case neither convergence, nor correctness in the case of convergence is guaranteed. Nonetheless, BP has gained popularity as it seems to remain effective in many cases of interest, even when the underlying graph is ‘far’ from being a tree. However, the theoretical understanding of BP (and its new relative survey propagation) when applied to CSPs is poor.

Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ‘planted’ solution; thus, we obtain the first rigorous result on BP for graph colouring in the case of a complex graphical structure (as opposed to trees). In particular, the analysis shows how belief propagation breaks the symmetry between the 3! possible permutations of the colour classes.

Copyright © Cambridge University Press 2009

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