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In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when ]> improving the previous best bound of λ< e − 1. This is almost tight as reconstruction is known to hold when λ > (e + o(1))ln 2k. We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees. [ABSTRACT FROM AUTHOR]

Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on n nodes and maximum degree d given observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using Θ(d logn) samples which is optimal up to a multiplicative constant. Our results seem to be the first results for general models that guarantee that the generating model is reconstructed. Furthermore, we provide an explicit O(dnd + 2 logn) running time bound. In cases where the measure on the graph has correlation decay, the running time is O(n2 logn) for all fixed d. In the full-length version we also discuss the effect of observing noisy samples. There we show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some cases, models with hidden nodes can also be recovered. [ABSTRACT FROM AUTHOR]