  In this paper, we disclose the statistical behavior of the max-product
algorithm configured to solve a maximum a posteriori (MAP) estimation problem
in a network of distributed agents. Specifically, we first build a distributed
hypothesis test conducted by a max-product iteration over a binary-valued
pairwise Markov random field and show that the decision variables obtained are
linear combinations of the local log-likelihood ratios observed in the network.
Then, we use these linear combinations to formulate the system performance in
terms of the false-alarm and detection probabilities. Our findings indicate
that, in the hypothesis test concerned, the optimal performance of the
max-product algorithm is obtained by an optimal linear data-fusion scheme and
the behavior of the max-product algorithm is very similar to the behavior of
the sum-product algorithm. Consequently, we demonstrate that the optimal
performance of the max-product iteration is closely achieved via a linear
version of the sum-product algorithm which is optimized based on statistics
received at each node from its one-hop neighbors. Finally, we verify our
observations via computer simulations.
