  A number of problems in statistical physics and computer science can be
expressed as the computation of marginal probabilities over a Markov random
field. Belief propagation, an iterative message-passing algorithm, computes
exactly such marginals when the underlying graph is a tree. But it has gained
its popularity as an efficient way to approximate them in the more general
case, even if it can exhibits multiple fixed points and is not guaranteed to
converge. In this paper, we express a new sufficient condition for local
stability of a belief propagation fixed point in terms of the graph structure
and the beliefs values at the fixed point. This gives credence to the usual
understanding that Belief Propagation performs better on sparse graphs.
