  Intrinsic brain activity is characterized by highly structured co-activations
between different regions, whose origin is still under debate. In this paper,
we address the question whether it is possible to unveil how the underlying
anatomical connectivity shape the brain's spontaneous correlation structure. We
start from the assumption that in order for two nodes to exhibit large
covariation, they must be exposed to similar input patterns from the entire
network. We then acknowledge that information rarely spreads only along an
unique route, but rather travels along all possible paths. In real networks the
strength of local perturbations tends to decay as they propagate away from the
sources, leading to a progressive attenuation of the original information
content and, thus, of their influence. We use these notions to derive a novel
analytical measure, $\mathcal{T}$ , which quantifies the similarity of the
whole-network input patterns arriving at any two nodes only due to the
underlying topology, in what is a generalization of the matching index. We show
that this measure of topological similarity can indeed be used to predict the
contribution of network topology to the expected correlation structure, thus
unveiling the mechanism behind the tight but elusive relationship between
structure and function in complex networks. Finally, we use this measure to
investigate brain connectivity, showing that information about the topology
defined by the complex fabric of brain axonal pathways specifies to a large
extent the time-average functional connectivity observed at rest.
