  Correlations in the signal observed via functional Magnetic Resonance Imaging
(fMRI), are expected to reveal the interactions in the underlying neural
populations through hemodynamic response. In particular, they highlight
distributed set of mutually correlated regions that correspond to brain
networks related to different cognitive functions. Yet graph-theoretical
studies of neural connections give a different picture: that of a highly
integrated system with small-world properties: local clustering but with short
pathways across the complete structure. We examine the conditional independence
properties of the fMRI signal, i.e. its Markov structure, to find realistic
assumptions on the connectivity structure that are required to explain the
observed functional connectivity. In particular we seek a decomposition of the
Markov structure into segregated functional networks using decomposable graphs:
a set of strongly-connected and partially overlapping cliques. We introduce a
new method to efficiently extract such cliques on a large, strongly-connected
graph. We compare methods learning different graph structures from functional
connectivity by testing the goodness of fit of the model they learn on new
data. We find that summarizing the structure as strongly-connected networks can
give a good description only for very large and overlapping networks. These
results highlight that Markov models are good tools to identify the structure
of brain connectivity from fMRI signals, but for this purpose they must reflect
the small-world properties of the underlying neural systems.
