  The brain is a highly complex system. Most of such complexity stems from the
intermingled connections between its parts, which give rise to rich dynamics
and to the emergence of high-level cognitive functions. Disentangling the
underlying network structure is crucial to understand the brain functioning
under both healthy and pathological conditions. Yet, analyzing brain networks
is challenging, in part because their structure represents only one possible
realization of a generative stochastic process which is in general unknown.
Having a formal way to cope with such intrinsic variability is therefore
central for the characterization of brain network properties. Addressing this
issue entails the development of appropriate tools mostly adapted from network
science and statistics. Here, we focus on a particular class of maximum entropy
models for networks, i.e. exponential random graph models (ERGMs), as a
parsimonious approach to identify the local connection mechanisms behind
observed global network structure. Efforts are reviewed on the quest for basic
organizational properties of human brain networks, as well as on the
identification of predictive biomarkers of neurological diseases such as
stroke. We conclude with a discussion on how emerging results and tools from
statistical graph modeling, associated with forthcoming improvements in
experimental data acquisition, could lead to a finer probabilistic description
of complex systems in network neuroscience.
