  The global functional brain network (graph) is more suitable for
characterizing brain states than local analysis of the connectivity of brain
regions. Therefore, graph-theoretic approaches are the natural methods to study
the brain. However, conventional graph theoretical analyses are limited due to
the lack of formal statistical methods for estimation and inference for random
graphs. For example, the concept of correlation between two vectors of graphs
is yet not defined. The aim of this article to introduce a notion of
correlation between graphs. In order to develop a framework to infer
correlation between graphs, we assume that they are generated by mathematical
models and that the parameters of the models are our random variables. Then, we
define that two vectors of graphs are independent whether their parameters are
independent. The problem is that, in real world, the model is rarely known, and
consequently, the parameters cannot be estimated. By analyzing the graph
spectrum, we showed that the spectral radius is highly associated with the
parameters of the graph model. Based on it, we constructed a framework for
correlation inference between graphs and illustrate our approach in a
functional magnetic resonance imaging data composed of 814 subjects comprising
529 controls and 285 individuals diagnosed with autism spectrum disorder (ASD).
Results show that correlations between default-mode and control, default-mode
and somatomotor, and default-mode and visual sub-networks are higher ($p<0.05$)
in ASD than in controls.
