  Node role explainability in complex networks is very difficult, yet is
crucial in different application domains such as social science, neurosciences
or computer science. Many efforts have been made on the quantification of hubs
revealing particular nodes in a network using a given structural property. Yet,
in several applications, when multiple instances of networks are available and
several structural properties appear to be relevant, the identification of node
roles remains largely unexplored. Inspired by the node automorphically
equivalence relation, we define an equivalence relation on graph nodes
associated with any collection of nodal statistics (i.e. any functions on the
node-set). This allows us to define new graph global measures: the power
coefficient, and the orthogonality score to evaluate the parsimony and
heterogeneity of a given nodal statistics collection. In addition, we introduce
a new method based on structural patterns to compare graphs that have the same
vertices set. This method assigns a value to a node to determine its role
distinctiveness in a graph family. Extensive numerical results of our method
are conducted on both generative graph models and real data concerning human
brain functional connectivity. The differences in nodal statistics are shown to
be dependent on the underlying graph structure. Comparisons between generative
models and real networks combining two different nodal statistics reveal the
complexity of human brain functional connectivity with differences at both
global and nodal levels. Using a group of 200 healthy controls connectivity
networks, our method computes high correspondence scores among the whole
population, to detect homotopy, and finally quantify differences between
comatose patients and healthy controls.
