Autorrelation and cross-relation of graphs and networks

The concepts of auto- and cross-correlation play a key role in several areas, including signal processing and analysis, pattern recognition, multivariate statistics, as well as physics in general, as these operations underlie several real-world structures and dynamics. In the present work, the concept of multiset similarity, more specifically the coincidence similarity index, is used as the basis for defining operations between a same network, or two distinct networks, which will be respectively called autorrelation and cross-relation. In analogous manner to the autocorrelation and cross-correlation counterparts, which are defined in terms of inner products between signals, the two operations suggested here allow the comparison of the similarity of nodes and graphs respectively to successive displacements along the neighborhoods of each of the constituent nodes, which therefore plays a role that is analogue to the lag in the class correlation. In addition to presenting these approaches, this work also illustrates their potential respectively to applications for the characterization of several model-theoretic and real world networks, providing a comprehensive description of the specific properties of each analyzed structure. The possibility of analyzing the obtained individual autorrelation signatures in terms of their respective coincidence similarity networks is also addressed and illustrated.