Communicability cosine distance: similarity and symmetry in graphs/networks

Estrada, E.
Computational and Applied Mathematics 43, (2024)

A distance based on the exponential kernel of the adjacency matrix of a graph and representing
how well two vertices connect to each other in a graph is defined and studied. This
communicability cosine distance (CCD) is a Euclidean spherical distance accounting for the
cosine of the angles spanned by the position vectors of the graph vertices in this space. The
Euclidean distance matrix (EDM) of CCD is used to quantify the similarity between vertices
in graphs and networks as well as to define a local vertex invariant—a closeness centrality
measure, which discriminate very well vertices in small graphs. It allows to distinguish all
nonidentical vertices, also characterizing all identity (asymmetric) graphs–those having only
the identity automorphism–among all connected graphs of up to 9 vertices. It also characterizes
several other classes of identity graphs. We also study real-world networks in term
of both the discriminating power of the new centrality on their vertices as well as in ranking
their vertices. We analyze some dictionary networks as well as the network of copurshasing
of political books, remarking some of the main advantages of the new approaches studied

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