Subgraph centrality and walk-regularity [article]

Eric Horton, Kyle Kloster, Blair D. Sullivan
2019 arXiv   pre-print
Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between subgraph centrality, which is defined using the matrix exponential f(x) = (x), and the walk structure of a network. In a walk-regular graph, the number of closed walks of each length must be the same for all nodes, implying uniform f-subgraph centralities for any
more » ... (or maximum f-walk entropy). We consider when non--walk-regular graphs can achieve maximum entropy, calling such graphs entropic. For parameterized measures, we are also interested in which values of the parameter witness this uniformity. To date, only one entropic graph has been identified, with only two witnessing parameter values, raising the question of how many such graphs and parameters exist. We resolve these questions by constructing infinite families of entropic graphs, as well as a family of witnessing parameters with a limit point at zero.
arXiv:1804.05966v3 fatcat:vwbqpvjuozczxaibjlc7v2e77a