2 research outputs found
A Class of Graph-Geodetic Distances Generalizing the Shortest-Path and the Resistance Distances
A new class of distances for graph vertices is proposed. This class contains
parametric families of distances which reduce to the shortest-path, weighted
shortest-path, and the resistance distances at the limiting values of the
family parameters. The main property of the class is that all distances it
comprises are graph-geodetic: if and only if every path
from to passes through . The construction of the class is based on
the matrix forest theorem and the transition inequality.Comment: 14 pages. Discrete Applied Mathematic
On the limiting behavior of parameter-dependent network centrality measures
We consider a broad class of walk-based, parameterized node centrality
measures for network analysis. These measures are expressed in terms of
functions of the adjacency matrix and generalize various well-known centrality
indices, including Katz and subgraph centrality. We show that the parameter can
be "tuned" to interpolate between degree and eigenvector centrality, which
appear as limiting cases. Our analysis helps explain certain correlations often
observed between the rankings obtained using different centrality measures, and
provides some guidance for the tuning of parameters. We also highlight the
roles played by the spectral gap of the adjacency matrix and by the number of
triangles in the network. Our analysis covers both undirected and directed
networks, including weighted ones. A brief discussion of PageRank is also
given.Comment: First 22 pages are the paper, pages 22-38 are the supplementary
material