Hubs-biased resistance distances on graphs and networks

TL;DR

This paper introduces two new effective resistance measures on graphs, based on hubs-biased navigation models, and explores their mathematical properties, spectral relations, and implications for real-world networks.

Contribution

It defines hubs-biased resistance distances, proves their Euclidean nature, relates them to Laplacian pseudoinverses, and formulates conjectures about their Kirchhoff indices compared to standard resistance.

Findings

Resistance distances are squared Euclidean distances.

Spectral properties relate resistance measures to Laplacian pseudoinverses.

Conjecture: hubs-repelling Kirchhoff index is larger than standard, which is larger than hubs-attracting.

Abstract

We define and study two new kinds of "effective resistances" based on hubs-biased -- hubs-repelling and hubs-attracting -- models of navigating a graph/network. We prove that these effective resistances are squared Euclidean distances between the vertices of a graph. They can be expressed in terms of the Moore-Penrose pseudoinverse of the hubs-biased Laplacian matrices of the graph. We define the analogous of the Kirchhoff indices of the graph based of these resistance distances. We prove several results for the new resistance distances and the Kirchhoff indices based on spectral properties of the corresponding Laplacians. After an intensive computational search we conjecture that the Kirchhoff index based on the hubs-repelling resistance distance is not smaller than that based on the standard resistance distance, and that the last is not smaller than the one based on the hubs-attracting resistance distance. We also observe that in real-world brain and neural systems the efficiency of standard random walk processes is as high as that of hubs-attracting schemes. On the contrary, infrastructures and modular software networks seem to be designed to be navigated by using their hubs.