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.
Highlights
Random walk and diffusive models are ubiquitous in mathematics, physics, biology and social sciences, in particular when the random walker moves through the vertices and edges of a graph G = (V, E) [35, 1, 12, 8, 36]
We have introduced the concept of hubs-biased resistance distance
These Euclidean distances are based on graph Laplacians which consider the edges e = (v, w) of a graph weighted by the degrees of the vertices v and w in a double orientation of that edge
Summary
Random walk and diffusive models are ubiquitous in mathematics, physics, biology and social sciences, in particular when the random walker moves through the vertices and edges of a graph G = (V, E) [35, 1, 12, 8, 36] In this scenario a random walker at the vertex j ∈ V of G at time t can move to any of the nearest neighbors of j with equal probability at time t + 1 [35]. We will refer hereafter to this scenario as the “hubs-attracting” one Due to the relation between resistance distance and commute time of random walks on graphs, we study here the efficiency of hubs attracting/repelling diffusive processes on graphs. We observe that certain classes of real-world networks, such as brain/neuronal networks and electronic circuits, have normal random walks as efficient as the hubs-attracting one, while others, like infrastructural networks, can reduce their average commuting times by 300% by using the hubs-attracting mechanism
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