Resistance distance distribution in large sparse random graphs

Abstract

We consider an Erdős–Rényi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N → ∞ one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N → ∞ in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c = 4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation 2 / c 3 . At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N → ∞ distribution suggested by the numerics.

An empirical comparison of connectivity-based distances on a graph and their computational scalability

“…Typically, when distance is discussed in the context of graphs and complex networks, it refers to shortest paths between nodes (e.g., [14,10,4,2]). Another approach in the literature consists of borrowing distance measures used in Euclidian space, like the Euclidian or cosine distances (e.g., [46,19]) for example.…”

“…On the topic of shortest path distance, Akara-pipattana et al [2] stated the following: "While intuitive and visual, this notion of distance is limited in that it does not fully capture the ease or difficulty of reaching point j from point i by navigating the graph edges. It does not say whether there is only one path of minimal length or many such paths, whether these paths can be straightforwardly located, or whether alternative paths are considerably or only slightly longer."…”

Random matrices with row constraints and eigenvalue distributions of graph Laplacians

“…A theory of eigenvalue distributions of random matrix ensembles with linear row constraints can be effectively developed using methods derived from statistical field theory [5,6]. These methods have a respectable history of applications to random matrix problems, see [3,4,[7][8][9][10][11][12][13][14][15][16][17][18][19][20] for a sampler of related literature. More specifically, we shall focus on supersymmetry-based techniques [21,22] that introduce auxiliary integrals over anticommuting variables and rewrite the eigenvalue density (or more precisely, the matrix resolvent) in terms of an integral admitting a saddle point evaluation at large N. Fyodorov and Mirlin proposed in [8,9] a powerful variation of this method (see [10,19] for further developments) that introduces functional integration into the game and derives very effectively an integral saddle point equation for the matrix resolvent in ensembles that are not easily tractable by other methods.…”