Continuous mean distance of a weighted graph

TL;DR

This paper introduces the continuous mean distance for weighted graphs, providing methods for efficient computation and exploring its relation to the traditional discrete mean distance.

Contribution

It presents the first algorithms to compute the continuous mean distance in quadratic time and analyzes its properties and convergence behavior.

Findings

Quadratic-time algorithms for continuous mean distance computation.

Structural results enabling faster computation for specific graph classes.

Analysis of convergence between discrete and continuous mean distances.

Abstract

We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a natural measure of the compactness of the graph, and has been intensively studied, together with several variants, including its version for weighted graphs. The continuous analog of the (discrete) mean distance is the mean of the distances between all pairs of points on the edges of the graph. Despite being a very natural generalization, to the best of our knowledge this concept has been barely studied, since the jump from discrete to continuous implies having to deal with an infinite number of distances, something that increases the difficulty of the parameter. In this paper we show that the continuous mean distance of a weighted graph can be computed in time quadratic in the number of edges, by two different methods that apply fundamental concepts in discrete algorithms and computational geometry. We also present structural results that allow a faster computation of this continuous parameter for several classes of weighted graphs. Finally, we study the relation between the (discrete) mean distance and its continuous counterpart, mainly focusing on the relevant question of the convergence when iteratively subdividing the edges of the weighted graph.