Far-out Vertices In Weighted Repeated Configuration Model

TL;DR

This paper studies the distribution of distant vertices in weighted random graphs with a fixed degree sequence, revealing their impact on global network properties and proposing a conjecture related to minimal spanning trees.

Contribution

It provides a convergence result for the distribution of far-out vertices and introduces a conjecture linking these vertices to the longest edge in the minimal spanning tree.

Findings

Convergence result for the distribution of far-out vertices.

Insight into the role of distant vertices in global graph properties.

A conjecture relating far-out vertices to minimal spanning tree edges.

Abstract

We consider an edge-weighted uniform random graph with a given degree sequence (Repeated Configuration Model) which is a useful approximation for many real-world networks. It has been observed that the vertices which are separated from the rest of the graph by a distance exceeding certain threshold play an important role in determining some global properties of the graph like diameter, flooding time etc., in spite of being statistically rare. We give a convergence result for the distribution of the number of such far-out vertices. We also make a conjecture about how this relates to the longest edge of the minimal spanning tree on the graph under consideration.