Relations between average shortest path length and another centralities in graphs

TL;DR

The paper investigates how the average shortest path length $L(G)$ relates to various centralities and local properties in graphs, especially geodetic graphs, with emphasis on measures such as the Watts–Strogatz clustering coefficient $C_{WS}$, radiality, and stress centrality $Str(i)$. It derives exact local and global relations, notably $L(N(i)) = 2 - c_i$ and $C_{WS}(G) = 2 - \frac{1}{n} \sum_i L(N(i))$, and establishes a universal bound $L(G) \le 1 + \frac{1}{n(n-1)} \sum_i Str(i)$, which becomes an equality for geodetic graphs: $L(G) = 1 + \frac{1}{n(n-1)} \sum_i Str(i)$. Additional results connect $L(G)$ to radiality and to clustering via subgraph considerations and vertex deletions, with the star graph illustrating sharpness of the bounds. These findings provide exact characterizations linking global path length to centrality-based descriptors, aiding analysis of small-world properties and network structure.

Abstract

Relations between average shortest path length and average clustering coefficient, radiality, closeness and stress centralities were obtained for simple graphs.

Theorems & Definitions (16)

• Definition 1
• Lemma 1
• Corollary 1
• Lemma 2
• Theorem 1
• Corollary 2
• Corollary 3
• Theorem 2
• Lemma 3
• Lemma 4