Relations between average clustering coefficient and another centralities in graphs

TL;DR

The paper addresses how the average clustering coefficient $C_{WS}$ relates to the global clustering coefficient $C$ and several centrality measures in simple graphs. It develops a rigorous framework of definitions and proves exact equalities and bounds, notably $E_{loc}(G)=\frac{1}{2}(1+C_{WS}(G))$, and inequalities linking $C_{WS}$ with $Str$, $BC_{loc}$, and local radiality. It further explores the relationship between $C_{WS}$ and $C$ under degree-ordering, with equality in regular graphs and constructed scenarios where $C_{WS}$ can exceed or fall below $C$. The results provide a theoretical toolkit for analyzing how clustering interacts with various centrality notions in networks, with implications for understanding small-world properties and network structure.

Abstract

Relations between average clustering coefficient and global clustering coefficient, local efficiency, radiality, closeness, betweenness and stress centralities were obtained for simple graphs.

Theorems & Definitions (12)

• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Lemma 2
• Theorem 4
• Lemma 3
• Theorem 5
• Theorem 6