Tuning the Clustering Coefficient of Generalized Circulant Networks

TL;DR

This paper investigates how to explicitly tune the clustering coefficient in static generalized circulant networks, providing formulas and constructions that enable precise control over network clustering properties.

Contribution

The authors derive explicit formulas for the clustering coefficient of generalized circulant graphs and introduce methods to modify these graphs to achieve desired clustering levels.

Findings

Explicit clustering coefficient formulas for circulant graphs.

Methods to modify graphs to tune clustering coefficients.

Construction of graph pairs with identical size and links but different clustering.

Abstract

Apart from the role the clustering coefficient plays in the definition of the small-world phenomena, it also has great relevance for practical problems involving networked dynamical systems. To study the impact of the clustering coefficient on dynamical processes taking place on networks, some authors have focused on the construction of graphs with tunable clustering coefficients. These constructions are usually realized through a stochastic process, either by growing a network through the preferential attachment procedure, or by applying a random rewiring process. In contrast, we consider here several families of static graphs whose clustering coefficients can be determined explicitly. The basis for these families is formed by the $k$-regular graphs on $N$ nodes, that belong to the family of so-called circulant graphs denoted by $C_{N,k}$. We show that the expression for the clustering coefficient of $C_{N,k}$ reported in literature, only holds for sufficiently large $N$. Next, we consider three generalizations of the circulant graphs, either by adding some pendant links to $C_{N,k}$, or by connecting, in two different ways, an additional node to some nodes of $C_{N,k}$. For all three generalizations, we derive explicit expressions for the clustering coefficient. Finally, we construct a family of pairs of generalized circulant graphs, with the same number of nodes and links, but with different clustering coefficients.