Clustering in complex networks. I. General formalism

TL;DR

This paper introduces a comprehensive theoretical framework for analyzing clustering in complex networks, emphasizing edge multiplicity and three-vertex correlations to better characterize transitive relations and classify networks.

Contribution

It develops new metrics and formalism for clustering, extending existing concepts, and applies them to real networks to classify them into weak and strong transitivity groups.

Findings

Networks can be classified into weak and strong transitivity classes.

Edge multiplicity correlates with the sharing of triangles among edges.

New metrics effectively characterize transitive relations in networks.

Abstract

We develop a full theoretical approach to clustering in complex networks. A key concept is introduced, the edge multiplicity, that measures the number of triangles passing through an edge. This quantity extends the clustering coefficient in that it involves the properties of two --and not just one-- vertices. The formalism is completed with the definition of a three-vertex correlation function, which is the fundamental quantity describing the properties of clustered networks. The formalism suggests new metrics that are able to thoroughly characterize transitive relations. A rigorous analysis of several real networks, which makes use of the new formalism and the new metrics, is also provided. It is also found that clustered networks can be classified into two main groups: the {\it weak} and the {\it strong transitivity} classes. In the first class, edge multiplicity is small, with triangles being disjoint. In the second class, edge multiplicity is high and so triangles share many edges. As we shall see in the following paper, the class a network belongs to has strong implications in its percolation properties.