Intensity and coherence of motifs in weighted complex networks

TLDR

The local structure of unweighted networks is traditionally quantified by counting subgraph occurrences, with the clustering coefficient representing a special case of this approach. This study extends that framework to weighted networks. We define subgraph intensity as the geometric mean of link weights and coherence as the ratio of this geometric mean to the corresponding arithmetic mean, enabling weighted motif scores and clustering coefficients. Applying these measures to financial and metabolic networks shows that incorporating weights can substantially alter conclusions drawn from unweighted analyses.

Abstract

The local structure of unweighted networks can be characterized by the number of times a subgraph appears in the network. The clustering coefficient, reflecting the local configuration of triangles, can be seen as a special case of this approach. In this paper we generalize this method for weighted networks. We introduce subgraph "intensity" as the geometric mean of its link weights "coherence" as the ratio of the geometric to the corresponding arithmetic mean. Using these measures, motif scores and clustering coefficient can be generalized to weighted networks. To demonstrate these concepts, we apply them to financial and metabolic networks and find that inclusion of weights may considerably modify the conclusions obtained from the study of unweighted characteristics.