Characterizing motifs in weighted complex networks

The local structure of unweighted complex networks can be characterized by the occurrence frequencies of subgraphs in the network. Frequently occurring subgraphs, motifs, have been related to the functionality of many natural and man-made networks. Here, we generalize this approach for weighted networks, presenting two novel measures: the intensity of a subgraph, defined as the geometric mean of its link weights, and the coherence, depicting the homogeneity of the weights. The concept of motif scores is then generalized to weighted networks using these measures. We also present a definition for the weighted clustering coefficient, which emerges naturally from the proposed framework. Finally, we demonstrate the concepts by applying them to financial and metabolic networks.