Motif Mining in Weighted Networks

Unexpectedly frequent subgraphs, known as motifs, can help in characterizing the structure of complex networks. Most of the existing methods for finding motifs are designed for unweighted networks, where only the existence of connection between nodes is considered, and not their strength or capacity. However, in many real world networks, edges contain more information than just simple node connectivity. In this paper, we propose a new method to incorporate edge weight information in motif mining. We think of a motif as a subgraph that contains unexpected information, and we define a new significance measurement to assess this subgraph exceptionality. The proposed metric embeds the weight distribution in subgraphs and it is based on weight entropy. We use the g-trie data structure to find instances of k-sized subgraphs and to calculate its significance score. Following a statistical approach, the random entropy of subgraphs is then calculated, avoiding the time consuming step of random network generation. The discrimination power of the derived motif profile by the proposed method is assessed against the results of the traditional unweighted motifs through a graph classification problem. We use a set of labeled ego networks of co-authorship in the biology and mathematics fields. The new proposed method is shown to be feasible, achieving even slightly better accuracy. Since it does not require the generation of random networks, it is also computationally faster, and because we are able to use the weight information in computing the motif importance, we can avoid converting weighted networks into unweighted ones.