Motif Mining in Weighted Networks

Unexpectedly frequent sub graphs, 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 sub graph that contains unexpected information, and we define a new significance measurement to assess this sub graph exceptionality. The proposed metric embeds the weight distribution in sub graphs and it is based on weight entropy. We use the g-trie data structure to find instances of $k$-sized sub graphs and to calculate its significance score. Following a statistical approach, the random entropy of sub graphs 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. Furthermore, we are able to be quicker by not having to generate random networks, and we are able to use the weight information in computing the motif importance, avoiding the need for converting weighted networks into unweighted ones.