The Generalized Feed-forward Loop Motif: Definition, Detection and Statistical Significance

Network motifs play an important role in the qualitative analysis and quantitative characterization of networks. The feed-forward loop is a semantically important and statistically highly significant motif. In this paper, we extend the definition of the feed-forward loop to subgraphs of arbitrary size. To avoid the complexity of path enumeration, we define generalized feed-forward loops as pairs of source and target nodes that have two or more internally disjoint connecting paths. Based on this definition, we formally derive an approach for the detection of this generalized motif. Our quantitative analysis demonstrates that generalized feed-forward loops up to a certain path length are statistically significant. Loops of greater size are statistically underrepresented and hence an anti-motif.