Establish the expected number of induced motifs on unlabeled graphs through analytical models

Complex networks are usually characterized by the presence of small and recurrent patterns of interactions between nodes, called network motifs. These small modules can help to elucidate the structure and the functioning of complex systems. Assessing the statistical significance of a pattern as a motif in a network G is a time consuming task which entails the computation of the expected number of occurrences of the pattern in an ensemble of random graphs preserving some features of G, such as the degree distribution. Recently, few models have been devised to analytically compute expectations of the number of non-induced occurrences of a motif. Less attention has been payed to the harder analysis of induced motifs. Here, we illustrate an analytical model to derive the mean number of occurrences of an induced motif in an unlabeled network with respect to a random graph model. A comprehensive experimental analysis shows the effectiveness of our approach for the computation of the expected number of induced motifs up to 10 nodes. Finally, the proposed method is helpful when running subgraph counting algorithms to get the number of occurrences of a topology become unfeasible.