Fig. 5

Examples of correlation between subgraphs. a Diagram depicting the Pearson correlation coefficients between n_S(H) and n_S(H^′) for pairs of 3-node subgraphs H and H^′. The input network is the S. cerevisiae TF network, and only coefficients of magnitude greater than 0.5 are shown. b An example of correlation in the frequency of induced subgraphs F and H in the set \( \mathbbm{S}(G) \) of graphs similar to the E. coli TF network, with a Pearson correlation coefficient of −0.999. This is calculated by computing \( \mathbbm{S}(G) \) using an algorithm based on the switching method, and then counting the number of times each of the induced subgraphs F and H appear in each graph in \( \mathbbm{S}(G) \). The resulting frequencies are then plotted on the vertical and horizontal axes respectively to produce the figure shown. Note that motifs are “induced subgraphs”, i.e. they are described by the pattern created by both their edges and their non-edges. Thus, an instance of F is not a special case of H. Had motifs been defined by non-induced subgraphs, this would have created many trivial positive correlations between the frequencies of motifs and the frequencies of their own sub-motifs. Our example shows that the choice of non-induced subgraphs is not enough to avoid correlations