If i1, i2, ik are k dierent indices from 1, n,then is connected and the colours of G, denoted by C, are specifically m1, mk. Ik corresponds, then, towards the set of attainable positions for the occurrence of a motif of size k. Figure 2 provides an instance of a motif and its occurrences. Variety of Occurrences. We introduce the random indicator variable Y which equals a single if motif m occurs at High quality of Approximation. To measure this high quality, we adopted two criteria, the Kolmogorov Smirnov distance which measures the maximal dierence in between the empir ical cumulative distribution function F and also the cdf of the standard or the Polya Aeppli distribution. The closer to 0 the KS distance, the better the approximation. 1 minus the empirical cdf calculated in the 99% and 99. 9% quantiles of the standard or of the Polya Aeppli distribution.
The closer to 1% and 0. 1% these values, the much better the approximation. Outcomes. Results for dierent values of n and p are extremely similar. We only present right here the ones corresponding to n 500 and P. 01 since these selelck kinase inhibitor values are very close to those observed in actual instances like the metabolic network of E. coli as deemed in Lacroix et al. Nonetheless, all outcomes are presented in the supplementary material. We can rst notice just by eye that the standard distribution seems satisfactory for frequent motifs however the rarer the motif, the worse the goodness of t. The Polya Aeppli distribution appears to t fairly appropriately the count distribution whatever the motif. These initial impres sions are emphasised when we appear at the Kolmogorov Smirnov distances.
The ones for the Polya Aeppli distribution are constantly smaller than those for the PI3K normal distribution and often significantly smaller sized. Actually, the distance for the standard distribution is rather large for very rare motifs ten. If we now focus on the distribution tails by looking at the empirical probabilities to exceed the 99% or 99. 9% quantiles qN and qP A, we can also notice that they are closer to 1% or 0. 1% for the Polya Aeppli distribution than for the typical distribution. For exceptionally rare motifs, quantiles qP A for each 99% and 99. 9% could not be appropriately calculated mainly because the corresponding Polya Aeppli distribution is both discrete and concentrated about 0. The values for the empirical tails supplied within the table are for that reason not meaningful in such instances, but thanks to the really compact KS distances, we can check that the approximation is still very good.
Lastly, observe that most of the time the standard distribution underestimates the quantile leading to false positives. 5. Discussion and Conclusion Within this paper, we proposed a new approach to assess the exceptionality of coloured motifs in networks which usually do not need to carry out simulations. Certainly, we have been in a position to establish analytical formulae for the imply and also the variance of the count of a coloured motif in an Erd os Renyi random graph model.