Calculation of number of motifs on three nodes using random sampling of frames in networks with directed links

The task of development of efficient algorithms for estimating the frequency of occurrence of non-isomorphic connected subnets (motifs) on a given number of nodes is an important task of network theory. Combinatorial and logical nature of this problem makes the calculation time-consuming and/or causes high consumption of RAM when estimating networks with hundreds of thousands of nodes. In order to solve the problem this paper develops a random sampling of frames method (MSF), based on a statistical approach, and an algorithm to estimate the occurrence of 3-motifs in networks with directed links is proposed. We suggest implementing the algorithm with the help of parallel computing. The results of numerical data experiments are given. When comparing the developed algorithm with other known algorithms its significant advantages in terms of accuracy, speed and consumption of RAM are revealed in some cases.