Abstract
The problem of calculating the frequencies of network motifs on three and four nodes in large networks is considered. Telecommunications networks, cell molecular networks are investigated. The sizes of the investigated networks are hundreds of thousands of nodes and connections. These networks are represented in the form of directed and undirected simple graphs. Exact calculating requires huge computational resources for such large graphs. A method for calculating the frequencies of network motifs using the Monte Carlo method with control of an accuracy of calculations is proposed. The proposed effective method minimizes the value of the coefficient of variation.
Highlights
A conception of «network motifs» was proposed by Milo et al in Science [1]
Network motifs are represented as connected subgraphs with a given number of vertices
100 thousand motifs were sampled by Rand-ESU method and 200 thousand frames were sampled by method [11] of sampling frame (MSF)
Summary
A conception of «network motifs» was proposed by Milo et al in Science [1]. They pointed out that large technical, social and molecular networks are natural to explore bottom-up. Dotted lines are edges that can be found 4-motive using the frame number of different sample spaces and various statistical experiments, which are used to calculate k-motifs) is equal to the number of different spanning trees with k vertices. If a frame or a set of frames (as in the case of 4-motifs) let us to detect all motifs with a given number k of vertices, statistical experiments with an probable sample of such frames allow us to calculate the frequencies of all possible k-motifs. The calculation of the frequencies of 4-motifs requires two different statistical experiments according to different spanning trees with 4 vertices (see figure 3). There are 218 (directed graph) and 11 (undirected graph) isomorphism classes in graphs containing 4 vertices
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