# Motif detection speed up by using equations based on the degree sequence

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## Abstract

To identify so-called *network motifs*, the *fixed degree sequence model* (fdsm) is usually used. For any real-world network, the fdsm is defined as the set of all graphs with the same degree sequence that do not have multiple edges between nodes or self-loops. A subgraph is called a network motif if it occurs statistically significantly often compared to its expected occurrence in the model. However, approximating this value by sampling from the fdsm is computationally expensive and does not scale for large networks. Thus, in this article, we propose a set of equations, based on the degree sequence and a simple independence assumption, to estimate the occurrence of a set of subgraphs in the fdsm. Based on a range of real-world networks, we show that these equations approximate the values in the fdsm very well, except only two data sets. We then propose an efficient way to characterize those data sets in which the equations can be used as an approximation to the fdsm.

## Keywords

Simple Graph Multiple Edge Network Motif Degree Sequence Closed Formula## Notes

### Acknowledgments

This work was funded by the DFG SPP 1736. We thank the official and unofficial reviewers for their helpful comments and insights.

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