An algorithm to automatically generate the combinatorial orbit counting equations
 Author
 Ine Melckenbeeck, P. Audenaert (UGent) , Tom Michoel, Didier Colle (UGent) and Mario Pickavet (UGent)
 Organization
 Abstract
 Graphlets are small subgraphs, usually containing up to five vertices, that can be found in a larger graph. Identification of the graphlets that a vertex in an explored graph touches can provide useful information about the local structure of the graph around that vertex. Actually finding all graphlets in a large graph can be timeconsuming, however. As the graphlets grow in size, more different graphlets emerge and the time needed to find each graphlet also scales up. If it is not needed to find each instance of each graphlet, but knowing the number of graphlets touching each node of the graph suffices, the problem is less hard. Previous research shows a way to simplify counting the graphlets: instead of looking for the graphlets needed, smaller graphlets are searched, as well as the number of common neighbors of vertices. Solving a system of equations then gives the number of times a vertex is part of each graphlet of the desired size. However, until now, equations only exist to count graphlets with 4 or 5 nodes. In this paper, two new techniques are presented. The first allows to generate the equations needed in an automatic way. This eliminates the tedious work needed to do so manually each time an extra node is added to the graphlets. The technique is independent on the number of nodes in the graphlets and can thus be used to count larger graphlets than previously possible. The second technique gives all graphlets a unique ordering which is easily extended to name graphlets of any size. Both techniques were used to generate equations to count graphlets with 4, 5 and 6 vertices, which extends all previous results. Code can be found at https://github.com/IneMelckenbeeck/equationgenerator and https://github.com/IneMelckenbeeck/graphletnaming.
 Keywords
 IBCN
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU7241900
 MLA
 Melckenbeeck, Ine, et al. “An Algorithm to Automatically Generate the Combinatorial Orbit Counting Equations.” PLOS ONE, vol. 11, no. 1, 2016, doi:10.1371/journal.pone.0147078.
 APA
 Melckenbeeck, I., Audenaert, P., Michoel, T., Colle, D., & Pickavet, M. (2016). An algorithm to automatically generate the combinatorial orbit counting equations. PLOS ONE, 11(1). https://doi.org/10.1371/journal.pone.0147078
 Chicago authordate
 Melckenbeeck, Ine, P. Audenaert, Tom Michoel, Didier Colle, and Mario Pickavet. 2016. “An Algorithm to Automatically Generate the Combinatorial Orbit Counting Equations.” PLOS ONE 11 (1). https://doi.org/10.1371/journal.pone.0147078.
 Chicago authordate (all authors)
 Melckenbeeck, Ine, P. Audenaert, Tom Michoel, Didier Colle, and Mario Pickavet. 2016. “An Algorithm to Automatically Generate the Combinatorial Orbit Counting Equations.” PLOS ONE 11 (1). doi:10.1371/journal.pone.0147078.
 Vancouver
 1.Melckenbeeck I, Audenaert P, Michoel T, Colle D, Pickavet M. An algorithm to automatically generate the combinatorial orbit counting equations. PLOS ONE. 2016;11(1).
 IEEE
 [1]I. Melckenbeeck, P. Audenaert, T. Michoel, D. Colle, and M. Pickavet, “An algorithm to automatically generate the combinatorial orbit counting equations,” PLOS ONE, vol. 11, no. 1, 2016.
@article{7241900, abstract = {{Graphlets are small subgraphs, usually containing up to five vertices, that can be found in a larger graph. Identification of the graphlets that a vertex in an explored graph touches can provide useful information about the local structure of the graph around that vertex. Actually finding all graphlets in a large graph can be timeconsuming, however. As the graphlets grow in size, more different graphlets emerge and the time needed to find each graphlet also scales up. If it is not needed to find each instance of each graphlet, but knowing the number of graphlets touching each node of the graph suffices, the problem is less hard. Previous research shows a way to simplify counting the graphlets: instead of looking for the graphlets needed, smaller graphlets are searched, as well as the number of common neighbors of vertices. Solving a system of equations then gives the number of times a vertex is part of each graphlet of the desired size. However, until now, equations only exist to count graphlets with 4 or 5 nodes. In this paper, two new techniques are presented. The first allows to generate the equations needed in an automatic way. This eliminates the tedious work needed to do so manually each time an extra node is added to the graphlets. The technique is independent on the number of nodes in the graphlets and can thus be used to count larger graphlets than previously possible. The second technique gives all graphlets a unique ordering which is easily extended to name graphlets of any size. Both techniques were used to generate equations to count graphlets with 4, 5 and 6 vertices, which extends all previous results. Code can be found at https://github.com/IneMelckenbeeck/equationgenerator and https://github.com/IneMelckenbeeck/graphletnaming.}}, articleno = {{e0147078}}, author = {{Melckenbeeck, Ine and Audenaert, P. and Michoel, Tom and Colle, Didier and Pickavet, Mario}}, issn = {{19326203}}, journal = {{PLOS ONE}}, keywords = {{IBCN}}, language = {{eng}}, number = {{1}}, title = {{An algorithm to automatically generate the combinatorial orbit counting equations}}, url = {{http://doi.org/10.1371/journal.pone.0147078}}, volume = {{11}}, year = {{2016}}, }
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