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How Symmetric Are Real-World Graphs? A Large-Scale Study

Ball, Fabian 1; Geyer-Schulz, Andreas 1
1 Karlsruher Institut für Technologie (KIT)

Abstract:

The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the “network redundancy” measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.


Verlagsausgabe §
DOI: 10.5445/IR/1000080151
Veröffentlicht am 12.02.2018
Originalveröffentlichung
DOI: 10.3390/sym10010029
Scopus
Zitationen: 16
Web of Science
Zitationen: 14
Dimensions
Zitationen: 19
Cover der Publikation
Zugehörige Institution(en) am KIT Fakultät für Wirtschaftswissenschaften – Institut für Informationswirtschaft und Marketing (IISM)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2073-8994
urn:nbn:de:swb:90-801519
KITopen-ID: 1000080151
Erschienen in Symmetry
Verlag MDPI
Band 10
Heft 1
Seiten 29
Bemerkung zur Veröffentlichung Gefördert durch den KIT-Publikationsfonds
Schlagwörter graph symmetry; graph automorphism groups; symmetry analysis; real-world networks
Nachgewiesen in Web of Science
Dimensions
Scopus
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