Next Article in Journal
A Distributed Snapshot Protocol for Efficient Artificial Intelligence Computation in Cloud Computing Environments
Previous Article in Journal
Acknowledgement to Reviewers of Symmetry in 2017
Open AccessArticle

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

Karlsruhe Institute of Technology, Institute of Information Systems and Marketing, Kaiserstr. 12, 76131 Karlsruhe, Germany
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(1), 29; https://doi.org/10.3390/sym10010029
Received: 22 December 2017 / Revised: 9 January 2018 / Accepted: 10 January 2018 / Published: 16 January 2018
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. View Full-Text
Keywords: graph symmetry; graph automorphism groups; symmetry analysis; real-world networks graph symmetry; graph automorphism groups; symmetry analysis; real-world networks
Show Figures

Figure 1

MDPI and ACS Style

Ball, F.; Geyer-Schulz, A. How Symmetric Are Real-World Graphs? A Large-Scale Study. Symmetry 2018, 10, 29.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Search more from Scilit
 
Search
Back to TopTop