Invariant Graph Partition Comparison Measures
Karlsruhe Institute of Technology, Institute of Information Systems and Marketing, Kaiserstr. 12, 76131 Karlsruhe, Germany
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Symmetry 2018, 10(10), 504; https://doi.org/10.3390/sym10100504
Received: 10 September 2018 / Revised: 10 October 2018 / Accepted: 11 October 2018 / Published: 15 October 2018
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived.