Complete and Lipschitz continuous invariants of graphs under geometric isomorphism in R^n
Keywords: Euclidean graph, geometric isomorphism, complete SE(n)-invariant, Lipschitz continuous metric, molecule
TL;DR: This paper develops complete invariants with Lipschitz continuous metrics for embedded graphs under geometric isomorphism (or rigid motion) in any Euclidean spaces, illustrated on the QM9 database of 130K+ molecules.
Abstract: Euclidean graphs embedded in R^n with unordered vertices and straight-line edges represent important real objects such as molecules whose atoms are connected by chemical bonds. Many real objects preserve their properties under any rigid motion from the special Euclidean group SE(n).Embedded graphs were previously distinguished under such rigid motion or geometric isomorphism in R^n.
Experimental noise motivates new Lipschitz continuous invariants so that perturbations of all vertices up to epsilon change the invariants up to a constant multiple of epsilon in a suitable metric, whose running time should polynomially depend on the number of unordered vertices.
We developed new complete invariants that are stable under noise, form a natural hierarchy, and distinguish all chemically different graphs in the QM9 database of 130K+ molecules within a few hours on a modest desktop.
Supplementary Material: zip
Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Submission Number: 10565
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