Abstract
For a symmetric bounded measurable function W on [0, 1]2 and a simple graph F, the homomorphism density
can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables.
The main motivation for this result comes from the theory of convergent graph sequences. A sequence (G n ) of dense graphs is said to be convergent if the probability, t(F, G n ), that a random map from V(F) into V(G n ) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]2. Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.
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L.L.’s research supported in part by OTKA Grant No. 67867.
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Borgs, C., Chayes, J. & Lovász, L. Moments of Two-Variable Functions and the Uniqueness of Graph Limits. Geom. Funct. Anal. 19, 1597–1619 (2010). https://doi.org/10.1007/s00039-010-0044-0
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DOI: https://doi.org/10.1007/s00039-010-0044-0