{"id":158057,"date":"2008-10-01T00:00:00","date_gmt":"2008-10-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/moments-of-two-variable-functions-and-the-uniqueness-of-graph-limits\/"},"modified":"2018-10-16T19:57:15","modified_gmt":"2018-10-17T02:57:15","slug":"moments-of-two-variable-functions-and-the-uniqueness-of-graph-limits","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/moments-of-two-variable-functions-and-the-uniqueness-of-graph-limits\/","title":{"rendered":"Moments of Two-Variable Functions and the Uniqueness of Graph Limits"},"content":{"rendered":"<div class=\"asset-content\">\n<p>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 \u201cmoment&#8221; 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 (Gn) of dense graphs is said to be convergent if the probability, t(F;Gn), that a random map from V (F) into V (Gn) 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.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 \u201cmoment&#8221; 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 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