Ipari és társadalomtudományi alkalmazott matematika
Matematika
For a symmetric bounded measurable function W on [0, 1](2) and a simple graph F, the
homomorphism density t(F,W) = integral([0,1]V(F)) Pi(ij is an element of E(F)) W(x(i),x(j))dx.
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.
