We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural "limit object," namely a symmetric measurable function W : [0, 1]2 → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. We also characterize graph parameters that are obtained as limits of subgraph densities by the "reflection positivity" property. Along the way we introduce a rather general model of random graphs, which seems to be interesting on its own right.
- Convergent graph sequence
- Graph homomorphism
- Quasirandom graph
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics