Forcing Generalized Quasirandom Graphs Efficiently
| Authors | |
|---|---|
| Year of publication | 2023 |
| Type | Article in Proceedings |
| Conference | European Conference on Combinatorics, Graph Theory and Applications |
| MU Faculty or unit | |
| Citation | |
| Web | https://journals.muni.cz/eurocomb/article/view/35604 |
| Doi | http://dx.doi.org/10.5817/CZ.MUNI.EUROCOMB23-070 |
| Keywords | graph limits; quasirandomness; stochastic block model |
| Description | We study generalized quasirandom graphs whose vertex set consists of q parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most (10q)^q+q vertices; subsequently, Lovász refined the argument to show that graphs with 4(2q+3)^8 vertices suffice. Our results imply that the structure of generalized quasirandom graphs with q>=2 parts is forced by homomorphism densities of graphs with at most 4q^2-q vertices, and, if vertices in distinct parts have distinct degrees, then 2q+1 vertices suffice. The latter improves the bound of 8q-4 due to Spencer. |
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