# Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing

@article{Borgs2007ConvergentSO,
title={Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing},
author={Christian Borgs and Jennifer T. Chayes and L{\'a}szl{\'o} Mikl{\'o}s Lov{\'a}sz and Vera T. S{\'o}s and Katalin Vesztergombi},
year={2007},
volume={219},
pages={1801-1851}
}
• Published 31 January 2007
• Mathematics
563 Citations
• Mathematics
• 2012
We consider sequences of graphs (Gn) and dene various notions of convergence related to these sequences including \left-convergence," dened in terms of the densities of homomorphisms from small
We consider sequences of large graphs which have certain convergent graph parameters. Many important graph parameters like the edge density may be represented asymptotically as homomorphism
• Mathematics, Computer Science
Random Struct. Algorithms
• 2017
This paper introduces a new notion of convergence of sparse graphs, which it is called Large Deviations or LD‐convergence, and which is based on the theory of large deviations, and establishes several previously unknown relationships between the other notions of convergence.
• Mathematics
Random Struct. Algorithms
• 2010
The convergence of the spectrum of large random graphs to the Spectrum of a limit infinite graph is analyzed and a new formula for the Stieljes transform of the spectral measure of such graphs is derived.
• Mathematics, Computer Science
• 2014
It is proved that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing.
• Mathematics
Internet Math.
• 2012
It is shown that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a “quasimonotonicity” property defined by a certain functional tending to zero.
• Computer Science, Mathematics
ESAIM: Control, Optimisation and Calculus of Variations
• 2020
The problem of subdividing a large graph in communities with a minimal amount of cuts can be approached in terms of graphons and the Γ-limit of the cut functional, and the resulting variational principles can be used to obtain insights into the bisection problem for large graphs, known to be NP-complete.
• Mathematics, Computer Science
• 2022
A measure–theoretic representation of matrices is introduced, and it is shown that such pseudo-metric is a metric on the subspace of adjacency or Laplacian matrices for graphs, thereby obtaining a metric for isomorphism classes of graphs.