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},
  journal={Advances in Mathematics},
  year={2007},
  volume={219},
  pages={1801-1851}
}

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References

SHOWING 1-10 OF 56 REFERENCES

Graph limits and parameter testing

We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if

Limits of dense graph sequences

Strong independence of graph copy functions

Let H be a finite graph on v vertices . We define a function CH , with domain the set of all finite graphs, by letting cH(G) denote the fraction of subgraphs of G on v vertices isomorphic to H. Our

Strong Independence of Graphcopy Functions

Let H be a finite graph on v vertices . We define a function CH , with domain the set of all finite graphs, by letting cH(G) denote the fraction of subgraphs of G on v vertices isomorphic to H. Our

Asymptotic Enumeration of Spanning Trees

  • R. Lyons
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 2005
It is shown that tree entropy is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs, which is also expressed using random walks.

Moments of Two-Variable Functions and the Uniqueness of Graph Limits

AbstractFor a symmetric bounded measurable function W on [0, 1]2 and a simple graph F, the homomorphism density $$t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$$ can be thought

Efficient Testing of Large Graphs

This theorem is used to prove that first order graph properties not containing a quantifier alternation of type "/spl forall//spl exist/" are always testable, while it is shown that some properties containing this alternation are not.

Regular Partitions of Graphs

Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph

A combinatorial characterization of the testable graph properties: it's all about regularity

One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.

Every monotone graph property is testable

It is shown that any monotone graph property can be tested with one-sided error, and with query complexity depending only on ε, and this result implies the testability of well-studied graph properties that were previously not known to be testable.
...