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} }
563 Citations
Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics
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Limits of locally–globally convergent graph sequences
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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.
Monotone Graph Limits and Quasimonotone Graphs
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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.
A measure-theoretic representation of graphs
- 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.
References
SHOWING 1-10 OF 56 REFERENCES
Graph limits and parameter testing
- MathematicsSTOC '06
- 2006
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…
Strong independence of graph copy functions
- Mathematics
- 1978
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
- Mathematics
- 1979
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
- Mathematics, Computer ScienceCombinatorics, 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
- Mathematics
- 2008
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
- Mathematics40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999
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
- Mathematics
- 1975
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
- Mathematics, Computer ScienceSTOC '06
- 2006
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
- Mathematics, Computer ScienceSTOC '05
- 2005
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.