Convergent sequences of sparse graphs (status report)

1. Convergent sequences of sparse graphs (status report) László Lovász Eötvös University, Budapest

2. For dense graphs: Left-convergence (homomorphisms from “small” graphs) Right-convergence (homomorphisms into “small” graphs) Distance of two graphs (optimal overlay; convergentCauchy) Limit of a convergent sequence (2-variable functions, reflection positive graph parameters, ergodic measures on countable graphs) Approximation by bounded-size graphs (Szemerédi Lemma, sampling) Parameters “continuous at infinity” (parameter testing, spectrum) For sparse graphs?

3. Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

4. Weighted version:

5. Gn is left-convergent if converges  connectedF Gn: sequence of graphs with degrees D

6. All possible neighborhoods with radius r Gn is left-convergent if converges for all r Equivalent definition: G  ( .1 .2 .13 .27 .2 0 .1 0 )

7. Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

8. Gn is right-convergent if is convergent q1Hin a neighborhood of Jq Jq: complete graph Kq with loops

9. Left-convergent  q2D is convergent. Number of q-colorings Right-convergent  left-convergent Borgs-Chayes-Kahn-L

10. is convergent if H is connected nonbipartite. Gn:nn discrete torus Long-range interaction between colors

11. infinite sum! Key to the proof: Mayer expansion where

12. : Dobrushin Lemma The expansions are convergent if H-Jq is small enough

13. Mayer expansion: where

14. Sample Lemma: Let F1,…FN beall connected graphs on at most q nodes. Then the matrix is nonsingular. : Erdős-L-Spencer

15. Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

16. ?

17. Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

18. The limit object

19. The limit object ?

20. The limit object Benjamini – Schramm: probability distribution on rooted countable graphs with degrees D with “unimodularity” condition

21. frequency of this neighborhood All possible neighborhoods with radius 0: x0 x11 x12 x13 x14 radius 1: x121 x122 x123 x124 radius 2: x1241 x1242 x1243 x1244 radius 3: +further equations

22. The limit object Benjamini – Schramm: probability distribution on rooted countable graphs with degrees D with “unimodularity” condition R.Kleinberg – L: bounded degree graph on [0,1] with “measure-preserving” condition

23. The limit object complete binary trees

24. Elek: “graphing”: measure-preserving involution The limit object Benjamini – Schramm: probability distribution on rooted countable graphs with degrees D with “unimodularity” condition R.Kleinberg – L: bounded degree graph on [0,1] with “measure-preserving” condition

25. Open problem: are all these limit objects?

26. expander same subgraph densities expander expander This notion of limit (or convergence) is not enough... Does not see global structure

27. This notion of limit (or convergence) is not enough... Girth of Gn tends to   Gn tends to union of trees Bollobás et al. “W-random” with probabilities W(x,y)/n tends to union of trees Does not distinguish graphs with large girth

28. This notion of limit (or convergence) is not enough... have the same limit Does not see the geometry or topology of the graphs

29. Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

30. For every r,D1 and 0 there is a q(r,,D) such that for every graph G with degrees D there is a graph H with degrees D and with q nodes such that for all for all connected graphs F with r nodes Easy observation: Thanks to Noga, Nati,... A construction for H? Effective bound on q?

31. Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

32. If (Gn) is a convergent sequence of connected graphs, then is convergent. Lyons Other parameters “continuous at infinity” T(G): number of spanning trees

33. Perfect matching G. Kun If (Gn) is a convergent sequence of bipartite graphs with perfect matchings  Limit graphing has a measurable perfect matching If (Gn) is a convergent sequence of bipartite graphs with maximum matching < (1-)|V(Gn)|/2  Limit graphing has no measurable perfect matching What about non-measurable?