Local and global convergence in bounded degree graphs László Lovász Eötvös Loránd University, Budapest Joint work with Christian Borgs, Jennifer Chayes and Jeff Kahn December Dedicated to the Memory of Oded Schramm
December The Benjamini-Schramm limit G: simple graph with all degrees ≤ D B G (v,r)= {nodes at distance ≤ r from node v} v random uniform node B G (v,r) random graph in A r P G (A)= P(B G (v,r)≈A) A r = {simple rooted graphs with all degrees ≤ D and radius ≤r } (G 1,G 2,…) convergent: is convergent for all A
December The Benjamini-Schramm limit A1A1 A2A2 A3A3 …
December The Benjamini-Schramm limit = {maximal paths from } = {rooted countable graphs with degrees ≤D} A = {maximal paths through A} A = { -algebra generated by the A } P: probability measure on ( ,A) P has some special properties…
December Other limit constructions
December Other limit constructions ?
December Other limit constructions Measure preserving graph: G=([0,1],E) (a) all degrees ≤D (b) X [0,1] Borel N(X) is Borel (c) X,Y [0,1] Borel R.Kleinberg – L
December Other limit constructions Graphing: G=([0,1],E) Elek
December Homomorphism functions Weighted version: Probability that random map V(G) V(H) is a hom
December Homomorphism functions Examples: hom(G, ) = # of independent sets in G
December Homomorphism functions We know we know
December Homomorphism functions
December Left and right convergence very large graph counting edges, triangles,... spectra,... counting colorations, stable sets,... statistical physics,... maximum cut,...
December Left and right convergence ?
December Examples
December Examples Fekete’s Lemma convergence
December Examples
December Examples
December Examples
December Examples Construct auxiliary graph G : H connected nonbipartite G connected nonbipartite
December Examples
December Left and right convergence
December Analogy: the dense case Left-convergence (homomorphisms from “small” graphs) Right-convergence (homomorphisms into “small” graphs) Distance of two graphs (optimal overlay; convergent Cauchy) Limit objects (2-variable functions) Approximation by bounded-size graphs (Szemerédi Lemma, sampling) Parameters “continuous at infinity” (parameter testing)
December 2009 Limit objects 24 Borgs, Chayes,L,Sós,Vesztergombi
For every convergent graph sequence (G n ) there is a graphon such that December Limit objects LS Conversely, for every graphon W there is a graph sequence (G n ) such that LS W is essentially unique (up to measure-preserving transformation). BCL
December Amenable (hyperfinite) limits o(n) edges (n) nodes Small cut decomposition:
December Amenable (hyperfinite) limits {G 1,G 2,…} amenable (hyperfinite): Can be decomposed into bounded pieces by small cut decomposition.
December Amenable graphs and hyperfinite limits For a convergent graph sequence, hyperfiniteness is reflected by the limit. Schramm Every minor-closed property is testable for graphs with bounded degree. Benjamini-Schramm-Shapira
December Regularity Lemma? -homogeneous: small cut decomposition, each piece H satisfies Every sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces by small cuts. Elek – Lippner Angel - Szegedy
December Regularity Lemma? Easy observation: 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 Alon A construction for H ? Effective bound on q ?