Graph limit theory: an overview

Graph limit theory: an overview László Lovász Eötvös Loránd University, Budapest IAS, Princeton

Limit theories of discrete structures rational numbers trees graphs digraphs hypergraphs permutations posets abelian groups metric spaces Aldous, Elek-Tardos Diaconis-Janson Elek-Szegedy Kohayakawa Janson Szegedy Gromov Elek

Common elements in limit theories sampling sampling distance limiting sample distributions combined limiting sample distributions limit object overlay distance regularity lemma applications trees graphs digraphs hypergraphs permutations posets abelian groups metric spaces

Limit theories for graphs Dense graphs: Borgs-Chayes-L-Sós-Vesztergombi L-Szegedy Inbetween: distances Bollobás-Riordan regularity lemma Kohayakawa-Rödl, Scott LaplacianChung Bounded degree graphs: Benjamini-Schramm, Elek

Left and right data very large graph counting colorations, stable sets, statistical physics, maximum cut, ... countingedges, triangles, ... spectra, ...

Dense graphs: convergence distribution of k-samples is convergent for everyk t(F,G):Probability that random mapV(F)V(G) preserves edges (G1,G2,…) convergent:Ft(F,Gn) is convergent

Dense graphs: limit objects W0 = {W: [0,1]2[0,1], symmetric, measurable} "graphon" GnW : F: t(F,Gn)  t(F,W)

Graphs to graphons 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 G AG WG

Dense graphs: basic facts For every convergent graph sequence(Gn) there is a WW0 such thatGnW. Is thistheonlyuseful notion of convergence of densegraphs? Conversely, W(Gn) such thatGnW. W is essentially unique (up to measure-preserving transformation).

Bounded degree: convergence Local : neighborhood sampling Benjamini-Schramm Global : metric space Gromov Local-global : Hatami-L-Szegedy Right-convergence,… Borgs-Chayes-Gamarnik

Graphings Graphing: bounded degree graph G on [0,1] such that:  E(G) is a Borel set in [0,1]2  measure preserving: 0 A B 1 degB(x)=2

Graphings Every Borel subgraph of a graphing is a graphing. Every graph you ever want to construct from a graphing is a graphing D=1: graphing  measure preserving involution G is a graphing  G=G1…  Gk measure preserving involutions (k2D-1)

Graphings: examples V(G) = circle x- x E(G) = {chords with angle } x+

Graphings: examples V(G) = {rooted 2-colored grids} E(G) = {shift the root}

Graphings: examples x+ x- x x x- x+ bipartite? disconnected?

Graphings and involution-invariant distributions x: random point of [0,1] Gx: connected component of G containing x Gx is a random connected graph with bounded degree This distribution is "invariant" under shifting the root. Everyinvolution-invariantdistributioncan be representedby a graphing. Elek

Graph limits and involution-invariant distributions graphs,graphings, orinv-invdistributions (Gn) locally convergent: Cauchy in d Gn G:d(Gn,G)  0 (n  ) inv-inv distribution

Graph limits and involution-invariant distributions Everylocallyconvergentsequence of bounded-degreegraphs has a limiting inv-invdistribution. Benjamini-Schramm Is everyinv-invdistributionthe limit of a locallyconvergentgraphsequence? Aldous-Lyons

Local-global convergence (Gn) locally-globally convergent: Cauchy in dk Gn G:dk(Gn,G)  0 (n  ) graphing

Local-global graph limits Everylocally-globallyconvergentsequence of bounded-degreegraphs has a limit graphing. Hatami-L-Szegedy

Convergence: examples Gn: random 3-regular graph Fn: random 3-regular bipartite graph Hn: GnGn Expander graphs Largegirthgraphs

Convergence: examples Local limit:Gn, Fn, Hn  rooted 3-regular treeT Containsrecentresultthatindependence ratio is convergent. Bayati-Gamarnik-Tetali Conjecture: (Gn), (Fn) and (Hn) are locally-globally convergent.

Convergence: examples Local-global limit:Gn, Fn, Hn tend to different graphings Conjecture: Gn T{0,1}, where V(T) = {rooted 2-colored trees} E(G) = {shift the root}

Local-global convergence: dense case Everyconvergentsequence of graphs is Cauchy indk L-Vesztergombi

Regularity lemma Given an arbitrarily large graph G and an >0, decompose G into f() "homogeneous" parts. (,)-homogeneous graph: SE(G), |S|<|V(G)|, all connected components of G-S with > |V(G)| nodes have the same neighborhood distribution (up to ).

Regularity lemma nxn grid is (, 2/18)-homogeneous. >0 >0 bounded-degGS E(G), |S|<|V(G)|, st. allcomponents of G-S are (,)-homogeneous. Angel-Szegedy, Elek-Lippner

Regularity lemma Given an arbitrarily large graph G and an >0, find a graph H of size at most f() such that G and H are -close in sampling distance. Frieze-Kannan "Weak" Regularity Lemma  suffices in the dense case. f() exists in the bounded degree case. Alon

Extremal graph theory It is undecidablewhether holdsforeverygraphG. Hatami-Norin It is undecidablewhetherthereis a graphingwith almost allr-neighborhoodsin a givenfamilyF . Csóka

Extremal graph theory: dense graphs Kruskal-Katona Razborov 2006 Fisher Goodman Bollobás Mantel-Turán Lovász-Simonovits 1 0 1/2 2/3 3/4 1

Extremal graph theory: D-regular D3/8 Harangi 0 D2/6

Graph limit theory: an overview