?W H A T I S . . .

a Graphon?Daniel Glasscock

Large graphs are ubiquitous in mathematics, anddescribing their structure is an important goal ofmodern combinatorics. The most famous descrip-tion is given by Szemerédi’s Regularity Lemma(1978), which asserts that the vertices of any largeenough finite graph may be partitioned into setsof roughly equal size in such a way that the edgesbetween most of the sets appear to be nearlyrandom.

One way to study large, finite objects is to passfrom sequences of larger and larger such objectsto ideal limiting objects. When done properly,properties of the limiting objects reflect propertiesof the finite objects which approximate them, andvice versa.

Graphons, short for graph functions, are thelimiting objects for sequences of large, finitegraphs with respect to the so-called cut metric.They were introduced and developed by C. Borgs,J. T. Chayes, L. Lovász, V. T. Sós, B. Szegedy,and K. Vesztergombi in [1] and [3]. Graphonsarise naturally wherever sequences of large graphsappear: extremal graph theory, property testingof large graphs, quasi-random graphs, randomnetworks, the thermodynamic limit of statisticalphysics systems, et cetera.

Let’s begin with a few definitions and a motivat-ing example. A graph G is a set of vertices V(G)and a set of edges E(G) whose elements are pairsof distinct vertices. A graph homomorphism fromH to G is a map from V(H) to V(G) that preservesedge adjacency; that is, for every edge {v,w} inE(H), the edge {ϕ(v),ϕ(w)} is in E(G). Denoteby hom(H,G) the number of homomorphismsfrom H to G. For example, hom( ,G) = |V(G)|,hom( ,G) = 2|E(G)|, and hom( ,G) is 6 timesthe number of triangles in G. Normalizing by

Daniel Glasscock is a graduate student in mathematics atThe Ohio State University. His email address is glasscock.4@osu.edu.

DOI: http://dx.doi.org/10.1090/noti1204

the total number of possible maps, we get thehomomorphism density of H into G,

t(H,G) = hom(H,G)|V(G)||V(H)| ,

the probability that a randomly chosen map fromV(H) to V(G) preserves edge adjacency. Thisnumber also represents the density of H as asubgraph in G asymptotically as n = |V(G)| → ∞.For example, t( ,G) = 2|E(G)|/n2, while thedensity of edges in G is 2|E(G)|/n(n − 1); thesetwo expressions are nearly the same when n islarge.

Consider the following problem from extremalgraph theory:

How many 4-cycles must there be in a graph withedge density at least 1/2?

It is easy to see that there are at most on theorder of n4 4-cycles in any graph with n vertices.A theorem of Erdos says that graphs with at leasthalf the number of possible edges have at least onthe order of n4 4-cycles. More precisely, for anygraph G,

t( ,G) ≥ t( ,G)4;

in particular, if t( ,G) ≥ 1/2, then t( ,G) ≥ 1/16.In light of this, the problem can be reformulatedas a minimization one: Minimize t( ,G) over finitegraphs G satisfying t( ,G) ≥ 1/2. With somework, it can be shown that no finite graph G witht( ,G) ≥ 1/2 achieves the minimum t( ,G) =1/16.

It’s useful at this point to draw an analogy witha problem from elementary analysis: Minimizex3 − 6x over rational numbers x satisfying x ≥ 0.This polynomial has a unique minimum on [0,∞)at x =

√2, so the best we can do over the rationals

is show that the polynomial achieves valuesapproaching this minimum along a sequenceof rationals approaching

√2. We know well to

avoid this complication by completing the rationalnumbers to the reals and realizing the limit ofsuch a sequence as

√2.

46 Notices of the AMS Volume 62, Number 1

There is a sequence of finite graphs with edgedensity at least 1/2 and 4-cycle density approaching1/16. Let Rn be an instance of a random graph on nvertices where each edge is decided independentlywith probability 1/2. Throwing away those Rn’s forwhich t( , Rn) < 1/2, the 4-cycle density in theremaining graph sequence converges almost surelyto 1/16. Following the

√2 analogy, we should look

to realize the limit of this sequence of finite graphsand understand how it solves the minimizationproblem at hand.

What might the limit of this sequence of randomgraphs (Rn)n be? From the adjacency matrix of alabeled graph, construct the graph’s pixel pictureby turning the 1’s into black squares, erasing the0’s, and scaling to the unit square [0,1]2:

0 1 0 11 0 1 00 1 0 11 0 1 0

-→

Pixel pictures can be seen to “converge” graphically;those of larger and larger random graphs withedge probability 1/2, regardless of how they arelabeled, seem to converge to a gray square, theconstant 1/2 function on [0,1]2.

The constant 1/2 function on [0,1]2 is anexample of a labeled graphon. A labeled graphon isa symmetric, Lebesgue-measurable function from[0,1]2 to [0,1] (modulo the usual identificationalmost everywhere); such functions can be thoughtof as edge-weighted graphs on the vertex set [0,1].An unlabeled graphon is a graphon up to relabeling,where a relabeling is the result of applying aninvertible, measure-preserving transformation tothe [0,1] interval. Note that any pixel picture is alabeled graphon, meaning that (labeled) graphs are(labeled) graphons.

As another example of this convergence, con-sider the growing uniform attachment graphsequence (Gn)n defined inductively as follows.Let G1 = . For n ≥ 2, construct Gn from Gn−1

by adding one new vertex, then including eachedge not already in E(Gn−1) independently withprobability 1/n. It turns out that this graph se-quence converges almost surely to the graphon1 −max(x, y). (Since matrices are indexed with(0,0) in the top left corner, so too are graphons.)

There are two natural ways to label a completebipartite graph, and each suggests a different limitgraphon for the complete bipartite graph sequence.Both sequences of labeled graphons in fact havethe same limit, as indicated in the diagram; thereader is encouraged to return to this exampleafter we define this convergence more precisely.

Homomorphism densities extend naturally tographons. For a finite graph G, the density t( ,G)can be computed by giving each vertex of G a massof 1/n and integrating the edge indicator functionover all pairs of vertices. In exactly the same way,the edge density t( ,W) of a labeled graphon Wis ∫

[0,1]2W(x, y) dxdy,

and the 4-cycle density t( ,W) is∫[0,1]4

W(x1, x2)W(x2, x3)

W(x3, x4)W(x4, x1) dx1dx2dx3dx4.

It is straightforward from here to write the expres-sion for the homomorphism density t(H,W) of afinite graph H into a graphon W . This allows usto see how the constant graphon W ≡ 1/2 solvesthe minimization problem: t( ,W) = 1/2 whilet( ,W) = 1/16.

To see the space of graphons as the completionof the space of finite graphs and make graphon con-vergence precise, define the cut distance δ�(W,U)between two labeled graphons W and U by

infϕ,ψ

supS,T

∣∣∣∣ ∫S×T

W(ϕ(x),ϕ(y)

)−U

(ψ(x),ψ(y)

)dxdy

∣∣∣∣,where the infimum is taken over all relabelings ϕof W and ψ of U , and the supremum is taken overall measurable subsets S and T of [0,1]. The cutdistance first measures the maximum discrepancybetween the integrals of two labeled graphons overmeasurable boxes (hence the �) of [0,1]2, thenminimizes that maximum discrepancy over allpossible relabelings. (It is possible to define the cutdistance between two finite graphs combinatorially,without any analysis, but the definition is quiteinvolved.)

The infimum in the definition of the cut distancemakes δ� well defined on the space of unlabeledgraphons, but it is not yet a metric. Graphons W

January 2015 Notices of the AMS 47

and U for which t(H,W) = t(H,U) for all finitegraphsH are called weakly isomorphic ; it turns outthat W and U are weakly isomorphic if and only ifδ�(W,U) = 0. The cut distance becomes a genuinemetric on the space G of unlabeled graphonsup to weak isomorphism. The examples of pixelpicture convergence above provide examples ofconvergent sequences and their limits in G (up toweak isomorphism).

We conclude by highlighting a few foundationalresults about graphons. The accompanying pagereferences are from Lovász’s book [2], to whichthe interested reader is encouraged to refer formore details.

Theorem 1 (Prop. 11.32, p. 185). Every graphon isthe δ�-limit of a sequence of finite graphs.

To approximate a labeled graphon W by a finitelabeled graph, let S be a set of n randomly chosenpoints from [0,1], then construct a graph on S inwhich the edge {si , sj} is included with probabilityW(si , sj). With high probability (as |S| → ∞), thislabeled graph approximates W well in cut distance.

Theorem 2 (Thm. 9.23, p. 149). The space (G, δ�)is compact.

This implies that (G, δ�) is a complete metricspace. Combining this fact with Theorem 1, wesee that the space of graphons is the completionof the space of finite graphs with the cut metric!This theorem also demonstrates the way in whichgraphons provide a bridge between different formsof Szemerédi’s Regularity Lemma: Theorem 2 maybe deduced from a weak form of the lemma,while a stronger regularity lemma follows from thecompactness of G.

Theorem 3 (Lem. 10.23, p. 167). For every finitegraph H, the map t(H, ·) : G → [0,1] is Lipschitzcontinuous.

Theorems 2 and 3 combine with elementaryanalysis to show that minimization problems inextremal graph theory (such as the one consideredabove) are guaranteed to have solutions in thespace of graphons. These graphon solutions pro-vide “templates,” via Theorem 1, for approximatesolutions in the space of finite graphs.

References1. C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, and

K. Vesztergombi, Convergent sequences of densegraphs. I. Subgraph frequencies, metric propertiesand testing, Adv. Math. 219 (2008), no. 6, 1801–1851.MR 2455626 (2009m:05161)

2. L. Lovász, Large Networks and Graph Limits, AmericanMathematical Society Colloquium Publications, vol. 60,American Mathematical Society, Providence, RI, 2012.MR 3012035

3. L. Lovász and B. Szegedy, Limits of dense graph se-quences, J. Combin. Theory Ser. B 96 (2006), no. 6,933–957. MR 2274085 (2007m:05132)

48 Notices of the AMS Volume 62, Number 1