GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS

Transcription

1 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS PERSI DIACONIS AND SVANTE JANSON Abstract. We evelo a clear connection between efinetti s theorem for exchangeable arrays (work of Alous Hoover Kallenberg) an the emerging area of grah limits (work of Lovász an many coauthors). Along the way, we translate the grah theory into more classical robability. S:intro seca 1. Introuction DeFinetti s rofoun contributions are now woven into many arts of robability, statistics an hilosohy. Here we show how eveloments from efinetti s work on artial exchangeability have a irect link to the recent eveloment of a limiting theory for large grahs. This introuction first recalls the theory of exchangeable arrays (Section 1.1). Then, the subject of grah limits is outline (Section 1.2). Finally, the link between these ieas, which forms the bulk of this aer, is outline (Section 1.3) Exchangeability, Partial Exchangeability an Exchangeable Arrays. Let {X i } 1 i < be a sequence of binary ranom variables. They are exchangeable if P (X 1 = e 1,, X n = e n ) = P (X 1 = e σ(1),, X n = e σ(n) ) for all n, ermutations σ an all e i {0, 1}. The celebrate reresentation theorem says Theorem 1.1. (efinetti) If {X i } 1 i < is a binary exchangeable sequence, then (i) With robability 1, X = lim 1 n (X X n ) exists. (ii) If µ(a) = P {X A}, then for all n, e i 1 i n for s = e e n. P (X 1 = e 1,, X n = e n ) = 1 0 x s (1 x) n s µ(x) (1.1) eqp1 It is natural to refine an exten efinetti s theorem to allow more general observables (X i with values in a Polish sace) an other notions of symmetry (artial exchangeability). A efinitive treatment of these eveloments is Date: February 22, 2007 (tyeset November 21, ). 1

2 2 PERSI DIACONIS AND SVANTE JANSON given in [12]. Of interest here is the extension of efinetti s theorem to two-imensional arrays. Definition. Let {X ij } 1 i, j < be binary ranom variables. They are jointly exchangeable if P (X ij = e ij, 1 i, j n) = P (X ij = e σ(i)τ(j) 1 i, j n) for all n, all ermutations σ, τ an all e ij {0, 1}. The question of two-imensional versions of efinetti s theorem uner joint exchangeability arose from the statistical roblems of two-way analysis of variance. Early workers execte a version of (1.1) with erhas a twoimensional integral. The robabilist Davi Alous [1] an the logician Douglas Hoover [11] foun that the answer is more comlicate. Define a ranom binary array {X ij } as follows: Let U i, V j 1 i, j < be ineenent an uniform in [0, 1]. Let W (x, y) be a function from [0, 1] 2 to [0, 1]. Let X ij be 1 or 0 as a W (U i, V j ) coin comes u heas or tails. Let P W be the robability istribution of {X ij } 1 i, j <. The family {X ij } is jointly exchangeable because of the symmetry of the construction. The Alous Hoover theorem says that any jointly exchangeable binary array is a mixture of such P W : Theorem 1.2. (Alous Hoover) Let X = {X ij } 1 i, j < be a jointly exchangeable binary array. Then, there is a robability µ such that P {X A} = P W (A)µ(W ). secb The uniqueness of µ resiste unerstaning; if Ŵ is obtaine from W by a measure-reserving change of each variable, clearly the associate rocess { X ij } has the same joint istribution as {X ij }. Using moel theory, Hoover was able to show that this was the only source of non-uniqueness. A robabilist s roof was finally foun by Kallenberg [12, Sect. 7.6 has etails an references]. These results hol for higher imensional arrays with X ij taking values in a Polish sace with minor change [12, Cha. 7]. The escrition above has not mentione several elegant results of the theory. In articular, Kallenberg s sreaable version of the theory relaces invariance uner a grou by invariance uner subsequences. A variety of tail fiels may be introuce to allow characterizing when W takes values in {0, 1} [8, Sect. 4]. Much more general notions of artial exchangeability are stue in [9] Grah Limits. Large grahs, both ranom an eterministic, aboun in alications. They arise from the internet, social networks, gene regulation, ecology an in mathematics. It is natural to seek an aroximation theory: What oes it mean for a sequence of grahs to converge? When can a large comlex grah be aroximate by a small grah?

3 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 3 In a sequence of aers, Laszlo Lovász with coauthors (liste here in orer of frequency) V. T. Sós, B. Szegey, C. Borgs, J. Chayes, K. Vesztergombi, A. Schrijver, M. Freeman, have eveloe a beautiful, unifying limite theory. This shes light on toics such as grah homomorhisms, Szemerei s regularity lemma, quasi-ranom grahs, grah testing an external grah theory. Their theory has been eveloe for ense grahs (number of eges comarable with the square of number of vertices) but arallel theories for sarse grahs are beginning to emerge [4]. Roughly, a growing sequence of finite grahs G n converges if, for any fixe grah F, the roortion of coies of F in G n converges. Section 2 below has recise efinitions. Examle 1.3. Define a robability istribution on grahs on n-vertices as follows. Fli a θ-coin for each vertex (iviing vertices into boys an girls ). Connect two boys with robability. Connect two girls with robability. Connect a boy an a girl with robability. Thus, if = = 1, = 0, we have a ranom biartite grah. If = = 0, = 1, we have two isjoint comlete grahs. If = =, we have the Erös Renyi moel. As n grows, these moels generate a sequence of ranom grahs which converge almost surely to a limiting object escribe below. If a sequence of grahs converges, what oes it converge to? For exchangeable ranom grahs (efine below), there is a limiting object which may be thought of as a robability measure on infinite ranom grahs. Suose W (x, y) = W (y, x) is a function from [0, 1] 2 [0, 1]. Choose {U i } 1 i < ineenent uniformly istribute ranom variables on [0, 1]. Form an infinite ranom grah by utting an ege from i to j with robability W (U i, U j ). This measure on grahs (or alternatively W ) is the limiting object. For the boys an girls examle above, W may be icture as θ θ 0 1 The theory eveloe shows that various roerties of G n can be well aroximate by calculations with the limiting object. There is an elegant characterization of these continuous grah roerties with alications to algorithms for grah testing (Does this grah contain an Eulerian cycle?) or arameter estimation (What is an aroximation to the size of the maximum cut?). There is a ractical way to fin useful aroximations to a large grah by grahs of fixe size [5]. This aer contains a useful review of the current state of the theory with roofs an references.

4 4 PERSI DIACONIS AND SVANTE JANSON secc We have sketche the theory for unweighte grahs. There are generalizations to grahs with weights on vertices an eges, to biartite, irecte an hyergrahs. The sketch leaves out many nice eveloments. For examle, the useful cut metric between grahs [19] an connections to statistical hysics Overview of the Present Paer. There is an aarent similarity between the measure P W of the Alous Hoover theorem an the limiting object W from grah limits. Roughly, working with symmetric W gives the grah limit theory; working with general W gives irecte grahs. The main results of this aer make these connections recise. Basic efinitions are in Section 2 which introuces a robabilist s version of grah convergence equivalent to the efinition using grah homomorhisms. Section 3 uses the well-establishe theory of weak convergence of a sequence of robability measures on a metric sace to get roerties of grah convergence. Section 4 carries things over to infinite grahs. The main results aear in Section 5. This introuces exchangeable ranom grahs an gives a one-to-one corresonence between infinite exchangeable ranom grahs an the sace of roer grah limits (Theorem 5.3). A useful characterization of the extreme oints of the set of exchangeable ranom grahs is in Theorem 5.5. These results are translate to the equivalence between roer grah limits an the Alous Hoover theory in Section 6. The equivalence involves symmetric W (x, y) an a single ermutation σ taking W (U i, U j ) to W (U σ(i), U σ(j) ). The original Alous Hoover theorem, with erhas non-symmetric W (x, y) an W (U i, V j ) to W (U σ(i), V τ(j) ) translates to a limit theorem for bi-artite grahs. This is eveloe in Section 7. The extensions to weighte grahs are covere by allowing X ij to take general values in the Alous Hoover theory. The extension to hyergrahs follows from the Alous Hoover theory for higher-imensional arrays. Desite these arallels, the theories have much to contribute to each other. The algorithmic, grah testing, Szemerei artitioning ersective is new to exchangeability theory. Inee, the boys an girls ranom grah was introuce to stuy the sychology of vision in Diaconis Freeman (1981). As far as we know, its grah theoretic roerties have not been stuie. The various eveloments aroun shell-fiels in exchangeability, which characterize 0/1 W (x, y), have yet to be translate into grah-theoretic terms. Acknowlegements. This lecture is an extene version of a talk resente at the 100th anniversary of efinetti s birth in Rome, We thank the organizers. A large art of this research was comlete uring a visit by Janson to the Université e Nice...??? The research was artly insire by lectures by an iscussions with Christian Borgs an Jennifer Chayes uring the Oberwolfach meeting Combinatorics, Probability an Comuting, hel in November, 2006.???

5 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 5 Sef 2. Definitions an basic roerties All grahs will be simle. Infinite grahs will be imortant in later sections, but will always be clearly state to be infinite; otherwise, grahs will be finite. We enote the vertex an ege sets of a grah G by V (G) an E(G), an the numbers of vertices an eges by v(g) := V (G) an e(g) := E(G). We consier both labelle an unlabelle grahs; the labels will be the integers 1,..., n, where n is the number of vertices in the grah. A labelle grah is thus a grah with vertex set [n] := {1,..., n} for some n 1; we let L n enote the set of the 2 (n 2) labelle grahs on [n] an let L := n=1 L n. An unlabelle grah can be regare as a labelle grah where we ignore the labels; formally, we efine U n, the set of unlabelle grahs of orer n, as the quotient set L n / = of labelle grahs moulo isomorhisms. We let U := n=1 U n = L/ =, the set of all unlabelle grahs. Note that we can, an often will, regar a labelle grah as an unlabelle grah. If G is an (unlabelle) grah an v 1,..., v k is a sequence of vertices in G, then G(v 1,..., v k ) enotes the labelle grah with vertex set [k] where we ut an ege between i an j if v i an v j are ajacent in G. We allow the ossibility that v i = v j for some i an j. (In this case, there is no ege ij because there are no loos in G.) We let G[k], for k 1, be the ranom grah G(v 1,..., v k ) obtaine by samling v 1,..., v k uniformly at ranom among the vertices of G, with relacement. In other wors, v 1,..., v k are ineenent uniformly istribute ranom vertices of G. For k v(g), we further let G[k] be the ranom grah G(v 1,..., v k ) where we samle v 1,..., v k uniformly at ranom without relacement; the sequence v 1,..., v k is thus a uniformly istribute ranom sequence of k istinct vertices. If F an G are two grahs, we efine, following [7] an [19], first assuming that F is labelle an with k = v(f ), t(f, G) := P ( F G[k] ). (2.1) t Note that both F an G[k] are grahs on [k], so the relation F G[k] is well-efine as containment of labelle grahs on the same vertex set, i.e. as E(F ) E(G[k]). Although the relation F G[k] may een on the labelling of F, the robability in (2.1) oes not, by symmetry, so t(f, G) is really well efine for unlabelle F an G. (The efinition by Lovász an Szegey [19], an also in [7], is actually state ifferently; they efine t(f, G) as the roortion of grah homomorhisms F G among all maings V (F ) V (G). This is eviently equivalent to (2.1).) With F, G an k as in (2.1), we further efine, again using the notation of [7] but stating the efinitions in ifferent but equivalent forms, t inj (F, G) := P ( F G[k] ) (2.2) tinj

6 6 PERSI DIACONIS AND SVANTE JANSON an t in (F, G) := P ( F = G[k] ), (2.3) tin rovie F an G are (unlabelle) grahs with v(f ) v(g). If v(f ) > v(g) we set t inj (F, G) := t in (F, G) := 0. Since the robability that a ranom samle v 1,..., v k of vertices in G contains some reeate vertex is k 2 /(2v(G)), it follows that [19] t(f, G) t inj (F, G) v(f )2. (2.4) a4 2v(G) Hence, when consiering asymtotics with v(g), it oes not matter whether we use t or t inj. Moreover, if F L k, then, as ointe out in [7] an [19], t inj (F, G) = t in (F, G) (2.5) a4a an, by inclusion-exclusion, t in (F, G) = F L k, F F F L k, F F ( 1) e(f ) e(f ) t inj (F, G). (2.6) a4b Hence, the two families {t inj (F, )} F U an {t in (F, )} F U of grah functionals contain the same information an can relace each other. The basic efinition of Lovász an Szegey [19] an Borgs, Chayes, Lovász, Sós an Vesztergombi [7] is that a sequence (G n ) of grahs converges if t(f, G n ) converges for every grah F. We can exress this by consiering the ma τ : U [0, 1] U efine by τ(g) := (t(f, G)) F U [0, 1] U. (2.7) tau Then (G n ) converges if an only if τ(g n ) converges in [0, 1] U, equie with the usual rouct toology. Note that [0, 1] U is a comact metric sace; as is well known, a metric can be efine by, for examle, ( (x F ), (y F ) ) := 2 i x Fi y Fi, (2.8) a5 i=0 where F 1, F 2,... is some enumeration of all unlabelle grahs. We efine U := τ(u) [0, 1] U to be the image of U uner this maing τ, an let U be the closure of U in [0, 1] U. Thus U is a comact metric sace. (For exlicit escritions of the subset U of [0, 1] U as a set of grah functionals, see Lovász an Szegey [19].) As ointe out in [19] an [7] (in equivalent terminology), τ is not injective; for examle, τ(k n,n ) is the same for all comlete biartite grahs K n,n. Nevertheless, as in [19] an [7], we can consier a grah G as an element of U by ientifying G an τ(g) (thus ientifying grahs with the same τ(g)), an then convergence of (G n ) as efine above is equivalent to convergence in U. The limit is thus an element of U, but tyically not a

7 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 7 grah in U. The main result of Lovász an Szegey [19] is a reresentation of the elements in U to which we will return in Section 6. Rmetric1 Remark 2.1. As sai above, U is a comact metric sace, an it can be given several equivalent metrics. One metric is the metric (2.8) inherite from [0, 1] U, which for grahs becomes (G, G ) = i 2 i t(f i, G) t(f i, G ). Another metric, shown by Borgs, Chayes, Lovász, Sós an Vesztergombi [7] to be equivalent, is the cut-istance δ, see [7] for efinitions. The ientification of grahs with the same image in U (i.e., with the same t(f, ) for all F ) is sometimes elegant but at other times inconvenient. It can be avoie if we instea let U + be the union of U an some one-oint set { } an consier the maing τ + : U [0, 1] U + = [0, 1] U [0, 1] efine by τ + (G) = ( τ(g), v(g) 1). (2.9) tau+ Then τ + is injective, because if τ(g 1 ) = τ(g 2 ) for two grahs G 1 an G 2 with the same number of vertices, then G 1 an G 2 are isomorhic an thus G 1 = G 2 as unlabelle grahs. (This can easily be shown irectly: it follows from (2.1) that G 1 [k] = G 2 [k] for every k, which imlies G 1 [k] = G2 [k] for every k v(g 1 ) = v(g 2 ); now take k = v(g 1 ). It is also a consequence of [7, Theorem 2.7 an Theorem 2.3 or Lemma 5.1].) Consequently, we can ientify U with its image τ + (U) [0, 1] U + an efine U [0, 1] U + as its closure. It is easily seen that a sequence (G n ) of grahs converges in U if an only if either v(g n ) an (G n ) converges in U, or the sequence (G n ) is constant from some n 0 on. Hence, convergence in U is essentially the same as the convergence consiere by by Lovász an Szegey [19], but without any ientification of non-isomorhic grahs of ifferent orers. Alternatively, we can consier τ inj or τ in efine by τ inj (G) := (t inj (F, G)) F U [0, 1] U, τ in (G) := (t in (F, G)) F U [0, 1] U. It is easy to see that both τ inj an τ in are injective maings U [0, 1] U. (If τ inj (F, G 1 ) = τ inj (F, G 2 ) for all F, we take F = G 1 an F = G 2 an conclue G 1 = G 2, using our secial efinition above when v(f ) > v(g).) Hence, we can again ientify U with its image an consier its closure U in [0, 1] U. Moreover, using (2.4), (2.5), an (2.6), it is easily shown that if (G n ) is a sequence of unlabelle grahs, then τ + (G n ) converges τ in (G n ) converges τ inj (G n ) converges. Hence, the three comactifications τ + (U), τ inj (U), τ in (U) are homeomorhic an we can use any of them for U. We let U := U \ U; this is the set of all limit objects of sequences (G n ) in U with v(g n ). (I.e., it is the set of all roer grah limits.)

8 8 PERSI DIACONIS AND SVANTE JANSON We will in the sequel refer to use U rather than U, thus not ientifying some grahs of ifferent orers, nor ientifying finite grahs with some limit objects in U. For every fixe grah F, the functions t(f, ), t inj (F, ) an t in (F, ) have unique continuous extensions to U, for which we use the same notation. We similarly exten v( ) 1 continuously to U by efining v(g) = an thus v(g) 1 = 0 for G U := U \ U. Then (2.4), (2.5) an (2.6) hol for all G U, where (2.4) means that t inj (F, G) = t(f, G), G U. (2.10) a4x or shaow? T1 Tcuoo Sconv Note that U is a comact metric sace. Different, equivalent, metrics are given by the embeings τ +, τ inj, τ in into [0, 1] U + an [0, 1] U. Another equivalent metric is, by Remark 2.1 an the efinition of τ +, δ (G 1, G 2 ) + v(g 1 ) 1 v(g 2 ) 1. We summarize the results above on convergence. Theorem 2.1. A sequence (G n ) of grahs converges in the sense of Lovász an Szegey [19] if an only if it converges in the comact metric sace U. Moreover, if v(g n ), the sequence (G n ) converges in this sense if an only if it converges in U. The rojection π : [0, 1] U + = [0, 1] U [0, 1] [0, 1] U mas τ + (G) to τ(g) for every grah G, so by continuity it mas U into U. For grah G U, π(g) = τ(g) is the object in U corresoning to G consiere above, an we will in the sequel enote this object by π(g); recall that this rojection U U is not injective. (We thus istinguish between a grah G an its ghost π(g) in U. Recall that when grahs are consiere as elements of U as in [19] an [7], certain grahs are ientifie with each other; we avoi this.) On the other han, an element G of U is by efinition etermine by τ(g) an v(g) 1, cf. (2.9), so the restriction π : U n U is injective for each n. In articular, π : U U is injective. Moreover, this ma is surjective because every element G U is the limit of some sequence (G n ) of grahs in U with v(g n ) ; by Theorem 2.1, this sequence converges in U to some element G, an then π(g ) = G. Since U is comact, the restriction of π to U is thus a homeomorhism, an we have the following theorem, saying that we can ientify the set U of roer grah limits with U. Theorem 2.2. The rojection π mas the set U := U \ U of roer grah limits homeomorhically onto U. 3. Convergence of ranom grahs A ranom unlabelle grah is a ranom element of U (with any istribution; we o not imly any articular moel). We consier convergence of a sequence (G n ) of ranom unlabelle grahs in the larger sace U; recall

9 T2 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 9 that this is a comact metric sace so we may use the general theory set forth in, for examle, Billingsley [2]., a.s. We use the stanar notations, for convergence in istribution, robability, an alsmost surely, resectively. We will only consier the case when v(g n ), at least in robability. (The reaer may think of the case when G n has n vertices, although that is not necessary in general.) We begin with convergence in istribution. Theorem 3.1. Let G n, n 1, be ranom unlabelle grahs an assume that v(g n ). The following are equivalent, as n. T2a (i) G n Γ for some ranom Γ U. T2b (ii) For every finite family F 1,..., F m of (non-ranom) grahs, the ranom variables t(f 1, G n ),..., t(f m, G n ) converge jointly in istribution. T2c (iii) For every (non-ranom) F U, the ranom variables t(f, G n ) converge in istribution. T2 (iv) For every (non-ranom) F U, the exectations E t(f, G n ) converge. If these roerties hol, then the limits in (ii), (iii) an (iv) are ( t(f i, Γ) ) m i=1, t(f, Γ) an E t(f, Γ), resectively. Furthermore, Γ U a.s. The same results hol if t is relace by t inj or t in. Proof. (i) (ii). Since U is a close subset of [0, 1] U +, convergence in istribution in U is equivalent to convergence of τ + (G n ) = ( (t(f, G n )) F U, v(g n ) 1) in [0, 1] U +, Since we assume v(g n ) 1 0, this is equivalent to convergence of (t(f, G n )) F U in [0, 1] U [2, Theorem 4.4], which is equivalent to convergence in istribution of all finite families (t(f i, G n )) m i=1. (ii) = (iii). Trivial. (iii) = (iv). Immeiate, since t is boune (by 1). (iv) = (ii). Let F 1,..., F m be fixe grahs an let l 1,..., l m be ositive integers. Let F be the isjoint union of l i coies of F i, i = 1,..., m. Then, for every G U, from the efinition of t, m t(f, G) = t(f i, G) l i, an hence E i=1 m t(f i, G) l i = E t(f, G). (3.1) b3 i=1 Consequently, if (iv) hols, then every joint moment E m i=1 t(f i, G) l i of t(f 1, G n ),..., t(f m, G n ) converges. Since t(f i, G n ) are boune (by 1), this imlies joint convergence in istribution by the metho of moments. The ientification of the limits is immeiate. Since v(g n ), (i) imlies that v(γ) = a.s., an thus Γ U.

10 10 PERSI DIACONIS AND SVANTE JANSON C2 Finally, it follows from (2.4), (2.5) an (2.6) that we can relace t by t inj or t in in (ii) an (iv), an the imlications (ii) = (iii) an (iii) = (iv) are immeiate for t inj an t in too. Secializing to the case of a non-ranom limit G U, we obtain the corresoning result for convergence in robability. Corollary 3.2. Let G n, n 1, be ranom unlabelle grahs such that v(g n ), an let G U. The following are equivalent, as n. C2a (i) G n G. C2c (ii) t(f, G n ) t(f, G) for every (non-ranom) F U. C2 (iii) E t(f, G n ) t(f, G) for every (non-ranom) F U. The same result hols if t is relace by t inj or t in. Note further that uner the same assumtions, it follows irectly from a.s. Theorem 2.1 that G n G if an only if t(f, G n ) a.s. t(f, G) for every F U. We observe another corollary to Theorem 3.1 (an its roof). C2aa Corollary 3.3. If Γ is a ranom element of U = U \ U = U, then, for every sequence F 1,..., F m of grahs, ossibly with reetitions, E m t(f i, Γ) = E t ( m i=1f i, Γ), (3.2) b3x i=1 where m i=1 F i enotes the isjoint union of F 1,..., F m. As a consequence, the istribution of Γ is uniquely etermine by the numbers E t(f, Γ), F U. Alternatively, the istribution of Γ is uniquely etermine by the numbers E t in (F, Γ), F U. Proof. Since U is ense in U U, there exists ranom unlabelle grahs a.s. G n such that G n Γ. In articular, G n Γ an v(g n ) (in fact, we may assume v(g n ) = n), so Theorem 3.1 an its roof aly, an (3.2) follows from (3.1) alie to G n by letting n. For the final statement, note that (3.2) shows that the exectations E t(f, Γ), F U, etermine all moments E m i=1 t(f i, Γ), an thus the joint istribution of t(f, Γ), F U, which is the same as the istribution of τ(γ) = ( t(f, Γ) ) F U [0, 1]U, an we have efine U such that we ientify Γ an τ(γ). Finally, the numbers E t in (F, Γ), F U, etermine all E t(f, Γ) by (2.5), recalling that t inj (F, Γ) = t(f, Γ) by (2.10). Remark 3.1. The numbers E t(f, Γ) for a ranom Γ U thus lay a role similar to the one laye by moments for a ranom variable. (An the relation between E t(f, Γ) an E t in (F, Γ) has some resemblance to the relation between moments an cumulants.)

11 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 11 Sinfinite TC1 4. Convergence to infinite grahs We will in this section consier also labelle infinite grahs with the vertex set N = {1, 2,... }. Let L enote the set of all such grahs. These grahs are etermine by their ege sets, so L can be ientifie with the ower set P(E(K )) of all subsets of the ege set E(K ) of the comlete infinite grah K, an thus with the infinite rouct set {0, 1} E(K ). We give this sace, an thus L, the rouct toology. Hence, L is a comact metric sace. It is sometimes convenient to regar L n for a finite n as a subset of L : we can ientify grahs in L n an L with the same ege set. In other wors, if G L n is a grah with vertex set [n], we a an infinite number of isolate vertices n + 1, n + 2,... to obtain a grah in L. Conversely, if H L is an infinite grah, we let H [n] L n be the inuce subgrah of H with vertex set [n]. If G is a (finite) grah, let Ĝ be the ranom labelle grah obtaine by a ranom labelling of the vertices of G by the numbers 1,..., v(g). (If G is labelle, we thus ignore the labels an ranomly relabel.) Thus Ĝ is a ranom finite grah with the same number of vertices as G, but as just sai, we can (an will) also regar Ĝ as a ranom grah in L. We use the same notation Ĝ also for a ranom (finite) grah G given a ranom labelling. Theorem 4.1. Let (G n ) be a sequence of ranom grahs in U an assume that v(g n ). Then the following are equivalent. (i) G n Γ in U for some ranom Γ U. (ii) Ĝn H in L for some ranom H L. If these hol, then P(H [k] = F ) = E t in (F, Γ) for every F L k. Furthermore, Γ U a.s. Proof. Let G be a labelle grah an consier the grah Ĝ [k], assuming k v(g). This ranom grah equals G[k] = G(v 1,..., v k ), where v 1,..., v k are k vertices samle at ranom without relacement as in Section 2. Hence, by (2.3), for every F L k, P(Ĝ [k] = F ) = t in (F, G), Alie to the ranom grah G n, this yiels if k v(g). E t in (F, G n ) P(Ĝn [k] = F ) E t in (F, G n ) + P ( v(g n ) < k ). (4.1) c3 By assumtion, P (v(g n ) < k) 0 as n, an it follows from (4.1) an Theorem 3.1 that G n Γ in U if an only if for every k 1 an every F L k. P(Ĝn [k] = F ) E t in (F, Γ) (4.2) c3a

12 finitary? Sexch LE TC1E TE 12 PERSI DIACONIS AND SVANTE JANSON Since L k is a finite set, (4.2) says that, for every k, Ĝ n [k] H k for some ranom grah H k L k with P(H k = F ) = E t in (F, Γ) for F L k. Since L has the rouct toology, this imlies Ĝn H in L for some ranom H L with H [k] = Hk. Conversely, if Ĝn above shows that H in L, then Ĝn [k] H [k] so the argument E t in (F, G n ) = P(Ĝn [k] = F ) + o(1) P(H [k] = F ) as n, for every F L k, an Theorem 3.1 yiels the existence of some ranom Γ U U with G n Γ an E t in (F, Γ) = P(H [k] = F ). 5. Exchangeable ranom grahs A ermutation σ : N N is finite if σ(n) = n for all sufficiently large n. Definition. A ranom infinite grah H L is exchangeable if its istribution is invariant uner every finite ermutation of the vertices. Lemma 5.1. Let H be a ranom infinite grah in L. Then the following are equivalent. (i) H is exchangeable. (ii) H [k] has a istribution invariant uner all ermutations of [k], for every k 1. (iii) P ( H [k] = F ) eens only on the isomorhism tye of F, an can thus be seen as a function of F as an unlabelle grah in U k, for every k 1. Proof. (i) = (ii). Immeiate. (ii) = (i). If σ is a finite ermutation of N, then σ restricts to a ermutation of [k] for every large k, an it follows that if H σ is H with the vertices ermute by σ, then, for all large k H σ [k] = H [k] σ = H [k], which imlies H [k] = H. (ii) (iii). Trivial. Theorem 5.2. The limit H is Theorem 4.1 is exchangeable. Proof. H satisfies Lemma 5.1(iii). Moreover, Theorem 4.1 imlies the following connection with ranom elements of U. Theorem 5.3. There is a one-to-one corresonence between istributions of ranom elements Γ U (or U ) an istributions of exchangeable ranom infinite grahs H L given by E t in (F, Γ) = P(H [k] = F ) (5.1) e2a

13 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 13 for every k 1 an every F L k, or, equivalently, for every F L. Furthermore, H [n] E t(f, Γ) = P(H F ) (5.2) e2b Γ in U as n. Proof. Note first that (5.1) an (5.2) are equivalent by (2.5) an (2.6), since t(f, Γ) = t inj (F, Γ) by (2.10), an H F if an only if H [k] F when F L k. Suose that Γ is a ranom element of U U. Since U is ense in U, there exist (as in the roof of Corollary 3.3) ranom unlabelle grahs G n such that G n a.s. Γ in U an thus v(g n ) a.s. Γ. Hence, an G n H for some ranom exchange- Theorems 4.1 an 5.2 show that Ĝn able infinite grah H satisfying (5.1). Furthermore, (5.1) etermines the istribution of H [k] for every k, an thus the istribution of k. Conversely, if H is an exchangeable ranom infinite grah, let G n = H [n]. By Lemma 5.1(ii), the istribution of each G n is invariant uner ermutations of the vertices, so if Ĝn is G n with a ranom (re)labelling, we have Ĝn = G n. Since G n H in L (because L has a rouct toology), we thus have Ĝn H in L, so Theorem 4.1 alies an shows the existence of a ranom Γ U such that G n Γ an (5.1) hols. Finally (5.1) etermines the istribution of Γ by Corollary 3.3. Ras CE Remark 5.1. Moreover, H [n] converges a.s. to some ranom variable Γ U, because t in (F, H [n] ), n v(f ), is a reverse martingale for every F Γ. Alternatively, this follows by concentration estimates from the reresentation in Section 6, see Lovász an Szegey [19, Theorem 2.5]. Corollary 5.4. There is a one-to-one corresonence between elements Γ of U = U an extreme oints of the set of istributions of exchangeable ranom infinite grahs H L. This corresonence is given by I hoe you agree it is a reverse martingale. Do we nee more etails? for every F L. Furthermore, H [n] t(f, Γ) = P(H F ) (5.3) ce a.s. Γ in U as n. Proof. The extreme oints of the set of istributions on U are the oint masses, which are in one-to-one corresonence with the elements of U. We can characterize these extreme oint istributions of exchangeable ranom infinite grahs as follows. TE2 te2a Theorem 5.5. Let H be an exchangeable ranom infinite grah. Then the following are equivalent. (i) The istribution of H is an extreme oint in the set of exchangeable istributions in L.

14 14 PERSI DIACONIS AND SVANTE JANSON te2b (ii) If F 1 an F 2 are two (finite) grahs with isjoint vertex sets V (F 1 ), V (F 2 ) N, then P(H F 1 F 2 ) = P(H F 1 ) P(H F 2 ). te2b (iii) The restrictions H [k] an H [k+1, ) are ineenent for every k. te2c (iv) Let F n be the σ-fiel generate by H [n, ). Then the tail σ-fiel n=1 F n is trivial, i.e., contains only events with robability 0 or 1. Proof. (i) = (ii). By Corollary 5.4, H corresons to some (non-ranom) Γ U such that P(H F ) = t(f, Γ) (5.4) e6 for every F L. We have efine L such that a grah F L is labelle by 1,..., v(f ), but both sies of (5.4) are invariant uner relabelling of F by arbitrary ositive integers; the left han sie because H is exchangeable an the right han sie because t(f, Γ) only eens on F as an unlabelle grah. Hence (5.4) hols for every finite grah F with V (F ) N. Furthermore, since Γ is non-ranom, Corollary 3.3 yiels t(f 1 F 2, Γ) = t(f 1, Γ)t(F 2, Γ). Hence, P(H F 1 F 2 ) = t(f 1 F 2, Γ) = t(f 1, Γ)t(F 2, Γ) = P(H F 1 ) P(H F 2 ). (ii) = (iii). By inclusion-exclusion, as for (2.3), (ii) imlies that if 1 k < l <, then for any grahs F 1 an F 2 with V (F 1 ) = {1,..., k} an V (F 2 ) = {k + 1,..., k + l}, the events H [k] = F 1 an H {k+1,...,l} = F 2 are ineenent. Hence H [k] an H {k,...,l} are ineenent for every l > k, an the result follows. (iii) = (iv). Suose A is an event in the tail σ-fiel n=1 F n. Let Fn be the σ-fiel generate by H [n]. By (iii), A is ineenent of Fn for every n, an thus of the σ-fiel F generate by Fn, which equals the σ-fiel F 1 generate by H. However, A F 1, so A is ineenent of itself an thus P(A) = 0 or 1. (iv) = (i). Let F L k for some k an let F n be F with all vertices shifte by n. Consier the two inicators I = 1[H F ] an I n = 1[H F n ]. Since I n is F n -measurable, P(H F F n ) = E(II n ) = E ( E(I F n )I n ). (5.5) e70 Moreover, E(I F n ), n = 1, 2,..., is a reverse martingale, an thus ( ) E(I F n ) E I F n = E I a.s. Hence, ( E(I F n ) E I ) I n 0 a.s., an by ominate convergence ( (E(I E Fn ) E I ) ) I n 0. Consequently, (5.5) yiels n=1 P(H F F n ) = E I E I n + o(1) = P(H F ) P(H F n ) + o(1).

15 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 15 Moreover, since H is exchangeable, P(H F F n ) (for n v(f )) an P(H F n ) o not een on n, an we obtain as n P(H F F k ) = P(H F ) 2. (5.6) e7 Let Γ be a ranom element of U corresoning to H as in Theorem 5.3. By (5.2) an (3.2), (5.6) can be written E t(f, Γ) 2 = ( E t(f, Γ) ) 2. Hence the ranom variable t(f, Γ) has variance 0 so it is a.s. constant. Since this hols for every F L, it follows that Γ is a.s. constant, i.e., we can take Γ non-ranom, an (i) follows by Corollary 5.4. Sre 6. Reresentations of grah limits an exchangeable grahs As sai in the introuction, the exchangeable infinite ranom grahs were??? characterize by Alous [1], see also Kallenberg [12], an the grah limits in U = U were characterize in a very similar way by Lovász an Szegey [19]. We can now make the connection between these two characterizations exlicit. Let W be the set of all measurable functions W : [0, 1] 2 [0, 1] an let W s be the subset of symmetric functions. For every W W s, we efine an infinite ranom grah G(, W ) L as follows: we first choose a sequence X 1, X 2,... of i.i.. ranom variables uniformly istribute on [0, 1], an then, given this sequence, for each air (i, j) with i < j we raw an ege ij with robability W (X i, X j ), ineenently for all airs (i, j) with i < j (conitionally given {X i }). Further, let G(n, W ) be the restriction G(, W ) [n], which is obtaine by the same construction with a finite sequence X 1,..., X n. It is evient that G(, W ) is an exchangeable infinite ranom grah, an the result by Alous [1] is that every exchangeable infinite ranom grah is obtaine as a mixture of such G(, W ); in other wors as G(, W ) with a ranom W. Consiering again a eterministic W W s, it is evient that Theorem 5.5(ii) hols, an thus Theorem 5.5 an Corollary 5.4 show that G(, W ) corresons to an element Γ W U. Moreover, by Theorem 5.3 an Remark 5.1, G(n, W ) Γ W a.s. as n, an (5.3) shows that if F L k, then t(f, Γ W ) = P ( F G(k, W ) ) = W (x i, x j ) x 1... x k. (6.1) tuw [0,1] k ij E(F ) an Hoover? The main result of Lovász an Szegey [19] is that every element of U = U can be obtaine as Γ W satisfying (6.1) for some W W s. It is now clear that the reresentation theorems by Alous [1] an Lovász an Szegey [19] are connecte by Theorem 5.3 an Corollary 5.4 above, an that one characterization easily follows from the other.

16 16 PERSI DIACONIS AND SVANTE JANSON Remark 6.1. The reresentation by W is far from unique. Clearly, any measure reserving bijection σ of [0, 1] onto itself transforms W into a function W σ with the same G(, W ) an Γ W. For a recise result, see Alous [1]?? Borgs, Chayes, Lovász, Sós an Vesztergombi [7] call an element W W s a grahon. They further efine a seuometric (calle the cut-istance) on W s an show that if we consier the quotient sace Ŵs obtaine by ientifying elements with cut-istance 0, we obtain a comact metric sace, an the maing W Γ W yiels a bijection Ŵs U = U, which further more is a homeomorhism. Remark 6.2. As remarke in Lovász an Szegey [19], we can more generally consier a symmetric measurable function W : S 2 [0, 1] for any robability sace (S, µ), an efine G(, W ) as above with X i i.i.. ranom variables in S with istribution µ. This oes not give any new limit objects G(, W ) or Γ W, since we just sai that every limit object is obtaine from some W W s, but they can sometimes give useful reresentations. An interesting case is when W is the ajacency matrix of a (finite) grah G, with S = V (G) an µ the uniform measure on S; we thus let X i be i.i.. ranom vertices of G an G(n, W ) equals the grah G[n] efine in Section 2. It follows from (6.1) an (2.1) that t(f, Γ W ) = t(f, G) for every F U, an thus Γ W = G as elements of U. In other wors, Γ W U = π(g), the ghost of G in U = U. Remark 6.3. For the asymtotic behavior of G(n, W ) in another, sarse, case, with W eening on n, see [3]. Sbi 7. Biartite grahs The efinitions an results above have analogues for biartite grahs, which we give in this section, leaving some etails to the reaer. The roofs are straightforwar analogues of the ones given above an are omitte. A biartite grah will be a grah with an exlicit biartition; in other wors, a biartite grah G consists of two vertex sets V 1 (G) an V 2 (G) an an ege set E(G) V 1 (G) V 2 (G); we let v 1 (G) := V 1 (G) an v 2 (G) := V 2 (G) be the numbers of vertices in the two sets. Again we consier both the labelle an unlabelle cases; in the labelle case we assume the labels of the vertices in V j (G) are 1,..., v j (G) for j = 1, 2. Let Bn L 1 n 2 be the set of the 2 n 1n 2 labelle biartite grahs with vertex sets [n 1 ] an [n 2 ], an let B n1 n 2 be the quotient set Bn L 1 n 2 / = of unlabelle biartite grahs with n 1 an n 2 vertices in the two arts; further, let B L := n 1,n 2 1 BL n 1 n 2 an B := n 1,n 2 1 B n 1 n 2. We let G[k 1, k 2 ] be the ranom grah in Bk L 1 k 2 obtaine by samling k j vertices from V j (G) (j = 1, 2), uniformly with relacement, an let, rovie k j v j (G), G[k 1, k 2 ] be the corresoning ranom grah obtaine by samling without relacement. We then efine t(f, G), t inj (F, G) an

17 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 17 t in (F, G) for (unlabelle) biartite grahs F an G in analogy with (2.1) (2.3). Then (2.4) (2.6) still hol, mutatis mutanis; for examle, t(f, G) t inj (F, G) v 1(F ) 2 2v 1 (G) + v 2(F ) 2. (7.1) a4xx 2v 2 (G) In analogy with (2.7), we now efine τ : B [0, 1] B by τ(g) := (t(f, G)) F B [0, 1] B. (7.2) taub We efine B := τ(b) [0, 1] B to be the image of B uner this maing τ, an let B be the closure of B in [0, 1] B ; this is a comact metric sace. Again, τ is not injective; we may consier a grah G as an element of B by ientifying G an τ(g), but this imlies ientification of some grahs of ifferent orers an we refer to avoi it. We let B + be the union of B an some two-oint set { 1, 2 } an consier the maing τ + : B [0, 1] B+ = [0, 1] B [0, 1] [0, 1] efine by τ + (G) = ( τ(g), v 1 (G) 1, v 2 (G) 1). (7.3) tab+ Then τ + is injective an we can ientify B with its image τ + (B) [0, 1] B+ an efine B [0, 1] B+ as its closure; this is a comact metric sace. The functions t(f, ), t inj (F, ), t in (F, ) an v j ( ) 1, for F B an j = 1, 2, have unique continuous extensions to B. We let B := {G B : v 1 (G) = v 2 (G) = }; this is the set of all limit objects of sequences (G n ) in B with v 1 (G n ), v 2 (G n ). By (7.1), t inj (F, G) = t(f, G) for every G B an every F B. The rojection π : B B restricts to a homeomorhism B = B. Remark 7.1. Note that in the biartite case there are other limit objects too in B; in fact, B can be artitione into B, B, an the sets B n, B n, for n = 1, 2,..., where, for examle, B n1 is the set of limits of sequences (G n ) of biartite grahs such that v 2 (G n ) but v 1 (G n ) = n 1 is constant. We will not consier such egenerate limits further here, but we remark that in the simlest case n 1 = 1, a biartite grah in B1n L 2 can be ientifie with a subset of [n 2 ], an an unlabelle grah in B 1n2 thus with a number in m {0,..., n 2 }, the number of eges in the grah, an it is easily seen that a sequence of such unlabelle grahs with n 2 converges in B if an only if the roortion m/n 2 converges; hence we can ientify B 1 with the interval [0,1]. We have the following basic result, cf. Theorem 2.1. T1B Theorem 7.1. Let (G n ) be a sequence of biartite grahs with v 1 (G n ), v 2 (G n ). Then the following are equivalent. T1Bt (i) t(f, G n ) converges for every F B. T1Btinj (ii) t inj (F, G n ) converges for every F B. T1Btin (iii) t in (F, G n ) converges for every F B. (iv) G n converges in B.

18 T2B 18 PERSI DIACONIS AND SVANTE JANSON In this case, the limit G of G n belongs to B an the limits in (i), (iii) an (iii) are t(f, G), t inj (F, G) an t in (F, G). For convergence of ranom unlabelle biartite grahs, the results in Section 3 hol with trivial changes. Theorem 7.2. Let G n, n 1, be ranom unlabelle biartite grahs an assume that v 1 (G n ), v 2 (G n ). The following are equivalent, as n. T2Ba (i) G n Γ for some ranom Γ B. T2Bb (ii) For every finite family F 1,..., F m of (non-ranom) biartite grahs, the ranom variables t(f 1, G n ),..., t(f m, G n ) converge jointly in istribution. T2Bc (iii) For every (non-ranom) F B, the ranom variables t(f, G n ) converge in istribution. T2B (iv) For every (non-ranom) F B, the exectations E t(f, G n ) converge. If these roerties hol, then the limits in (ii), (iii) an (iv) are ( t(f i, Γ) ) m i=1, t(f, Γ) an E t(f, Γ), resectively. Furthermore, Γ B a.s. The same results hol if t is relace by t inj or t in. C2B Corollary 7.3. Let G n, n 1, be ranom unlabelle biartite grahs such that v 1 (G n ), v 2 (G n ), an let G B. The following are equivalent, as n. C2Ba (i) G n G. C2Bc (ii) t(f, G n ) t(f, G) for every (non-ranom) F B. C2B (iii) E t(f, G n ) t(f, G) for every (non-ranom) F B. The same result hols if t is relace by t inj or t in. TC1B As above, the istribution of Γ is uniquely etermine by the numbers E t(f, Γ), F B. Let B L enote the set of all labelle infinite biartite grahs with the vertex sets V 1 (G) = V 2 (G) = N. B L is a comact metric sace with the natural rouct toology. If G is a biartite grah, let Ĝ be the ranom labelle biartite grah obtaine by ranom labellings of the vertices in V j (G) by the numbers 1,..., v j (G), for j = 1, 2. This is a ranom finite biartite grah, but we can also regar it as a ranom element of B L by aing isolate vertices. Definition. A ranom infinite biartite grah H B L is exchangeable if its istribution is invariant uner every air of finite ermutations of V 1 (H) an V 2 (H). Theorem 7.4. Let (G n ) be a sequence of ranom grahs in B an assume that v 1 (G n ), v 2 (G n ). Then the following are equivalent. (i) G n Γ in B for some ranom Γ B. (ii) Ĝn H in B L for some ranom H B. L

19 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 19 If these hol, then P(H [k1 ] [k 2 ] = F ) = E t in (F, Γ) for every F B L k 1 k 2. Furthermore, Γ B a.s., an H is exchangeable. TEB Theorem 7.5. There is a one-to-one corresonence between istributions of ranom elements Γ B (or B ) an istributions of exchangeable ranom infinite grahs H B L given by E t in (F, Γ) = P(H [k1 ] [k 2 ] = F ) (7.4) for every k 1, k 2 1 an every F B L k 1 k 2, or, equivalently, for every F B L. Furthermore, H [n1 ] [n 2 ] E t(f, Γ) = P(H F ) (7.5) Γ in B as n 1, n 2. CEB Corollary 7.6. There is a one-to-one corresonence between elements Γ of B = B an extreme oints of the set of istributions of exchangeable ranom infinite grahs H B L. This corresonence is given by for every F B L. Furthermore, H [n1 ] [n 2 ] t(f, Γ) = P(H F ) (7.6) Γ in B as n 1, n 2. TE2B te2ba te2bb Remark 7.2. We have not checke whether H [n1 ] [n 2 ] Γ in B as n 1, n 2. This hols at least for a subsequence (n 1 (m), n 2 (m)) with both n 1 (m) an n 2 (m) non-ecreasing because then t inj (F, H [n1 ] [n 2 ]) is a reverse martingale. Theorem 7.7. Let H be an exchangeable ranom infinite biartite grah. Then the following are equivalent. (i) The istribution of H is an extreme oint in the set of exchangeable istributions in B. L (ii) If F 1 an F 2 are two (finite) biartite grahs with the vertex sets V j (F 1 ) an V j (F 2 ) isjoint subsets of N for j = 1, 2, then P(H F 1 F 2 ) = P(H F 1 ) P(H F 2 ). The construction in Section 6 takes the following form; note that there is no nee to assume symmetry of W. For every W W, we efine an infinite ranom biartite grah G(,, W ) B L as follows: we first choose two sequence X 1, X 2,... an Y 1, Y 2,... of i.i.. ranom variables uniformly istribute on [0, 1], an then, given these sequences, for each air (i, j) N N we raw an ege ij with robability W (X i, Y j ), ineenently for all airs (i, j). Further, let G(n 1, n 2, W ) be the restriction G(,, W ) [n1 ] [n 2 ], which is obtaine by the same construction with finite sequences X 1,..., X n1 an Y 1,..., Y n2. It is evient that G(,, W ) is an exchangeable infinite ranom biartite grah. Furthermore, it satisfies Theorem 7.7(ii). Theorem 7.5 an a.s. Do we care about this? I m too lazy to state the analogues of Theorem 5.5(iii) an (iv) correctly

20 20 PERSI DIACONIS AND SVANTE JANSON Corollary 7.6 yiel a corresoning element Γ W B = B such that G(n 1, n 2, W ) Γ W as n 1, n 2 an, for every F Bk L 1 k 2, t(f, Γ W ) = W (x i, y j ) x 1... x k1 y 1... y k2. (7.7) tuww [0,1] k 1 +k 2 ij E(F ) The result by Alous [1] in the non-symmetric case is that every exchangeable infinite ranom biartite grah is obtaine as a mixture of such G(,, W ); in other wors as G(,, W ) with a ranom W. By Theorem 7.5 an Corollary 7.6 above, this imlies (an is imlie by) the fact that every element of B equals Γ W for some (non-unique) W W; the biartite version of the characterization by Lovász an Szegey [19]. References Alous [1] D. Alous, Reresentations for artially exchangeable arrays of ranom variables. J. Multivar. Anal. 11, , Bill [2] P. Billingsley, Convergence of Probability Measures. Wiley, New York, SJ178 [3] B. Bollobás, S. Janson an O. Rioran, The hase transition in inhomogeneous ranom grahs. Ranom Struct. Alg., to aear, BR [4] B. Bollobás an O. Rioran, BCL3 [5] C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, B. Szegey an K. Vesztergombi, Grah limits an arameter testing. STOC, BCL2 [6] C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós an K. Vesztergombi, Counting grah homomorhisms. In Toics in Discrete Mathematics (e. M. Klazar, J. Kratochvil, M. Loebl, J. Matousek, R. Thomas, P. Valtr), Sringer, New York, BCL1 [7] C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós an K. Vesztergombi, Convergent sequences of ense grahs I: Subgrah frequencies, metric roerties an testing. II: Multiway cuts an statistical hysics. Prerint, January htt://arxiv.org/math.co/ DF1981 [8] P. Diaconis an D. Freeman, On the statistics of vision: The Julesz conjecture. J. Math. Psychol. 24, , DF1984 [9] P. Diaconis an D. Freeman, Partial exchangeability an sufficiency. In Statistics: Alications an New Directions (es. J. K. Jhosh an J. Roy), Inian Statistical Institute, Calcutta, Free [10] M. Freeman, L. Lovász an A. Schrijver, Reflection ositivity, rank connectivity, an homomorhism of grahs. J. Amer. Math. Soc. 20, 37 51, Hoover [11] D. Hoover, Relations on Probability Saces an Arrays of Ranom Variables. Prerint, Institute for Avance Stuy, Princeton, NJ, Kallenberg:exch [12] O. Kallenberg, Probabilistic Symmetries an Invariance Princiles. Sringer, New York, LL2006 [13] L. Lovász, The rank of connection matrices an the imension of grah algebras. Eur. J. Comb. 27, , 2006.

21 GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS 21 LL2007 LSos LSzcont LSzszem LSztest LSz [14] L. Lovász, Connection matrices. In Combinatorics, Comlexity, an Chance: A Tribute to Dominic Welsh (e. G. Grimmet an C. McDiarmi), Oxfor University Press, Oxfor, [15] L. Lovász an V. T. Sós, Generalize quasiranom grahs. J. Comb. Theory B. [16] L. Lovász an B. Szegey, Contractors an connectors in grah algebras. J. Comb. Theory B. [17] L. Lovász an B. Szegey, Szemeréi s Lemma for the analyst. J. Geom. Func. Anal. 17, , [18] L. Lovász an B. Szegey, Testing roerties of grahs an functions. [19] L. Lovász an B. Szegey, Limits of ense grah sequences. J. Comb. Theory B 96, , htt://arxiv.org/math.co/ Stanfor an Nice??? Deartment of Mathematics, Usala University, PO Box 480, SE Usala, Sween aress: svante.janson@math.uu.se URL: htt:// svante/