ON CONVERGENCE FOR GRAPHEXES

Transcription

1 ON CONVERGENCE FOR GRAPHEXES SVANTE JANSON Abstract. We stuy four ifferent notions of convergence for graphexes, recently introuce by Borgs, Chayes, Cohn an Holen, an by Veitch an Roy. We give some properties of them an some relations between them. We also exten results by Veitch an Roy on convergence of empirical graphons. 1. Introuction The theory of graph limits for ense graphs an the representation of such graph limits by classical graphons has evelope over the last ecae an has been very succesful, see e.g. the book by Lovász [14]. Here, a classical graphon is a symmetric measurable function W : S 2 [0, 1], where S is a probability space; without loss of generality one can take S = [0, 1]. There have been many ifferent attempts to fin corresponing results for sparse graphs. One recent approach has been through ranom graphs efine by certain iscrete exchangeable ranom measures on R 2 +. Exchangeable ranom measures on R 2 + were characterize by Kallenberg [11, 13], an his general construction of such measures can be interprete as a construction of ranom graphs. Note that the classical theory of graph limits an graphons on [0, 1] can be erive from the relate characterizations by Alous an Hoover of exchangeable arrays, see [1; 7], although this was not the original metho or motivation. This use of Kallenberg s construction was first one by Caron an Fox [6] in a special case, an extene by Herlau, Schmit an Mørup [8], Veitch an Roy [15] an Borgs, Chayes, Cohn an Holen [3]. We will here use the version by Veitch an Roy [15], escribe in etail in Section 2.2 below. It uses a graphex, which is a triple I, S, W, where the most interesting part is W which is a graphon, but a graphon in a new more general sense; W is efine on the infinite measure space R + instea of [0, 1]. More generally, graphons can be efine on any σ-finite measure space S. This was evelope by Borgs, Chayes, Cohn an Holen [3]. However, one of their results is that it is possible to take S = R + without loss of generality, an we will in the present paper only consier this case following [15]. Having efine graphons an graphexes, it is natural to efine a topology on them, an thus a notion of convergence. For classical graphons, there are several quite ifferent ways to efine convergence, but they are all equivalent, see e.g. Borgs, Chayes, Lovász, Sós an Vesztergombi [4, 5]. This important fact is closely relate to the fact that the space of classical graphons, moulo equivalence, is compact. Date: 21 February, Partly supporte by the Knut an Alice Wallenberg Founation. 1

2 2 SVANTE JANSON In the present, more general, context, there are also several possibilities, but, unforunately, they are not equivalent. It seems not yet clear which notions of convergence that will turn out to be useful in applications, an it seems that several possibilities ought to be stuie more. The present paper is a small contribution to this. We consier in this paper four ifferent notions of convergence for graphexes an graphons. Two of them, enote GP an GS, were efine by Veitch an Roy [16], base on convergence in istribution of the corresponing ranom graphs; we stress which is implicit in [16] that both convergences are metric, i.e., can be efine by pseuometrics. The two other notions of convergence apply only to the special case of integrable graphons; they use the pseuometrics δ an δ s efine by Borgs, Chayes, Cohn an Holen [3] see also [10]. See Section 3 for etaile efinitions. The four metrics stuie in the present are not the only possible ones. In particular, we o not consier the left convergence stuie in [3]. We show that for integrable graphons, convergence in δ δ s implies convergence GP GS. We conjecture that the converses o not hol, but we leave that as an open problem. Each graphex efines a ranom graph process G s W s 0, see Section 2.2 below. Borgs, Chayes, Cohn an Holen [3, Theorem 2.23] show that for any integrable graphon W, the empirical graphon efine by the ranom graph G s W a.s. converges to W in the metric δ s as s. Similarly, Veitch an Roy [16] show that for any graphex W, the empirical graphon efine by G s W converges to the graphex W in GP after suitable stretching an in GS as s ; however, they prove this only for a sequence s k, an in general only with convergence of probability. We exten their theorems to convergence for the full family G s with a continuous parameter s ; moreover, we show a.s. convergence. See Theorems 5.1 an 5.3. Furthermore, in orer to prove this result, Veitch an Roy [16] first show a relate convergence result for a ranomly relabelle version of the ranom graph, again for sequences s k. Again we improve their result to convergence for the continuous parameter s Theorem 4.1. In both cases, we make only some minor technical improvements in the proof; the proofs are thus essentially ue to [16]. 2. Notation an preliminaries Much of the notation follows Veitch an Roy [15, 16], but there are various moifications an aitions for our purposes. λ enotes Lebesgue measure in one or several imensions. R + := [0,, the set of non-negative real numbers. If S is a measurable space, then PS is the set of probability measures on S. If X is a ranom variable in some measurable space S, then LX enotes the istribution of X; thus LX PS. If S is a metric or metrizable space; we equip PS with the usual weak topology, see e.g. [2] or [12, Chapter 4]. Note that if S is a Polish space i.e., it can be given a complete an separable metric, then PS is Polish too, see [2, Appenix III]. We enote convergence in istribution of ranom variables in S by ; recall that X n X means LX n LX in PS.

3 ON CONVERGENCE FOR GRAPHEXES 3 If furthermore S is a locally compact Polish space = locally compact secon countable Hausorff space, then MS is the set of locally finite Borel measures on S. We will only use S = R 2 + an subsets thereof. We equip MS with the vague topology, which makes MS into a Polish space, see [12, Appenix A.2 an Theorem A.2.3]. Furthermore, N S is the subset of integer-value measures in MS, i.e., the set of all locally finite sums of unit point masses δ x, an we let N s S be the subset of simple integer-value measures, i.e., locally finite sums of istinct unit point masses. It is easily seen that N S an N s S are measurable subsets of MS Graphs an ajacency measures. We consier both unlabelle an labelle graphs; in the labelle case, each vertex is labelle with a real number in R +, an these labels are suppose to be istinct. The graphs may be finite or countably infinite, but we always assume that there are no isolate vertices; thus a graph G is etermine by its ege set EG. We furthermore consier only graphs that are simple in the sense that there are no multiple eges; in general we allow loops but in many applications we o not have any. We enote the vertex set an ege set of a graph G by V G an EG, an let vg := V G an eg := EG be the nembers of vertices an eges. If Γ is a labelle graph, then the corresponing unlabelle graph, obtaine by ignoring the labels, is enote GΓ. Conversely, if G is an unlabelle graph an s > 0, then Lbl s G is the ranom labelle graph obtaine by labelling the vertices by ranom i.i.. labels that are U0, s, i.e., uniformly istribute in 0, s. Note that this yiels istinct labels a.s., so we may assume that the labels are istinct as require above. If G is a labelle graph, we efine Lbl s G in the same way; thus relabelling the vertices ranomly regarless of their original labels. In other wors, then Lbl s G := Lbl s GG. When G is a ranom graph labelle or not, Lbl s G is efine by taking the labelling inepenent of G. If Γ is a labelle graph, we represent the ege set EΓ of Γ an thus the graph Γ itself by the measure ξ = ξγ := δ x,y 2.1 x,y R + :x,y EΓ on R 2 +; an ege between two istinct vertices labelle x an y is thus represente by the two point masses δ x,y + δ y,x, while a loop if such exist at a vertex labelle x is represente by δ x,x. We consier unirecte graphs, an thus the enpoints of an ege have to be treate symmetrically. Note that Γ is etermine as a labelle graph by ξ. If Γ is a labelle graph, an ξ the corresponing measure, then for r 0, Γ r enotes the inuce subgraph of Γ obtaine by first eliminating all vertices with labels > r, an any eges incient to such a vertex, an then also removing all remaining vertices that have become isolate. In other wors, we keep the eges whose enpoints both have labels r, an the enpoints of these eges. We let ξ r enote the corresponing measure on R 2 +, an note that this is just the restriction of ξ to [0, r] 2.

4 4 SVANTE JANSON A labelle graph Γ is locally finite if Γ r is finite for each r < ; equivalently, ξγ is a locally finite measure. We consier only locally finite graphs. We say that a measure ξ on R 2 + is an ajacency measure if it is given by 2.1 for some locally finite labelle graph Γ. Hence, a measure is an ajacency measure if an only if it is a symmetric measure in N s R 2 +. We enote the set of ajacency measures by N s,s R 2 +, an let Ĝ be the set of locally finite labelle graphs. Thus, 2.1 efines a 1 1 corresponence Γ ξγ between the sets Ĝ an N s,s R 2 + of locally finite labelle graphs an ajacency measures. We give the set of ajacency measures N s,s R 2 + the subspace topology as a subset of MR 2 +, an give Ĝ the corresponing topology inuce by the corresponence 2.1. Thus N s,s R 2 + an Ĝ are metric spaces, an a sequence Γ n Γ in Ĝ if an only if ξγ n ξγ in MR 2 +, i.e., in the vague topology. Returning to unlabelle graphs, we let G be the set of finite or countably infinite unlabelle graphs, an G f the subset of finite unlabelle graphs. Then G f is countable, an we give it the iscrete topology. We o not efine a topology on G Graphons, graphexes an ranom graphs. A graphon is in the present context a symmetric, measurable function W : R 2 + [0, 1] that satisfies the integrability conitions, where µ W x := 0 W x, y y: i µ W x < for a.e. x an λ{x : µ W x > 1} < ; ii R 2 + W x, y1{µ W x 1}1{µ W y 1} x y < ; iii R + W x, x <. Note that these conitions are satisfie if W is integrable, but they are also satisfie for some non-integrable W. A graphex is a triple W = I, S, W, where I 0 is a non-negative real number, S : R + R + is measurable with S 1 integrable an W is a graphon. Let W be the set of all graphexes. Each graphex W W efines a ranom ajacency measure ξ = ξw, an thus a corresponing ranom labelle graph Γ = ΓW, by the following construction; see further Veitch an Roy [15, 16] an Kallenberg [13]: Take realizations of inepenent unit-rate Poisson processes Ξ = {θ j, ϑ j } j on R 2 +, Ξ i = {σ ij, χ ij } j on R 2 + for i N, an Ξ = {ρ j, ρ j, η j} j on R 3 +. We regar θ i, σ ij, ρ i an ρ j as potential vertex labels, while ϑ j, χ ij an η j can be regare as types of the corresponing labels. Given W = I, S, W an these realizations, an a family of i.i.. ranom variables ζ i,j U0, 1 inepenent of them, efine the ajacency measure ξw = i,j 1{ζ i,j W ϑ i, ϑ j }δ θi,θ j + j,k + k 1{χ jk Sϑ j } δ θj,σ jk + δ σjk,θ j 1{η k I} δ ρk,ρ k + δ ρ k,ρ k. 2.2

5 ON CONVERGENCE FOR GRAPHEXES 5 In other wors, the corresponing ranom labelle graph ΓW is efine to have the following eges, with all ranom choices inepenent: G1 θ i, θ j with probability W ϑ i, ϑ j for each pair i, j with i j, G2 θ j, σ jk for each j an k with χ jk Sϑ j G3 ρ k, ρ k for each k such that η k I. Equivalently, we can efine Γ by starting with the Poisson process Ξ = θ j, ϑ j an first efine the eges in G1, an then a for each j a star with centre in θ j an peripheral vertices labelle by a Poisson process {σ jk } k on R + with intensity Sϑ j, an finally a eges ρ k, ρ k accoring to a Poisson process with intensity 2I in {x, x R 2 + : x < x }. Again all ranom choices are inepenent. It follows from more general results by Kallenberg [11], [13, Theorem 9.24] that this construction yiels all jointly exchangeable ranom ajacency measures, provie we allow the graphex W to be ranom; see Veitch an Roy [15] for the present context. See also [3, Theorem 2.21] for a relate result. The conitions i iii are precisely the conitions neee to guarantee that the constructe measure ξ a.s. is locally finite an thus an ajacency measure, see [15] an [13, Proposition 9.25]. Note that the eges of type G3 are inepenent of everything else an a.s. isolate; they form a ust of little interest. Also the stars prouce by G2 are of minor interest. One therefore often takes S = 0 an I = 0. A graphex 0, 0, W can be ientifie with the graphon W, an we write ξw = ξ0, 0, W an ΓW = Γ0, 0, W. In this case, the construction uses only the Poisson process θ j, ϑ j an gives the eges in G1; see also [3] at least when W is integrable. We write for convenience Γ s W = ΓW s, an we are particularly intereste in the graph value process Γ s W s 0. Note that this is an increasing process of finite labelle graphs, where Γ 0 is empty an Γ r is an inuce subgraph of Γ s whenever 0 r s. Furthermore, Γ = s 0 Γ s, so the typically infinite graph Γ or the measure ξγ an the process Γ s etermine each other. We consier also the corresponing processes ξ s W := ξγ s W = ξw s of finite ajacency measures an G s W := GΓ s W of finite unlabelle graphs. It follows from the construction above, that for every fixe s > 0, given the unlabelle graph G s W, the vertex labels on the labelle graph Γ s W are i.i.. an U0, s; in other wors, conitione on G s W, an therefore also unconitionally, Γ s W = Lbl s G s W The classical case of graphons on [0, 1]. As sai above, the classical theory of graphons consiers graphons W : [0, 1] 2 [0, 1] efine on [0, 1] 2 ; we can ientify them with graphons efine on R 2 + that vanish outsie [0, 1] 2. As sai above, we furthermore ientify the graphon W an the graphex 0, 0, W. There is no ust an no ae stars in the classical theory. In the classical theory, one efines for each n N a ranom graph G n W with n vertices by taking n i.i.. ranom numbers ϑ i U0, 1 an conitionally on these variables, letting there be an ege ij with probability

6 6 SVANTE JANSON W ϑ i, ϑ j for each pair ij. Furthermore, there are no loops. We may impose this by assuming that W vanishes on the iagonal := {x, x}. For convenience, we tacitly assume this; note that reefining a graphon W to be 0 on is equivalent to ignoring loops. On the other han, in the construction above of Γ s W, G2 an G3 o not appear, an for G1 we can ignore every θ i, ϑ i with ϑ i > 1 since then W ϑ i, ϑ j = 0 for every j. In the construction of Γ s W an G s W, we thus consier points θ i, ϑ i [0, s] [0, 1]. Let the number of these points be N s := Ξ[0, s] [0, 1] Pos. It follows that conitione on N s, the ranom graph G s W constructe above equals G Ns W, with all isolate vertices elete. 3. Graphex equivalence an convergence In this section, we efine the four ifferent types of convergence that we consier in the present paper Convergence of corresponing ranom graphs: GP. The construction in Section 2 efines a ranom labelle graph ΓW an a corresponing ranom ajacency measure ξw for every graphex W. This efines a map Ψ : W PMR 2 + by ΨW := LξW, the istribution of ξw. Unfortunately, Ψ is not injective, i.e., a graphex W is not uniquely etermine by the istribution of ξw. We say that two graphexes W an W are equivalent if LξW = LξW ; we enote this by W = W. Let W := W/ =, the set of equivalence classes; then Ψ can be regare as an injection W PMR 2 +, an we can ientify W with its image an equip W with the subspace topology inherite from PMR 2 +. Since MR 2 + is a Polish space, PMR 2 + is metrizable, an thus the topology of W can be efine by a metric GP. There are many possible choices GP but we assume that one is chosen; we will not istinguish an explicit choice. Moreover, we can also regar GP as efine on W; this makes W into a pseuometric space, with W = W GP W, W = Convergence in the pseuometric GP is enote GP by Veitch an Roy [16]. They actually use vi in Theorem 3.1 below as the efinition. Convergence GP can be characterize as follows, which is at least implicit in [16] but state explicitly here for easy reference. Note that since the topology is metric, it suffices to consier convergence of sequences; the theorem extens immeiately to, e.g., convergence of families with a continuous parameter. Theorem 3.1. Let W n, n 1, an W be graphexes. Then the following are equivalent, as n. i W n GP W. ii GP W n, W 0. iii LξW n LξW in PMR 2 +. iv ξw n ξw in MR 2 +. v ξ s W n ξ s W in MR 2 + for every s <.

7 ON CONVERGENCE FOR GRAPHEXES 7 vi G s W n G s W in G f, for every s <. In v vi, it suffices to consier s in a given unboune subset of R +, for example s N. Proof. i ii iii hols by the efinitions above. iii iv hols by the efinition of convergence in istribution. iv = v. Let, Ŝ be the class of boune measurable subsets of R 2 +, an for a ranom measure ξ, let Ŝξ := {B Ŝ : ξ B = 0 a.s.}. By [12, Theorem 16.16], a sequence of ranom measures ξ n ξ if an only if, for every finite sequence B 1,..., B k Ŝ, ξn B 1,..., ξ n B k ξb 1,..., ξb k. 3.2 Let Q s := [0, s] 2. The ranom graph ΓW has a.s. no label s, an thus the ranom measure ξw has a.s. no mass at Q s, i.e., Q s ŜξW. Hence, if B Ŝξ sw, then B Q s ŜξW. It follows, see 3.2, that if ξw n ξw an B 1,..., B k Ŝξ sw, then ξs W n B j k j=1 = ξw n B j Q s k j=1 ξwb j Q s k j=1 = ξ s WB j k j= Consequently, ξ s W n ξ s W by [12, Theorem 16.16] again. Alternatively, this implication follows by the continuous mapping theorem since a similar argument shows that the mapping ξ ξ s from MR 2 + to itself is continuous at every ξ with ξ Q s = 0. v = iv. Let B 1,..., B k ŜξW, an let s be so large that j B j Q s. Then ξ s B j = ξb j an thus ξ s W n ξ s W implies, again using [12, Theorem 16.16], ξw n B j k ξwb j=1 j k j=1. Hence, ξw n ξw. v vi. Fix s > 0. By 2.3, ξ s W = ξlbl s G s W, an similarly for W n. Hence the equivalence follows from Lemma 3.2 below, taken from [16]. Lemma 3.2 [16, Lemma 4.11]. If G n, n 1, an G are any ranom graphs in G f, an s > 0, then G n G in G f if an only if Lbl s G n Lbl s G in Ĝ, i.e., if an only if ξlbl sg n ξlbl s G in MR 2 +. We will not repeat the proof of [16, Lemma 4.11], but we note that it is can be interprete as efining a map Φ s : PG f PMR 2 +, by taking for µ PG f a ranom graph G µ, an efining Φ s µ := LξLbl s G PMR 2 +. It is then shown that Φ s is continuous, injective an proper. Finally, any continuous an proper map to a metric space is close, an a continuous an close injection is a homeomorphism onto a close subset Cut metric: δ. The invariant cut metric δ is efine only for integrable graphons. It is the stanar metric for classical graphons, see e.g. [4], [14], [9]. The efinition was extene to graphons efine on arbitrary

8 8 SVANTE JANSON σ-finite measure spaces by Borgs, Chayes, Cohn an Holen [3], to which we refer for etails. Here we only consier graphons on R +, an then the results simplify as follows, see [3]. First, for an integrable function F on R 2 +, we efine its cut norm by F := sup F x, y µx µy, 3.4 T,U T U If W is a graphon an ϕ : R R is measure preserving, let W ϕ x, y := W ϕx, ϕy. Then, for two integrable graphons W 1 an W 2, efine δ W 1, W 2 := inf W ϕ 1 ϕ 1,ϕ 1 W ϕ 2 2, taking the infimum over all pairs of measure preserving maps ϕ 1, ϕ 2 : R + R +. Moreover, it is shown in [3, Proposition 4.3c] that δ W 1, W 2 := inf ϕ W ϕ 1 W 2, 3.6 taking the infimum over all measure-preserving bijections ϕ : R + R +. By [3, Theorem 2.22], δ W 1, W 2 = 0 if an only if W 1 = W2, i.e., W 1 an W 2 are equivalent in the sense efine above. For characterizations of this, see [3] an [10]. If W n an W are integrable graphons, we write W n δ W as n if δ W n, W 0. We shall show that this is stronger than GP. We begin with the special case of classical graphons, where we have equivalence. Lemma 3.3. Suppose that W n, n 1, an W are graphons with support on [0, 1] 2. Also suppose that they all vanish on the iagonal := {x, x}. Then the following are equivalent, as n : i δ W n, W 0 ii for every s 0, G s W n G s W. iii W n GP W. Proof. i = ii. Using the notation in Section 2.3, i implies an is equivalent to G N W n G N W as n for every N N, see e.g. [4; 7]. Hence the same hols if we let N = N s Pos be ranom, an since the space of finite graphs is iscrete, the result hols also if we remove all isolate vertices from the graphs, which yiels ii, see Section 2.3. ii iii. By Theorem 3.1. ii = i. The implication i = ii shows that the map W LG s W s 0 PG f R + is continuous. Since the space of equivalence classes of classical graphons is compact, it suffices to show that the map is injective. The only complication is cause by the removal of isolate vertices. Thus, let H be a finite graph without isolate vertices, an let H + mk 1 be H with m isolate vertices ae. Then, with v := vh, PG s W = H = P G Ns W = H + mk 1 m=0 = e s m=0 s m+v m + v! P G m+v W = H + mk

9 ON CONVERGENCE FOR GRAPHEXES 9 Hence, if W an W are two classical graphons an PG s W = H = PG s W = H, then multiplication by e s an ientification of the coefficients of the power series in 3.7 show that PG m+v W = H + mk 1 = PG m+v W = H+mK 1 for every m 0. Consequently, G s W = G s W for every s > 0 implies that G N W = G N W for every N, an thus W = W an δ W, W = 0. Theorem 3.4. Suppose that W n, n 1, an W are integrable graphons. If W n δ W as n, then W n GP W. Proof. Assume δ W n, W 0. By 3.6, we may then replace each W n by an equivalent Wn ϕn such that W n W 0; note that the measurable rearrangement Wn ϕn efines the same ranom graphs as W n in istribution, i.e., ξwn ϕn = ξw n, an thus GP W n, Wn ϕn = 0, see 3.1. In the sequel we thus assume ε n := W n W 0. Define the truncations W n N := W n 1 [0,N] 2 an W N := W 1 [0,N] 2. Let ε > 0 an fix a large N such that W N W L 1 < ε. Then, W N n W N W n W = ε n as n. Consequently, by Lemma 3.3 an a rescaling, for every s 0, Furthermore, W n W n N λ = G s W n N G s W N as n. 3.9 W W N λ + W n W λ R 2 + \[0,N]2 < ε + 2 W n W = ε + 2ε n G s W an G s W N iffer only if the labelle graph Γ s W contains some ege with at least one label > N. The expecte number of such eges is E eg s W eg s W N = s2 W W N λ, an thus P G s W G s W N s 2 an similarly, using 3.10, R 2 + R 2 + W W N λ < s 2 ε, 3.12 P G s W n G s W n N s 2 W n W n N λ < s 2 ε + 2ε n R 2 + Consequently, if f : G f [0, 1] is any function, then by 3.9 an 3.13, lim sup E fg s W n E fg s W n lim sup E fgs W n N E fg s W N + s 2 2ε + 2ε n n = 2s 2 ε. Since ε > 0 is arbitrary, this shows E fg s W n E fg s W, an thus G s W n G s W as n. The result follows by Theorem 3.1.

10 10 SVANTE JANSON Remark 3.5. In Lemma 3.3ii, it is not necessary to assume the conition G s W n G s W for every s 0. In fact, it suffices to assume this for s in an arbitrary interval non-empty interval a, b, or even more generally, for s in any infinite set having a cluster point in [0, ; this follows by the same proof an the uniqueness theorem for analytic functions. Moreover, the argument suggests that it might suffice to assume ii for a single s > 0. We state this as an open problem. By the proof above, this is equivalent to the problem whether G s W = G s W for the ranom graphs without isolate vertices is equivalent to the corresponing equality in istribution of the ranom graphs G Pos W an G Pos W with isolate vertices. Problem 3.6. In Lemma 3.3, oes G s W n G s W for a single s > 0 imply W n GP W? We can also ask whether this hols for general graphexes an not just for classical graphons. We leave this too as open problems. Problem 3.7. In Theorem 3.1, oes G s W n G s W for, say, 0 < s < 1 imply W n GP W? Problem 3.8. In Theorem 3.1, oes G s W n G s W for a single s > 0 imply W n GP W? 3.3. Stretche convergence: δ s an GS. We efine, as [3] an [16], given a graphon W or more generally a graphex W = I, S, W an a real number c > 0, the stretche graphon or graphex by W c x, y := W x/c, y/c or W c := c 2 I, S c, W c where further S c x := csx/c. It follows easily from the construction of the ranom graphs above that a stretche graphex efines the same ranom graph process G r W up to a change of parameter: G r W c r = Gcr W r Borgs, Chayes, Cohn an Holen [3] efine the stretche cut metric δ s by, for two nonzero integrable graphons W 1 an W 2, δw s c 1, W 2 := δ W 1 1, W c where c i := W i 1/2 L 1. We write W n δ s W as n if δ s W n, W 0. Since δ W n, W 0 implies W n L 1 W L 1, it is easily seen that, for any integrable graphons W n an W, W n δ s W W cn n δ W c for some constants c n, c > Moreover, noting that the ranom graph process r G r W is increasing an right-continuous an thus calag, an thus has a countable number of jumps τ k, Veitch an Roy [16] consier for any non-zero graphex W the sequence G τk W k of ifferent finite an unlabelle graphs that appear in {G r W : 0 r < }; they efine for W n, W W := W \ {0}, W n GS W if G τk W n k=1 G τk W k=

11 ON CONVERGENCE FOR GRAPHEXES 11 This too is a metric convergence. In analogy with the efinition of GP above, we can efine a map Ψ : W \ {0} PG f by ΨW = L G τk W k 1, an by fixing a metric on PG f, we efine a pseuometric GS on W. Then obviously W n GS W GS W n, W It follows by 3.14 that G τk W k = Gτk W c k for any stretching of a graphex W, an thus GS W, W c = 0, cf. [16, Corollary 5.5]. Furthermore, it follows from Theorem 3.1 that, assuming W, W W, see [16, Lemma 5.6] for a etaile proof, W n GP W = W n GS W Consequently, we have the following partial analogue of Lemma 3.9 [16]. If W n, W W an there exist c n, c > 0 such that W cn n GP W c, then W n GS W. Problem Does the converse to Lemma 3.9 hol? We conjecture so. We also have a result corresponing to Theorem 3.4 for the stretche metrics. Theorem Suppose that W n, n 1, an W are integrable non-zero graphons. If W n δ s W as n, then W n GS W. Proof. By 3.16, we have W n cn δ W c for some constants c n an c, an the result follows by Theorem 3.4 an Lemma 3.9. Problem Does the converses of Theorems 3.4 an 3.11 hol. We conjecture not. 4. Ranom relabellings We show in this section that [16, Theorem 4.3] extens to convergence as s through the set of all positive real numbers. Theorem 4.1 Extension of [16, Theorem 4.3]. Let W be a graphex an let G s := GΓ s W, s 0, be the corresponing process of unlabelle graphs. Then, a.s., as s, Lbl s G s G s ΓW in Ĝ, i.e., LLbl sg s G s LΓW in PĜ. Equivalently, ξlbl s G s G s ξw in MR 2 +. Note that Theorem 4.1 not only generalizes [16, Theorem 4.3]; also [16, Theorem 4.4] is an immeiate corollary. In the proof we use the following lemma, which extens a stanar lemma use by [16] from a parameter s N to a parameter s Q +. We guess that also the version here is known, but since we o not know a reference, we give a simple proof for completeness. Lemma 4.2. Assume that X s is a ranom variable for each s Q + such that X s X a.s. as s, an further a.s. X s Y for some ranom variable Y with E Y <. Let F s s Q+ be a ecreasing family of σ-fiels an let F := s Q + F s. Then E X s F s E X F a.s. as s with s Q +.

12 12 SVANTE JANSON Proof. Let Z t := sup s t X s X, where as in the rest of the proof we consier only s Q +. The assumption implies a.s. X Y an thus 0 Z t 2Y. Moreover, Z t 0 a.s. as t by assumption. Hence, E Z t < an E Z t 0 as t by ominate convergence. Fix t. For s t we have X s X Z t, an thus E X s X F s E Z t F s a.s. Consequently, using the convergence theorem for reverse martingales, [12, Theorem 7.23], a.s., lim sup s Hence, for every t 0, E X s X F s lim sup E Z t F s = E Zt F. 4.1 s E lim sup E X s X F s E E Zt F = E Zt. 4.2 s However, we have shown that E Z t 0 as t, an thus 4.2 implies E lim sup E X s X F s = s Consequently, a.s., E X s X F s 0 as s, an thus E X s X F s E X s X F s Furthermore, by the reverse martingale convergence theorem again an 4.4, a.s., E X s F s = E X Fs + E Xs X F s E X F Proof of Theorem 4.1. Let τ k k be the jump times, where G s G s. Then G s is constant for s [τ k, τ k+1, an it is easily seen that the istribution LLbl s G s G s is a continuous function of s [τ k, τ k+1. It follows that it suffices to prove the result as s through the countable set of rational numbers. Thus, in the sequel of the proof, we assume that s Q +, an we consier limits as s through Q +. Except for this, we follow the proof of [16, Theorem 4.3]; for completeness we repeat most of the arguments. First, by [16, Lemma 4.2], a consequence of [12, Theorems ], it suffices to prove that if U is a finite union of rectangles with rational coorinates, then, with ξ s := ξlbl s G s an ξ := ξw, Lξ s U G s L ξu a.s. 4.6 as s. Note that [16, Lemma 4.2], although state for sequences, immeiately extens to a parameter s Q + or even R + since convergence in PMR 2 + can be efine by a metric. Next, 4.6 means that for every function f : Z + R + of the form fx = 1 {k} x = 1{x = k}, E fξ s U G s E fξu a.s. 4.7 Thus, fix such a U an f or any boune f : Z + R +, an fix some large enough r such that U [0, r] 2. For s Q +, let Γ s be the partially labelle graph obtaine from ΓW by forgetting all labels in [0, s] but keeping larger labels. Let F s be the σ-fiel generate by Γ s. Conitione

13 ON CONVERGENCE FOR GRAPHEXES 13 on Γ s, ΓW is obtaine by ranomly relabelling the unlabelle vertices by labels in [0, s]; note that the unlabelle part of Γ s is G s. Hence, if s > r, E fξu F s = E fξlbls G s G s = E fξs U G s. 4.8 Define as in [16] U t := U + t, t an, for s > r, X r s := 1 s r s r 0 fξu t t. 4.9 Since t Y t := fξu t is a stationary stochastic process, which is r- epenent in the sense that {Y t } t t0 is inepenent of {Y t } t>t0 +r for every t 0, it follows from the ergoic theorem [12, Corollary 10.9] that X s r a.s. E fξu as s. Lemma 4.2 now shows that E X r s F s a.s. E fξu Furthermore, by symmetry, for every s an every t < s r, E fξu t F s = E fξu Fs a.s., an thus 4.9 implies E X s r F s = E fξu Fs a.s We obtain 4.7 by combining 4.8, 4.11 an 4.10, which completes the proof. 5. Convergence of empirical graphons If G is any finite graph, an s > 0, we efine as in [16] the stretche empirical graphon ŴG,s as the graphon obtaine from the ajacency matrix of G by replacing each vertex by an interval of length 1/s. This assumes some orering of the vertices, but ifferent orerings give equivalent graphons. In other wors, using the notation of Section 3.3, Ŵ G,s = Ŵ 1/s G where Ŵ G := ŴG,1. Note that ŴG,s only takes the values {0, 1}. We ientify as usual the graphon ŴG,s with the graphex 0, 0, ŴG,s. Borgs, Chayes, Cohn an Holen [3, Theorem 2.23] show that if W is a non-zero integrable graphon, then a.s. δŵgsw s, W 0 an implicitly, or as a consequence δ ŴGsW,s, W 0 as s. Similarly, Veitch an Roy [16, Theorems 4.8 an 4.12] show convergence in GP for a general graphex; however, only for sequences s k. We show in this section an extension of their result to convergence as s through the set of all positive real numbers. Theorem 5.1 Extension of [16, Theorems 4.8 an 4.12]. Let W be any graphex an let G s := GΓ s W, s 0, be the corresponing process of unlabelle graphs. Then, Ŵ Gs,s GP W a.s. as s. Proof. Again we follow the proof in [16] with some moifications. Let 0 < r s. Given G s we efine two ranom inuce subgraphs X r s an M r s of G s ; in X r s we select each vertex of G s with probability r/s, an in M r s we take Por/s copies of each vertex in G s thus allowing repetitions; in both cases this is one inepenently for all vertices. In both cases, we then a the eges inuce by G s an remove any isolate vertex.

14 14 SVANTE JANSON It follows that, conitionally given G s, Lbl r X r s = Lbl s G s r. Hence, by Theorems 4.1 an 3.1 together with 2.3, a.s. as s, Lblr X r s = G s Lbls G s r G s ΓW r = Γ r W = Lbl r G r W. Consequently, using Lemma 3.2, X s r G s G r W. 5.1 On the other han, still conitionally given G s, the ranom graph G r ŴG s,s constructe as in Section 2.2 from the empirical graphon ŴG s,s is precisely M r s. Hence, M s = r G s Gr ŴG s,s G s. 5.2 Lemma 5.2 below shows that, still conitionally given G s, there exists a 2r/svX s r. coupling of X r s an M r s such that P M r s X r s X r s Since trivially also the probability is at most 1, this yiels, for any constant A > 0, P M s r P M s r X s r X s r X s r 2Ar/s + 1{vX s r > A} an thus G s 2Ar/s + P vx s r > A G s. 5.3 Moreover, 5.1 implies that vx s r G s vg r W = vγ r W 5.4 a.s. as s, an thus P vx r s a.s. > A G s P vγ r W > A. 5.5 It follows from 5.3 an 5.5, that for every fixe r an A, a.s., lim sup P M r s X s r G s 0 + P vγr W > A. 5.6 s The left-han sie oes not epen on A, an as A, the right-han sie tens to 0. Hence, P M s r X s r G s a.s. as s, for every fixe r. Combining 5.7 with 5.1 an 5.2, we obtain Gr ŴG s,s G s G r W 5.8 a.s. as s, for every fixe r, an thus a.s. for every r N. Consequently, the result follows by Theorem 3.1. Lemma 5.2. Given a finite graph G an p [0, 1], let X p an M p be the ranom inuce subgraphs of G obtaine by inepenently taking Bep an Pop copies of each vertex of G, respectively, an then eliminating all resulting isolate vertices. Then X p an M p may be couple such that PM p X p X p 2pvX p. Proof. Let, for i V G, I i Bep an Y i Pop be ranom variables with the pairs I i, Y i i V G inepenent, let X p an M p be the inuce subgraphs of G obtaine by taking I i or Y i copies of each vertex i, respectively, an let X p an M p be the subgraphs of X p an M p obtaine by eleting all isolate vertices.

15 ON CONVERGENCE FOR GRAPHEXES 15 We couple I i an Y i such that for each i, I i = 0 = Y i = 0. This is possible since PY i = 0 = e p 1 p = PI i = 0, an we have by a simple calculation P Y i I i I i = 1 = PY i I i PI i = 1 = 2p pe p = 21 e p 2p. 5.9 p Using this coupling, the vertices of Mp form a subset of the vertices of Xp, possibly with some repetitions, an it follows that if vertex i is selecte for X p but is isolate, an thus oes not appear in X p, then it also will not appear in M p, since even if it appears in M p it will be isolate there. Hence, if X p M p, there must be some vertex i X p such that Y i I i. Consequently, by 5.9, P X p M p X p P Y i I i I i = 1 2pvX p i V X p As in [16], we obtain a corollary on stretche convergence of unstretche empirical graphons, cf. Section 3.3. Theorem 5.3 Extension of [16, Theorem 5.72]. If W is any non-zero graphex, then ŴG τk W,1 GS W. Proof. Immeiate by Theorem 5.1 an Lemma 3.9. Remark 5.4. Note that even if W = I, S, W is a general graphex with non-zero ust an star components I an S, Theorem 5.1 an [16, Theorem 4.12] yiel convergence to W of a sequence of graphons, where the ust an star components are taken to be 0. This shows that for finite r, it is not possible to istinguish between isolate eges or stars coming from I an S an isolate eges or stars coming from the graphon part W. Note that in contrast, for the infinite ranom graph ΓW, there is an obvious ifference between eges prouce by I, S an W : they have a.s. 2, 1 an 0 enpoints of egree 1, respectively. Acknowlegement I thank Daniel Roy for an interesting iscussion at the Isaac Newton Institute for Mathematical Sciences uring the programme Theoretical Founations for Statistical Network Analysis in 2016 EPSCR Grant Number EP/K032208/1. References [1] Tim Austin. On exchangeable ranom variables an the statistics of large graphs an hypergraphs. Probab. Surv , [2] Patrick Billingsley. Convergence of Probability Measures. Wiley, New York, [3] Christian Borgs, Jennifer T. Chayes, Henry Cohn & Nina Holen. Sparse exchangeable graphs an their limits via graphon processes. Preprint, arxiv: v2

16 16 SVANTE JANSON [4] Christian Borgs, Jennifer T. Chayes, László Lovász, Vera T. Sós & Katalin Vesztergombi. Convergent sequences of ense graphs I: Subgraph frequencies, metric properties an testing, Avances in Math , [5] Christian Borgs, Jennifer T. Chayes, László Lovász, Vera T. Sós & Katalin Vesztergombi. Convergent sequences of ense graphs II. Multiway cuts an statistical physics. Ann. of Math , no. 1, [6] François Caron & Emily B. Fox. Sparse graphs using exchangeable ranom measures. Preprint, arxiv: v3 [7] Persi Diaconis & Svante Janson. Graph limits an exchangeable ranom graphs. Ren. Mat. Appl , no. 1, [8] Tue Herlau, Mikkel N. Schmit & Morten Mørup, Completely ranom measures for moelling block-structure networks. Preprint, arxiv: v3 [9] Svante Janson. Graphons, cut norm an istance, rearrangements an coupling. New York J. Math. Monographs, 4, [10] Svante Janson. Graphons an cut metric on σ-finite measure spaces. Preprint, arxiv: [11] Olav Kallenberg. Exchangeable ranom measures in the plane. J. Theoret. Probab , no. 1, [12] Olav Kallenberg. Founations of Moern Probability. 2n e., Springer, New York, [13] Olav Kallenberg. Probabilistic Symmetries an Invariance Principles. Springer, New York, [14] László Lovász. Large Networks an Graph Limits. American Mathematical Society, Provience, RI, [15] Victor Veitch & Daniel M. Roy. The class of ranom graphs arising from exchangeable ranom measures. Preprint, arxiv: v1 [16] Victor Veitch & Daniel M. Roy. Sampling an estimation for sparse exchangeable graphs. Preprint, arxiv: v1 Department of Mathematics, Uppsala University, PO Box 480, SE Uppsala, Sween aress: svante.janson@math.uu.se URL: