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Limits of random trees II∗ 4 1 Attila Deák† 0 [email protected] 2 g August 7, 2014 u A 6 Abstract ] R Local convergence of bounded degree graphs was introduced by Ben- P jamini and Schramm. This result was extended further by Lyons to . bounded average degree graphs. In this paper we study the convergence h of random tree sequences with given degree distributions. Denote by Dn at theset ofpossible degree sequencesofalabeled treeon nnodes. Let Dn m bearandomvariableonDn andT(Dn)beauniformrandomlabeledtree [ with degree sequence Dn. We show that the sequence T(Dn) converges in probability if and only if Dn → D = (D(i))∞i=1, where D(i) ∼ D(j), 3 E(D(1))=2 and D(1) is a random variable on N+. v 6 Keywords: sparse graph limits, random trees 9 7 3 1 Introduction . 1 0 In recent years the study of the structure and behavior of real world networks 4 has received wide attention. The degree sequence of these networks appear to 1 have special properties (like power law degree distribution). Classical random : v graphmodels (likethe Erdo˝s-Rényimodel)haveverydifferentdegreesequence. i X An obvious solution is to study a random graph with given degree sequence. More generally generate a random graph with a degree sequence from a family r a of degree sequences. In [7] Chatterjee, Diaconis and Sly studied random dense graphs(graphswhosenumberofedgesiscomparabletothesquareofthenumber of vertices) with a given degree sequence. It is not always easy to generate a truly random graph with a given degree sequence. There is a fairly large literature on the configuration model (for the exactdefinitionofthemodelsee[5]),whereforagivendegreesequenceforeach nodei weconsiderd stubs andtakearandompairingofthe stubs andconnect i the corresponding nodes with an edge. This model creates the required degree distribution, but gives a graph with possible loops and parallel edges. ∗MSC2010SubjectClassification: 05C80 †MTA-ELTE”NumericalAnalysisandLargeNetworks”ResearchGroup 1 Anotionofconvergencefor(dense)graphsequenceswasdevelopedbyBorgs, Chayes, Lovász, So´s and Vesztergombi in [6]. The limit objects were described by Lovász and Szegedy in [12]. Using this limit theory, the authors in [7] described the structure of random (dense) graphs from the configuration model. They defined the convergence of degree sequences and for convergent degree sequencestheygaveasufficientconditiononthedegreesequence,whichimplies the convergence of the random graph sequence (taken from the configuration model). What can we say if the graphs we want to study are sparse (the number of edgesiscomparabletothenumberofvertices)andnotdense? Isthereasimilar characterization for sparse graphs with given degree sequence? We establish a characterizationforrandomtreeswithgiven(possiblyrandom)degreesequence. Therearevariouslimittheoriesandconvergencenotionsfortreesintroducedby Aldous[1]andbyElekandTardos[10]. Weusethenotionofconvergenceintro- duced for bounded degree graphs (that is the degree of each vertex is bounded abovebysomeuniformconstantd)firstintroducedbyBenjaminiandSchramm [3]. This notion was extended by Lyons [13] to bounded averagedegree graphs. In [8] the author described the behavior of a random tree sequence with a givendegreedistribution. Inthispaperweextendthisresultandproveasimilar characterizationasin[7]forrandomtreeswithgivendegreesequence. Wedefine theconvergenceofdegreesequencesandgiveanecessaryandsufficientcondition on the degree sequence, which implies the convergence of the tree sequence T(D ) in the sense of Lyons [13]. In the case of convergence we describe the n limit object. Thispaperisorganizedasfollows: InSection2,wegivethebasicdefinitions and notations. In Section 3, we describe the basic properties and the limit of a sequence of random degree sequences. At the end of the section we state our main theorem. In Section 4, we deal with labeled homomorphisms and in Section 5, we describe the limit object. 2 Basic definitions and notations 2.1 Random weak limit of graph sequences Let G = G(V,E) be a finite simple graph on n nodes. For S ⊆ V(G) denote by G[S] the subgraph of G spanned by the vertices v ∈ S. For a finite simple graphG onn nodes, letB (v,R)be the rootedR-ballaroundthe nodev, also G called as the R-neighborhood of v, that is the subgraph induced by the nodes at distance at most R from v: B (v,R)=G[{u∈V(G):dist (u,v)≤R}]. G G Two rooted graphs G ,G are rooted isomorphic if there is an isomorphism 1 2 between them which maps the root of G to the root of G . Given a positive 1 2 integer R, a finite rooted graph F and a probability distribution ρ on rooted 2 graphs,let p(R,F,ρ) denote the probability that the graphF is rooted isomorphic to the R-ball around the root of a rooted graph chosen with distribution ρ. It is clear that p(R,F,ρ) depends only on the component of the root of the graph chosen from ρ. So we will assume that ρ is concentrated on connected graphs. Forafinite graphG,letU(G)denotethe distributiongraphs.edgraphs obtained by choosing a uniform random vertex of G as root of G. It is easy to see, that for any finite graph G we have |{v ∈V(G):B (v,R) is rooted isomorhpic to F}| G p(R,F,U(G))= . |V(G)| Definition1. Let(G )beasequenceoffinitegraphsonnnodes,ρaprobability n distribution on infinite rooted graphs. We say that the random weak limit of G n is ρ, if for any positive integer R and finite rooted graph F, we have lim p(R,F,U(G ))=p(R,F,ρ). (1) n n→∞ IfG isasequenceofrandomfinitegraphs,thenp(R,F,U(G ))isarandom n n variable, so by convergence we mean convergence in probability. Definition 2. Let (G ) be a sequence of random finite graphs on n nodes, ρ a n probability distribution on infinite rooted graphs. We say that the random weak limit of G is ρ, if ∀ǫ>0,R∈N+ and finite rooted graph F, we have n lim P(|p(R,F,U(G ))−p(R,F,ρ)|>ǫ)=0. (2) n n→∞ The formal meaning of this formula is, that the statistics p(R,F,U(G )) as n random variables are concentrated. 2.2 Other notations We will denote random variables with bold characters. For a probability space (Ω,B,µ) and A∈ B denote by I(A) the indicator variable of the event A. De- note by D the set of possible degree sequences of a labeled tree on n nodes. n Throughout the paper we consider labeled trees on n nodes unless stated otherwise. Let D be a random variable on D . Denote by T(D ) the uniform n n n random tree on n labeled nodes, with degree sequence D . Denote the de- n gree sequence of a tree T by D = (D (i))n . For a given degree sequence T T i=1 D =(D(i))n there are i=1 n−2 D(1)−1,D(2)−1,··· ,D(n)−1 (cid:18) (cid:19) labeled trees with degree sequence D. It follows that for an arbitrary tree T P(D =D ) P(T(D )=T)= n T . n n−2 D (1)−1,D (2)−1,··· ,D (n)−1 (cid:18) T T T (cid:19) 3 If it does not cause any confusion, we will use D,T instead of D ,T(D ) n T n respectively. AfiniterootedgraphGwithrootv issaidtobel-deepifthelargestdistance from the root is l, that is l= max dist(v,u). u∈V(G) Denote by Ul the set of equivalence classes of finite unlabeled l-deep rooted graphswithrespecttoroot-preservingisomorphisms. LetTl beanl-deeprooted x tree on k nodes with root x. Denote the vertices at distance i from the root by T , and let t = |T | (t is 1, t is the degree of the root). For every finite i i i 0 1 graphG, p(R,F,U(G)) induces a probability measure on UR which we call the R-neighborhoodstatisticsofG. IfGisatreethenp(R,F,U(G))isconcentrated on rooted trees. Let T be the set of all countable, connected infinite rooted trees. For an infinite rooted tree T ∈ T denote by T(R) the R-neighborhood of the root of T. For an R-deep rooted tree F define the set T(F)={T ∈T :T(R) is rooted isomorphic to F}. Let F be the sigma-algebra generated by the sets (T(F)) , where F is an F arbitrary finite rooted tree. (T,F) is a probability field. We call a probability measure µ on T an infinite rooted random tree. Every infinite random tree µ has the property that for any F ∈UR: p(R,F,µ)= p(R+1,H,µ). (3) H∈UR+X1, H(R)∼=F Actuallyeverydistributiononrootedinfinitegraphshastheaboveproperty. Note that if we want to prove the convergence of a random tree sequence to a certain limit distribution ρ, then we need to have the convergence of the neighborhood densities and also (3), the consistency of these densities, which ensuresthat ρ will be concentratedoninfinite rootedtrees. These together will imply (2). 3 Limits of degree sequences Consider a random degree sequence D = (D (i))n and construct a labeled n n i=1 tree T(D ) with uniform distribution given the degree sequence. We want to n describe the limit of T(D ) as n → ∞. We give a characterization of the n degree sequences for which T(D ) has an infinite random tree as a limit. To n describe the model and the limit, we need to define and understand the limit of a random degree sequence D . Here we only deal with degree sequences of n trees. We further assume that D is an exchangeable sequence, that is for any n σ ∈S we have n (D (i))n ∼(D (σ(i)))n . n i=1 n i=1 4 Exchangeability is a way to eliminate exceptional vertices and allows us to use the limit theory of exchangeable sequences. For more on exchangeable random variables we refer to [2]. Definition 3. We say that an exchangeable sequence D is convergent and n D →D, where D is a random infinite sequence, if for every k ∈N we have n (D (i))k →P (D(i))k . n i=1 i=1 It is easy to see, that if D is exchangeable and D → D then D is also n n an exchangeable sequence. The following theorem of Hewitt and Savage( see [11]), which is a generalization of de Finetti’s theorem, describes the limits of exchangeable sequences. Theorem 1. Let X be a random infinite exchangeable array. Then X is a mixture of infinite dimensional iid distributions X = λdp(λ), ZIID where p is a distribution on infinite dimensional IID distributions λ. As a result we have that the limit of an exchangeable degree sequence is an infinite exchangeablesequenceandsoa mixture ofIID distributions. Note that if D is not exchangeable then we can take a random permutation σ ∈S and n n define the exchangeable degree sequence D˜ (i)=D (σ(i)). n n Lemma1. LetX beaninfiniteexchangeablerandomsequence. Furtherassume that we have P(X(1)=i,X(2)=i)=P(X(1)=i)P(X(2)=i). Then X is an infinite IID distribution (p is concentrated on one distribution). Proof: From Jensen’s inequality we have that 2 λ(i)2dp(λ)≥ λ(i)dp(λ) . (4) ZIID (cid:18)ZIID (cid:19) Also from Theorem 1 we have λ(i)2dp(λ)=P(X(1)=i,X(2)=i)= ZIID 2 =P(X(1)=i)P(X(2)=i)= λ(i)dp(λ) (cid:18)ZIID (cid:19) It follows that in (4) equality holds which means that p is a degenerate distribution and so proves our lemma. We will see that if T(D ) is convergent, then D satisfies the assumptions n n in Lemma 1. So for a convergent random tree sequence T(D ) the limit of the n degree sequence D needs to be an infinite IID distribution. n 5 Example 1. Let X be a uniform random element of [n]. Consider the degree sequence n−1, if i=X D(i)= 1, otherwise (cid:26) Let St = T(D ) be the star-graph on n nodes. The limit degree sequence is n n just the constant 1 vector 1 = (1,1,···). Obviously in the limit the expected degree of a node is 1. It is not hard to see, that if F is not a single edge, then u(R,F,St )=0 for every n>|V(F)|. Thus there is no limit distribution ρ on n infinite graphs such that P(|p(R,F,U(St )−p(R,F,ρ)|>ǫ)→0 for every F. n Example 1 shows that if only the average degree is bounded, too many unboundeddegreeverticesdestroyconvergence. Asthe averagedegreeofatree onnnodesis2n−1,onewouldexpectthatinthelimitdistributiontheexpected n degree of a node is 2, that is E(D(i))=2 for every i. It turns out that it is enough to have that the degree sequence converges and E(D(i))=2 holds ∀i. Now we are ready to state our main theorem which describes the degree sequence of convergentrandom tree sequences. Theorem 2. Let D be a sequence of random degree sequences (D ∈ D ). n n n The random tree sequence T(D ) is convergent and converges to an infinite n random tree if and only if D →D, where D =(D ,D ,···) is an infiniteIID n 0 0 sequence and E(D )=2. 0 4 Labeled subgraph densities To proveconvergencewe need to understand the neighborhoodstatistics of the random tree T(D ). First we will count subgraph densities and then relate n them to neighborhood statistics. For fixed unlabeled graphs F and G denote by |{φ:φ is an injective homomorphism from F to G}| inj(F,G)= |V(G)| the normalized number of copies of F in G. We call F the test graph. We call inj(F,G) the injective density of F in G. For bounded degree graphs the convergenceofinjectivedensitiesforeveryF isequivalenttothe convergenceof neighborhood densities for every H rooted finite graph. For bounded average degreegraphssubgraphstatistics maybe unbounded. For the randomstartree St we have n (n−1)(n−2) inj( ,St )= . n n To avoid unbounded subgraph statistics we add a further structure to the test graph F. We call a pair (F,r) a numbered graph, where r = (r )V(F) and i i=1 r ∈ N. We call r the remainder degree of the node i ∈V(F). Let (F,r) be a i i 6 numbered graphand φ be a homomorphismfromF to a graphG. We say that φ is a labeled homomorphism if φ is a homomorphism and D (φ(v))=D (v)+r , ∀v ∈V(F). G F v Let |{φ:φ is an injective labeled homomorphism from F to G}| inj ((F,r),G) = lab |V(G)| bethenormalizednumberofnumberedcopiesofF inG. Firstwewanttoderive properties of degree sequences D for which inj ((F,r),T(D )) is convergent n lab n for every finite graph F and remainder degrees r. Then in Section 5 we will turn to the convergence of neighborhood statistics. Remark 1. Theconvergenceofinj (.,G )forevery(F,r)doesnotimplythe lab n randomweakconvergenceofG ingeneral. inj ((F,R),St )isconvergentfor n lab n every (F,r), but as we saw earlier St is not a convergent tree sequence. n Remark 2. Let (F,r) be an arbitrary numbered graph on k nodes. One can easily see that inj ((F,r),G) is uniformly bounded for every G. lab Proof.: To see this we will bound the number of ways we can construct an injective labeled homomorphism ψ from (F,r) to G. Let R = max{r }. i If we define ψ(1) = v ∈ V(G), then D (ψ(1)) = r . There are at most G 1 DG(ψ(1))DF(1) = r1DF(1) ≤ Rk possibilities for ψ(u)’s (u ∈ NF(1)), where N (1)isthesetofneighborsof1inF. Followingthisideawegetthatforevery F v there are at most (Rk)k possible ways to extend ψ, given ψ(1) = v. Hence there are at most nRk2 injective labeled homomorphisms from F to G and the remark follows. For an arbitrary numbered tree (T,r), and φ:V(T)7→[n] let I ((T,r),φ)=I({φ is an injective labeled homomorphism of T to T }) n n X(T,r) = I ((T,r),φ)=n·inj ((T,r),T(D )). (5) n n lab n φ:V(XT)7→[n] We define I ((F,r),φ),X(F,r) similarly for a numbered forest (F,r). If it does n n notcauseanyconfusion,wewillomitrfromtheformulasaboveanduseI (T,φ), n XT,I (F,φ)andXF insteadtosimplifynotation. Forrandomgraphsequences n n n G by the convergence of inj (.,G ) we mean convergence in probability. n lab n Let D be a random degree sequence and T = T(D ) be the associated n n n random tree sequence. Let (T,r) be a numbered tree. As inj ((T,r),T ) is lab n bounded, we have that inj ((T,r),T ) is convergent for every (T,r) if and lab n only if we have that XT D2 n =D2(inj ((T,r),T ))→0. (6) lab n n (cid:18) (cid:19) 7 We will use this formula to prove properties of the degree sequence. We can expand the above formula using (5) XT 1 D2 n = E(I (T,ψ)I (T,φ))− n n2 n n (cid:18) (cid:19) (cid:16)ψ,φ:VX(T)7→[n] − E(I (T,ψ))E(I (T,φ)) →0. (7) n n ψ,φ:VX(T)7→[n] (cid:17) The following two lemmas will establish a connection between the degree sequence and the probabilities P(I (T,φ) = 1). Then we will use (7) to prove n that the degree sequence satisfies the conditions in Lemma 1. Remark 3. As the degreesequenceis exchangeablewehavethatforanyψ,φ: V(T)7→[n] P(I (T,φ)=1)=P(I (T,ψ)=1). n n Let T be an arbitrary tree on k nodes. For a random degree sequence D and n φ:V(T)7→[n] let D =(D (φ(i)))k . φ n i=1 Lemma 2. Let D ∈ D be a random degree sequence and T = T(D ). n n n n Let F be an arbitrary forest on m (m ≤ n) nodes with remainder degrees r = (r ,··· ,r ). Let R = r and denote by C ,C ,··· ,C the connected 1 m i i 1 2 c components of F. The probability that an arbitrary φ:V(T)7→[n] is an injec- P tive labeled homomorphism is (n−m+c−2)! P(I (F,φ)=1)= H(r,F)P(D =D ), n φ T (n−2)! where H(r,F)= c r (DF(j)+rj−1)! is a constant depend- i=1 j∈Ci j j∈Ci (rj!) ing only on F and thehre(cid:16)mainder d(cid:17)egrees r. i Q P Q Proof: We may assume that φ(i) = i,∀i ∈ V(F). Let R = r . Fix a i j∈Ci j degree sequence D = (D(i))n . It follows from the Pru˝fer sequence that the number of trees realizing thiis=1degree sequence is n−2 P . We need D(1)−1,···,D(n)−1 to count the trees with degree sequence D which have F spanned by the first (cid:0) (cid:1) m nodes and the remainder degree condition holds. Contract every connected component C of F to a single vertex u . Also contract the images of these i i components in T(D ). We get a tree on n−m+c nodes with degree sequence n D′ =(R ,R ,··· ,R ,D(m+1),··· ,D(n)). 1 2 c There are n−m+c R −1,R −1,··· ,R −1,D(m+1)−1,··· ,D(n)−1 (cid:18) 1 2 c (cid:19) trees realizing the degree sequence D′. For each connected component C we i can connect the R edges to the vertices in i R ! i r ! j∈Ci j Q 8 ways. It follows that the number of labeled trees realizing the degree sequence D and having F on the first m vertices is c n−m+c−2 R ! i . R −1,··· ,R −1,D(m+1)−1,··· ,D(n)−1 r ! (cid:18) 1 c (cid:19)i=1" j∈Ci j # Y Q From this it follows that P(I (F,φ)=1|D =D)= n n n−m+c−2 c R −1,··· ,R −1,D(m+1)−1,··· ,D(n)−1 R ! = (cid:18) 1 c (cid:19) i . (8) n−2 r ! i=1 j∈Ci j D(1)−1,··· ,D(n)−1 Y (cid:18) (cid:19) Q NotethatthedegreesequenceD shouldbe suchthatD(i)=D (i)+r , i= F i 1,··· ,mholdsforthefirstmdegrees. Weneedtosumthisprobabilityforevery possible degree sequence. In our case we sum over degree sequences for which D(i) = D (i)+r , i = 1,··· ,m holds. As in equation (8) the right hand side F i does not depend on D(i), i>m, we have P(I (F,φ)=1)= n c (n−m+c−2)! (D (j)+r −1)! = R F j P(D =D ). i φ T (n−2)!  (r !)  j iY=1 jY∈Ci   If we take H(r,F) = c R (DF(j)+rj−1)! , we get the desired equa- i=1 i j∈Ci (rj!) tion. h i Q Q Let(F ,r ),(F ,r )betwolabeledgraphs,φ:V(F )7→[n]andψ :V(F )7→ 1 1 2 2 1 2 [n]. We denote by F the graph obtained by identifying nodes i∈V(F ), j ∈ 1,2 1 V(F ) if and only if φ(i)=ψ(j). We can define remainder degrees r on F 2 1,2 1,2 in a straightforwardway if φ(i)=ψ(j)⇒r (i)=r (j). 1 2 Lemma 3. Let D ∈ D be a random degree sequence and T = T(D ). Let n n n n F ,F be two forests on m and m nodes (m ,m ≤n) with remainder degrees 1 2 1 2 1 2 r ,r . Let φ : V(F ) 7→ [n] and ψ : V(F ) 7→ [n]. If F is a forest and we 1 2 1 2 1,2 can define r , then let m =|V(F )|, c ={the number of components of 1,2 1,2 1,2 1,2 F } and R = r (i). We have 1,2 1,2 V(F1,2) 1,2 P P(I (F ,φ)=1|I (F ,ψ)=1)= n 1 n 2 (n−m +c −2)!H(r ,F ) 1,2 1,2 1,2 1,2 P(D =D |D =D ). (9) (n−m +c −2)! H(r ,F ) φ F1 ψ F2 2 2 2 2 Proof: The proof follows immediately from the definition of conditional probability. 9 Let D be a degree sequence and T = T(D ) be the associated random n n n tree. Assume that inj ((T,r),T ) is convergent. Then by (6) we have that lab n D2(XT/n)→0. For any tree T on k nodes we have n XT 1 D2 n = E(I (T,φ)I (T,ψ))−E(I (T,φ))E(I (T,ψ)) n n2 n n n n (cid:18) (cid:19) φ,ψ:VX(T)7→[n](cid:16) (cid:17) (10) Now if we split the sum by the size ofthe intersectionof φ(V(T))and ψ(V(T)) and use Remark 3, we have XT 1 k D2 n = n(n−1)·...·(n−2k+i+1)· n n2 (cid:18) (cid:19) i=0 |φ(V(T))∩Xψ(V(T))|=i · E(I (T,φ)I (T,ψ))−E(I (T,φ))E(I (T,ψ)) (11) n n n n From Lemma 2 and 3 w(cid:0)e can easily derive that the order of the term(cid:1)s corresponding to i 6= 0 is O(1). It follows that the condition D2(XT) → 0 is n n equivalent to (n−1)·...·(n−2k+1) (P(I (T,φ)I (T,ψ))−P(I (T,φ))P(I (T,ψ)))→0. n n n n n Using again Lemma 2 and 3 we can easily derive the following: XT ∀T, D2 n →0⇔∀φ,ψ :V(T)7→[n], φ(V(T))∩ψ(V(T))=∅ n (cid:18) (cid:19) P(D =D ,D =D )→P(D =D )P(D =D ) (12) φ T ψ T φ T ψ T The following corollary is an easy application of Lemma 1 and (12). Corollary1. Thelabeledsubgraphdensitiesofarandom treesequenceconverge in probability if and only if the corresponding degree sequence converges to an infinite IID sequence. Remark 4. The formula in Lemma 2 yields an easy result on the probability that two vertices i,j with degrees d ,d are connected: i j d +d −2 P(ij ∈E(T(D )) | D (i)=d ,D (j)=d )= i j . n n i n j n−2 Similarlyforagivenedgeij ∈E(T(D ))the degreedistributionofthevertices n i and j can be expressed: P(D (i)=d ,D (j)=d | ij ∈E(T(D )))= n i n j n n d +d −2 i j P(D (i)=d ,D (j)=d ) n i n j n−2 2 10