Diameters in preferential attachment models

Transcription

1 Diameers in preferenial aachmen models Sander Dommers and Remco van der Hofsad Gerard Hooghiemsra January 20, 200 Absrac In his paper, we invesigae he diameer in preferenial aachmen PA- models, hus quanifying he saemen ha hese models are small worlds. The models sudied here are such ha edges are aached o older verices proporional o he degree plus a consan, i.e., we consider affine PA-models. There is a subsanial amoun of lieraure proving ha, quie generally, PA-graphs possess power-law degree sequences wih a power-law exponen τ > 2. We prove ha he diameer of he PA-model is bounded above by a consan imes log, where is he size of he graph. When he power-law exponen τ exceeds 3, hen we prove ha log is he righ order, by proving a lower bound of his order, boh for he diameer as well as for he ypical disance. This shows ha, for τ > 3, disances are of he order log. For τ 2, 3, we improve he upper bound o a consan imes log log, and prove a lower bound of he same order for he diameer. Unforunaely, his proof does no exend o ypical disances. These resuls do show ha he diameer is of order log log. These bounds parially prove predicions by physiciss ha he ypical disance in PA-graphs are similar o he ones in oher scale-free random graphs, such as he configuraion model and various inhomogeneous random graph models, where ypical disances have been shown o be of order log log when τ 2, 3, and of order log when τ > 3. Inroducion In he pas decade, many examples have been found of real-world complex neworks ha are small worlds and scale free. The small-world phenomenon saes ha disances in neworks are small. The scale-free phenomenon saes ha he degree sequences in hese neworks saisfy a power law. See [3, 24, 39] for reviews on complex neworks, and [5] for a more exposiory accoun. Thus, hese complex neworks are no a all like classical random graphs see [4, 9, 35] and he references herein, paricularly since he classical models do no have power-law degrees. As a resul, hese empirical findings have ignied enormous research on random graph models ha do obey power-law degree sequences. See [] for he mos general inhomogeneous random graph models, as well as a review of he models under invesigaion. Exensive discussions of various scale-free random graph models are given in [2, 25]. While hese models have power-law degree sequences, hey do no explain why many complex neworks are scale free. A possible explanaion was given by Barabási and Alber [6] by a phenomenon called preferenial aachmen PA. Preferenial aachmen models he growh of he nework in such a way ha new verices are more likely o add heir edges o already presen verices having a high degree. For Eindhoven Universiy of Technology, Deparmen of Mahemaics and Compuer Science, P.O. Box 53, 5600 MB Eindhoven, The Neherlands. S.Dommers@ue.nl, rhofsad@win.ue.nl Delf Universiy of Technology, Elecrical Engineering, Mahemaics and Compuer Science, P.O. Box 503, 2600 GA Delf, The Neherlands. G.Hooghiemsra@ewi.udelf.nl

2 example, in a social nework, a newcomer is more likely o ge o know a person who is socially acive, and, herefore, already has a high number of acquainances high degree. Ineresingly, PA-models wih so-called affine PA rules have power-law degree sequences, and, herefore, preferenial aachmen offers a convincing explanaion why many real-world neworks possess his propery. There is a large amoun of lieraure sudying such models. See e.g. [2, 0, 2, 3, 4, 5, 7, 22] and he references herein. The lieraure primarily focusses on hree main quesions. The firs key quesion for PA-models is o prove ha such random graphs are indeed scale free [2, 0, 2, 3, 7, 22], by proving ha heir degree sequence indeed obeys a power law wih a cerain power-law exponen τ > 2. The second key quesion for PA-models is heir vulnerabiliy, for example o deliberae aack [3] or o he spread of a disease [7]. The hird key quesion for PA-models is o show ha he resuling models are small worlds by invesigaing he disances in hem. See in paricular [5] for a resul on he diameer for a PA-model wih power-law exponen τ = 3. In non-rigorous work, i is ofen suggesed ha many of he scale-free models, such as he configuraion model, he inhomogeneous random graph models in [] and he PA-models, have similar properies for heir disances. Disances in he configuraion model have been shown o depend on he number of finie momens of he degree disribuion. Similar resuls are rue for he so-called rank- inhomogeneous random graph see e.g. [8, 9, 26, 40]. The naural quesion is, herefore, wheher he same applies o preferenial aachmen models. This is he main goal of he presen paper, in which we invesigae he diameer of scale-free PA-models. The remainder of his secion is organized as follows. We firs inroduce he models ha we will invesigae in his paper. Then we give he main resuls and conclude wih a discussion of universaliy in power-law random graphs. In his paper, we invesigae he diameer in some PA-models. The models ha we invesigae produce a graph sequence or graph process {G m,δ }, which, for fixed or 2, yields a graph wih verices and m edges for some given ineger m. In he sequel, we shall denoe he verices of G m,δ by m,..., m. When m is clear from he conex, we will leave ou he superscrip and wrie [] {, 2,..., }. We shall consider hree sligh variaions of he PA-model, which we shall denoe by models a, b and c, respecively. a The firs model is an exension of he Barabási-Alber model formulaed rigorously in [7]. We sar wih G,δ consising of a single verex wih a single self-loop. We denoe he degree of verex i a ime by D i, where a self-loop increases he degree by 2. Then, for m =, and condiionally on G,δ, he growh rule o obain G,δ + is as follows. We add a single verex + having a single edge. This edge is conneced o a second end poin, which is equal o + wih probabiliy proporional o + δ, and o a verex i G,δ wih probabiliy proporional o D i + δ, where δ is a parameer of he model. Thus, P + i G,δ = +δ 2+δ++δ, for i = +, D i +δ 2+δ++δ, for i [].. The model wih ineger m >, is defined in erms of he model for m = as follows. We sar wih G,δ m, wih δ = δ/m. Then we idenify he verices, 2..., m in G,δ m o be verex m in G m,δ, and for < j, he verices j m +,..., jm in G,δ m o be verex j m in G m,δ ; in paricular he degree D j m of verex j m in G m,δ is equal o he sum of he degrees of he verices j m +,..., jm in G,δ m. This defines he model for ineger m. Observe ha he range of δ is [ m,. The resuling graph G m,δ has precisely m edges and verices a ime, bu is no necessarily conneced. For δ = 0 we obain he original model sudied in [7], and furher sudied in [3, 4, 5]. The exension o δ 0 is crucial in our seing, as we shall explain in more deail below. 2

3 b The second model is idenical o he one above, apar from he fac ha no self-loops are allowed for m =. We sar again wih he definiion for m =. To preven a self-loop in he firs sep, we le G,δ undefined, and sar from G,δ 2, which is defined by he verices and 2 joined ogeher by 2 edges. Then, for 2, we define, condiionally on G,δ, he growh rule o obain G,δ + as follows. For δ, P + i G,δ = D i + δ, for i [] δ The model wih m > is again defined in erms of he model for m =, in precisely he same way as in model a. This model is sudied in deail in [25], and he model wih m = corresponds o scale-free rees as sudied in e.g. [6, 37, 38, 42]. c In he hird model, and condiionally on G m,δ, he end poins of each of he m edges of verex +, are chosen independenly, and are equal o a verex i m G m,δ, wih probabiliy proporionally o D i m + δ, where δ m. We sar again from G m,δ 2, wih he verices m and 2 m joined ogeher by 2m, m, edges. Since he end poin of he edges are chosen independenly we can give he definiion of {G m,δ } 2, for m, in one sep. For j m, P j h edge of + m is conneced o i m G m,δ = D i m + δ, for i []..3 2m + δ In his model, as is he case in model b, he graph G m,δ is a conneced random graph wih precisely verices and m edges. This model was sudied in [23, 36]. Remark.. In models a and b for m >, he choice of δ = δ/m is such ha in he resuling graph G m,δ, where m verices in G m are grouped ogeher o a single verex in G m,δ, he end poins of he added edges are chosen according o he degree plus he consan δ. Remark.2. For m =, he models b and c are he same. This fac will be used laer on. The growh rules in..3 are indeed such ha verices wih high degree are more likely o arac edges of new verices. One would expec he models a c o behave quie similarly, as is known rigorously for he scale-free behaviour, where he asympoic degree disribuion is known o be equal in models a c. As i urns ou, he affine PA mechanism in..3 gives rise o power-law degree sequences. Indeed, in [23], i was proved ha for model c, he degree sequence is close o a power law wih exponen τ = 3 + δ/m. For model a and δ = 0, his was proved in [7], while in [22], power-law degree sequences for PA-models wih affine PA mechanisms are proved in raher large generaliy. We see ha, by varying he parameers m, δ > m, we can obain any power-law exponen τ > 2, which is he reason for inroducing he parameer δ in..3. However, here is no inrinsic reason for he affine PA mechanism. For resuls on PA-models in he non-affine case, see e.g., [4, 44]. In general, such models do no produce power laws. The goal in his paper is o sudy he diameer in he above models, as a firs sep owards he sudy of disances in PA-models and he verificaion of he predicion ha disances behave similarly in various scale-free random models see also Secion.2 below. In he following secion, we describe our precise resuls.. Bounds on he diameer in preferenial aachmen models In his secion, we presen he diameer resuls for he PA-models a c. The diameer of a graph G is defined as diamg = max {dis Gi, j dis G i, j < },.4 i,j G 3

4 where dis G i, j denoes he graph disance beween verices i, j G. We prove ha, for all δ > m, he diameer of G m,δ is bounded by a consan imes log. When δ = 0, we adap he argumen in [5] o prove ha he diameer is bounded from below by ε log log log. For δ > 0, his lower bound is improved o a consan imes log, while, for δ < 0, we prove ha he diameer is bounded above and below by a consan imes log log. This esablishes a phase ransiion for he diameer of PA-models when δ changes sign. We now sae he precise resuls, which shall all hold for each of he models a c simulaneously. In he resuls below, for a sequence of evens {A }, we wrie ha A occurs wih high probabiliy whp when lim PA =. Theorem.3 A log upper bound on he diameer. Fix m and δ > m. Then, here exiss a consan c = c m, δ > 0 such ha whp, he diameer of G m,δ is a mos c log. When m =, so ha he graphs are in fac rees, here is a sharper resul proved by Piel [42], which, in paricular, implies Theorem.3 for model b. In his case, Piel shows ha he heigh of he ree, which is equal o he maximal graph disance beween verex and any of he oher verices, grows like +δ γ2+δ log + o, where γ solves he equaion γ + + δ + log γ = 0..5 This proves ha he diameer is a leas as large, and suggess ha he diameer has size 2+δ γ2+δ log + o. Scale-free rees have received subsanial aenion in he lieraure, we refer o [6, 42] and he references herein. I is no hard o see ha a similar resul as proved in [42] also follows for models a and c. This is proved when δ = 0 in [6], where i is shown ha he diameer in model a has size γ log, where γ is he soluion of.5 when δ = 0. Thus, we see ha he log upper bound in Theorem.3 is sharp, a leas for m =. I is no hard o exend he upper bound o m 2. In paricular, for model b, he upper bound for m 2 immediaely follows from he upper bound for m =. For models a and c, he exension is no as rivial, bu he proof is fairly sraighforward, and will be omied here. To see an implicaion of [42] for model a, we noe ha C, he number of conneced componens of G,δ in model a, has disribuion C = + I I, where I i is he indicaor ha he i h edge connecs o iself, so ha {I i } i=2 are independen indicaor variables wih PI i = = + δ 2 + δi + + δ..6 As a resul, C / log converges in probabiliy o + δ/2 + δ <, so ha whp here exiss a larges conneced componen of size a leas / log. Condiionally on having size s, he law of any conneced componen in model a is equal in disribuion o he law of he graph G,δ s + in model b, apar from he fac ha he verices and 2 in G,δ s + are idenified hus creaing a double self-loop and he verices are relabeled by order of appearance. In paricular, condiionally on having size s, he law of he diameer of he conneced componen in model a equals ha of G,δ s + in model b. This close connecion beween he wo models allows one o ransfer resuls for model b o model a when m =. Theorem.4 A log lower bound on he diameer for δ > 0. Fix m and δ > 0. Then, here exiss c 2 = c 2 m, δ > 0, such ha whp, he diameer of G m,δ is a leas c 2 log. Theorems.3.4 imply ha, for δ > 0 and whp, diamg m,δ = Θlog. Theorems.3.4 indicae ha disances in PA-models are similar o he ones in oher scale-free models for τ > 3. We shall discuss his analogy in more deail below. As we shall see in Secion 2.2, he proof of Theorem.4 also 4

5 reveals ha, whp, he ypical disance in G m,δ, which is he disance beween wo uniformly chosen conneced verices in he graph, is bounded from below by c 2 log. We conjecure ha, for δ > 0, a limi resul holds for he consan in fron of he log. In is saemen, we wrie dis G v, v 2 for he graph disance in he graph G beween wo verices v, v 2 []. Then, he ypical disance in a graph G is defined by dis G V, V 2 where V, V 2 [] are wo uniformly chosen independen verices. Conjecure.5 Convergence in probabiliy for δ > 0. Fix m and δ > 0. Then, he diameer diamg m,δ / log and he ypical disance dis G V, V 2 / log converge in probabiliy o posiive and differen consans. We now urn o he case where δ m, 0 and hence τ = 3 + δ/m 2, 3: Theorem.6 A log log upper bound on he diameer for δ < 0. Fix m 2 and assume ha δ m, 0. Then, for every σ > /3 τ and wih C G = he diameer of G m,δ is, whp, bounded above by C G log log, as. 4 log τ 2 + 4σ log m,.7 In his resul, we do no obain a sharp resul in erms of he consan. However, he proof suggess 4 ha for mos pairs of verices he disance should be equal o log τ 2 log log + o. When m =, Theorem.6 does no hold see he discussion below Theorem.3. We nex discuss he lower bound on he diameer for δ m, 0: Theorem.7 A log log lower bound on he diameer. Fix m 2 and δ > m. Then, he diameer of G m,δ is, whp, bounded below by log log, for all ε 0,. ε log m Unforunaely, he proof of Theorem.7 does no allow for an exension o ypical disances, and, hus, we have no maching lower bound for his. We finally conjecure ha, for δ m, 0, a limi resuls holds for he consan in fron of he log log : Conjecure.8 Convergence in probabiliy for δ < 0. Fix m 2 and δ m, 0. Then, he diameer diamg m,δ / log log and he ypical disance dis G V, V 2 / log log converge in probabiliy o posiive and differen consans..2 Discussion of universaliy of disances in power-law random graphs Theorems.3.7 prove ha he diameer in PA-models wih a power-law degree sequence denoed by τ undergoes a phase ransiion as τ changes from τ 2, 3 o τ > 3. The resuls idenify he order of growh of he diameer of hree relaed models of affine PA models as he size of he graph ends o infiniy. We do no obain he righ consans. For he ypical disance, we obain a similar phase ransiion, and again he resuls idenify he correc asympoics for τ > 3, bu, for τ 2, 3 we miss a maching lower bound. In non-rigorous work, i is ofen suggesed ha he disances are similarly behaved in he various scale-free random graph models, such as he configuraion model or various models wih condiional independence of edges as in []. For power-law random graphs, his informal saemen can be made precise by conjecuring ha disances have he same leading order growh in graphs wih he same powerlaw degree exponen. This, however, is no correc for he diameer of such power-law random graphs, since he diameer depends sensiively on he deails of he graph, such as he proporion of verices wih 5

6 degrees and 2. See [29] and [34] for resuls showing ha for he configuraion model wih power-law degree exponen τ 2, 3, he diameer can be of order log or of order log log depending on he proporion of verices wih degrees and 2, where is he size of he graph. Similarly, in inhomogeneous random graphs wih power-law degree exponen τ 2, 3 he diameer is always of order log see e.g. [], while he ypical disances can be of order log log see e.g. [8, 9]. Thus, we shall inerpre he physiciss predicion by conjecuring ha he leading order growh of he ypical disances of various power-law random graphs depends only on he power-law degree exponen τ 2, 3. The resuls on disances are mos complee for he configuraion model CM, see e.g. [27, 29, 32, 33, 43]. In he CM, here are various cases depending on he ails of he degree disribuion. When he degrees have infinie mean, hen ypical disances are bounded [27], when he degrees have finie mean bu infinie variance, ypical disances grow proporionally o log log [33, 43], where is he size of he graph, while, for finie variance degrees, he ypical disances grow proporionally o log [32]. Similar resuls for models wih condiionally independen edges exis, see e.g. [, 8, 26, 40], bu paricularly in he regime τ 2, 3, he resuls are no ha srong. Thus, for hese classes of models, disances are quie well undersood. If he disances in PA-models are similar o he ones in e.g. he CM, hen we should have ha he disances are of order log when τ > 3, i.e., δ > 0, while hey should be of order log log when τ 2, 3, i.e., for δ < 0. In PA-models wih a linear growh of he number of edges, infinie mean degrees canno arise, which explains why τ > 2 for PA-models. An aemp in he direcion of creaing PA-models wih power-law exponen τ, 2 can be found in [23], where a preferenial aachmen model is presened in which a random number of edges per new verex is added. In his model, i is shown ha he degrees again obey a power law wih exponen equal o τ = min{3 + δ µ, τ w}, where τ w is he power-law exponen for he number of edges added and µ he expeced number of added edges per verex. Thus, when τ w, 2, infinie mean degrees can arise. This model is furher sudied in [8], where a wealh of resuls for various PA-models can be found. There are few resuls on disances in PA-models. In [5], i was proved ha in model a and for log log log δ = 0, for which τ = 3, he diameer of he graph of size is equal o + o. Unforunaely, he maching resul for he CM has no been proved, so ha his does no allow us o verify wheher he models have similar disances. The resuls saed above subsaniae he physiciss predicion, since, for δ > 0 for which τ 3,, he ypical disances are of order log, while, for δ < 0, for which τ 2, 3, hey are bounded above by log log. A relaed resul on PA-models in he spiri of [22] can be found in [20], where a similar phase ransiion as in his paper is proved, in he case where he number of edges grows a leas log +ε imes as fas as he number of verices. I would be of ineres o improve he bounds presened in his paper up o he consan in fron of he log and log log, respecively. Due o he dynamical naure of PA-models, his is more involved for PA-models han i is for saic models such as he CM and inhomogeneous random graphs. This paper is organized as follows. In Secion 2, we prove he log lower bound for he diameer saed in Theorem.4. In Secion 3 and Secion 4, we prove he log log upper bound and he log log lower bound, on he diameer for δ < 0, of Theorem.6 and Theorem.7, respecively. 2 A log lower bound on he diameer for δ > 0: Proof of Theorem.4 In his secion, we prove Theorem.4 by exending he argumen in [5] from δ = 0 o δ > 0. We shall also exend he lower bound for δ = 0 o models b and c. For model c, denoe by {g, j = s}, j m, 2. he even ha a ime he j h edge of verex is aached o he earlier verex s <. For models a and b, his even means ha in {G,δ m} he edge from verex m + j is aached o one of he 6

7 verices ms +,..., ms. I is a direc consequence of he definiion of PA-models ha he even 2. increases he preference for verex s, and hence decreases in a relaive way he preference for he verices u, u, u s. I should be inuiively clear ha anoher way of expressing his effec is o say ha, for differen s s 2, he evens {g, j = s } and {g 2, j 2 = s 2 } are negaively correlaed. In order o sae such a resul, we inroduce some noaion. For ineger n s and i =,..., n s, we denoe by E s = n s i= { gi, j i = s }, 2.2 he even ha a ime i he j h i edge of verex i is aached o he earlier verex s, for all i =,..., n s. We will sar by proving ha for each k and all possible choices of i, j i, he evens E s, for differen s, are negaively correlaed: Lemma 2. Negaive correlaion of aachmen evens. For disinc s, s 2,..., s k, k P E si i= k PE si. 2.3 i= Proof. We will use inducion on he larges edge number presen in he evens E s. Here, for an even {g, j = s}, we le he edge number be m + j, which is he order of he edge when we consider he edges as being aached in sequence. The inducion hypohesis is ha 2.3 holds for all k and all choices of i, j i such ha max i,s m i + j i e, where inducion is performed wih respec o e. We iniialize he inducion for e = m in models a and b and for e = 2m in model c. We noe ha for his choice of e, he inducion hypohesis holds rivially, since everyhing is deerminisic. This iniializes he inducion. To advance he inducion, we assume ha 2.3 holds for all k and all choices of i, j i such ha max i,s m i + j i e. Clearly, for k and i, j i such ha max i,s m i + j i e, he bound follows from he inducion hypohesis, so we may resric aenion o he case ha max i,s m i +j i = e. We noe ha here is a unique choice of, j such ha m + j = e. In his case, here are again wo possibiliies. Eiher here is exacly one choice of s and i, j i such ha i =, j i = j, or here are a leas wo of such choices. In he laer case, we immediaely have ha k i= E si =, since he e h edge can only be conneced o a unique verex. Hence, here is nohing o prove. Thus, we are lef o invesigae he case where here exiss unique s and i, j i such ha i =, j i = j. Denoe by E s = n s i=: i,j i,j he resricion of E s o he oher edges. Then we can wrie { gi, j i = s }, 2.4 k E si = { g, j = s } E s k E si. 2.5 i= i=:s i s By consrucion, all he edge numbers of he evens in E s k i=:si s E s i are a mos e. Thus, we obain k [ k ] P E si E I[E s E si ]P e g, j = s, 2.6 i= i=:s i s where P e denoes he condiional probabiliy given he edge aachmens up o he e s edge connecion, and where, for an even A, I[A] denoes he indicaor of A. 7

8 We now firs rea model c, for which we have ha P e g, j = s = D s + δ 2m + δ. 2.7 We wish o use he inducion hypohesis. For his, we noe ha D s = m + I[g, j = s]. 2.8,j : We noe ha each of he erms in 2.8 has edge number sricly smaller han e and occurs wih a nonnegaive muliplicaive consan. As a resul, we may use he inducion hypohesis for each of hese erms. Thus, we obain, using also m + δ 0, ha, k 2m + δ P E si i= We can recombine o obain k P i= [ E si E m + δpe s +,j : k i=:s i s PE si PE s {g, j = s} I[E s] D s + δ ] k 2m + δ i=:s i s k i=:s i s PE si. 2.9 PE si, 2.0 and he advancemen is compleed when we noe ha [ E I[E s] D s + δ ] = PE s. 2. 2m + δ The proofs for models a and b are somewha simpler, since he evens E si can be reformulaed in erms of he graph process {G,δ }. We nex give he probabiliies of E s when n s 2; we omi he proof, since i is a simple adapaion o ha in [5]. Lemma 2.2 Connecions in PA-models. There exis absolue consans M, M 2, such ha i for each j m, and > s, P g, j = s M a s a, 2.2 and ii for 2 > > s, and any j, j 2 m, M 2 P g, j = s, g 2, j 2 = s 2 a, 2.3 s2a where a = m 2m+δ. We combine he resuls of Lemmas 2. and 2.2 ino he following corollary, yielding an upper bound for he probabiliy of he exisence of a pah. In is saemen, we call a pah Γ = s 0, s,..., s l self-avoiding when s i s j for all 0 i < j l. We use he noaion x y = minx, y and x y = maxx, y. Again, we omi he proof for deails, see [5]. Corollary 2.3 Pah probabiliies in PA-models. Le Γ = s 0, s,..., s l be a self-avoiding pah of lengh l consising of he l + unordered verices s 0, s,..., s l, hen here exiss an absolue consan C > 0 such ha l P Γ G m,δ m 2 C l i=0 s i s i+ a a. 2.4 s i s i+ 8

9 2. Lower bound on he diameer for δ = 0 I follows from 2.4 ha for δ = 0, The furher proof ha 2.5 implies ha for δ 0, l P Γ G m,δ m 2 C l. 2.5 si s i=0 i+ L = log log3cm 2 log, 2.6 is a lower bound for he diameer of G m,δ, is idenical o he proof of [5, Theorem 5, p. 4], wih n replaced by. This exends he lower bound for δ = 0 for model a in [5] o models b c. 2.2 The lower bound on disances for δ > 0 We nex improve he bound in he previous secion in he case when δ > 0, in which case a = m/2m+δ < /2. From he above discussion, we conclude ha P dis Gm,δ, = k c k s k j=0 s j s j+ a a, 2.7 s j s j+ where c = m 2 C, and where he sum is over s = s 0,..., s k wih s k =, s 0 =, s l for all l =,..., k and s l s n for all l n, since we may assume ha our pah s 0,..., s k is self-avoiding. Define f k i, = s k j=0 s j s j+ a a, 2.8 s j s j+ where now he sum is over s = s 0,..., s k wih s k =, s 0 = i, s l for all l =,..., k and s l s n for all l n, so ha P dis Gm,δ i, = k c k f k i,. 2.9 We sudy he funcion f k i, in he following lemma: Lemma 2.4 A bound on f k. Fix a < /2. Then, for every b > a such ha a + b <, here exiss a C a,b > 0 such ha, for every i < and all k, f k i, Ck a,b i b b Proof. We prove he lemma using inducion on k. To iniialize he inducion hypohesis, we noe ha, for i < and every b a, f i, = i a i a = i a a = a b = i i i b. 2.2 b This iniializes he inducion hypohesis as long as C a,b. To advance he inducion hypohesis, noe ha we have he recursion relaion f k i, i s= s a i a f k s, + 9 s=i+ i a s a f k s,. 2.22

10 We now bound each of hese wo conribuions, making use of he inducion hypohesis. For he firs sum, we bound i s= s a i a f k s, Ca,b k i s= s a i a s b b = since a + b <. For he second sum, we bound s=i+ i a s a f k s, Ca,b k = Ck a,b i a b s=i+ s=i+ i a s a s b s Ck i a,b i a b s= + Ck b a,b s=+ a+b + Ck a,b i a b s=+ s a+b Ca,b k a b i b, 2.23 b i a s a b s b 2.24 s 2 a b since + b a >, 2 a b >, b > a and /i a /i b. We conclude ha when f k i, Ck a,b i b b b a + 2 a b This advances he inducion hypohesis, and complees he proof. Using Lemma 2.4 and 2.9, we obain ha C k a,b C k a,b b a i b b + a b i b b, Ck a,b i b, 2.25 b C a,b = b a a b As a resul, P dis Gm,δ, = k cc a,b k b P diamg m,δ k P dis Gm,δ, k cc a,b k+ b = o, 2.28 cc a,b b whenever k log cc a,b log. We conclude ha here exiss c 2 = c 2 m, δ such ha, wih high probabiliy diam G m,δ c 2 log. We nex exend he above discussion o ypical disances. Lemma 2.5 Typical disances for δ > 0. Fix m and δ > 0. Le H = dis A, A 2 be he disance beween wo uniformly chosen verices. Then, for c 2 = c 2 m, δ > 0 sufficienly small, whp, H c 2 log. Proof. For c 2 = c 2 m, δ > 0, define B # { i, j [] : i < j : dis Gm,δ i, j c 2 log }, 2.29 where #{A} denoes he cardinaliy of he se A. By Lemma 2.4, wih K = log cc a,b 2 and a < b < a, and for all i < j, P dis Gm,δ i, j = k c k f k i, j ekk i b j b 0

11 As a resul, and hus, using also j i= i b j b / b, I now suffices o noe ha P dis Gm,δ i, j c 2 log Kc 2 e K i b j b e K, 2.3 E[B ] O i<j Kc 2 i b j b = OKc PH c 2 log = E [ I[dis Gm,δ A, A 2 c 2 log ] ] = 2E[B ] + 2 = o, 2.33 by 2.32, for every c 2 > 0 such ha Kc 2 + < 2. Noe ha 2.6 is also a lower bound on ypical disances in case δ = 0, which can be proved as above. 3 A log log upper bound on he diameer: Proof of Theorem.6 The proof of Theorem.6 is divided ino wo key seps. In he firs, in Theorem 3., we bound he diameer of he core which consiss of he verices wih degree a leas a cerain power of log. This argumen is close in spiri o he argumen in [8] or [43] used o prove bounds on he ypical disance for he inhomogeneous random graph and he configuraion model, respecively, bu subsanial adapaions are necessary o deal wih preferenial aachmen. Afer his, in Theorem 3.6, we derive a bound on he disance beween verices wih a small degree and he core. We sar by defining and invesigaing he core of he PA-model. In he sequel, i will be convenien o prove Theorem.6 for 2 raher han for. Clearly, his does no make any difference for he resuls. We make use of some echnical resuls, saed in he appendix. 3. The diameer of he core We recall ha τ = 3 + δ/m, so ha m < δ < 0 corresponds o τ 2, 3. We ake σ > /3 τ = m/δ > and define he core Core o be Core = { i [] : D i log σ}, 3. i.e., all he verices which a ime have degree a leas log σ. For A [], we wrie diam A = max i,j A dis G m,δ i, j. 3.2 Then, diam 2 Core is bounded in he following heorem: Theorem 3. The diameer of he core. Fix m 2 and δ m, 0. For every σ > /3 τ, whp, 4 log log diam 2 Core + o log τ The proof of Theorem 3. is divided ino several smaller seps. We sar by proving ha he diameer diam 2 Inner, where Inner = { } i [] : D i u, and where u = 2τ log 2, 3.4

12 is, whp, bounded. The choice of u is a echnical one: u is he larges value l so ha, whp, he oal degree of verices wih degree exceeding l can be bounded from below by l 2 τ, see Lemma A.. In Proposiion 3.2, we will show ha he diameer of Inner is bounded. Afer his, we will show ha he disance from any verex in he core Core o he inner core Inner can be bounded by a fixed consan imes log log. This also shows ha diam 2 Core is bounded by a differen consan imes log log. We now give he deails. Proposiion 3.2 The diameer of he inner core. Fix m 2 and δ m, 0. Then whp, diam 2 Inner 2τ 3 τ Proof. We firs inroduce he imporan noion of a -connecor beween a verex i [] and a se of verices A []. This noion will play a crucial role hroughou he proof. We say ha he verex j [2] \ [] is a -connecor beween i and A if one of he firs wo edges inciden o j connecs o i and he oher of he firs wo edges inciden o j connecs o a verex in A. Thus, when here exiss a -connecor beween i and A, he disance beween i and A in G m,δ 2 is a mos 2. We coninue he analysis by firs considering model c. We noe ha for a se of verices A and a verex i wih degree a ime equal o D i, we have ha, condiionally on G m,δ, he probabiliy ha j [2] \ [] is a -connecor for i and A is a leas D A + δ A D i + δ [22m + δ] 2 ηd AD i 2, 3.6 where in he inequaliy, we use ha D i m, and we le η = m + δ 2 /2m2m + δ 2 > 0, while, for any A [], we wrie D A = i A D i. 3.7 Noe ha for fixed j [2] \ [] he lower bound 3.6 holds independenly of he fac wheher he oher verices are -connecors or no. We now give a coupling proof which shows ha a subse of size n = of he se Inner has, whp, a bounded diameer. Lemma A. in he appendix shows ha, whp, Inner conains a leas verices. Denoe he firs verices of Inner by I. For each pair i, i 2 I and each j [2] \ [], he probabiliy ha j is a -connecor for i, i 2 is, by 3.6, a leas ηu 2 2 τ τ 2 = η 2 log log 2 = q, 3.8 independenly of he fac wheher he oher verices are -connecors or no. In he coupling we inend o compare he se I and all pairs of verices of he se I, which are -conneced by some j [2] \ [] wih a so-called mulinomial random graph H n. The graph H n has n verices and we idenify he e = n n /2 /2 pairs of verices, which we number from o e in an arbirary order, wih e cells of a mulinomial experimen wih rials and probabiliies given by p k = q, k e, p 0 = e q. 3.9 We can represen he rials by independen random vecors N, N 2,..., N, where N j = N j,, N j,2,..., N j,e, j, 3.0 wih disribuion PN j = i = q, PN j = 0 = e q, 3. 2

13 where i is he i h uni vecor of lengh e, and 0 he null vecor. If cell k of he mulinomial experimen is no empy, i.e., if j= N j,k > 0, hen we draw he edge wih number k in he graph H n, if he cell is empy hen his edge is lef ou. Noe ha cell 0 is jus an overflow cell, which couns he number of rials ha no resuled in one of he cells, 2,..., e. By he saemen in 3.8 he disance in G m,δ 2 beween any wo verices in I is a mos wo imes he disance beween he corresponding verices in H n. In Lemma A.2 of he appendix we will show ha he diameer of H n is a mos he diameer of a uniform Erdős-Rényi graph Gn, m, wih n verices and m edges, where m = 2 e q. 3.2 From [35, Secion.4] we conclude ha he above menioned uniform Erdős-Rényi graph Gn, m is asympoically equivalen wih he classical binomial Erdős-Rényi graph Gn, λ, where he edge probabiliy λ is defined by λ = q τ 2 2 log Nex, we show ha diamgn, λ is, whp, bounded by τ 3 τ +. For his we use he resuls in [9, Corollaries 0. and 0.2], which give sharp bounds on he diameer of an Erdős-Rényi random graph. Indeed, his resuls imply ha if p 2 n 2 log n and n 2 p, hen diamgn, p = 2, whp, while, for d 3, if log n/d 3 log log n and p d n d 2 log n, while p d n d 2 2 log n, hen diamgn, p = d, whp. In our case, n = n = /2 and p = λ, which implies ha, whp, diamgn, p = τ 3 τ +. We herefore obain ha he diameer of I in G m,δ2 is, whp, bounded by diam 2 I 2τ 3 τ We finally show ha for any i Inner \ I, he probabiliy ha here does no exis a -connecor connecing i and I is small. Indeed, since D I u and D i u, he menioned probabiliy is bounded above by ηd ID i { exp 2 ηd } { } ID i exp ηu2 exp η log τ 2 = o, 3.5 for τ < 3. Thus, whp, such a verex i does no exis. This proves ha whp he disance beween any verex i Inner \ I and I is bounded by 2, and, ogeher wih he above bound on diam 2 I we hus obain 3.5. Proposiion 3.3 Disance from he core o he inner core. Fix m 2 and δ m, 0. Wih high probabiliy, he inner core Inner can be reached from any verex in he core Core using no more han 2 log log log τ 2 edges in G m,δ 2. More precisely, whp, Proof. For k, we define max i Core min dis Gm,δ 2i, j j Inner 2 log log log τ N k = {i [] : D i u k }, 3.7 wih u defined in 3.4, and where we define u k, for k 2, recursively, so ha for any verex i [] wih degree a leas u k, he probabiliy ha here is no -connecor for he verex i and he se N k, condiionally on G m,δ, is iny. According o 3.6 and A. in he appendix, his probabiliy is a mos ηd N k D i 2 exp { 3 ηb 2 τ u k uk } = o 2, 3.8

14 for some B > 0, when we define u k = D log τ 2, u k 3.9 wih D exceeding 2ηB and is sufficienly large so ha u k u. The following lemma idenifies u k : Lemma 3.4 Idenificaion of u k. For each k N, u k = D a k log b k c k, 3.20 where a k = τ 2k τ 2k, b k = 3 τ 3 τ 2 τ 2k, c k = τ 2k 2τ. 3.2 Proof. We leave he sraighforward inducion proof o he reader. Then, he key sep in he proof of Proposiion 3.3 is he following lemma: Lemma 3.5 Conneciviy beween N k and N k. Fix m 2 and δ m, 0. Then, uniformly in k, he probabiliy ha here exiss an i N k ha is no a disance a mos wo from N k in G m,δ 2 is o. Proof. I follows from 3.8 ha he probabiliy in he saemen is by Boole s inequaliy bounded by exp ηb[u k ] 2 τ u k = o 2 = o We now complee he proof of Proposiion 3.3. Fix k = log log log τ As a resul of Lemma 3.5, we have ha he disance beween N k and Inner = N is a mos 2k. Therefore, Proposiion 3.3 follows when we can show ha Core = {i : D i log σ } N k = {i : D i u k }, 3.24 so ha i suffices o prove ha log σ u k, for any σ > /3 τ. This follows rivially for large from he explici represenaion of u k given by Lemma 3.4. Proof of Theorem 3.. We noe ha whp diam 2 Core 2τ 3 τ k, 3.25 where k is given in 3.23, and where we have made use of Proposiions 3.2 and 3.3. This proves Theorem 3.. 4

15 3.2 Connecing he periphery o he core In his secion, we exend he resuls of he previous secion and, in paricular, sudy he disance beween he verices no in he core Core and he core. The main resul is he following heorem: Theorem 3.6 Connecing he periphery o he core. Fix m 2 and δ m, 0. For every σ > /3 τ, whp, he maximal disance beween any verex and Core in G m,δ 2 is bounded from above by 2σ log log / log m. Togeher wih Theorem 3., Theorem 3.6 proves he main resul in Theorem.6. The proof of Theorem 3.6 consiss of wo key seps. The firs key sep in Proposiion 3.7 saes ha he disance beween any verex in [] and he core Core is bounded by a consan imes log log. The second key sep in Proposiion 3.0 shows ha he disance beween any verex in [2] \ [] and [] is bounded by anoher consan imes log log. Proposiion 3.7 Connecing half of he periphery o he core. Fix m 2 and δ m, 0. For every σ > /3 τ, whp, he disance beween any verex in [] and he core Core in G m,δ 2 is bounded from above by σ log log / log m. Proof. We sar from a verex i [] and will show ha he probabiliy ha he disance beween i and Core is a leas σ log log / log m is o. This proves he claim. For his, we explore he neighborhood of i as follows. From i, we connec is m 2 edges. Then, successively, we connec he m edges from each of he a mos m verices ha i has conneced o and have no ye been explored. We coninue in he same fashion. We call he arising process when we have explored up o disance k from he iniial verex i he k-exploraion ree of verex i. When we never connec wo edges o he same verex, hen he number of verices we can reach wihin k seps is precisely equal o m k. We call an even where an edge connecs o a verex which already was in he exploraion ree a collision. When k increases, he probabiliy of a collision increases. However, he probabiliy ha here exiss a verex for which more han l collisions occur in is k-exploraion ree, where l, before i his he core is small, as we prove now: Lemma 3.8 A bound on he probabiliy of muliple collisions. Fix m 2 and δ m, 0. Fix l, b 0, ] and ake k σ log log / log m. Then, for every verex i [], he probabiliy ha is k-exploraion ree has a leas l collisions before i his Core [ b ] is bounded above by v T k i m l log 2σl/ bl Proof. Take i [] \ [ b ] and consider is k-exploraion ree T k i. Since we add edges afer ime b he denominaor in.-.3 is a leas b. Moreover, before hiing he core, any verex in he k-exploraion ree has degree a mos log σ. Hence, for l =, he probabiliy menioned in he saemen of he lemma is a mos D v + δ b log σ b mk+ log σ b, 3.27 v T k i where he bound follows from δ < 0 and #{v T k i } m k+. For general l his upper bound becomes: m k+ log σ l When k = σ log log / log m, we have ha m kl = log σl. Therefore, he claim in Lemma 3.8 holds. b 5

16 We nex prove ha here exiss a b > 0 such ha, whp, [ b ] is a subse of he core. Noe ha in his lemma he condiions m 2 or δ m, 0 are no necessary. Lemma 3.9 Early verices have large degrees whp. Fix m. There exiss a b > 0 such ha, for every σ > /3 τ, whp, min j b D j log σ. As a resul, whp, [ b ] Core. We defer he proof of Lemma 3.9 o Secion A.3 of he appendix. Now we are ready o complee he proof of Proposiion 3.7: Proof of Proposiion 3.7. By combining Lemmas 3.8 and 3.9, he probabiliy ha here exiss an i [] for which he exploraion ree T k i has a leas l collisions before hiing he core is o, whenever l > /b, since, by Boole s inequaliy, i is bounded by m l log 2σl/ bl = m l log 2σl bl+ = o i= When he k-exploraion ree his he core, hen we are done. When he k-exploraion ree from a verex i does no hi he core, bu has less han l collisions, hen here are a leas m k l verices in k-exploraion ree. Indeed, when we have a mos l collisions, he size of he k-exploraion ree is minimal when all edges of he roo connec o he same verex v, all edges of v connec o he same verex v 2, ec. Ieraing his a mos l levels deep yields a ree wih a leas m k l verices. When k = σ log log / log m 2, m k l log σ+o. The oal degree of he core is, by A. in he appendix, a leas i Core D i Blog τ 2σ, 3.30 for some B > 0. The probabiliy ha here does no exis a -connecor beween he k-exploraion ree and he core is, by 3.6 and 3.30, bounded above by exp since σ > /3 τ. This complees he proof. { ηblog τ 2σ log σ+o } = o, 3.3 Proposiion 3.0 Connecing he remaining periphery. Fix m 2 and δ m, 0. For every σ > /3 τ, whp, he maximal disance beween any verex and [] in G m,δ 2 is bounded from above by σ log log / log m. Proof. Take k = σ log log / log m, and j [2]\[] wih disance larger han k o he se of verices []. We now apply Lemma 3.8 wih replaced by 2 and leing l = 2 and b = b 0, such ha 2 b =, o conclude ha wih probabiliy exceeding o, he k-exploraion ree of j has a mos collision before i his Core 2 []. We can hence conclude ha wih probabiliy exceeding o, here are a leas m k = m m k verices in [2] \ [] a disance precisely equal o k from our saring verex j. Denoe hese verices by i,..., i mk. We consider case c, he proof for a and b is similar. Noe ha, uniformly in s [2] \ [], i= D i s + δ 2m + δs Hence, P l [m k ] such ha dis Gm,δ 2i l, Core 2 [] 2 m k = o,

17 since m k = m log m σ, wih σ > /3 τ >. Therefore, any verex j [2] \ [] is, whp, wihin 2 disance k + from Core 2 []. Proposiion A.3 shows ha, whp he se Core 2 [], so ha, whp, Core 2 [] = [] and he proposiion follows. Proof of Theorem 3.6. Proposiion 3.0 saes ha whp every verex in G m,δ 2 is wihin disance σ log log / log m of [] and Proposiion 3.7 saes ha whp every verex in [] is a mos disance σ log log / log m from he core Core. This shows ha every verex in G m,δ 2 is whp wihin disance 2σ log log / log m from he core. Proof of Theorem.6. Theorem 3.6 saes ha every verex in G m,δ 2 is wihin disance 4 log log log τ 2 2σ log log log m he core Core. Theorem 3. saes ha he diameer of he core is a mos + o, so ha he diameer of G m,δ 2 is a mos C G log log, where C G is given in.7, because we can choose any σ > /3 τ. This complees he proof of Theorem.6. 4 A log log lower bound on he diameer: Proof of Theorem.7 We will again prove his heorem for ime 2 raher han ime. To show ha he diameer of he graph is, whp, a leas k, we will sudy, a ime 2, he k-exploraion rees T k i of verices i [2]\[] as defined above. We shall call he ree T k i proper if he following condiions hold: The k-exploraion ree has no collisions; of All verices of T k i are in [2]\[]; No oher verex connecs o a verex in T k i. When such a ree exiss in G m,δ 2 for a cerain verex i hen we know ha he diameer is a leas k, since he disance beween he roo of he ree i and he verices a deph k is exacly k; here canno be a shorer roue. To prove ha a proper k-exploraion ree exiss in G m,δ 2, we will use a second momen mehod. Le T k m2 be he se of all possible k-exploraion rees ha can exis in G m,δ 2 and saisfy he firs wo condiions. Noe ha he order in which he edges are added maers: if wo edges are added in a differen order, hen he arising exploraion ree will be considered a differen ree. Le Z k m,δ 2 be he number of proper k-exploraion rees in G m,δ 2, i.e., Z k m,δ 2 = T T k m2 I[T G m,δ 2 and T is proper]. 4. Here he even ha all edges of T have been formed in G m,δ 2 is denoed by T G m,δ 2. In Secion 4. we will invesigae he firs momen of Z k m,δ 2 and prove he following: Proposiion 4. Expeced number of proper rees ends o infiniy. Fix m 2 and δ > m. Le k = log log, wih 0 < ε <. Then ε log m [ ] lim E Z k m,δ 2 =. 4.2 The variance of Z k m,δ 2 will be he subjec of Secion 4.2, where we will prove he following: 7

18 Proposiion 4.2 Concenraion of he number of proper rees. Fix m 2, δ > m and le 0 k log log log m. Then here exiss a consan c m,δ > 0, such ha, for sufficienly large, Var Z k m,δ 2 log 2 [ ] 2 [ ] c m,δ E Z k m,δ 2 + E Z k m,δ We use hese wo proposiions o prove Theorem.7: Proof of Theorem.7. We firs use he Chebychev inequaliy o obain ha P diamg m,δ 2 < k P Z k m,δ 2 = 0 Var Z k m,δ 2 [ ] E Z k m,δ 2 By Proposiion 4.2, he righ-hand side of 4.4 is, for some consan c m,δ > 0, a mos by Proposiion 4.. c m,δ log 2 4. The firs momen of he number of proper rees + [ ] E Z k m,δ 2 = o, 4.5 Le B T denoe he even ha no verex ouside a ree T connecs o a verex in his ree. We can hen wrie ha he expeced number of proper k-exploraion rees in G m,δ 2 equals [ ] E Z k m,δ 2 = P T G m,δ 2 and T is proper T T k m2 = T T k m2 = T T k m2 P T is proper T G m,δ 2 P T G m,δ 2 P B T T G m,δ 2 P T G m,δ We will firs give a lower bound on he probabiliy ha a given k-exploraion ree exiss in he graph a ime 2. For convenience we will wrie a m,δ = m+δ 32m+δ. Lemma 4.3 Lower bound on exisence probabiliy. Fix m 2, δ > m and k 0. Given a proper k-exploraion ree T T k m2, hen, for sufficienly large, where m k = mk+ m. P T G m,δ 2 am,δ m k, 4.7 Proof. Since every verex is added before ime 2, he denominaor in..3 is a mos 32m + δ. The degree of all verices already in he graph is a leas m, so he probabiliy ha a cerain given edge is formed is a leas m + δ 32m + δ = a m,δ. 4.8 Since exacly m k edges have o be formed o form he given ree T, we have ha P T G m,δ 2 am,δ m k

19 We will now give a lower bound on he probabiliy ha no oher verex connecs o a given ree. log log Lemma 4.4 No oher verex connecs o T. Fix m 2, δ > m and 0 k log m. Given a proper k-exploraion ree T T k m2, hen, for sufficienly large, and wriing m δ = m + + δ >, P B T T G m,δ 2 m δm k+ m. 4.0 Remark 4.5. In P B T T G m,δ 2, B T makes a claim abou edges no in T, while he even T G m,δ 2 saes ha all edges in T are formed in our random graph process. Thus condiioning on T G m,δ 2 gives informaion only abou inside edges. Proof. Firs noe ha for k m δ m k+ log log log m and sufficienly large, m δm k+ m δ m log. So 0. Furher noe ha verices in [] canno connec o a verex in T, since T [2]\[]. In he remainder of he proof we will refer o ouside edges as hose edges ha do no belong o T, of which here are exacly m m k added afer ime. For A a se of verices, le E n A denoe he even ha he n-h ouside edge added afer ime connecs o a verex in A and le E n A be he complemen of E n A. We use inducion on he number of ouside edges ha did no connec o he ree T, i.e., we show ha: n P E i T T G m,δ 2 m δm k+ n, 4. i= by inducion on n = 0,..., m m k. For n = 0 he above holds, because boh sides equal. Now assume ha he above holds for 0 n < m m k, hen n+ P E i T T G m,δ 2 i= n n = P E n+ T E i T {T G m,δ 2} P E i T T G m,δ 2 i= i= n P E n+ T E i T {T G m,δ 2} m δm k+ n. 4.2 i= Since i is known ha a he ime ha he n + -s ouside edge afer ime is added, no oher ouside edge has conneced o a verex in he ree, we know ha he degree of all verices in he ree a ha momen is a mos m +. Furher, since his edge is added afer ime, he denominaor of..3 will be a leas. Thus, he righ-hand side of 4.2 is a leas i T m + + δ m δm k+ n m δm k+ m δm k+ n = m δm k+ n+, 4.3 where he inequaliy holds because here are less han m k+ verices in he ree. Applying he above o n = m m k, we obain ha P B T T G m,δ 2 m δm k+ m m k m δm k+ m

20 We finally give a lower bound on he number of possible proper k-exploraion rees ha can be formed. I should be noed ha when a verex i connecs o a verex j, we will always have ha i > j. So when exploring a verex i in he exploraion ree, all m verices his verex connecs o have a smaller label han i. Lemma 4.6 Number of proper rees. Fix m 2 and 0 k he number of possible proper k-exploraion rees a ime 2 is a leas m k = mk+ m. log log log m. Then, for sufficienly large, /m k+ m k, where we recall ha log log Proof. For sufficienly large and k log m, mk+ m log. Since he k-exploraion ree of a verex i has o be proper, here are no collisions, so he number of verices in he ree equals #{v T k i } = m k. 4.5 For any subse X [2]\[] wih #{v X} = m k here exiss a leas one possible proper k-exploraion ree. To see his, firs order he verex labels in descending order. Le he firs verex, i.e. he verex wih he larges label, be he roo of he ree. Then le he nex m verices be he verices a disance from he roo, he nex m 2 verices be he verices a disance 2 from he roo, eceera, unil he las m k verices which will be a disance k from he roo. This way, all verices will connec o m verices wih a smaller label, i.e., verices ha were already in he graph when he verex was added, so his is a possible proper k-exploraion ree wih all verices in X. The number of subses of [2]\[] of size m k is m k which is a leas m k m k m k m k+, 4.6 where we used ha for b a we have ha a ib b ia for all 0 i < b, so ha a b a i a b = b b i. 4.7 b i=0 We can now combine he hree bounds above o ge a lower bound on he expeced number of proper k-exploraion rees. Corollary 4.7 Lower bound on expeced number of proper rees. Fix m 2, δ > m and 0 k. Then, for sufficienly large, log log log m [ ] E Z k m,δ 2 a m,δ am,δ m k+ m k+ m δm k+ m

21 Proof. Using he bounds from Lemmas 4.3, 4.4 and 4.6 we ge ha [ ] E Z k m,δ 2 = P B T T G m,δ 2 P T G m,δ 2 T T k m 2 #{T T k m2} m δm k+ m k m k+ m δm k+ m k+ am,δ m k+ m δm k+ a m,δ m am,δ m k m am,δ m k m. 4.9 The facor in he corollary above urns ou o be crucial for he remainder of he proof. This facor arises from he fac ha here is exacly one edge less in a proper k-exploraion ree han here are verices. We can now show ha he expeced number of k-exploraion rees ends o infiniy, for k = ε log m log log, wih 0 < ε <. Proof of Proposiion 4.. Firs noe ha for k = hen use Corollary 4.7 o ge ha since am,δ m k+ [ ] lim E Z k m,δ 2 lim m k+ = am,δ mlog ε mlog ε a m,δ ε log m log log, wih 0 < ε <, mk = log ε. We can am,δ, and m k+ m k+ mm δm k+ m =, 4.20 mm δm k+ m m m e m2 m δ log ε. 4.2 I is easy o see ha he same argumen can be applied o k = log log log m log log log log m. 4.2 The second momen of he number of proper rees In his secion we will invesigae he variance of Z k m,δ 2. To shoren he noaion, for a k-exploraion ree T T k m2, le F T denoe he even ha T G m,δ 2 and T is proper. Then, he variance of he number of proper k-exploraion rees in G m,δ 2 is given by Var Z k m,δ 2 = Var I[T G m,δ 2 and T is proper] T T k m2 = Var I[F T ] = Cov I[F T ], I[F T ] T T k m 2 T,T T k m 2 = P F T F T P F T P F T + P F T P F T T,T T k m2 T T k T =T m 2 We sar by sudying he erms of he firs sum in he following lemma. 2

22 Lemma 4.8 Weak dependence of ree occurrences. Fix m 2, δ > m and 0 k T, T T k m2 wih T T. Then, for sufficienly large, log log log m. Le P F T F T P F T P F T + 2m δm log 2m log P F T P F T Proof. When T T, a leas one edge of one of he rees will connec o a verex in he oher ree, so he rees T and T canno boh be proper. Thus, for T T, rivially 4.23 holds. For T T =, we have o ake a closer look a he probabiliies involved. All hree probabiliies in he lemma are a produc over all edges of he probabiliy ha eiher he edge does no connec o any of he verices in he rees or he probabiliy ha he edge makes a prescribed connecion in one of he rees. Le E j,s A denoe he even ha he j-h edge of verex s connecs o a verex in A, wih E j,s i = E j,s {i}. Le E j,s A be he complemen of E j,s A. We have ha PE j,s A = i A PE j,s i, 4.24 because he evens on he righ-hand side are disjunc. These probabiliies are given by he growh rules..3. Suppose ha he j-h edge, j m, of a verex 0 should no connec o a verex in T T. Then in P F T F T, here will be a facor P E j,0 T T = P E j,0 T T = P E j,0 i i T T In P F T P F T, here will be a facor P E j,0 i P E j,0 i i T i T I is easy o see ha x y x y for x, y 0, so 4.26 is a leas as big as When he j-h edge, j m, of a verex 0, + 0 2, should connec o a verex h T, hen in P F T F T here will only be a facor P E j,0 h, 4.27 since i will hen auomaically no connec o a verex in T. In P F T P F T, however, here will be a facor P E j,0 h P E j,0 i i T When we muliply 4.28 by i T P E j, 0 i we obain precisely By symmery, he same holds when an edge should connec o a verex in T. Since he degree of he verices in he rees is a mos m +, he edges of ineres are added afer ime and here are less han m k+ verices in he ree, we have ha P E j,0 i i T m δm k