Distances in random graphs with infinite mean degrees

Transcription

1 Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree sequence of wic te tail of te distribution function F is regularly varying wit exponent τ [1, 2]. In particular, te degrees ave infinite mean. Suc random graps can serve as models for complex networks were degree power laws are observed. Te minimal number of edges between two arbitrary nodes, also called te grap distance or te opcount, is investigated wen te size of te grap tends to infinity. Te paper is part of a sequel of tree papers. Te oter two papers study te case were τ 2, 3, and τ 3,, respectively. Te main result of tis paper is tat te grap distance for τ 1, 2 converges in distribution to a random variable wit probability mass exclusively on te points 2 and 3. We also consider te case were we condition te degrees to be at most N α for some α > 0, were N is te size of te grap. For fixed k {0, 1, 2,...} and α suc tat τ + k 1 < α < τ + k 1 1, te opcount converges to k + 3 in probability, wile for α > τ 1 1, te opcount converges to te same limit as for te unconditioned degrees. Te proofs use extreme value teory. Key words and prases: Complex networks, Extreme value teory, Internet, random graps. AMS 2000 Subject classifications: Primary 60G70, Secondary 05C80. 1 Introduction Te study of complex networks as attracted considerable attention in te past decade. Tere are numerous examples of complex networks, suc as co-autorsip and citation networks of scientists, te World-Wide Web and Internet, metabolic networks, etc. Te topological structure of networks affects te performance in tose networks. For instance, te topology of social networks affects te spread of information and disease see e.g., [17, 20], wile te performance of traffic in Internet depends eavily on te topology of te Internet. Measurements on complex networks ave sown tat many real networks ave similar properties. A first example of suc a fundamental network property is te fact tat typical distances between nodes are small. Tis is called te small world penomenon, see te pioneering work of Watts [21], and te references terein. In Internet, for example, messages cannot use more tan a tresold of pysical links, and if te distances in Internet would be large, ten service would simply break down. Tus, te grap of te Internet as evolved in suc a way tat typical distances are relatively small, even toug te Internet is rater large. Delft University of Tecnology, Electrical Engineering, Matematics and Computer Science, P.O. Box 5031, 2600 GA Delft, Te Neterlands. H.vandenEsker@ewi.tudelft.nl, G.Hoogiemstra@ewi.tudelft.nl Department of Matematics and Computer Science, Eindoven University of Tecnology, P.O. Box 513, 5600 MB Eindoven, Te Neterlands. rofstad@win.tue.nl EURANDOM, P.O. Box 513, 5600 MB Eindoven, Te Neterlands. znamenski@eurandom.nl 1

2 A second, maybe more surprising, property of many networks is tat te number of nodes wit degree n falls off as an inverse power of n. Tis is called a power law degree sequence. In Internet, te power law degree sequence was first observed in [8]. Te observation tat many real networks ave te above properties as incited a burst of activity in network modelling. Most of te models use random graps as a way to model te uncertainty and te lack of regularity in real networks. See [3, 17] and te references terein for an introduction to complex networks and many examples were te above two properties old. Te current paper presents a rigorous derivation for te random fluctuations of te grap distance between two arbitrary nodes also called te opcount in a grap wit i.i.d. degrees aving infinite mean. Te model wit i.i.d. degrees is a variation of te configuration model, wic was originally proposed by Newman, Strogatz and Watts [18], were te degrees originate from a given deterministic sequence. Te observed power exponents are in te range from τ = 1.5 to τ = 3.2 see [3, Table II] or [17, Table II]. In a previous paper [10], te case τ > 3 was investigated, wile te case τ 2, 3 was studied in [11]. Here we focus on te case τ 1, 2, and study te typical distances between arbitrary connected nodes. In a fortcoming paper [12], we will survey te results from te different cases for τ, and investigate te connected components of te random graps. Tis section is organized as follows. In Section 1.1, we start by introducing te model, and in Section 1.2, we state our main results. In Section 1.3, we explain euristically ow te results are obtained. Finally, we describe related work in Section Te model Consider an i.i.d. sequence D 1, D 2,..., D N. Assume tat L N = N j=1 D j is even. Wen L N is odd, ten we increase D N by 1, i.e., we replace D N by D N + 1. Tis cange will make ardly any difference in wat follows, and we will ignore it in te sequel. We will construct a grap in wic node j as degree D j for all 1 j N. We will later specify te distribution of D j. We start wit N separate nodes and incident to node j, we ave D j stubs wic still need to be connected to build te grap. Te stubs are numbered in an arbitrary order from 1 to L N. We continue by matcing at random te first stub wit one of te L N 1 remaining stubs. Once paired, two stubs form an edge of te grap. Hence, a stub can be seen as te left or te rigt alf of an edge. We continue te procedure of randomly coosing and pairing te next stub and so on, until all stubs are connected. Te probability mass function and te distribution function of te nodal degree are denoted by PD 1 = j = f j, j = 1, 2,..., and F x = f j, 1.1 were x is te largest integer smaller tan or equal to x. Our main assumption will be tat x j=1 x τ 1 [1 F x] 1.2 is slowly varying at infinity for some τ 1, 2. Tis means tat te random variables D j obey a power law wit infinite mean. 1.2 Main results We define te grap distance H N between te nodes 1 and 2 as te minimum number of edges tat form a pat from 1 to 2, were, by convention, tis distance equals if 1 and 2 are not connected. Observe tat te grap distance between two randomly cosen nodes is equal in distribution to H N, because te nodes are excangeable. In tis paper, we will present two separate teorems for te case τ 1, 2. We also consider te boundary cases τ = 1 Teorem 1.3 and τ = 2 Remark 1.4. In Teorem 1.1, we take te 2

3 sequence D 1, D 2,..., D N of i.i.d. copies of D wit distribution F, satisfying 1.2, wit τ 1, 2. Te result is tat te grap distance or opcount converges in distribution to a limit random variable wit mass p = p F, 1 p, on te values 2, 3, respectively. In te paper te abbreviation wp, means tat te involved event appens wit probability converging to 1, as N. Teorem 1.1 Fix τ 1, 2 in 1.2 and let D 1, D 2,..., D N be a sequence of i.i.d. copies of D. Ten, lim PH N = 2 = 1 lim PH N = 3 = p F 0, N N One migt argue tat including degrees larger tan N 1 is artificial in a network wit N nodes. In fact, in many real networks, te degree is bounded by a pysical constant. Terefore, we also consider te case were te degrees are conditioned to be smaller tan N α, were α is an arbitrary positive number. Of course, we cannot condition on te degrees to be at most M, were M is fixed and independent on N, since in tis case, te degrees are uniformly bounded, and tis case is treated in [10]. Terefore, we consider cases were te degrees are conditioned to be at most a given power of N. Te result wit conditioned degrees appears in te Teorem 1.2. It turns out tat for α > 1/τ 1, te conditioning as no influence in te sense tat te limit random variable is te same as tat for te unconditioned case. Tis is not so strange, since te maximal degree is of order N 1/τ 1, so tat te conditioning does noting in tis case. However, for fixed k N {0} and α suc tat 1/τ +k < α < 1/τ +k 1, te grap distance converges to a degenerate limit random variable wit mass 1 on te value k + 3. It would be of interest to extend te possible conditioning scemes, but we will not elaborate furter on it in tis paper. In te teorem below, we write D N for te random variable D conditioned on D < N α. Tus, PD N = j = f j PD < N α, 0 j < N α. 1.4 Teorem 1.2 Fix τ 1, 2 in 1.2 and let D N 1, D N 2,..., D N N D N. be a sequence of i.i.d. copies of i For k N {0} and α suc tat 1/τ + k < α < 1/τ + k 1, ii If α > 1/τ 1, ten were p F 0, 1 is defined in Teorem 1.1. lim PH N = k + 3 = N lim PH N = 2 = 1 lim PH N = 3 = p F, 1.6 N N Te boundary case τ = 1 and τ = 2 are treated in Teorem 1.3 and Remark 1.4, below. We will prove tat for τ = 1, te opcount converges to te value 2. For τ = 2, we sow by presenting two examples, tat te limit beavior depends on te beavior of te slowly varying tail x[1 F x]. Teorem 1.3 For τ = 1 in 1.2 and let D 1, D 2,..., D N be a sequence of i.i.d. copies of D. Ten, lim PH N = 2 = N Remark 1.4 For τ = 2 in 1.2 and wit D 1, D 2,..., D N a sequence of i.i.d. copies of D, te limit beavior of H N depends on te slowly varying tail x[1 F x], x. In Section 4.2, we present two examples, were we ave, depending on te slowly varying function x[1 F x], different limit beavior for H N. We present an example wit lim N PH N k = 0, for all fixed integers k, as N, and a second example were H N {2, 3}, wp, as N. 3

4 1.3 Heuristics Wen τ 1, 2, we consider two different cases. In Teorem 1.1, te degrees are not conditioned, wile in Teorem 1.2 we condition on eac node aving a degree smaller tan N α. We now give a euristic explanation of our results. In two previous papers [10, 11], te cases τ 2, 3 and τ > 3 ave been treated. In tese cases, te probability mass function {f j } introduced in 1.1 as a finite mean, and te te number of nodes on grap distance n from node 1 can be coupled to te number of individuals in te n t generation of a brancing process wit offspring distribution {g j } given by g j = j + 1 µ f j, 1.8 were µ = E[D 1 ]. For τ 1, 2, as we are currently investigating, we ave µ =, and te brancing process used in [10, 11] does not exist. Wen we do not condition on D j being smaller tan N α, ten L N is te i.i.d. sum of N random variables D 1, D 2,..., D N, wit infinite mean. It is well known tat in tis case te bulk of te contribution to L N = N 1/τ 1+o1 comes from a finite number of nodes wic ave giant degrees te so-called giant nodes. A basic fact in te configuration model is tat two sets of stubs of sizes n and m are connected wp wen nm is at least of order L N. Since te giant nodes ave degree rougly N 1/τ 1, wic is muc larger tan L N, tey are all attaced to eac oter, tus forming a complete grap of giant nodes. Eac stub is wp attaced to a giant node, and, terefore, te distanc e between any two nodes is, wp, at most 3. In fact, tis distance equals 2 wen te two nodes are connected to te same giant node, and is 3 oterwise. In particular, for τ = 1, te quotient D N /L N converges to 1 in probability, and consequently te opcount converges to 2, in probability. Wen we truncate te distribution as in 1.4, wit α > 1/τ 1, we ardly cange anyting since witout truncation wp all degrees are below N α. On te oter and, if α < 1/τ 1, ten, wit truncation, te largest nodes ave degree of order N α, and L N N 1+α2 τ. Again, te bulk of te total degree L N comes from nodes wit degree of te order N α, so tat now tese are te giant nodes. Hence, for 1/τ < α < 1/τ 1, te largest nodes ave degrees muc larger tan L N, and tus, wp, still constitute a complete grap. Te number of giant nodes converges to infinity, as N. Terefore, te probability tat two arbitrary nodes are connected to te same giant node converges to 0. Consequently, te opcount equals 3, wp. If α < 1/τ, ten te giant nodes no longer constitute a complete grap, so tat te opcount can be greater tan 3. For almost every α < 1/τ, te opcount coverges to a single value. Te beavior of te opcount for te cases tat α = 1/τ + k for k N {0}, will be dependent on te slowly varying function in 1.2, as is te case for τ = 2. We do expect tat te opcount converges to at most 2 values in tese cases. Te proof in tis paper is based on detailed asymptotics of te sum of N i.i.d. random variables wit infinite mean, as well as on te scaling of te order statistics of suc random variables. Te scaling of tese order statistics is crucial in te definition of te giant nodes wic are described above. Te above considerations are te basic idea in te proof of Teorem 1.1. In te proof of Teorem 1.2, we need to investigate wat te conditioning does to te scaling of bot te total degree L N, as well as to te largest degrees. 1.4 Related work Te above model is a variation of te configuration model. In te usual configuration model one often starts from a given deterministic degree sequence. In our model, te degree sequence is an i.i.d. sequence D 1,..., D N wit distribution equal to a power law. Te reason for tis coice is tat we are interested in models for wic all nodes are excangeable, and tis is not te case wen 4

5 te degrees are fixed. Te study of tis variation of te configuration model was started in [18] for te case τ > 3 and studied by Norros and Reittu [19] in case τ 2, 3. For a survey of complex networks, power law degree sequences and random grap models for suc networks, see [3] and [17]. Tere, a euristic is given wy te opcount scales proportionally to log N, wic is originally from [18]. Te argument uses a variation of te power law degree model, namely, a model were an exponential cut off is present. An example of suc a degree distribution is f j = Cj τ e j/κ 1.9 for some κ > 0. Te size of κ indicates up to wat degree te power law still olds, and were te exponential cut off starts to set in. Te above model is treated in [10] for any κ <, but, for κ =, falls witin te regimes were τ 2, 3 in [11] and witin te regime in tis paper for τ 1, 2. In [18], te autors conclude tat since te limit as κ does not seem to converge, te average distance is not well defined wen τ < 3. In tis paper, as well as in [11], we sow tat te average distance is well defined, but it scales differently from te case were τ > 3. In [12], we give a survey to te results for te opcount in te tree different re gimes τ 1, 2, τ 2, 3 and τ > 3. Tere, we also prove results for te connectivity properties of te random grap in tese cases. Tese results assume tat te expected degree is larger tan 2. Tis is always te case wen τ 1, 2, and stronger results ave been sown tere. We prove tat te largest connected component as wp size N1 + o1. Wen τ 1, 3 2 we even prove tat te grap is wp connected. Wen τ > 3 2 tis is not true, and we investigate te structure of te remaining dust tat does not belong to te largest connected component. Te analysis makes use of te results obtained in tis paper for τ 1, 2. For instance, it will be crucial tat te probability tat two arbitrary nodes are connected converges to 1. Tere is substantial related work on te configuration model for te cases τ 2, 3 and τ > 3. References are included in te paper [11] for te case τ 2, 3, and in [10] for τ > 3. We again refer to te references in [12] and [3, 17] for more details. Te grap distance for τ 1, 2, tat we study ere, as, to our best knowledge, not been studied before. Values of τ 1, 2 ave been observed in networks of messages and networks were te nodes consist of software packages see [17, Table II], for wic our configuration model wit τ 1, 2 can possibly give a good model. In [1], random graps are considered wit a degree sequence tat is precisely equal to a power law, meaning tat te number of nodes wit degree n is precisely proportional to n τ. Aiello et al. [1] sow tat te largest connected component is of te order of te size of te grap wen τ < τ 0 = , were τ 0 is te solution of ζτ 2 2ζτ 1 = 0, and were ζ is te Riemann zeta function. Wen τ > τ 0, te largest connected component is of smaller order tan te size of te grap and more precise bounds are given for te largest connected component. Wen τ 1, 2, te grap is wp connected. Te proofs of tese facts use couplings wit brancing processes and strengten previous results due to Molloy and Reed [15, 16]. See also [1] for a istory of te problem and references predating [15, 16]. See [2] for an introduction to te matematical results of various models for complex networks also called massive graps, as well as a detailed account of te results in [1]. A detailed account for a related model can be found in [5] and [6], were links between nodes i and j are present wit probability equal to w i w j / l w l for some expected degree vector w = w 1,..., w N. Cung and Lu [5] sow tat wen w i is proportional to i 1 τ 1, te average distance log log N between pairs of nodes is proportional log N1+o1 wen τ > 3, and equal to 2 logτ 2 1+o1 wen τ 2, 3. In teir model, also τ 1, 2 is possible, and in tis case, similarly to τ 1, 3 2 in our paper, te grap is connected wp. Te difference between tis model and ours is tat te nodes are not excangeable in [5], but te observed penomena are similar. Tis can be understood as follows. Firstly, te actual degree vector in [5] sould be close to te expected degree vector. Secondly, for te expected degree vector, 5

6 we can compute tat te number of nodes for wic te degree is at least n equals {i : w i n} = {i : ci 1 τ 1 n} n τ+1, were te proportionality constant depends on N. Tus, one expects tat te number of nodes wit degree at least n decreases as n τ+1, similarly as in our model. In [6], Cung and Lu study te sizes of te connected components in te above model. Te advantage of working wit te expected degree model is tat different links are present independently of eac oter, wic makes tis model closer to te classical random grap Gp, N. 1.5 Organization of te paper Te main body of te paper consists of te proofs of Teorem 1.1 in Section 2 and te proof of Teorem 1.2 in Section 3. Bot proofs contain a tecnical lemma and in order to make te argument more transparent, we ave postponed te proofs of tese lemmas to te appendix. Section 4 contains te proof of Teorem 1.3 and two examples for te case τ = 2. Section 5 contains simulation results, conclusions and open problems. 2 Proof of Teorem 1.1 In tis section, we prove Teorem 1.1, wic states tat te opcount between two arbitrary nodes as wp a non-trivial distribution on 2 and 3. We start wit an outline of our proof. Below, we introduce an event A ε,n, suc tat wen A ε,n occurs, te opcount between two arbitrary nodes is eiter 2 or 3. We ten prove tat PA c ε,n < ε, for N N ε see Lemma 2.2 below. For tis we need a modification of te extreme value teorem for te k largest degrees, for all k N. We introduce D 1 D 2 D N, to be te order statistics of D 1,..., D N, so tat D 1 = min{d 1,..., D N }, D 2 is te second smallest degree, etc. Let u N be an increasing sequence suc tat lim N [1 F u N] = N It is well known tat te order statistics of te degrees, as well as te total degree, are governed by u N in te case tat τ 1, 2. Te following lemma sows tis in detail. In te lemma E 1, E 2,... is an i.i.d. sequence of exponential random variables wit unit mean and Γ j = E 1 + E E j, ence Γ j as a gamma distribution wit parameters j and 1. Trougout te paper, equality in distribution is denoted by te symbol d =, wereas d denotes convergence in distribution.. Lemma 2.1 Convergence in distribution of order statistics For any k N, LN, D N,..., D N k+1 u N u N u N d η, ξ 1,..., ξ k, as N, 2.2 were η, ξ 1,..., ξ k is a random vector wic can be represented by η, ξ 1,..., ξ k = d Γ 1/τ 1 j, Γ 1/τ 1 1,..., Γ 1/τ j=1 k Moreover, ξ k 0 in probability, as k

7 Proof. Because τ 1 0, 1, te proof is a direct consequence of [14, Teorem 1 ], and te continuous mapping teorem [4, Teorem 5.1], wic togeter yield tat on R R, equipped wit te product topology, we ave S # N, Z N d S #, Z, 2.5 were S # N = u 1 N L N, Z N = u 1 N D N,..., D 1, 0, 0,..., and Z j = Γ 1/τ 1 j, j 1. If we subsequently take te projection from R R R k+1, defined by πs, z = s, z 1,..., z k, 2.6 i.e., we keep te sum and te k largest order statistics, ten we obtain 2.2 and 2.3 from, again, te continuous mapping teorem. Finally, 2.4 follows because te series j=1 Z j converges almost surely. We need some additional notation. In tis section, we define te giant nodes as te k ε largest nodes, i.e., tose nodes wit degrees D N,..., D N k ε+1, were k ε is some function of ε, to be cosen below. We define A ε,n = B ε,n C ε,n D ε,n, 2.7 were i B ε,n is te event tat te stubs of node 1 and node 2 are attaced exclusively to stubs of giant nodes; ii C ε,n is te event tat any two giant nodes are attaced to eac oter; and iii D ε,n is defined as were q ε = min{n : 1 F n < ε/8}. D ε,n = {D 1 q ε, D 2 q ε }, Te reason for introducing te above events is tat on A ε,n, te opcount or grap distance is eiter 2 or 3. Indeed, on B ε,n, bot node 1 and node 2 are attaced exclusively to giant nodes. On te event C ε,n, giant nodes ave mutual grap distance 1. Hence, on te intersection B ε,n C ε,n, te opcount between node 1 and node 2 is at most 3. Te event D ε,n prevents tat te opcount can be equal to 1, because te probability on te intersection of {H N = 1} wit D ε,n can be bounded by qε/n 2 0 see te first part of te proof of Teorem 1.1 for details. Observe tat te expected number of stubs of node 1 is not bounded, since te expectation of a random variable wit distribution 1.2 equals +. Putting tings togeter we see tat if we can sow tat A ε,n appens wp, ten te opcount is eiter 2 or 3. Te fact tat A ε,n appens wp is te content of Lemma 2.2, were we sow tat PA c ε,n < ε, for N N ε. Finally, we observe tat te opcount between node 1 and 2 is precisely equal to 2, if at least one stub of node 1 and at least one stub of node 2 is attaced to te same giant node, and equal to 3 oterwise. Te events B ε,n and C ε,n depend on te integer k ε, wic we will take to be large for ε small, and will be defined now. Te coice of te index k ε is rater tecnical, and depends on te distributional limits of Lemma 2.1. Since L N /u N = D 1 + D D N /u N converges in distribution to te random variable η wit support 0,, we can find a ε, suc tat PL N < a ε u N < ε/36, N. 2.8 Tis follows since convergence in distribution implies tigtness of te sequence L N /u N [4, p. 9], so tat we can find a closed subinterval I 0,, wit PL N /u N I > 1 ε, N. 7

8 We next define b ε, wic is rater involved. It depends on ε, te quantile q ε, te value a ε defined above and te value of τ 1, 2 and reads ε 2 1 a 2 τ ε b ε =, q ε were te peculiar integer 2304 is te product of 8 2 and 36. Given b ε, we take k ε equal to te minimal k suc tat Pξ k b ε /2 ε/ It follows from 2.4 tat suc a number k exists. We ave now defined te constants tat we will use in te proof, and we next claim tat te probability of A c ε,n is at most ε: Lemma 2.2 Te good event as ig probability For eac ε > 0, tere exists N ε, suc tat PA c ε,n < ε, N N ε Te proof of tis lemma is rater tecnical and can be found in appendix A.1. We will now complete te proof of Teorem 1.1 subject to Lemma 2.2. Proof of Teorem 1.1. As seen in te discussion following te introduction of te event A ε,n, tis event implies te event {H N 3}, so tat PA c ε,n < ε induces tat te event {H N 3} occurs wit probability at least 1 ε. Te remainder of te proof consist of two parts. In te first part we sow tat P {H N = 1} A ε,n < ε. In te second part we prove tat lim P H N = 2 = p F, N for some 0 < p F < 1. Since ε is an arbitrary positive number, te above statements yield te content of te teorem. We turn to te first part. Te event {H N = 1} occurs iff at least one stub of node 1 connects to a stub of node 2. For j D 1, we denote by {[1.j] [2]} te event tat j t stub of node 1 connects to a stub of node 2. Ten, wit P N te conditional probability given te degrees D 1, D 2,..., D N, D 1 D 1 P{H N = 1} A ε,n E P N {[1.j] [2]} A ε,n E D 2 L N 1 1 {A ε,n } q2 ε N 1 < ε, j=1 j= for large enoug N, since L N N. We next prove tat lim N P H N = 2 = p, for some 0 < p < 1. Since by definition for any ε > 0, we ave tat max{pb c ε,n, PD c ε,n} PA c ε,n ε, PH N = 2 P {H N = 2} D ε,n B ε,n PH 1 N = PH N = 2 P{H N = 2} D ε,n B ε,n PB ε,n PB ε,n 2PBc ε,n + PD c ε,n PB ε,n 3ε 1 ε, uniformly in N, for N sufficiently large. If we sow tat lim P {H N = 2} D ε,n B ε,n = rε, 2.13 N 8

9 ten tere exists a double limit p F = lim ε 0 lim P {H N = 2} D ε,n B ε,n = lim P H N = 2. N N Moreover, if we can bound rε away 0 and 1, uniformly in ε, for ε small enoug, ten we also obtain tat 0 < p F < 1. In order to prove te existence of te limit in 2.13 we claim tat P N {H N = 2} D ε,n B ε,n can be written as te ratio of two polynomials, were eac polynomial only involves components of te vector DN u N,..., D N kε+1 u N, 1 u N Due to 2.2, tis vector converges in distribution to ξ 1,..., ξ kε, 0. Hence, by te continuous mapping teorem [4, Teorem 5.1, p. 30], we ave te existence of te limit We now prove te above claim. Indeed, te opcount between nodes 1 and 2 is 2 iff bot nodes are connected to te same giant node. For any 0 i D 1, 0 j D 2 and 0 k < k ε, let F i,j,k be te event tat bot te i t stub of node 1 and te j t stub of node 2 are connected to te node wit te N k t largest degree. Ten, conditionally on te degrees D 1, D 2,..., D N, D 1 D 2 k ε 1 P N {H N = 2} D ε,n B ε,n = P N F i,j,k Bε,N, i=1 j=1 k=0 were te rigt-and side can be written by te inclusion-exclusion formula, as a linear combination of terms P N F i1,j 1,k 1 F in,j n,k n B ε,n It is not difficult to see tat tese probabilities are ratios of polynomials of components of For example, P N F i,j,k B ε,n = D N k D N k 1 D N kε D N D N kε D N 1, 2.16 so tat dividing bot te numerator and te denominator of 2.16 by u 2 N, we obtain tat te rigt-and side of 2.16 is indeed a ratio of two polynomials of te vector given in Similar arguments old for general terms of te form in Hence, P N {H N = 2} D ε,n B ε,n itself can be written as a ratio of two polynomials were te polynomial in te denominator is strictly positive. Terefore, te limit in 2.13 exists. We finally bound rε from 0 and 1 uniformly in ε, for any ε < 1/2. Since te opcount between nodes 1 and 2 is 2, given B ε,n, if tey are bot connected to te node wit largest degree, ten and by 2.16 we ave rε P{H N = 2} D ε,n B ε,n E[P N F 1,1,0 B ε,n ], [ = lim P {H N = 2} D ε,n B ε,n lim E N N ] [ 2 ] 2 ξ = E[ 1 ξ 1 + +ξ kε E ξ1 η. ] D N D N 1 D N + +D N k ε On te oter and, conditionally on B ε,n, te opcount between nodes 1 and 2 is at least 3, wen all stubs of te node 1 are connected to te node wit largest degree, and all stubs of te node 2 9

10 are connected to te node wit te one but largest degree. Hence, for any ε < 1/2 and similarly to 2.16, we ave rε = lim P {H N = 2} D ε,n B ε,n 1 lim P {H N > 2} D ε,n B ε,n N N 1 lim P {H N > 2} D 1 N 2, N B ε,n [ D1 1 lim E D N 2i D2 ] D N 1 2i N D N + +D N kε+1 D 1 D N + +D N kε+1 D 2 1 {D 12, N i=0 i=0 } q 1 E[ ξ1 ξ 12 ] 2, η 2 because: i te event D 1 implies tat bot D 2,N 1 q 1 and D 2 q 1, ii te event B ε,n implies 2 2 tat all stubs of te normal nodes 1 and 2 are connected to stubs of giant nodes, iii Lemma 2.1 implies q 12 q 12 D lim N 2i D N 1 2i N E and iv ξ ξ kε η. Bot expectations D i=0 N + + D N k D ε+1 1 [ ξ q 12 1 ξ 2 = E ξ ξ kε ξ ξ kε [ ξ1 ] 2 E η and D i=0 N + + D N k ε+1 D 2 q 12 ], 2.17 [ ξ1 ξ q 12 ] 2 E, 2.18 are strictly positive and independent of ε. Hence, for any ε < 1/2, te quantity rε is bounded away from 0 and 1, were te bounds are independent of ε, and tus 0 < p F < 1. Tis completes te proof of Teorem 1.1 subject to Lemma Proof of Teorem 1.2 In Teorem 1.2, we consider te opcount in te configuration model wit degrees an i.i.d. sequence wit a truncated distribution given by 1.4, were D as distribution F satisfying 1.2. We distinguis two cases: i α < 1/τ 1 and ii α > 1/τ 1. Since part ii is simpler to prove tan part i, we start wit part ii. Proof of Teorem 1.2ii. We ave to prove tat te limit distribution of H N is a mixed distribution wit probability mass p F on 2 and probability mass 1 p F on 3, were p F is given by Teorem 1.1. As before, we denote by D 1, D 2,..., D N te i.i.d. sequence witout conditioning. We bound te probability tat for at least one index i {1, 2,..., N} te degree D i exceeds N α, by N N P {D i > N α } PD i > N α = NPD > N α = N [1 F N α ] N ε, i=1 i=1 for some positive ε, because α > 1/τ 1. We can terefore couple te i.i.d. sequence D N = D N 1, D N 2,..., D N N to te sequence D = D 1, D 2,..., D N, were te probability of a miscoupling, i.e., a coupling suc tat D N D, is at most N ε. Terefore, te result of Teorem 1.1 carries over to case ii in Teorem 1.2. Proof of Teorem 1.2i. Tis proof is more involved. We start wit an outline of te proof. Fix 1 τ + k < α < 1 τ + k 1, η 2

11 wit k N {0} and define M N = N n=1 From [9, Teorem 1, p. 281], te expected value of M N E[M N ] = D N n. 3.2 is given by N N α 1 F N α PD > i = N 1+α2 τ ln, 3.3 i=0 were N ln is slowly varying at infinity. In te sequel, we will use te same ln, for different slowly varying functions, so tat te value of ln may cange from line to line. For te outline, we assume tat M N as rougly te same size as E[M N ] in 3.3. Te proof consists of sowing tat PH N k + 2 = o1 and PH N > k + 3 = o1. We will sketc te proof of eac of tese results. To prove tat PH N k + 2 = o1, note tat wp te degrees of nodes 1 and 2 are bounded by q ε for some large q ε. Terefore, on tis event, te number of nodes tat can be reaced from node 1 in l 1 steps is at most q ε N l 2α, and te number of stubs attaced to nodes at distance l 1 is at most q ε N l 1α. Te probability tat one of tese stubs is attaced to a stub of node 2, making H N at most l, is of te order qεn 2 l 1α /M N. By 3.3 and te assumed concentration of M N, tis is at most qεlnn 2 l 3+τα 1 = o1, wenever α < 1/l 3+τ. Applying tis to l = k +2, we see tat tis probability is o1 if α < 1/k +τ 1. To prove tat PH N > k + 3 = o1, we use te notion of giant nodes in a similar way as in te proof of Teorem 1.1. Due to te conditioning on te degree, Lemma 2.1 no longer olds, so tat we need to adapt te definition of a giant node. In tis section, a giant node is a node wit degree D N, satisfying tat, for an appropriate coice of β, N β < D N N α. 3.4 Nodes wit degree at most N β will be called normal nodes, and we will denote by K N number of stubs of te normal nodes, i.e., te total Similarly to 3.3, we see tat K N = N n=1 D n N 1 N {D n N β }. 3.5 E[K N ] = N 1+β2 τ ln. 3.6 To motivate our coice of β, wic depends on te value of k, observe tat a node wit at least N β stubs, wic cooses exclusively oter giant nodes, in k+1 steps can reac approximately N k+1β oter nodes. Te number of stubs of N k+1β giant nodes is by definition at least N k+2β. Hence, if we take β suc tat M N N k+2β, or equivalently, by 3.3, 1 + α2 τ k + 2β, ten we basically ave all giant nodes on mutual distance at most k + 1, so tat te non-giant nodes 1 and 2, given tat tey bot connect to at least one giant node, are on distance at most k + 3. In te proof, we will see tat we can pick any β suc tat 1 + α2 τ k + 2 < β < α, were we use tat 1+α2 τ k+2 < α, precisely wen α > 1 τ+k. Having tis in mind, we coose β = α2 τ + α k + 2 Here ends te outline of te proof. 11

12 We now turn to te definition of te events involved. Tis part is similar, but not identical, to te introduction of A ε,n in 2.7, because giant nodes no longer are on mutual distance 1. We keep te same notation for te event B ε,n, te event tat te stubs of node 1 and 2 are attaced exclusively to stubs of giant nodes, altoug te definition of a giant node as been canged. We take tis sligt abuse of notation for granted. Te event D ε,n = {D 1 q ε, D 2 q ε }, were q ε = min{k : 1 F k < ε/8}, is identical to te definition in Section 2 below 2.7. We define G ε,n = B ε,n D ε,n H ε,n, 3.8 were H ε,n = {N 1+α2 τ ln M N N 1+α2 τ ln } {K N N 1+β2 τ ln}, 3.9 were ln, ln, ln are slowly varying at infinity. Te event H ε,n will enable us to control te distance between any pair of giant nodes, as sketced in te outline. Te following lemma is te counterpart of Lemma 2.2 in Section 2. Lemma 3.1 Te good event as ig probability For eac ε > 0, tere exists N ε, suc tat, for all N N ε, P G c ε,n < ε Te proof of Lemma 3.1 is rater tecnical and can be found in Appendix A.2. and Te remainder of te proof of Teorem 1.2 is divided into two parts, namely, te proofs of Indeed, if we combine te statements 3.11 and 3.12, ten P{H N k + 2} G ε,n < ε/2, 3.11 P{H N > k + 3} G ε,n < ε/ PH N = k + 3 = P{H N = k + 3} G ε,n + P {H N = k + 3} G c ε,n P{H N = k + 3} G ε,n ε = 1 P{H N > k + 3} G ε,n P{H N < k + 3} G ε,n ε > 1 2ε, and te conclusion of Teorem 1.2i is reaced. We will prove 3.11, 3.12 in two lemmas Lemma 3.2 Te distance is at least k + 3 on te good event For fixed k N {0}, and α as in 3.1, for eac ε > 0, tere exists an integer N ε, suc tat P{H N k + 2} G ε,n < ε/2, N N ε. Proof. Te inequality of te lemma is proved by a counting argument. We will sow tat for eac l {1, 2, 3,..., k + 2} for some δ l > 0. Since P{H N = l} G ε,n < N δ l, 3.14 k+2 P{H N k + 2} G ε,n P{H N = l} G ε,n k + 2N δ, l=1 were δ = min{δ 1,..., δ k+2 } > 0, te claim of te lemma follows if we coose N ε, suc tat k + 2Nε δ ε/2. 12

13 To prove tat P{H N = l} G ε,n < N δ l for any l k + 2, we note tat on G ε,n, te degrees of nodes 1 and 2 are bounded by q ε. Terefore, on G ε,n and using tat all degrees are bounded by N α, te number of nodes tat can be reaced from node 1 in l 1 steps is at most q ε N l 2α, and te number of stubs incident to nodes at distance l 1 from node 1 is at most q ε N l 1α. Wen H N = l, ten one of tese stubs sould be attaced to one of te at most q ε stubs incident to node 2. Denote by M l N te number of stubs tat are not part of an edge incident to a node at distance at most l 1 from node 1. Ten, conditionally on M l N and te fact tat node 2 is at distance at least l 1 from node 1, te stubs of node 2 will be connected to one of tese M l N stubs uniformly at random. More precisely, conditionally on M l N and te fact tat node 2 is at distance at least l 1 from node 1, te event {H N = l} occurs precisely wen a stub of node 2 is paired wit a stub attaced to a node at distance l 1 from node 1. We note tat, on G ε,n, M l N M N 2q ε N l 2α = M N 1 + o1 lnn 1+2 τα, 3.15 wen l 2α < τα, i.e., wen α < 1/l + τ 4. Since l k + 2 and α < 1/k + τ 1, te latter is always satisfied. Te probability tat one of te at most q ε stubs of node 2 is paired wit one of te stubs attaced to nodes at distance l 1 from node 1 is, on G ε,n and conditionally on M l N and te fact tat node 2 is at distance at least l 1 from node 1, bounded from above by q 2 εn l 1α M l N = q2 εn l 1α M N 1 + o1 lnn l 3+τα 1 < N δ l, 3.16 for all δ l < 1 l 3 + τα and N sufficiently large. Here, we use te lower bound on M N in 3.9. Applying tis to l = k + 2, wic gives te worst possible value of δ l, we see tat tis probability is bounded from above by N δ for any δ < 1 k + τ 1α. Since α < 1/k + τ 1, we ave tat 1 k + τ 1α > 0, so tat we can also coose δ > 0. We turn to te proof 3.12, wic we also formulate as a lemma: Lemma 3.3 Te distance is at most k + 3 on te good event Fix k N {0}, and α as in 3.1. For eac ε > 0 tere exists an integer N ε, suc tat, P{H N > k + 3} G ε,n < ε/2, N N ε. In te proof of Lemma 3.3, we need tat te number of giant nodes reacable from an arbitrary giant node in at most l steps, as a lower bound proportional to N lβ. We denote by Z l te set of all nodes wic are reacable in exactly l steps from a node : Z l = {n = 1, 2,..., N : d, n = l} for l {0, 1,...}, were d, n denotes te grap-distance between te nodes and n. Te number of giant nodes in Z l is denoted by El. Lemma 3.4 Growt of te number of giant nodes For eac ε > 0, α < 1/τ + k 1, l {0, 1,..., k} and β given by 3.7, P N =1 for sufficiently large N. { 1 ε l N lβ E l < N lα} { is giant} G ε,n > 1 N k e 3ε1 εs N sβ /16, s=1 13

14 Proof. Te upper bound N lα on E l is trivial, because eac node as less tan N α stubs. We will prove by induction wit respect to l, tat for l {0, 1,..., k}, and k fixed, N { P E l < 1 εl N lβ} l { is giant} G ε,n N e 3ε1 εs N sβ / Denote =1 F l ε,n = ten it suffices to prove tat N =1 s=1 { } {E l 1 εl N lβ } { is giant}, 3.18 PF l ε,n c F l 1 ε,n G ε,n Ne 3ε1 εl N lβ / Indeed, if 3.19 olds, ten 3.17 follows, by te induction ypotesis, as follows: N { P E l < 1 εl N lβ} { is giant} G ε,n =1 = PG ε,n F ε,n l c PG ε,n F ε,n l c F l 1 ε,n + PG ε,n F l 1 ε,n c l 1 Ne 3ε1 εl N lβ /16 + N e 3ε1 εs N sβ /16. s= For l = 0, 3.19 trivially olds. We terefore assume tat 3.19 is valid for l = m 1 and we will prove tat 3.19 olds for l = m. In tis paragrap we will work conditionally given te degrees D 1, D 2,..., D N. For a giant node, we consider only A N = E m 1 N β stubs of te nodes in Z m 1. To be more precise: we consider N β stubs of eac of te E m 1 giant nodes in Z m 1. We number tese stubs by i {1, 2,..., A N } and stub i will connect to a stub of a node n i. Ten we denote by r N,i, for i {1, 2,..., A N }, te probability tat stub i does not connect to a stub of a normal node. We denote by s N,i te probability tat stub i does not connect to a stub of a node in Z m 1 and te total number of stubs of tis set is at most N mα, and finally, we denote by t N,i,j te probability tat stub i does not connect to te giant node j previously selected by te stubs j {1, 2,..., i 1} for eac j tere are are most D N j N α of suc stubs. If none of te above attacments appens, ten we ave a matc wit a not previously found giant node, and we denote by q N,i te probability of suc a matc of stub i, i.e., i 1 q N,i = 1 r N,i s N,i t N,i,j. From te number of stubs mentioned between te parentesis, we can bound tis probability from below by q N,i 1 K N N mα i 1 N α M N M N M N Since, i 1 E m 1 N β N αm 1 N β N αm 1+β, and K N N 1+β2 τ ln, M N > lnn 1+α2 τ on G ε,n, we can bound 1 q N,i on G ε,n from above by j=1 j=1 1 q N,i lnn 1+β2 τ + N αm + N αm 1+β+α lnn 1+α2 τ. For sufficiently large N and uniformly in i, we ave tat 1 q N,i mα + β kα + β < k + 1α < 1 + α2 τ. < ε/2, because β < α, and 14

15 Introduce te binomially distributed random variable Y N wit parameters B N and ε/2, were B N = 1 ε m N mβ. On F m 1 ε,n, we ave tat A N = E m 1 N β B N, so tat te number of mismatces will be stocastically dominated by Y N. We need at least 1 εb N matces, so tat P{E m 1 ε m N mβ } {A N B N } G ε,n PY N < εb N We will now use te Janson inequality [13], wic states tat for any t > 0, P Y N E[Y N ] t 2 exp Since E[Y N ] = εb N /2, we obtain, wit t = εb N /2, t 2 2E[Y N ] + t/3 PY N < εb N P Y N E[Y N ] > εb N /2 2 exp Combining tis wit 3.22, and since tere are at most N giant nodes: PF m ε,n c F m 1 ε,n G ε,n NPY N εb N 2N exp εB N. 16 3ε1 εm N mβ Proof of Lemma 3.3. We start wit an outline. On te event G ε,n, eac stub of te nodes 1 and 2 is attaced to a stub of some giant node. Te idea is to sow tat wp te distance between any two giant nodes is at most k + 1. Tis implies tat te grap distance between nodes 1 and 2, intersected wit te event G ε,n is wp at most k + 3, and ence Lemma 3.3. We will extend te event G ε,n to include te main event in Lemma 3.4: were F k ε,n was defined in Ten I ε,n = G ε,n F k ε,n, 3.25 P{H N > k + 3} G ε,n P{H N > k + 3} I ε,n + P G ε,n F k ε,n c, 3.26 and te second term on te rigt and side of 3.26 can be bounded by ε/4 using Lemma 3.4. We use as indicated in te outline of te proof given above, tat P{H N > k + 3} I ε,n P { 1, 2 are giant} {d 1, 2 > k + 1} I ε,n 1, 2 1, 2 P{ 1, 2 are giant} {d 1, 2 > k + 1} I ε,n, 3.27 were te sum is taken over all pairs of nodes, and were, as before, d 1, 2 denotes te grapdistance between 1 and 2. Indeed, on I ε,n, te nodes 1 and 2 are connected to giant nodes, so tat wen H N > k + 3, tere must be giant nodes 1, 2 at mutual distance at least k + 1. Clearly for any pair of nodes 1 and 2, {d 1, 2 > k + 1} {d 1, 2 > k}, wic implies tat for any pair of nodes 1 and 2, P N {d 1, 2 > k + 1} { 1, 2 are giant} I ε,n P N {d1, 2 > k + 1} { 1, 2 are giant} I ε,n d 1, 2 > k. 15

16 On te event {d 1, 2 > k} { 1, 2 are giant}, te giant node 2 is not attaced to one of te nodes at distance k from te node 1. More precisely, te giant node 2 is not attaced to one of te k 1 l=0 Z l 1 nodes. We ave less tan M N k 1 l=0 El 1 N β stubs to coose from, and te event {d 1, 2 > k + 1} conditioned on {d 1, 2 > k} implies tat no stubs of te giant node 2 will attac to one of te at least E k 1 N β free stubs of Z k 1. Terefore, we ave, almost surely, P N {1, 2 are giant} {d 1, 2 > k + 1} G N ε,n d 1, 2 > k D N 1 2 i=0 exp 1 1 Ek 1 N β { M N E k 1 N β M N k 1 j=0 Ej D N 2 ε1 εk N βk+2 N 1+α2 τ ln 1 {Iε,N } } 1 {Iε,N } 1 N β 2i ε1 εk N βk+1 N 1+α2 τ ln N β { exp ε1 ε k N δ}, 3.28 were we used te inequality 1 x e x, x 0, in te one but last inequality, and were 0 < δ < βk α2 τ. If we substitute tis upper bound in te rigt and side of 3.27, ten we end up wit P{H N > k + 3} I ε,n N 2 exp ε1 ε k N δ < ε/2. Tis completes te proof of Lemma 3.3 and ence of Teorem Te cases τ = 1 and τ = Proof of Teorem 1.3 It is well known, see e.g. [7, 8.2.4], tat wen 1 F x is slowly varying, te quotient of te maximum and te sum of N i.i.d. random variables wit distribution F, converges to 1 in probability, i.e., D N L N 1, in probability. 4.1 Terefore, we obtain tat wp, bot node 1 and node 2 are connected to te node wit maximal degree, wic gives te stated result. 4.2 Two examples wit τ = 2 In te following two examples we sow tat for τ = 2, te limit opcount distribution is sensitive to te slowly varying function. Example 1. Let, for x 2, 2log F x = x log x Ten we sow tat for all k fixed, P H N > k = 1 + o1, as N. 4.3 We first prove 4.3 for k = 2. We sow tis in two steps. In te first step we sow tat for any ε > 0, tere exists v ε N suc tat wit probability at least 1 ε all nodes at distance at most 1 from nodes 1 and 2 ave degrees at most v ε. In te second step we sow tat tere exists N v N, 16

17 suc tat for any N N v, wit probability at least 1 ε, any two given nodes wit degrees at most v ε, are disconnected. Bot steps togeter clearly imply 4.3. Te second step is similar to 2.12, and is omitted ere. To obtain te first step we consider te event D ε,n, defined below 2.7. Ten, for any v N, te probability tat witin te first q ε stubs of node 1 or node 2 tere is a stub connected to a stub of node wit degree at least v + 1 is at most [ ] 2q ε N E D i 1 L {Di >v}. N i=1 It remains to sow tat te above expectation is at most ε/2 for some v = v ε large enoug. For tis, we need tat te first moment of te degree distribution for tis example is finite. Indeed, from 4.8 E[D 1 ] = 1 + x=2 2log 2 2 xlog x log 2 2 du = 1 + 2log 22 ulog u 2 log 2 Ten, from te Law of Large Numbers applied to L N = D D N, we obtain P L N µ ε N dy <. 4.4 y2 ε 12q ε, 4.5 for µ ε > E[D 1 ]. Due to 4.4, 4.5 and te Markov inequality [ E 2qε ] N L N i=1 D i1 {Di >v} ε 6 + 2q εp 2q N ε i=1 D i1 {Di >v} εl N 6 ε 6 + 2q εp L N µ ε + 2q ε P ε q2 ε εµ ε E D i 1 {Di >v} ε 2, 2q ε N i=1 D i 1 {Di >v} ε 6 µ εn for large enoug v, and ence we ave te second step, since PD c ε,n 2PD 1 > q ε ε/4. In a similar way we can sow tat, for any ε > 0, tere exists v ε N suc tat wit probability at least 1 ε all nodes at distance at most 2 from nodes 1 and 2 ave degrees at most v ε. Tis statement implies tat PH N > 4 1. Similarly, we obtain tat for any ε > 0 tere exists v ε N suc tat wit probability at least 1 ε all nodes at distance at most k from nodes 1 and 2 ave degrees at most v ε, wic implies tat for any fixed integer k, lim PH N > 2k = 1, 4.6 N i.e., te probability mass of H N drifts away to + as N. Tis beavior of H N for τ = 2, is in agreement wit te beavior of H N for te case τ 2, 3, see [11], were we sow, among oter tings, tigtness of te sequence Example 2. Let H N log log N logτ F x = c log xlog log x 1 log log x, x x, x N. 4.8 x were x is cosen suc tat for x x, te rigt side of 4.8 is a non-increasing function, and c is suc tat 1 F x = 1. We will sow tat P H N {2, 3} = 1 + o1, as N

18 0.6 N=10 N=10 N= probability opcount Figure 1: Empirical probability mass function of te opcount for τ = 1.8 and N = 10 3, 10 4, 10 5, for te unconditioned degrees. Tus, we see entirely different beavior as in te first example. Define giant nodes as nodes wit degree at least N 1 2 +δ, for some δ > 0, to be determined later on. Te nodes wit degree at most N 1 2 +δ 1 we call normal. Define te event A ε,n as in 2.7, were, in te definition of B ε,n, we use te above definition of te giant node. In Appendix A.3, we will prove te following lemma, wic is similar to Lemma 2.2: Lemma 4.1 For eac ε > 0, tere exists N ε, suc tat for all N N ε, PA c ε,n < ε We now complete te proof of 4.3 subject to Lemma 4.1, wic is straigtforward. By 2.12, we obtain tat P {H N = 1} A ε,n = o1. Moreover, wen A ε,n occurs, all stubs of nodes 1 and 2 are connected to giant nodes due to B ε,n, and te giant nodes form a complete grap due to C ε,n, so tat P {H N > 3} A ε,n = 0. Tis proves Simulation and conclusions To illustrate Teorems 1.1 and 1.2, we ave simulated our random grap wit degree distribution D = U 1 τ 1, were U is uniformly distributed over 0, 1. Tus, 1 F x = PU 1 τ 1 > x = x 1 τ, x = 1, 2, 3,... In Figure 1, we ave simulated te grap distance or opcount wit τ = 1.8 and te values of N = 10 3, 10 4, Te istogram is in accordance wit Teorem 1.1: for increasing values of N we see tat te probability mass is divided over te values H N = 2 and H N = 3, were te probability PH N = 2 converges. As an illustration of Teorem 1.2, we again take τ = 1.8, but now condition te degrees to be less tan N, so tat α = 1. Since in tis case τ 1 1 = 5 4, we expect from Teorem 1.2 case i, tat in te limit te opcount will concentrate on te value H N = 3. Tis is indeed te case as is sown in Figure 2. 18

19 0.8 N=10 N=10 N=10 N= probability opcount Figure 2: Empirical probability mass function of te opcount for τ = 1.8 and N = 10 3, 10 4, 10 5, 10 6, were te degrees are conditioned to be less tan N, 1 τ < α = 1 < 1 τ 1. Our results give convincing asymptotics for te opcount wen te mean degree is infinite, using extreme value teory. Some details remain open: i It is possible to compute upper and lower bounds on te value p F, based on Lemma 2.1. We presented two suc bounds in Tese bounds can be obtained from simulating te random variables Γ 1, Γ 2,... in 2.3. It sould be possible to obtain muc sarper upper and lower bounds, and possibly even numerical values, depending on te specific degree distribution F. ii In te boundary cases α = 1/τ + k, k N {0}, it is natural to conjecture tat te specific limit beavior of H N will depend on te slowly varying function, as is te case for τ = 2 and α > 1 τ 1 = 1 as described in Section 4.2. A Appendix. In te appendix we prove Lemma 2.2, Lemma 3.1 and Lemma 4.1. Te proofs of Lemma 3.1 and 4.1 are bot adaptations of te proof of Lemma 2.2 in Section A.1 below. A.1 Proof of Lemma 2.2 In tis section we restate Lemma 2.2 and ten give a proof. Lemma A.1.1 For eac ε > 0, tere exists N ε suc tat Proof. We start wit an outline of te proof. By 2.7, PA c ε,n < ε, N N ε. A.1.1 PA c ε,n PB c ε,n + PC c ε,n + PD c ε,n, A

20 and an obvious way to prove result A.1.1 would be to sow tat eac of te tree terms on te rigt-and side of A.1.2 is smaller tan ε/3. Tis direct approac is somewat difficult and instead we introduce an additional event E ε,n, wic controls te total degree L N in part c, te degree of te giant nodes in part b, and te total degree of all normal non-giant nodes in part a: { N kε } E ε,n = D n ε 8q ε L N a n=1 A.1.3 {D N k ε+1 c ε u N } b {L N d ε u N }, c were q ε is te ε-quantile of F used in te definition of D ε,n P ξ kε < c ε < ε/24 and Pη > d ε < ε/24, and were c ε, d ε > 0 are defined by respectively. Observe tat c ε is a lower quantile of ξ kε, wereas b ε defined in 2.9 and 2.10 is an upper quantile of ξ kε. Furtermore, d ε is an upper quantile of η, wereas a ε defined in 2.8 is a lower quantile of η. Intersection wit te additional event E ε,n, facilitates te bounding of bot B c ε,n and Cε,N. c Terefore, we write PA c ε,n PB c ε,n D ε,n E ε,n + PC c ε,n D ε,n E ε,n + PD c ε,n + PE c ε,n, A.1.4 and our strategy to prove te lemma is tat we sow tat eac of te four terms on te rigt-and side of A.1.4 is at most ε/4. Nodes 1 and 2 are connected to giant nodes only. On B c ε,n D ε,n at least one of te 2q ε stubs is attaced to a stub of te nodes D 1,..., D N kε. Hence, te first term on te rigt side of A.1.4 satisfies due to point a of E ε,n. PB c ε,n D ε,n E ε,n 2q ε E [ 1 L N N k ε n=1 D n 1 {Eε,N } ] ε/4, Te giant nodes form a complete grap. We turn to te second term of A.1.4. Recall tat C c ε,n induces tat no stubs of at least two giant nodes are attaced to one anoter. Since we ave at most N 2 pairs of giant nodes 1 and 2, te items b, c of E ε,n imply D 1 /2 1 PC c ε,n D ε,n E ε,n E N 2 D L N 2i 1 {1, 2 giant} i=0 N 2 1 c εu N d ε u N for large enoug N, because u N = N 1/τ 1+o1. cεu N /2 N 2 exp c2 εu N ε/4, 2d ε A.1.5 Nodes 1 and 2 ave small degree. Te tird term on te rigt-and side of A.1.4 is at most ε/4, because PDε,N c 2PD 1 > q ε 2ε/8 = ε/4. A

21 Te order statistics. It remains to estimate te last term on te rigt side of A.1.4. Clearly, P E c ε,n N kε P n=1 D n > ε 8q ε L N a +P D N k ε+1 < c ε u N b +P L N > d ε u N. c A.1.7 We will consequently sow tat eac term in te above expression is at most ε/12. Let a ε and b ε > 0 be as in 2.8 and 2.9, ten we can decompose te first term on te rigt-and side of A.1.7 as P N kε n=1 D n > ε L N 8q ε From te Markov inequality, N P D i 1 {Di <b εu N } > i=1 P L N < a ε u N + P N kε n=1 D n > ε a ε u N 8q ε P L N < a ε u N + P D N k ε+1 > b ε u N N +P D i 1 {Di <b εu N } > ε a ε u N. 8q ε i=1 ε a ε u N 8q ε A.1.8 8q εne [ ] D1 {D<bε u N }. A.1.9 εa ε u N Since 1 F x varies regularly wit exponent τ 1, we ave, by [9, Teorem 1b, p. 281], E [ b ] ε u N D1 {D<bε u N } = [1 F k] 22 τb ε u N [1 F b ε u N ], A.1.10 k=0 for large enoug N. Due to 2.1, for large enoug N, we ave also N [1 F u N ] 2. A.1.11 Substituting A.1.10 and A.1.11 in A.1.9, we obtain N P D i 1 {Di <b ε u N } > i=1 ε a ε u N 8q ε for large enoug N. From te regular variation of 1 F x, Hence te rigt-and side of A.1.12 is at most 16q εn2 τb ε u N [1 F b ε u N ] εu N a ε 32q ε2 τb ε [1 F b ε u N ], A.1.12 εa ε [1 F u N ] 1 F b ε u N lim = b ε 1 τ. N 1 F u N 64q ε 2 τ b ε 2 τ εa ε ε/36, for sufficiently large N, by te definition of b ε in 2.9. We now sow tat te second term on te rigt side of A.1.8 is at most ε/36. Since D N kε+1 /u N converges in distribution to ξ kε, we find from 2.10, P D N kε+1 > b ε u N P ξ kε > b ε /2 + ε/72 ε/36, 21