Random networks with sublinear preferential attachment: Degree evolutions
aa r X i v : . [ m a t h . P R ] J u l Random networks with sublinear preferentialattachment: Degree evolutions by Steffen Dereich , and Peter M¨orters Institut f¨ur Mathematik, MA 7-5, Fakult¨at IITechnische Universit¨at BerlinStraße des 17. Juni 13610623 BerlinGermany Department of Mathematical SciencesUniversity of BathClaverton DownBath BA2 7AYUnited Kingdom
Version of September 20, 2018
Summary.
We define a dynamic model of random networks, where new vertices are con-nected to old ones with a probability proportional to a sublinear function of their degree.We first give a strong limit law for the empirical degree distribution, and then have a closerlook at the temporal evolution of the degrees of individual vertices, which we describe interms of large and moderate deviation principles. Using these results, we expose an inter-esting phase transition: in cases of strong preference of large degrees, eventually a singlevertex emerges forever as vertex of maximal degree, whereas in cases of weak preference,the vertex of maximal degree is changing infinitely often. Loosely speaking, the transitionbetween the two phases occurs in the case when a new edge is attached to an existing vertexwith a probability proportional to the root of its current degree.
Keywords.
Barabasi-Albert model, sublinear preferential attachment, dynamic randomgraphs, maximal degree, degree distribution, large deviation principle, moderate deviationprinciple.
Primary 05C80 Secondary 60C05 90B15 Introduction
Dynamic random graph models, in which new vertices prefer to be attached to vertices withhigher degree in the existing graph, have proved to be immensely popular in the scientificliterature recently. The two main reasons for this popularity are, on the one hand, that thesemodels can be easily defined and modified, and can therefore be calibrated to serve as models forsocial networks, collaboration and interaction graphs, or the web graph. On the other hand, ifthe attachment probability is approximately proportional to the degree of a vertex, the dynamicsof the model can offer a credible explanation for the occurrence of power law degree distributionsin large networks.The philosophy behind these preferential attachment models is that growing networks are builtby adding nodes successively. Whenever a new node is added it is linked by edges to one or moreexisting nodes with a probability proportional to a function f of their degree. This function f ,called attachment rule , or sometimes weight function , determines the qualitative features of thedynamic network.The heuristic characterisation does not amount to a full definition of the model, and someclarifications have to be made, but it is generally believed that none of these crucially influencethe long time behaviour of the model.It is easy to see that in the general framework there are three main regimes: • the linear regime, where f ( k ) ≍ k ; • the superlinear regime, where f ( k ) ≫ k ; • the sublinear regime, where f ( k ) ≪ k .The linear regime has received most attention, and a major case has been introduced in themuch-cited paper Barab´asi and Albert (1999). There is by now a substantial body of rigorousmathematical work on this case. In particular, it is shown in Bollob´as et al. (2001), M´ori (2002)that the empirical degree distribution follows an asymptotic power law and in M´ori (2005) thatthe maximal degree of the network is growing polynomially of the same order as the degree ofthe first node.In the superlinear regime the behaviour is more extreme. In Oliveira and Spencer (2005) it isshown that a dominant vertex emerges, which attracts a positive proportion of all future edges.Asymptotically, after n steps, this vertex has degree of order n , while the degrees of all othervertices are bounded. In the most extreme cases eventually all vertices attach to the dominantvertex.In the linear and sublinear regimes Rudas et al. (2007) find almost sure convergence of theempirical degree distributions. In the linear regime the limiting distribution obeys a power law,whereas in the sublinear regime the limiting distributions are stretched exponential distributions.Apart from this, there has not been much research so far in the sublinear regime, which is themain concern of the present article, though we include the linear regime in most of our results.2pecifically, we discuss a preferential attachment model where new nodes connect to a randomnumber of old nodes, which in fact is quite desirable from the modelling point of view. More pre-cisely, the node added in the n th step is connected independently to any old one with probability f ( k ) /n , where k is the (in-)degree of the old node. We first determine the asymptotic degreedistribution, see Theorem 1.1, and find a result which is in line with that of Rudas et al. (2007).The result implies in particular that, if f ( k ) = ( k + 1) α for 0 α <
1, then the asymptoticdegree distribution ( µ k ) satisfies log µ k ∼ − − α k − α , showing that power law behaviour is limited to the linear regime. Under the assumption thatthe strength of the attachment preference is sufficiently weak, we give very fine results about theprobability that the degree of a fixed vertex follows a given increasing function, see Theorem 1.10and Theorem 1.12. These large and moderate deviation results, besides being of independent in-terest, play an important role in the proof of our main result. This result describes an interestingdichotomy about the behaviour of the vertex of maximal degree, see Theorem 1.7: • The strong preference case: If P n /f ( n ) < ∞ , then there exists a single dominant vertex–called persistent hub – which has maximal degree for all but finitely many times. However,only in the linear regime the number of new vertices connecting to the dominant vertex isgrowing polynomially in time. • The weak preference case: If P n /f ( n ) = ∞ , then there is almost surely no persistenthub. In particular, the index, or time of birth, of the current vertex of maximal degreeis a function of time diverging to infinity in probability. In Theorem 1.15 we provideasymptotic results for the index and degree of this vertex, as time goes to infinity.A rigorous definition of the model is given in Section 1.2, and precise statements of all theprincipal results follow in Section 1.3. At the end of that section, we also give a short overviewover the further parts of this paper. We now explain how precisely we define our preferential attachment model given a monotonicallyincreasing attachment rule f : { , , , . . . } −→ (0 , ∞ ) with f ( n ) n + 1 for all n ∈ Z + := { , , . . . } . At time n = 1 the network consists of a single vertex (labeled 1) without edges andfor each n ∈ N the graph evolves in the time step n → n + 1 according to the following rule • add a new vertex (labeled n + 1) and • insert for each old vertex m a directed edge n + 1 → m with probability f (indegree of m at time n ) n . The new edges are inserted independently for each old vertex. Note that the assumptionsimposed on f guarantee that in each evolution step the probability for adding an edge is smalleror equal to 1. Formally we are dealing with a directed network, but indeed, by construction, alledges are pointing from the younger to the older vertex, so that the directions can trivially berecreated from the undirected (labeled) graph. 3here is one notable change to the recipe given in Krapivsky and Redner (2001): We do notadd one edge in every step but a random number, a property which is actually desirable in mostapplications. Given the graph after attachment of the n th vertex, the expected number of edgesadded in the next step is 1 n n X m =1 f (cid:0) indegree of m at time n (cid:1) . This quantity converges, as n → ∞ almost surely to a deterministic limit λ , see Theorem 1.1.Moreover, the law of the number of edges added is asymptotically Poissonian with parameter λ .Observe that the out degree of every vertex remains unchanged after the step in which thevertex was created. Hence our principal interest when studying the asymptotic evolution ofdegree distributions is in the in degrees. We denote by Z [ m, n ], for m, n ∈ N , m n , the indegree of the m -th vertex after the insertionof the n -th vertex, and by X k ( n ) the proportion of nodes of indegree k ∈ Z + at time n , that is X k ( n ) = 1 n n X i =1 {Z [ i,n ]= k } . Moreover, denote µ k ( n ) = E X k ( n ), X ( n ) = ( X k ( n ) : k ∈ Z + ), and µ ( n ) = ( µ k ( n ) : k ∈ Z + ). Theorem 1.1 (Asymptotic empirical degree distribution) . (i) Let µ k = 11 + f ( k ) k − Y l =0 f ( l )1 + f ( l ) for k ∈ Z + ,which is a sequence of probability weights. Then, almost surely, lim n →∞ X ( n ) = µ in total variation norm.(ii) If f satisfies f ( k ) ηk + 1 for some η ∈ (0 , , then the conditional distribution of theoutdegree of the ( n + 1) st incoming node (given the graph at time n ) converges almostsurely in variation topology to the Poisson distribution with parameter λ : = h µ, f i . Remark 1.2.
The asymptotic degree distribution coincides with that in the random tree modelintroduced in Krapivsky and Redner (2001) and studied by Rudas et al. (2007), if f is chosenas an appropriate multiple of their weight function. This is strong evidence that these modelsshow the same qualitative behaviour, and that our further results hold mutatis mutandis forpreferential attachment models in which new vertices connect to a fixed number of old ones. Example 1.3.
Suppose f ( k ) ∼ γk α , for 0 < α < γ >
0, then a straight forward analysisyields that log µ k ∼ − k +1 X l =1 log (cid:0) l − α (cid:1) ∼ − γ − α k − α . Hence the asymptotic degree distribution has stretched exponential tails.4n order to analyse the network further, we scale the time as well as the way of counting theindegree. To the original time n ∈ N we associate an artificial timeΨ( n ) := n − X m =1 m ∼ log n, and to the original degree j ∈ Z + we associate the artificial degreeΦ( j ) := j − X k =0 f ( k ) . An easy law of large numbers illustrates the role of these scalings.
Proposition 1.4 (Law of large numbers) . For any fixed vertex labeled m ∈ N , we have that lim n →∞ Φ( Z [ m, n ])Ψ( n ) = 1 almost surely . Remark 1.5.
Since Ψ( n ) ∼ log n , we conclude that for any m ∈ N , almost surely,Φ( Z [ m, n ]) ∼ log n as n → ∞ . In particular, we get for an attachment rule f with f ( n ) ∼ γn and γ ∈ (0 , n ) ∼ γ log n which implies that log Z [ m, n ] ∼ log n γ , almost surely.Furthermore, an attachment rule with f ( n ) ∼ γn α for α < γ > Z [ m, n ] ∼ ( γ (1 − α ) log n ) − α . We denote by T := { Ψ( n ) : n ∈ N } the set of artificial times, and by S := { Φ( j ) : j ∈ Z + } theset of artificial degrees. From now on, we refer by time to the artificial time, and by (in-)degree to the artificial degree. Further, we introduce a new real-valued process ( Z [ s, t ]) s ∈ T ,t > via Z [ s, t ] := Φ( Z [ m, n ]) if s = Ψ( m ), t = Ψ( n ) and m n ,and extend the definition to arbitrary t by letting Z [ s, t ] := Z [ s, s ∨ max( T ∩ [0 , t ])]. For notationalconvenience we extend the definition of f to [0 , ∞ ) by setting f ( u ) := f ( ⌊ u ⌋ ) for all u ∈ [0 , ∞ ),and linearly interpolate Φ at the breakpoints S so thatΦ( u ) = Z u f ( v ) dv. We denote by L [0 , ∞ ) the space of c`adl`ag functions x : [0 , ∞ ) → R endowed with the topologyof uniform convergence on compact subsets of [0 , ∞ ). Proposition 1.6 (Central limit theorem) . In the case of weak preference, for all s ∈ T , (cid:16) Z [ s, s + ϕ ∗ κt ] − ϕ ∗ κt √ κ : t > (cid:17) ⇒ ( W t : t > , in distribution on L [0 , ∞ ) , where ( W t : t > is a standard Brownian motion and ( ϕ ∗ t ) t > is theinverse of ( ϕ t ) t > given by ϕ t = Z Φ − ( t )0 f ( u ) du. main result describes the behaviour of the vertex of maximal degree, and reveals an inter-esting dichotomy between weak and strong forms of preferential attachment. Theorem 1.7 (Vertex of maximal degree) . Suppose f is concave. Then we have the followingdichotomy: Strong preference. If ∞ X k =0 f ( k ) < ∞ , then with probability one there exists a persistent hub, i.e. there is a single vertex whichhas maximal indegree for all but finitely many times. Weak preference. If ∞ X k =0 f ( k ) = ∞ , then with probability one there exists no persistent hub and the time of birth, or index, ofthe current hub tends to infinity in probability. Remark 1.8.
Without the assumption of concavity of f , the assertion remains true in the weakpreference regime. In the strong preference regime our results still imply that, almost surely,the number of vertices, which at some time have maximal indegree, is finite. Remark 1.9.
In the weak preference case the information about the order of the vertices isasymptotically lost: as a consequence of the proof of Theorem 1.7, we have for two nodes s < s ′ in T that lim t →∞ P ( Z [ s, t ] > Z [ s ′ , t ]) = , a phenomenon reminiscent of propagation of chaos . Conversely, in the strong preference case,the information about the order is not lost completely and one haslim t →∞ P ( Z [ s, t ] > Z [ s ′ , t ]) > . Investigations so far were centred around typical vertices in the network. Large deviation princi-ples, as provided below, are the main tool to analyse exceptional vertices in the random network.Throughout we use the large-deviation terminology of Dembo and Zeitouni (1998) and, fromthis point on, the focus is on the weak preference case.Our aim is to determine the typical age and indegree evolution of the hub. For this purpose weassume that • f is regularly varying with index 0 α < , • for some η <
1, we have f ( j ) η ( j + 1) for all j ∈ Z + . (1)We set ¯ f := f ◦ Φ − , and recall from Lemma A.1 in the appendix that we can represent ¯ f as¯ f ( u ) = u α/ (1 − α ) ¯ ℓ ( u ) for u >
0, where ¯ ℓ is a slowly varying function. We denote by I [0 , ∞ ) thespace of nondecreasing functions x : [0 , ∞ ) → R with x (0) = 0 endowed with the topology ofuniform convergence on compact subintervals of [0 , ∞ ).6 heorem 1.10 (Large deviation principles) . Under assumption (1) , for every s ∈ T , the familyof functions (cid:16) κ Z [ s, s + κt ] : t > (cid:17) κ> satisfies large deviation principles on the space I [0 , ∞ ) , • with speed ( κ − α ¯ ℓ ( κ )) and good rate function J ( x ) = (R ∞ x α − α t [1 − ˙ x t + ˙ x t log ˙ x t ] dt if x is absolutely continuous, ∞ otherwise . • and with speed ( κ ) and good rate function K ( x ) = ( a f (0) if x t = ( t − a ) + for some a > , ∞ otherwise . Remark 1.11.
The large deviation principle states, in particular, that the most likely deviationfrom the growth behaviour in the law of large numbers is having zero indegree for some (unusuallylong) time, and after that time typical behaviour kicking in.More important for our purpose is a moderate deviation principle, which describes deviations ona finer scale. Similar as before, we denote by L (0 , ∞ ) the space of c`adl`ag functions x : (0 , ∞ ) → R endowed with the topology of uniform convergence on compact subsets of (0 , ∞ ), and alwaysuse the convention x := lim inf t ↓ x t . Theorem 1.12 (Moderate deviation principle) . Suppose (1) and that ( a κ ) is regularly varying,so that the limit c := lim κ ↑∞ a κ κ α − − α ¯ ℓ ( κ ) ∈ [0 , ∞ ) exists. If κ − α − α ¯ ℓ ( κ ) − ≪ a κ ≪ κ, then, for any s ∈ T , the family of functions (cid:16) Z [ s, s + κt ] − κta κ : t > (cid:17) κ> satisfies a large deviation principle on L (0 , ∞ ) with speed ( a κ κ α − − α ¯ ℓ ( κ )) and good rate function I ( x ) = (cid:26) R ∞ ( ˙ x t ) t α − α dt − c f (0) x if x is absolutely continuous and x , ∞ otherwise,where we use the convention / ∞ . Remark 1.13. If c = ∞ there is still a moderate deviation principle on the space of functions x : (0 , ∞ ) → R with the topology of pointwise convergence. However, the rate function I , whichhas the same form as above with 1 / ∞ interpreted as zero, fails to be a good rate function. Remark 1.14.
Under assumption (1) the central limit theorem of Proposition 1.6 can be statedas a complement to the moderate deviation principle: For a κ ∼ κ − α − α ¯ ℓ ( κ ) − , we have (cid:18) Z [ s, s + κt ] − κta κ : t > (cid:19) ⇒ (cid:16)q − α − α W t − α − α : t > (cid:17) . See Section 2.1 for details. 7ur final result describes weak limit laws for index and degree of the vertex of maximal degree.This result relies on the moderate deviation principle above.
Theorem 1.15 (Limit law for age and degree of the vertex of maximal degree) . Suppose f is regularly varying with index α < . Define s ∗ t to be the index of the hub at time t , and Z max t = Z [ s ∗ t , t ] to be the corresponding maximal indegree. One has, in probability, s ∗ t ∼ Z max t − t ∼
12 1 − α − α t − α − α ¯ ℓ ( t ) = 12 1 − α − α t ¯ f ( t ) . Moreover, in probability on L (0 , ∞ ) , lim t →∞ (cid:16) Z [ s ∗ t , s ∗ t + tu ] − tut − α − α ¯ ℓ ( t ) − : u > (cid:17) = (cid:16) − α − α (cid:0) u − α − α ∧ (cid:1) : u > (cid:17) . Remark 1.16.
In terms of the natural scaling, we get for the index m ∗ n of the hub and themaximal indegree Z max n at natural time n ∈ N that, in probability,log m ∗ n ∼
12 1 − α − α (log n ) − α − α ¯ ℓ (log n )and Z max n − Φ − (Ψ( n )) ∼
12 1 − α − α log n. The remainder of this paper is devoted to the proofs of these results. Rather than proving theresults in the order in which they are stated, we proceed by the techniques used. Section 2 isdevoted to martingale techniques, which in particular prove the law of large numbers, Propo-sition 1.4, and the central limit theorem, Proposition 1.6. We also prove a property of themartingale limit which is crucial in the proof of Theorem 1.7. Section 3 is using Markov chaintechniques and provides the proof of Theorem 1.1. In Section 4 we collect the large deviationtechniques, proving Theorem 1.10 and Theorem 1.12. Section 5 combines the various techniquesto prove our main result, Theorem 1.7, along with Theorem 1.15. An appendix collects theauxiliary statements from the theory of regular variation.
In this section we identify a martingale associated with the degree evolution of a vertex, andstudy its properties. This will be a vital tool in the further analysis of the network.
Lemma 2.1.
Fix s ∈ T and represent Z [ s, · ] as Z [ s, t ] = t − s + M t . hen ( M t ) t ∈ T ,t > s is a martingale. Moreover, the martingale converges if and only if Z ∞ f ( u ) du < ∞ , (2) and otherwise it satisfies the following functional central limit theorem: Let ϕ t := Z Φ − ( t )0 f ( v ) dv = Z t f ( v ) dv, and denote by ϕ ∗ : [0 , ∞ ) → [0 , ∞ ) the inverse of ( ϕ t ) ; then the martingales M κ := (cid:18) √ κ M s + ϕ ∗ κt (cid:19) t > for κ > converge in distribution to standard Brownian motion as κ tends to infinity. In any case theprocesses ( κ Z [ s, s + κt ]) t > converge, as κ ↑ ∞ , almost surely, in L [0 , ∞ ) to the identity. Proof.
For t = Ψ( n ) ∈ T we denote by ∆ t the distance between t and its right neighbour in T ,i.e. ∆ t = 1 n = 1Ψ − ( t ) . (3)One has E (cid:2) Z [ s, t + ∆ t ] − Z [ s, t ] (cid:12)(cid:12) G n (cid:3) = E (cid:2) Φ ◦ Z [ i, n + 1] − Φ ◦ Z [ i, n ] (cid:12)(cid:12) G n (cid:3) = f ( Z [ i, n ]) n × f ( Z [ i, n ]) = 1 n . Moreover, h M i t +∆ t − h M i t = (cid:16) − ¯ f ( Z [ s, t ])∆ t (cid:17) f ( Z [ s, t ]) ∆ t f ( Z [ s, t ]) ∆ t. (4)Observe that by Doob’s martingale inequality and the uniform boundedness of ¯ f ( · ) − one has a i := E [sup s t s +2 i +1 | M t | ] (cid:0) i/ log 2 i (cid:1) C i , where C = C ( f (0)) is a constant only depending on f (0). Moreover, by Chebyshev, one has P (cid:16) sup t > s +2 i | M t |√ t − s log( t − s ) > (cid:17) ∞ X k = i P (cid:16) sup s +2 k t s +2 k +1 M t ( t − s ) log ( t − s ) > (cid:17) ∞ X k = i E h sup s +2 k t s +2 k +1 M t ( t − s ) log ( t − s ) i ∞ X k = i a k < ∞ . Letting i tend to infinity, we conclude that almost surelylim sup t →∞ | M t |√ t − s log( t − s ) . (5)9n particular, we obtain almost sure convergence of ( κ Z [ s, s + κt ]) t > to the identity. As aconsequence of (4), for any ε >
0, there exists a random almost surely finite constant η = η ( ω, ε )such that, for all t > s , h M i t Z t − s f (Φ − ((1 − ε ) u )) du + η. Note that Φ : [0 , ∞ ) → [0 , ∞ ) is bijective and substituting (1 − ε ) κu by Φ( v ) leads to h M i t − ε Z Φ − ((1 − ε )( t − s ))0 f ( v ) dv + η − ε Z Φ − ( t − s )0 f ( v ) dv + η. Thus, condition (2) implies convergence of the martingale ( M t ).We now assume that ( ϕ t ) t > converges to infinity. Since ε > t →∞ h M i t ϕ t − s , almost surely.To conclude the converse estimate note that P t ∈ T (∆ t ) < ∞ so that we get with (4) and (5)that h M i t > Z t − s f (Φ − ((1 + ε ) u )) du − η >
11 + ε Z Φ − ( t − s )0 f ( v ) dv − η, for an appropriate finite random variable η . Therefore,lim t →∞ h M i t ϕ t − s = 1 almost surely. (6)The jumps of M κ are uniformly bounded by a deterministic value that tends to zero as κ tendsto ∞ . By a functional central limit theorem for martingales (see, e.g., Theorem 3.11 in Jacodand Shiryaev (2003)), the central limit theorem follows once we establish that, for any t > κ →∞ h M κ i t = t in probability , which is an immediate consequence of (6). (cid:3) Proof of Remark 1.14.
We suppose that f is regularly varying with index α < . By thecentral limit theorem the processes( Y κt : t >
0) := (cid:16) Z [ s, s + ϕ ∗ tϕ κ ] − ϕ ∗ tϕ κ p ϕ ( κ ) : t > (cid:17) for κ > W t ) as κ tends to infinity. For each κ > τ κt ) t > := ( ϕ κt /ϕ κ ). Using that ϕ is regularly varying with parameter − α − α we find uniform convergence on compacts:( τ κt ) → ( t − α − α ) =: ( τ t ) as κ → ∞ . Therefore, (cid:16) Z [ s, s + κt ] − κt p ϕ ( κ ) : t > (cid:17) = ( Y κτ κt : t > ⇒ ( W τ t : t > . and, as shown in Lemma A.1, ϕ κ ∼ − α − α κ − α − α ¯ ℓ ( κ ) − . (cid:3) .2 Absolute continuity of the law of M ∞ In the sequel, we consider the martingale ( M t ) t > s,t ∈ T given by Z [ s, t ] − ( t − s ) for a fixed s ∈ T in the case of strong preference. We denote by M ∞ the limit of the martingale. Proposition 2.2. If f is concave, then the distribution of M ∞ is absolutely continuous withrespect to Lebesgue measure. Proof.
For ease of notation, we denote Y t = Z [ s, t ], for t ∈ T , t > s . Moreover, we fix c > A t denote the event that Y u ∈ [ u − c, u + c ] for all u ∈ [ s, t ] ∩ T .Now observe that for two neighbours v − and v in SP ( Y t +∆ t = v ; A t ) = (1 − ¯ f ( v ) ∆ t ) P ( Y t = v ; A t ) + ¯ f ( v − ) ∆ t P ( Y t = v − ; A t ) . (7)Again we use the notation ∆ t = − ( t ) . Moreover, we denote ∆ ¯ f ( v ) = ¯ f ( v ) − ¯ f ( v − ). In the firststep of the proof we derive an upper bound for h ( t ) = max v ∈ S P ( Y t = v ; A t ) for t ∈ T , t > s. With (7) we conclude that P ( Y t +∆ t = v ; A t ) (1 − ∆ ¯ f ( v ) ∆ t ) h ( t ) . For w > ς ( w ) = max S ∩ [0 , w ]. Due to the concavity of f , we get that h ( t + ∆ t ) (1 − ∆ ¯ f ( ς ( t + c + 1)) ∆ t ) h ( t ) . Consequently, h ( t ) Y u ∈ [ s,t ) ∩ T (1 − ∆ ¯ f ( ς ( u + c + 1)) ∆ u )and using that log(1 + x ) x we obtain h ( t ) exp (cid:16) − X u ∈ [ s,t ) ∩ T ∆ ¯ f ( ς ( u + c + 1)) ∆ u (cid:17) . (8)We continue with estimating the sum Σ in the latter exponential:Σ = X u ∈ [ s,t ) ∩ T ∆ ¯ f ( ς ( u + c + 1)) ∆ u > Z ts ∆ ¯ f ( ς ( u + c + 1)) du. Next, we denote by f lin the continuous piecewise linear interpolation of f | N . Analogously, weset Φ lin ( v ) = R v f lin ( u ) du and ¯ f lin ( v ) = f lin ◦ (Φ lin ) − ( v ). Using again the concavity of f weconclude that Z ts ∆ ¯ f ( ς ( u + c + 1)) du > Z ts ( f lin ) ′ (Φ − ( u + c + 1)) du, and that f lin > f ⇒ Φ lin Φ ⇒ (Φ lin ) − > Φ − ⇒ ( f lin ) ′ ◦ (Φ lin ) − ( f lin ) ′ ◦ Φ − . > Z ts ( f lin ) ′ (Φ − ( u + c + 1)) du > Z ts ( f lin ) ′ ◦ (Φ lin ) − ( u + c + 1) du. For Lebesgue almost all arguments one has( ¯ f lin ) ′ = ( f lin ◦ (Φ lin ) − ) ′ = ( f lin ) ′ ◦ (Φ lin ) − · ((Φ lin ) − ) ′ = ( f lin ) ′ ◦ (Φ lin ) − · f lin ◦ (Φ lin ) − so that ( f lin ) ′ ◦ (Φ lin ) − = ( ¯ f lin ) ′ ¯ f lin = (log ¯ f lin ) ′ . Consequently, Σ > log ¯ f lin ( t + c + 1) − log ¯ f lin ( s + c + 1)Using that f lin > f and (Φ lin ) − > Φ − we finally get thatΣ > log ¯ f ( t + c + 1) − log c ∗ , where c ∗ is a positive constant not depending on t . Plugging this estimate into (7) we get h ( t ) c ∗ ¯ f ( t + c + 1) . Fix now an interval I ⊂ R of finite length and note that P ( M t ∈ I ; A t ) = P ( Y t ∈ t − s + I ; A t ) t − s + I ) ∩ S ∩ A t ] · h ( t ) . Now ( t − s + I ) ∩ S ∩ A t is a subset of [ t − c, t + c ] and the minimal distance of two distinctelements is bigger than f ( t + c ) . Therefore, t − s + I ) ∩ S ∩ A t ] | I | ¯ f ( t + c ) + 1, and P ( M t ∈ I ; A t ) c ∗ | I | + c ∗ ¯ f ( t + c ) . Moreover, for any open and thus immediately also for any arbitrary interval I one has P ( M ∞ ∈ I ; A ∞ ) lim inf t →∞ P ( M t ∈ I ; A t ) c ∗ | I | , where A ∞ = T t ∈ [ s, ∞ ) ∩ T A t . Consequently, the Borel measure µ c on R given by µ c ( E ) = E [1l A ∞ · E ( M ∞ )], is absolutely continuous with respect to Lebesgue measure. The distribution µ of M ∞ , i.e. µ ( E ) = E [1l E ( M ∞ )], can be written as monotone limit of the absolutely continuousmeasures µ c ( c ∈ N ), and it is thus also absolutely continuous. (cid:3) In this section we prove Theorem 1.1. For k ∈ Z + and n ∈ N let µ k ( n ) = E [ X k ( n )] and µ ( n ) = ( µ k ( n )) k ∈ Z + . We first show that ( µ ( n )) n ∈ N converges to µ = ( µ k ) k ∈ Z + as n tends toinfinity. We start by deriving a recursive representation for µ ( n ). For k ∈ Z + , E [ X k ( n + 1) | X ( n )] = 1 n + 1 (cid:16) n X i =1 E h − {Z [ i,n ]= k< Z [ i,n +1] } + 1l {Z [ i,n ]
0. Now h X ( n ) , f i h X ( n ) , [0 ,m ) · f i + h X a ( n ) , [ m, ∞ ) · f a i so that almost surely lim sup n →∞ h X ( n ) , f i h µ, [0 ,m ) · f i + h µ a , [ m, ∞ ) · f a i . Since m can be chosen arbitrarily large we conclude thatlim sup n →∞ h X ( n ) , f i h µ, f i . n →∞ E h n X m =1 ∆ Z [ m, n ] (cid:12)(cid:12)(cid:12) F n i = h µ, f i . Since, conditional on F n , P nm =1 ∆ Z [ m, n ] is a sum of independent Bernoulli variables withsuccess probabilities tending uniformly to 0, we finally get that L ( P nm =1 ∆ Z [ m, n ] |F n ) convergesin the weak topology to a Poisson distribution with parameter h µ, f i . In this section we derive tools to analyse rare events in the random network. We provide largeand moderate deviation principles for the temporal development of the indegree of a given vertex.This will allow us to describe the indegree evolution of the node with maximal indegree in thecase of weak preferential attachment. The large and moderate deviation principles are based onan exponential approximation to the indegree evolution processes, which we first discuss.
In order to analyze the large deviations of the process Z [ s, · ] (or Z [ m, · , ]) we use an approxi-mating process. We first do this on the level of occupation measures. For s ∈ T and 0 u < v we define T s [ u, v ) = sup { t ′ − t : Z [ s, t ] > u, Z [ s, t ′ ] < v, t, t ′ ∈ T } to be the time the process Z [ s, · ] spends in the interval [ u, v ). Similarly, we denote by T s [ u ] thetime spent in u . Moreover, we denote by ( T [ u ]) u ∈ S a family of independent random variableswith each entry T [ u ] being Exp ( f ( u ))-distributed, and denote T [ u, v ) := X w ∈ S u w Fix η ∈ (0 , , let s ∈ T and denote by τ the entry time into u of the process Z [ s, · ] . One can couple ( T s [ u ]) u ∈ S and ( T [ u ]) u ∈ S such that, almost surely, { ¯ f ( u )∆ τ η } | T s [ u ] − T [ u ] | (1 ∨ η ¯ f ( u ))∆ τ, where η is a constant only depending on η . Proof. We fix t ∈ T with ¯ f ( u )∆ t η . Note that it suffices to find an appropriate couplingconditional on the event { τ = t } . Let U be a uniform random variable and let F and ¯ F denotethe (conditional) distribution functions of T [ u ] and T s [ u ], respectively. We couple T [ u ] and T s [ u ]by setting T [ u ] = F − ( U ) and T s [ u ] = ¯ F − ( U ), where ¯ F − denotes the right continuous inverseof ¯ F . The variables T [ u ] and T s [ u ] satisfy the assertion of the lemma if and only if F (cid:0) v − (1 ∨ η ¯ f ( u ))∆ t (cid:1) ¯ F ( v ) F (cid:0) v + (1 ∨ η ¯ f ( u ))∆ t (cid:1) for all v > . (14)17e compute1 − ¯ F ( v ) = Y t w,w +∆ w t + vw ∈ T (cid:0) − ¯ f ( u )∆ w (cid:1) = exp X t w,w +∆ w t + vu ∈ T log (cid:0) − ¯ f ( u )∆ w (cid:1) Next observe that, from a Taylor expansion, for a suitably large η > 0, we have − ¯ f ( u )∆ w − η ¯ f ( u ) [∆ w ] log (cid:0) − ¯ f ( u )∆ w (cid:1) − ¯ f ( u )∆ w, so that1 − ¯ F ( v ) exp (cid:16) − ¯ f ( u ) X t w,w +∆ w t + vw ∈ T ∆ w (cid:17) exp (cid:0) − ¯ f ( u )( v − ∆ t ) (cid:1) = 1 − F ( v − ∆ t ) . This proves the left inequality in (14). It remains to prove the right inequality. Note that1 − ¯ F ( v ) > exp (cid:16) − X t w,w +∆ w t + vw ∈ T ( ¯ f ( u )∆ w + η ¯ f ( u ) [∆ w ] ) (cid:17) and X t w,w +∆ w t + vw ∈ T [∆ w ] ∞ X m =[∆ t ] − m t ] − − ∆ t. Consequently, 1 − ¯ F ( v ) > exp {− ¯ f ( u )( v + η ¯ f ( u )∆ t ) } = 1 − F ( v + η ¯ f ( u )∆ t ). (cid:3) As a direct consequence of this lemma we obtain an exponential approximation. Lemma 4.2. Suppose that, for some η < we have f ( j ) η ( j + 1) for all j ∈ Z + . If ∞ X j =0 f ( j ) ( j + 1) < ∞ , then for each s ∈ T one can couple T s with T such that, for all λ > , P (cid:0) sup u ∈ S | T s [0 , u ) − T [0 , u ) | > λ + √ K (cid:1) e − λ K , where K > is a finite constant only depending on f . Proof. Fix s ∈ T and denote by τ u the first entry time of Z [ s, · ] into the state u ∈ S . We couplethe random variables T [ u ] and T s [ u ] as in the previous lemma and let, for v ∈ S , M v = X u ∈ S u In distribution on the Skorokhod space, we have lim s ↑∞ ( Z [ s, s + t ]) t > = ( Z t ) t > . Proof. Recall that Lemma 4.1 provides a coupling between ( T s [ u ]) u ∈ S and ( T [ u ]) u ∈ S for anyfixed s ∈ T . We may assume that the coupled random variables ( ¯ T s [ u ]) u ∈ S and ( ¯ T [ u ]) u ∈ S aredefined on the same probability space for all s ∈ T (though this does not respect the jointdistributions of ( T s [ u ]) u ∈ S for different values of s ∈ T ). Denote by ( ¯ Z [ s, · ]) s ∈ T and ( ¯ Z t ) thecorresponding processes such that ¯ T s [0 , u ) + s = s + P v to ( ¯ Z t ) t > in the Skorokhod topology. (cid:3) Proposition 4.4. Uniformly in s , the processes • ( κ Z κt : t > κ> and ( κ Z [ s, s + κt ] : t > κ> ; • ( a κ ( Z κt − κt ) : t > κ> and ( a κ ( Z [ s, s + κt ] − κt ) : t > κ> ,are exponentially equivalent on the scale of the large, respectively, moderate deviation principles. Proof. We only present the proof for the first large deviation principle of Theorem 1.10 sinceall other statements can be inferred analogously.We let U δ ( x ) denote the open ball around x ∈ I [0 , ∞ ) with radius δ > d generating the topology of uniform convergence on compacts, and, for fixed η > K = { x ∈ I [0 , ∞ ) : I ( x ) η } with finitely many balls ( U δ ( x )) x ∈ I ,where I ⊂ K . Since every x ∈ I is continuous, we can find ε > x ∈ I andincreasing and right continuous τ : [0 , ∞ ) → [0 , ∞ ) with | τ ( t ) − t | ε , y ∈ U δ ( x ) ⇒ y τ ( · ) ∈ U δ ( x ) . For fixed s ∈ T we couple the occupation times ( T s [0 , u )) u ∈ S and ( T [0 , u )) u ∈ S as in Lemma 4.2,and hence implicitly the evolutions ( Z [ s, t ]) t > s and ( Z t ) t > . Next, note that Z [ s, s + · ] can be19ransformed into Z · by applying a time change τ with | τ ( t ) − t | sup u ∈ S | T s [0 , u ) − T [0 , u ) | .Consequently, P (cid:0) d (cid:0) κ Z [ s, s + κ · ] , κ Z κ · (cid:1) > δ (cid:1) P (cid:16) κ Z κ · [ x ∈ I U δ ( x ) (cid:17) + P (cid:16) sup u ∈ S | ¯ T s [0 , u ) − ¯ T [0 , u ) | > κε (cid:17) , and an application of Lemma 4.2 gives a uniform upper bound in s , namelylim sup κ →∞ sup s ∈ S κ − α ¯ ℓ ( κ ) log P (cid:0) d (cid:0) κ Z [ s, s + κ · ] , κ Z κ · (cid:1) > δ (cid:1) − η. Since η and δ > (cid:3) By the exponential equivalence, Proposition 4.4, and (Dembo and Zeitouni, 1998, Theorem 4.2.13)it suffices to prove the large and moderate deviation principles in the framework of the expo-nentially equivalent processes (15) constructed in the previous section.The first step in the proof of the first part of Theorem 1.10, is to show a large deviation principlefor the occupation times of the underlying process. Throughout this section we denote a κ := κ / (1 − α ) ¯ ℓ ( κ ) . We define the function ξ : R → ( −∞ , ∞ ] by ξ ( u ) = ( log − u if u < , ∞ otherwise . Its Legendre-Fenchel transform is easily seen to be ξ ∗ ( t ) = ( t − − log t if t > , ∞ otherwise. Lemma 4.5. For fixed u < v the family ( κ T [ κu, κv )) κ> satisfies a large deviation principlewith speed ( a κ ) and rate function Λ ∗ u,v ( t ) = sup ζ ∈ R [ tζ − Λ u,v ( ζ )] , where Λ u,v ( ζ ) = Z vu s α − α ξ ( ζs − α/ (1 − α ) ) ds. Proof. For fixed u < v denote by I κ = I [ u,v ) κ = { j ∈ Z + : Φ( j ) ∈ [ κu, κv ) } . We get, using ( S j )for the underlying sequence of Exp ( f ( j ))-distributed independent random variables,Λ κ ( θ ) := log E e θT [ κu,κv ) /κ = X j ∈ I κ log E e θκ S j = X j ∈ I κ log 11 − θκf ( j ) = X t ∈ Φ( I κ ) ξ (cid:16) θκf (Φ − ( t )) (cid:17) = Z ¯ I κ f (Φ − ( t )) ξ (cid:16) θκf (Φ − ( t )) (cid:17) dt, I κ = ¯ I [ u,v ) κ = S j ∈ I κ [Φ( j ) , Φ( j +1)). Now choose θ in dependence on κ as θ κ = ζκ / (1 − α ) ¯ ℓ ( κ )with ζ < u α/ (1 − α ) . Then Z ¯ I κ ¯ f ( t ) ξ (cid:16) θ κ κ ¯ f ( t ) (cid:17) dt = κ Z ¯ I κ /κ ¯ f ( κs ) ξ (cid:16) θ κ κ ¯ f ( κs ) (cid:17) ds = κ / (1 − α ) Z ¯ I κ /κ s α − α ¯ ℓ ( κs ) ξ (cid:16) ζ ¯ ℓ ( κ ) s α − α ¯ ℓ ( κs ) (cid:17) ds. Note that inf ¯ I κ /κ and sup ¯ I κ /κ approach the values u and v , respectively. Hence, we concludewith the dominated convergence theorem that one hasΛ κ ( θ κ ) ∼ κ / (1 − α ) ¯ ℓ ( κ ) Z vu s α − α ξ ( ζs α − α ) ds | {z } =Λ u,v ( ζ ) as κ tends to infinity. Now the G¨artner-Ellis theorem implies the large deviation principle forthe family ( T [ κu, κv )) κ> for 0 < u < v . It remains to prove the large deviation principle for u = 0. Note that E T [0 , κv ) = E X j ∈ I κ S j = Z ¯ I κ f (Φ − ( t )) 1 f (Φ − ( t )) dt ∼ κv and var( T [0 , κv )) = X j ∈ I κ var( S j ) = Z ¯ I κ f (Φ − ( t )) 1 f (Φ − ( t )) dt . f (0) κv. Consequently, T [0 ,κε ) κ converges in probability to ε . Thus for t < v P (cid:16) κ T [0 , κv ) t (cid:17) > P (cid:16) κ T [0 , κε ) (1 + ε ) ε (cid:17)| {z } → P (cid:16) κ T [ κε, κv ) t − (1 + ε ) ε (cid:17) and for sufficiently small ε > κ →∞ a κ log P (cid:16) κ T [0 , κv ) t (cid:17) > − Λ ∗ ε,v ( t − (1 + ε ) ε ) , while the upper bound is obvious. (cid:3) The next lemma is necessary for the analysis of the rate function in Lemma 4.5. It involves thefunction ψ defined as ψ ( t ) = 1 − t + t log t for t > Lemma 4.6. For fixed < x < x there exists an increasing function η : R + → R + with lim δ ↓ η δ = 0 such that for any u, v ∈ [ x , x ] with δ := v − u > and all w ∈ [ u, v ] , t > onehas (cid:12)(cid:12)(cid:12) Λ ∗ u,v ( t ) − w α − α t ψ (cid:0) δt (cid:1)(cid:12)(cid:12)(cid:12) η δ (cid:16) δ + t ψ (cid:0) δt (cid:1)(cid:17) . We now extend the definition of Λ ∗ continuously by setting, for any u > t > ∗ u,u ( t ) = u α − α t. For the proof of Lemma 4.6 we use the following fact, which can be verified easily.21 emma 4.7. For any ζ > and t > , we have | ξ ∗ ( ζt ) − ξ ∗ ( t ) | | ζ − | + | log ζ | + 2 | ζ − | ξ ∗ ( t ) . Proof of Lemma 4.6. First observe that γ δ := sup x with lim δ ↓ ¯ η δ = 0such that for all ζ ∈ [1 /γ δ , γ δ ] and t > | ξ ∗ ( ζt ) − ξ ∗ ( t ) | ¯ η δ (1 + ξ ∗ ( t )) . Consequently, one has for any δ > x < w, ¯ w < x with | w − ¯ w | δ and ζ ∈ [1 /γ δ , γ δ ] that | ¯ w α − α ξ ∗ ( ζt ) − w α − α ξ ∗ ( t ) | ¯ w α − α | ξ ∗ ( ζt ) − ξ ∗ ( t ) | + ξ ∗ ( t ) | ¯ w α − α − w α − α | c ¯ η δ (1 + ξ ∗ ( t )) + cδξ ∗ ( t ) , where c < ∞ is a constant only depending on x , x and α . Thus for an appropriate function( η δ ) δ> with lim δ ↓ η δ = 0 one gets | ¯ w α − α ξ ∗ ( ζt ) − w α − α ξ ∗ ( t ) | η δ (1 + ξ ∗ ( t )) . (16)Fix x < u < v < x and set δ := v − u . We estimate, for θ > δu α − α ξ ( θv − α/ (1 − α ) ) Λ u,v ( θ ) δv α − α ξ ( θu − α/ (1 − α ) ) , and the reversed inequalities for θ 0. Consequently,Λ ∗ u,v ( δt ) = sup θ [ θt − Λ u,v ( θ )] δ sup θ [ θt − u α − α ξ ( θv − α/ (1 − α ) )] ∨ δ sup θ [ θt − v α − α ξ ( θu − α/ (1 − α ) )]= δu α − α ξ ∗ (( v/u ) α − α t ) ∨ δv α − α ξ ∗ (( u/v ) α − α t ) . Since ( v/u ) α/ (1 − α ) and ( u/v ) α/ (1 − α ) lie in [1 /γ δ , γ δ ] we conclude with (16) that for w ∈ [ u, v )Λ ∗ u,v ( δt ) w α − α ξ ∗ ( t ) δ + η δ (1 + ξ ∗ ( t )) δ. To prove the converse inequality, observeΛ ∗ u,v ( t ) > (cid:16) δ sup θ [ θt − u α − α ξ ( θv − α/ (1 − α ) )] (cid:17) ∨ (cid:16) δ sup θ > [ θt − v α − α ξ ( θu − α/ (1 − α ) )] (cid:17) . Now note that the first partial Legendre transform can be replaced by the full one if t ( u/v ) α (1 − α ) .Analogously, the second partial Legendre transform can be replaced if t > ( v/u ) α (1 − α ) . Thuswe can proceed as above if t (1 /γ δ , γ δ ) and conclude thatΛ ∗ u,v ( t ) > w α − α ξ ∗ ( t ) δ − η δ (1 + ξ ∗ ( t )) δ. The latter estimate remains valid on (1 /γ δ , γ δ ) if x α/ (1 − α )1 ( ξ ∗ (1 /γ δ ) ∨ ξ ∗ ( γ δ )) η δ . Since γ δ tendsto 1 and ξ ∗ (1) = 0 one can make η δ a bit larger to ensure that the latter estimate is valid andlim δ ↓ η δ = 0. This establishes the statement. (cid:3) As the next step in the proof of Theorem 1.10 we formulate a finite-dimensional large deviationprinciple, which can be derived from Lemma 4.5.22 emma 4.8. Fix t < t < · · · < t p . Then the vector (cid:16) κ Z κt j : j ∈ { , . . . , p } (cid:17) satisfies a large deviation principle in { a · · · a p } ⊂ R p with speed a κ and rate function J ( a , . . . , a p ) = p X j =1 Λ ∗ a j − ,a j ( t j − t j − ) , with a := 0 . Proof. First fix 0 = a < a < · · · < a p . Observe that, whenever s j − < s j with s = 0, P (cid:0) κ Z κt j > a j > κ Z κs j for j ∈ { , . . . , p } (cid:1) > P (cid:0) s j − s j − < κ T [ a j − κ, a j κ ) t j − t j − for j ∈ { , . . . , p } (cid:1) . Moreover, supposing that 0 < t j − t j − − ( s j − s j − ) δ for a δ > 0, we obtain P (cid:0) a j κ Z κt j < a j + ε for j ∈ { , . . . , p } (cid:1) > P ( κ Z κt j > a j > κ Z κs j and T [ a j κ, ( a j + ε ) κ ) > δ for j ∈ { , . . . , p } (cid:1) By Lemma 4.5, given ε > A > 0, we find δ > κ large, P (cid:0) κ T [ a j κ, ( a j + ε ) κ ) < δ (cid:1) e − Aa κ . Hence, for sufficiently small δ we get with the above estimates thatlim inf κ →∞ a κ log P (cid:0) a j + ε > κ Z κt j > a j for j ∈ { , . . . , p } (cid:1) > lim inf κ →∞ a κ log P (cid:0) s j − s j − < κ T [ a j − κ, a j κ ) t j − t j − for j ∈ { , . . . , p } (cid:1) > − p X j =1 Λ ∗ a j − ,a j ( t j − t j − ) . Next, we prove the upper bound. Fix 0 = a . . . a p and 0 = b . . . b p with a j < b j ,and observe that by the strong Markov property of ( Z t ), P (cid:0) b j > κ Z κt j > a j for j ∈ { , . . . , p } (cid:1) = p Y j =1 P (cid:0) b j > κ Z κt j > a j (cid:12)(cid:12) b i > κ Z κt i > a i for i ∈ { , . . . , j − } (cid:1) p Y j =1 P (cid:0) κ T [ b j − κ, a j κ ) < t j − t j − κ T [ a j − κ, b j κ ) (cid:1) . Consequently, lim sup κ ↑∞ a κ log P (cid:0) b j > κ Z κt j > a j for j ∈ { , . . . , p } (cid:1) − p X j =1 r j , r j = Λ ∗ b j − ,a j ( t j − t j − ) if a j − b j − > t j − t j − , Λ ∗ a j − ,b j ( t j − t j − ) if b j − a j − t j − t j − , , otherwise.Using the continuity of ( u, v ) Λ ∗ u,v ( t ) for fixed t , it is easy to verify continuity of each r j of theparameters a j − , a j , b j − , and b j . Suppose now that ( a j ) and ( b j ) are taken from a predefinedcompact subset of R d . Then we have p X j =1 (cid:12)(cid:12) r j − Λ ∗ a j − ,a j ( t j − t j − ) (cid:12)(cid:12) ϑ (cid:0) max { b j − a j : j = 1 , . . . , p } (cid:1) , for an appropriate function ϑ with lim δ ↓ ϑ ( δ ) = 0. Now the upper bound follows with anobvious exponential tightness argument. (cid:3) We can now prove a large deviation principle in a weaker topology, by taking a projective limitand simplifying the resulting rate function with the help of Lemma 4.6. Lemma 4.9. On the space of increasing functions with the topology of pointwise convergencethe family of functions (cid:16) κ Z κt : t > (cid:17) κ> satisfies a large deviation principle with speed ( a κ ) and rate function J . Proof. Observe that the space of increasing functions equipped with the topology of pointwiseconvergence can be interpreted as projective limit of the spaces { a · · · a p } withthe canonical projections given by π ( x ) = ( x ( t ) , . . . , x ( t p )) for 0 < t < . . . < t p . By theDawson-G¨artner theorem, we obtain a large deviation principle with good rate function˜ J ( x ) = sup 0, and partitions t ′ = t n < · · · < t nn = t ′′ with δ n := sup j x ( t nj ) − x ( t nj − ) converging to 0. Assume n is sufficiently large suchthat η δ n ( t ′ ) α − α , with η as in Lemma 4.6. Then, n X j =1 Λ ∗ x ( t nj − ) ,x ( t nj ) ( t nj − t nj − ) > 12 ( t ′ ) α − α h n X j =1 ( t nj − t nj − ) ψ (cid:0) x ( t nj ) − x ( t nj − ) t nj − t nj − (cid:1)| {z } ( ∗ ) − ( x ( t ′′ ) − x ( t ′ )) i , (18)and ( ∗ ) is uniformly bounded as long as ˜ J ( x ) is finite. On the other hand also the finiteness ofthe right hand side of (17) implies uniform boundedness of ( ∗ ). Hence, either both expressionsin (17) are infinite or we conclude with Lemma 4.6 that for an appropriate choice of t nj ,sup t ′ = t < ··· The level sets of J are compact in I [0 , ∞ ) . Proof. We have to verify the assumptions of the Arzel`a-Ascoli theorem. Fix δ ∈ (0 , t > x ∈ I [0 , ∞ ) with finite rate J . We choose δ ′ ∈ (0 , δ ) with x t + δ ′ = ( x t + x t + δ ),denote ε = x t + δ − x t , and observe that J ( x ) > Z t + δt x α − α s [1 − ˙ x s + ˙ x s log ˙ x s ] ds > ( δ − δ ′ ) (cid:16) ε (cid:17) α − α Z t + δt + δ ′ [1 − ˙ x s + ˙ x s log ˙ x s ] dsδ − δ ′ . Here we used that x s > ε/ s ∈ [ t + δ ′ , t + δ ]. Next, we apply Jensen’s inequality to theconvex function ψ to deduce that J ( x ) > ( δ − δ ′ ) (cid:16) ε (cid:17) α − α ψ (cid:16) δ − δ ′ ε (cid:17) . Now assume that ε > δ . Elementary calculus yields J ( x ) > δ (cid:16) ε (cid:17) α − α ψ (cid:16) δ ε (cid:17) > (cid:16) ε (cid:17) − α log ε eδ . If we additionally assume ε > eδ , then we get ( J ( x ) / log δ − ) − α > ε . Therefore, in general x t + δ − x t max (cid:16) (cid:16) J ( x )log δ − (cid:17) − α , eδ (cid:17) . Hence the level sets are uniformly equicontinuous. As x = 0 for all x ∈ I [0 , ∞ ) this impliesthat the level sets are uniformly bounded on compact sets, which finishes the proof. (cid:3) We now improve our large deviation principle to the topology of uniform convergence on compactsets, which is stronger than the topology of pointwise convergence. To this end we introduce,for every m ∈ N , a mapping f m acting on functions x : [0 , ∞ ) → R by f m ( x ) t = x t j if t j := jm t < j +1 m =: t j +1 . (19) Lemma 4.11. For every δ > and T > , we have lim m →∞ lim sup κ ↑∞ a κ log P (cid:0) sup t T (cid:12)(cid:12) f m (cid:0) κ Z κ · (cid:1) t − κ Z κt (cid:12)(cid:12) > δ (cid:1) = −∞ . Proof. Note that P (cid:0) sup t T (cid:12)(cid:12) f m (cid:0) κ Z κ · (cid:1) t − κ Z κt (cid:12)(cid:12) > δ (cid:1) T m X j =0 P (cid:0) κ Z κt j +1 − κ Z κt j > δ (cid:1) . By Lemma 4.9 we havelim sup κ ↑∞ a κ log P (cid:0) κ Z κt j +1 − κ Z κt j > δ (cid:1) inf (cid:8) J ( x ) : x t j +1 − x t j > δ (cid:9) , j , as m ↑ ∞ . (cid:3) Proof of the first large deviation principle in Theorem 1.10. We apply (Dembo andZeitouni, 1998, Theorem 4.2.23), which allows to transfer the large deviation principle from thetopological Hausdorff space of increasing functions with the topology of pointwise convergence, tothe metrizable space I [0 , ∞ ) by means of the sequence f m of continuous mappings approximatingthe identity. Two conditions need to be checked: On the one hand, using the equicontinuity ofthe sets { I ( x ) η } established in Lemma 4.10, we easily obtainlim sup m →∞ sup J ( x ) η d (cid:0) f m ( x ) , x (cid:1) = 0 , for every η > 0, where d denotes a suitable metric on I [0 , ∞ ). On the other hand, by Lemma 4.11,we have that ( f m ( κ Z κ · )) are a family of exponentially good approximations of ( κ Z κ · ). (cid:3) The proof of the second large principle can be done from first principles. Proof of the second large deviation principle in Theorem 1.10. For the lower boundobserve that, for any T > ε > P (cid:0) sup t T | κ Z κt − ( t − a ) + | < ε (cid:1) > P (cid:0) Z κa = 0 (cid:1) P (cid:0) sup a t T | κ ( Z κt − Z κa ) − ( t − a ) | < ε (cid:1) , and recall that the first probability on the right hand side is exp {− κ af (0) } and the secondconverges to one, by the law of large numbers. For the upper bound note first that, by the firstlarge deviation principle, for any ε > A ⊂ { J ( x ) > ε } ,lim sup κ ↑∞ κ log P (cid:0) κ Z κ · ∈ A (cid:1) = −∞ . Note further that, for any δ > T > 0, there exists ε > J ( x ) ε impliessup t T | x − y | < δ , where y t = ( t − a ) + for some a ∈ [0 , T ]. Then, for θ < f (0), P (cid:0) sup t T | κ Z κt − y | δ (cid:1) P (cid:0) Z κa δκ (cid:1) = P (cid:0) T [0 , κδ ] > κa (cid:1) e − κaθ Y Φ( j ) κδ E exp (cid:8) θS j (cid:9) = e − κaθ exp X Φ( j ) κδ log 11 − θf ( j ) , and the result follows because the sum on the right is bounded by a constant multiple of κδ . (cid:3) Recall from the beginning of Section 4.2 that it is sufficient to show Theorem 1.12 for the ap-proximating process Z defined in (15). We initially include the case c = ∞ in our consideration,and abbreviate b κ := a κ κ α − − α ¯ ℓ ( κ ) ≪ κ α − α ¯ ℓ ( κ ) , so that we are looking for a moderate deviation principle with speed a κ b κ .27 emma 4.12. Let u < v , suppose that f and a κ are as in Theorem 1.12 and define I [ u,v ) = Z vu s − α − α ds = − α − α (cid:16) v − α − α − u − α − α (cid:17) . Then the family (cid:18) T [ κu, κv ) − κ ( v − u )) a κ (cid:19) κ> satisfies a large deviation principle with speed ( a κ b κ ) and rate function I [ u,v ) ( t ) = I [ u,v ) t if u > or t c I [0 ,v ) f (0) , c f (0) t − I [0 ,v ) ( c f (0)) if u = 0 and t > c I [0 ,v ) f (0) . Proof. Denoting by Λ κ the logarithmic moment generating function of b κ ( T [ κu, κv ) − κ ( v − u )),observe thatΛ κ ( θ ) = log E exp (cid:8) θb κ (cid:0) T [ κu, κv ) − κ ( v − u ) (cid:1)(cid:9) = X w ∈ S ∩ [ κu,κv ) log E exp (cid:8) θb κ (cid:0) T [ w ] (cid:1)(cid:9) − θκb κ ( v − u )= X w ∈ S ∩ [ κu,κv ) ξ (cid:16) θb κ ¯ f ( w ) (cid:17) − θκb κ ( v − u ) = Z I κ ¯ f ( w ) ξ (cid:16) θb κ ¯ f ( w ) (cid:17) dw − θκb κ ( v − u ) , (20)where I κ = { w > ι ( w ) ∈ [ κu, κv ) } and ι ( w ) = max S ∩ [0 , w ]. Since κu inf I κ <κu + ( f (0)) − and κv sup I κ < κv + ( f (0)) − we get (cid:12)(cid:12)(cid:12) Λ κ ( θ ) − Z I κ (cid:2) ¯ f ( w ) ξ (cid:16) θb κ ¯ f ( w ) (cid:17) − θb κ (cid:3) dw (cid:12)(cid:12)(cid:12) θb κ f (0) . (21)Now focus on the case u > 0. A Taylor approximation gives ξ ( w ) = w + (1 + o (1)) w , as w ↓ Z I κ (cid:2) ¯ f ( w ) ξ (cid:16) θb κ ¯ f ( w ) (cid:17) − θb κ (cid:3) dw ∼ Z I κ f ( w ) dw × θ b κ ∼ κ − α − α ¯ ℓ ( κ ) Z vu w − α − α dw × θ b κ = a κ b κ I [ u,v ) θ . Together with (21) we arrive at Λ κ ( θ ) ∼ a κ b κ I [ u,v ) θ . Now the G¨artner-Ellis theorem implies that the family (( T [ κu, κv ) − κ ( v − u )) /a κ ) satisfies a largedeviation principle with speed ( a κ b κ ) having as rate function the Fenchel-Legendre transform of I [ u,v ) θ which is I [ u,v ) .Next, we look at the case u = 0. If θ > c f (0) then Λ κ ( θ ) = ∞ for all κ > 0, so assume thecontrary. The same Taylor expansion as above now gives¯ f ( w ) ξ (cid:16) θb κ ¯ f ( w ) (cid:17) − θb κ ∼ θ b κ ¯ f ( w )28s w ↑ ∞ . In particular, the integrand in (20) is regularly varying with index − α − α > − κ ( θ ) ∼ θ b κ κ − α − α ¯ ℓ ( κ ) Z v s − α/ (1 − α ) ds = a κ b κ I [0 ,v ) θ . (22)Consequently, lim κ →∞ a κ b κ Λ κ ( θ ) = ( I [0 ,v ) θ if θ < c f (0) , ∞ otherwise.The Legendre transform of the right hand side is I [0 ,v ) ( t ) = ( I [0 ,v ) t if t c I [0 ,v ) f (0) , c f (0) t − I [0 ,v ) (cid:0) c f (0) (cid:1) if t > c I [0 ,v ) f (0) . Since I [0 ,v ) is not strictly convex the G¨artner-Ellis Theorem does not imply the full large deviationprinciple. It remains to prove the lower bound for open sets ( t, ∞ ) with t > c I [0 ,v ) f (0). Fix ε ∈ (0 , u ) and note that, for sufficiently large κ , P (cid:0) ( T [ κu, κv ) − κv ) /a κ > t (cid:1) > P (cid:0) ( T [ κε, κv ) − κ ( v − ε )) /a κ > c I [0 ,v ) f (0) (cid:1) × P (cid:0) ( T (0 , κε ) − κε ) /a κ > − ε (cid:1)| {z } → P (cid:0) T [0] /a κ > t − c I [0 ,v ) f (0) + ε (cid:1) . so that by the large deviation principle for (( T [ κε,κv ) − κ ( v − ε )) /a κ ) and the exponential distri-bution it follows thatlim inf κ →∞ a κ b κ log P (cid:0) ( T [0 , κv ) − κv ) /a κ > t (cid:1) > − ( c I [0 ,v ) f (0)) I [ ε,v ) − ( t − c I [0 ,v ) f (0) + ε ) c f (0) . Note that the right hand side converges to − I [0 ,v ) ( t ) when letting ε tend to zero. This establishesthe full large deviation principle for (( T [0 , κv ) − κv ) /a κ ). (cid:3) We continue the proof of Theorem 1.12 with a finite-dimensional moderate deviation principle,which can be derived from Lemma 4.12. Lemma 4.13. Fix t < t < · · · < t p . Then the vector (cid:16) a κ (cid:0) Z κt j − κt j (cid:1) : j ∈ { , . . . , p } (cid:17) satisfies a large deviation principle in R p with speed a κ b κ and rate function I ( a , . . . , a p ) = p X j =1 I [ t j − ,t j ) ( a j − − a j ) , with a := 0 . Proof. We note that, for −∞ a ( j ) < b ( j ) ∞ , we have (interpreting conditions on the rightas void, if they involve infinity) P (cid:0) a ( j ) a κ Z κt j − κt j < b ( j ) a κ for all j (cid:1) = P (cid:0) T [0 , κt j + a κ a ( j ) ) κ t j , T [0 , κt j + a κ b ( j ) ) > κ t j for all j (cid:1) . 29o continue from here we need to show that the random variables T [0 , κt + a κ b ) and T [0 , κt )+ a κ b are exponentially equivalent in the sense thatlim κ →∞ a − κ b − κ log P (cid:0)(cid:12)(cid:12) T [0 , κt + a κ b ) − T [0 , κt ) − a κ b (cid:12)(cid:12) > a κ ε (cid:1) = −∞ . (23)Indeed, first let b > 0. As in Lemma 4.12, we see that for any t > θ ∈ R , a − κ b − κ log E exp (cid:8) θb κ (cid:0) T [ κt, κt + a κ b ) − a κ b (cid:1)(cid:9) −→ , (24)Chebyshev’s inequality gives, for any A > P (cid:0) T [0 ,κt + a κ b ) − T [0 , κt ) − a κ b > a κ ε (cid:1) e − Aa κ b κ E exp (cid:8) Aε b κ (cid:0) T [ κt, κt + a κ b ) − a κ b (cid:1)(cid:9) . A similar estimate can be performed for P ( T [0 , κt + a κ b ) − T [0 , κt ) − a κ b < − a κ ε ), and theargument also extends to the case b < 0. From this (23) readily follows.Using Lemma 1.12 and independence, we obtain a large deviation principle for the vector (cid:16) a κ (cid:0) T [ κt j − , κt j ) − κ ( t j − t j − ) (cid:1) : j ∈ { , . . . , p } (cid:17) , with rate function I ( a , . . . , a p ) = p X j =1 I [ t j − ,t j ) ( a j ) . Using the contraction principle, we infer from this a large deviation principle for the vector (cid:16) a κ (cid:0) T [0 , κt j ) − κ t j (cid:1) : j ∈ { , . . . , p } (cid:17) with rate function I ( a , . . . , a p ) = p X j =1 I [ t j − ,t j ) ( a j − a j − ) . Combining this with (23) we obtain that a − κ b − κ log P (cid:0) T [0 , κt j + a κ a ( j ) ] < κ t j , T [0 , κt j + a κ b ( j ) ] > κ t j for all j (cid:1) ∼ a − κ b − κ log P (cid:0) − a κ b ( j ) < T [0 , κt j ] − κ t j < − a κ a ( j ) for all j (cid:1) , and (observing the signs!) the required large deviation principle. (cid:3) We may now take a projective limit and arrive at a large deviation principle in the space P (0 , ∞ )of functions x : (0 , ∞ ) → R equipped with the topology of pointwise convergence. Lemma 4.14. The family of functions (cid:16) a κ (cid:0) Z κt − κt (cid:1) : t > (cid:17) κ> satisfies a large deviation principle in the space P (0 , ∞ ) , with speed a κ b κ and rate function I ( x ) = (cid:26) R ∞ ( ˙ x t ) t α − α dt − c f (0) x if x is absolutely continuous and x . ∞ otherwise. roof. Observe that the space of functions equipped with the topology of pointwise convergencecan be interpreted as the projective limit of R p with the canonical projections given by π ( x ) =( x ( t ) , . . . , x ( t p )) for 0 < t < . . . < t p . By the Dawson-G¨artner theorem, we obtain a largedeviation principle with rate function˜ I ( x ) = sup We need to verify that, denoting by k · k the supremum norm on any compact subsetof (0 , ∞ ), for every δ > m →∞ lim sup κ →∞ a − κ b − κ log P (cid:0) k ¯ Z ( κ ) − f m ( ¯ Z ( κ ) ) k > δ (cid:1) = −∞ . The crucial step is to establish that, for sufficiently large κ , for all j > P (cid:16) sup t j − t 0, the right hand side exceeds for sufficientlylarge κ , thus proving (25).With (25) at our disposal, we observe that, for some integers n > n > m and the chosen compact subset of (0 , ∞ ), P (cid:0) k ¯ Z ( κ ) − f m ( ¯ Z ( κ ) ) k > δ (cid:1) n X j = n P (cid:16) sup t j − t We apply (Dembo and Zeitouni, 1998, Theorem 4.2.23) to transferthe large deviation principle from the topological Hausdorff space P (0 , ∞ ) to the metrizablespace L (0 , ∞ ) using the sequence f m of continuous functions. There are two conditions to bechecked for this, on the one hand that ( f m ( ¯ Z ( κ ) )) m ∈ N are exponentially good approximations of( ¯ Z ( κ ) ), as verified in Lemma 4.16, on the other hand thatlim sup m →∞ sup I ( x ) η d (cid:0) f m ( x ) , x (cid:1) = 0 , for every η > 0, where d denotes a suitable metric on L (0 , ∞ ). This follows easily from theequicontinuity of the set { I ( x ) η } established in Lemma 4.15. Hence the proof is complete. (cid:3) In this section we prove Theorem 1.7 and Theorem 1.15. The key to the proof is Proposition 5.1 which shows that, in the strong preference case, thedegree of a fixed vertex can only be surpassed by a finite number of future vertices. The actualformulation of the result also contains a useful technical result for the weak preference case.Recall that ϕ t = R t f ( v ) dv , and let t ( s ) = sup { t ∈ S : 4 ϕ t s } , for s > . Moreover, we let ϕ ∞ = lim t →∞ ϕ t , which is finite exactly in the strong preference case. In thiscase t ( s ) = ∞ eventually. Proposition 5.1. For any fixed η > , almost surely only finitely many of the events A s := (cid:8) ∃ t ′ ∈ [ s, t ( s )) ∩ T : Z [ s, t ′ ] > t ′ − η (cid:9) , for s ∈ T , occur. For the proof we identify a family of martingales and then apply the concentration inequalityfor martingales, Lemma A.3. For s ∈ T , let ( ¯ T su ) u ∈ S be given by ¯ T su = u − T s [0 , u ), where T s [ u, v )is the time spent by the process Z [ s, · ] in the interval [ u, v ).The following lemma is easy to verify. 33 emma 5.2. Let ( t i ) i ∈ Z + be a strictly increasing sequence of nonnegative numbers with t = 0 and lim i →∞ t i = ∞ . Moreover, assume that λ > is fixed such that λ ∆ t i := λ ( t i − t i − ) ,for all i ∈ N , and consider a discrete random variable X with P (cid:0) X = t i (cid:1) = λ ∆ t i i − Y j =1 (1 − λ ∆ t j ) for i ∈ N . Then E [ X ] = 1 λ and var( X ) λ . With this at hand, we can identify the martingale property of ( ¯ T su ) u ∈ S . Lemma 5.3. For any s ∈ S , the process ( ¯ T su ) u ∈ S is a martingale with respect to the naturalfiltration ( G u ) . Moreover, for two neighbours u < u + in S , one has var (cid:0) ¯ T su + | G u (cid:1) f ( u ) . Proof. Fix two neighbours u < u + in S and observe that given G u (or given the entry time T s [0 , u ) + s into state u ) the distribution of T s [ u ] is as in Lemma 5.2 with λ = ¯ f ( u ). Thus thelemma implies that E [ ¯ T su + |G u ] = ¯ T su + 1 f ( u ) − E [ T s [ u ] | G u ] = ¯ T su so that ( ¯ T su ) is a martingale. The variance estimate of Lemma 5.2 yields the second assertion. (cid:3) Proof of Proposition 5.1. We fix η > /f (0) and u ∈ S with ¯ f ( u ) > 2. We consider P ( A s )for sufficiently large s ∈ T . More precisely, s needs to be large enough such that t ( s ) > u and s − η − u > p s/ 2. We denote by σ the first time t in T for which Z [ s, t ] > t − η , if such a timeexists, and set σ = ∞ otherwise.We now look at realizations for which σ ∈ [ s, t ( s )) or, equivalently, A s occurs. We set ν = Z [ s, σ ].Since the jumps of Z [ s, · ] are bounded by 1 /f (0) we conclude that ν σ − η + 1 /f (0) σ. Conversely, T s [0 , ν ) + s is the entry time into state ν and thus equal to σ ; therefore, ν = Z [ s, σ ] > T s [0 , ν ) + s − η, and thus ¯ T sν = ν − T s [0 , ν ) > s − η. Altogether, we conclude that A s ⊂ (cid:8) ∃ u ∈ [0 , t ( s )) ∩ S : ¯ T su > s − η (cid:9) . By Lemma 5.3 the process ( ¯ T su ) u ∈ S is a martingale. Moreover, for consecutive elements u < u + of S that are larger than u , one hasvar( ¯ T su + |G u ) = 1¯ f ( u ) , ¯ T su + − ¯ T su f ( u ) , and ¯ T su u . λ s = s − η − u − p ϕ t ( s ) > 0, that P ( A s ) P (cid:0) sup u ∈ [0 ,t ( s )) ∩ S ¯ T su > s − η (cid:1) P (cid:0) sup u ∈ [ u ,t ( s )) ∩ S ¯ T su − ¯ T su > s − η − u (cid:1) (cid:16) − λ s ϕ t ( s ) + λ s / (cid:17) , where we use that X u ∈ S ∩ [0 ,t ( s )) f ( u ) = ϕ t ( s ) . As ϕ t ( s ) s/ 4, we obtain lim sup − s log P ( A s ) > . Denoting by ι ( t ) = max[0 , t ] ∩ T , we finallyget that X s ∈ T P ( A s ) Z ∞ e s P ( A ι ( s ) ) ds < ∞ , so that by Borel-Cantelli, almost surely, only finitely many of the events ( A s ) s ∈ T occur. (cid:3) Proof of Theorem 1.7. We first consider the weak preference case and fix s ∈ T . Recallthat ( Z [ s, t ] − ( t − s )) t > s and ( Z [0 , t ] − t ) t > are independent and satisfy functional central limittheorems (see Theorem 1.6). Thus ( Z [ s, t ] − Z [0 , t ]) t > s also satisfies a central limit theorem,i.e. an appropriately scaled version converges weakly to the Wiener process. Since the Wienerprocess changes its sign almost surely for arbitrarily large times, we conclude that Z [ s, t ] will belarger, respectively smaller, than Z [0 , t ] for infinitely many time instances. Therefore, s is not apersistent hub, almost surely. This proves the first assertion.In the strong preference case recall that ϕ ∞ < ∞ . For fixed η > 0, almost surely, only finitelymany of the events ( A s ) s ∈ T occur, by Proposition 5.1. Recalling that Z [0 , t ] − t has a finite limit,we thus get that almost surely only finitely many degree evolutions overtake the one of the firstnode. It remains to show that the limit points of ( Z [ s, t ] − t ) for varying s ∈ T are almost surelydistinct. But this is an immediate consequence of Proposition 2.2. (cid:3) From now on we assume that the attachment rule f is regularly varying with index α < , andwe represent f and ¯ f as f ( u ) = u α ℓ ( u ) and ¯ f ( u ) = u α − α ¯ ℓ ( u ) for u > . Moreover, we fix a κ = κ − α − α ¯ ℓ ( κ ) − . For this choice of ( a κ ) the moderate deviation principle, Theorem 1.12, leads to the speed ( a κ ), inother words the magnitude of the deviation and the speed coincide. The proof of Theorem 1.15is based on the following lemma. Lemma 5.4. Fix u < v and define I κ as I κ = T ∩ [ a κ u, a κ v ) . Then, for all ε > , lim κ →∞ P (cid:16) max s ∈ I κ Z [ s, κ ] ∈ κ + a κ h − v + q − α − α v − ε, − u + q − α − α v + ε i(cid:17) = 1 . roof. Our aim is to analyze the random variable max s ∈ I κ Z [ s, κ ] for large κ . We fix ζ > − u and observe that P (max s ∈ I κ Z [ s, κ ] < κ + a κ ζ ) = Y s ∈ I κ P ( Z [ s, κ ] < κ + a κ ζ ) ( P ( Z [ s max , κ ] < κ + a κ ζ ) I κ = P ( T s max [0 , κ + a κ ζ ) + s max > κ ) I κ > P ( Z [ s min , κ ] < κ + a κ ζ ) I κ = P ( T s min [0 , κ + a κ ζ ) + s min > κ ) I κ , (26)where s min and s max denote the minimal and maximal element of I κ .Next, we observe that lim κ →∞ s max /a κ = v and lim κ →∞ s min /a κ = u . Consequently, we candeduce from the moderate deviation principle, Lemma 4.12, together with Lemma 4.2, thatlog P ( T s max [0 , κ + a κ ζ ) + s max κ ) = log P (cid:16) T s max [0 , κ + a κ ζ ) − κ − a κ ζa κ − s max a κ − ζ (cid:17) ∼ − a κ I [0 , ( − v − ζ ) = − a κ 12 1 − α − α ( v + ζ ) (27)and analogously thatlog P ( T s min [0 , κ + a κ ζ ) + s min κ ) ∼ − a κ 12 1 − α − α ( u + ζ ) . Next we prove that P (max s ∈ I κ Z [ s, κ ] < κ + a κ ζ ) tends to 0 when ζ < − v + q − α − α v .If ζ < − u , then the statement is trivial since by the moderate deviation principle P ( Z [ s min , κ ] <κ + a κ ζ ) tends to zero. Thus we can assume that ζ > − u . By (26) one has P (cid:0) max s ∈ I κ Z [ s, κ ] < κ + a κ ζ (cid:1) exp (cid:8) I κ log (cid:0) − P ( T s max [0 , κ + a κ ζ ) + s max κ ) (cid:1)(cid:9) . and it suffices to show that the term in the exponential tends to −∞ in order to prove theassertion. The term satisfies I κ log (cid:0) − P ( T s max [0 , κ + a κ ζ ) + s max κ ) (cid:1) ∼ − I κ P ( T s max [0 , κ + a κ ζ ) + s max κ )= − exp n a κ h a κ log I κ + 1 a κ log P ( T s max [0 , κ + a κ ζ ) + s max κ ) | {z } =: c κ io . Since a κ log I κ converges to v , we conclude with (27) thatlim κ →∞ c κ = v − − α − α ( v + ζ ) . Now elementary calculus implies that the limit is bigger than 0 by choice of ζ . This implies thefirst part of the assertion.It remains to prove that P (max s ∈ I κ Z [ s, κ ] < κ + a κ ζ ) tends to 1 for ζ > − u + q − α − α v . Now P (cid:0) max s ∈ I κ Z [ s, κ ] < κ + a κ ζ (cid:1) > exp n I κ log (cid:0) − P ( T s min [0 , κ + a κ ζ ) + s min κ ) (cid:1)o I κ log(1 − P ( T s min [0 , κ + a κ ζ ) + s min κ )) ∼ − exp (cid:0) a κ (cid:2) a κ log I κ + 1 a κ log P ( T s min [0 , κ + a κ ζ ) + s min κ ) | {z } =: c κ (cid:3)(cid:1) . We find convergence lim κ →∞ c κ = v − − α − α ( u + ζ ) and (as elementary calculus shows) the limit is negative by choice of ζ . (cid:3) For s ∈ T and κ > Z ( s,κ ) = ( ¯ Z ( s,κ ) t ) t > the random evolution given by¯ Z ( s,κ ) t = Z [ s, s + κt ] − κta κ . Moreover, we let z = ( z t ) t > = (cid:16) − α − α (cid:0) t − α − α ∧ (cid:1)(cid:17) t > . Proof of Theorem 1.15. By Lemma 5.4 the maximal indegree is related to theunimodal function h defined by h ( u ) = − u + q − α − α u, for u > h attains its unique maximum in u max = 12 1 − α − α and h ( u max ) = u max . We fix c > − α − α , let ζ = max[ h ( u max ) − h ( u max ± ε )] and decompose the set [0 , u max − ε ) ∪ [ u max + ε, c ) into finitelymany disjoint intervals [ u i , v i ) i ∈ J , with mesh smaller than ζ/ 3. Then for the hub s ∗ κ at time κ > P ( s ∗ κ ∈ a κ [ u max − ε, u max + ε )) > P (cid:16) max s ∈ a κ [ u max − ε,u max ) ∩ T Z [ s, κ ] > κ + a κ ( h ( u max ) − ζ/ (cid:17) × Y i ∈ J P (cid:16) max s ∈ a κ [ u i ,v i ) ∩ T Z [ s, κ ] κ + a κ ( h ( v i ) + ζ/ (cid:17) × P (cid:16) max s ∈ [ c a κ , ∞ ) ∩ T Z [ s, κ ] κ (cid:17) . (28)By Lemma 5.4 the terms in the first two lines on the right terms converge to 1. Moreover, byProposition 5.1 the third term converges to 1, if for all sufficiently large κ and κ + = min[ κ, ∞ ) ∩ S ,one has 4 ϕ κ + ca κ . This is indeed the case, since one has κ + κ + f (0) − so that byLemma A.1, 4 ϕ κ + ∼ − α − α a κ . The statement on the size of the maximal indegree is now animmediate consequence of Lemma 5.4. We now prove that (an appropriately scaled version of) the evolution of a hub typicallylies in an open neighbourhood around z . 37et U denote an open set in L (0 , ∞ ) that includes z and denote by U c its complement in L (0 , ∞ ).Furthermore, we set A ε = n x ∈ L (0 , ∞ ) : max t ∈ [ , x t > u max − ε ) o for ε > 0. We start by showing that z is the unique minimizer of I on the set A . Indeed,applying the inverse H¨older inequality gives, for x ∈ A with finite rate I ( x ), I ( x ) > Z ˙ x t t α − α dt > (cid:16)Z | ˙ x t | dt (cid:17) (cid:16)Z t − α − α dt (cid:17) − > 12 1 − α − α = u max = I ( z ) . Moreover, one of the three inequalities is a strict inequality when x = z . Recall that, byLemma 4.15, I has compact level sets. We first assume that one of the entries in U c ∩ A has finite rate I . Since U c ∩ A is closed, we conclude that I attains its infimum on U c ∩ A .Therefore, I ( U c ∩ A ) := inf { I ( x ) : x ∈ U c ∩ A } > I ( z ) = u max . Conversely, using again compactness of the level sets, giveslim ε ↓ I ( U c ∩ A ε ) = I ( U c ∩ A ) . Therefore, there exists ε > I ( U c ∩ A ε ) > I ( z ). Certainly, this is also true if U c contains no element of finite rate.From the moderate deviation principle, Theorem 1.12, together with the uniformity in s , see Propo-sition 4.4, we infer thatlim sup κ →∞ a κ max s ∈ T log P (cid:0) ¯ Z ( s,κ ) ∈ U c ∩ A ε (cid:1) − I ( U c ∩ A ε ) < − I ( z ) . (29)It remains to show that P ( ¯ Z s ∗ κ ,κ ∈ U c ) converges to zero. For ε > κ , P ( ¯ Z s ∗ κ ,κ ∈ U c ) P ( s ∗ κ a κ [ u max − ε, u max + ε ])+ P (cid:0) max t ∈ [ , ¯ Z s ∗ κ ,κt u max − ε ) , s ∗ κ ∈ a κ [ u max − ε, u max + ε ] (cid:1) + P (cid:0) ¯ Z s ∗ κ ,κ ∈ U c , max t ∈ [ , ¯ Z s ∗ κ ,κt > u max − ε ) , s ∗ κ ∈ a κ [ u max − ε, u max + ε ] (cid:1) . By the first part of the proof the first and second term in the last equation tend to 0 for any ε > 0. The last term can be estimated as follows P (cid:0) ¯ Z s ∗ κ ,κ ∈ U c , ¯ Z s ∗ κ ,κt > u max − ε ) , s ∗ κ ∈ a κ [ u max − ε, u max + ε ] (cid:1) X s ∈ T ∩ a κ [ u max − ε,u max + ε ] P ( ¯ Z ( s,κ ) ∈ U c ∩ A ε ) . (30)Moreover, log (cid:0) T ∩ a κ [ u max − ε, u max + ε ] (cid:1) ∼ a κ ( u max + ε ). Since, for sufficiently small ε > I ( U c ∩ A ε ) > u max + ε we infer from (29) that the sum in (30) goes to zero. (cid:3) Appendix A.1 Regularly varying attachment rules In the following we assume that f : [0 , ∞ ) → (0 , ∞ ) is a regularly varying attachment rule withindex α < 1, and represent f as f ( u ) = u α ℓ ( u ), for u > 0, with a slowly varying function ℓ . Lemma A.1. 1. One has Φ( u ) ∼ − α u − α ℓ ( u ) as u tends to infinity and ¯ f admits the representation ¯ f ( u ) = f ◦ Φ − ( u ) = u α − α ¯ ℓ ( u ) , for u > , where ¯ ℓ is again a slowly varying function.2. If additionally α < , then ϕ u = Z u f ( u ) du ∼ − α − α u − α − α ¯ ℓ ( u ) . Proof. The results follow from the theory of regularly variation, and we briefly quote therelevant results taken from Bingham et al. (1987). The asymptotic formula for Φ is an immediateconsequence of Karamata’s theorem, Theorem 1.5.11. Moreover, by Theorem 1.5.12, the inverseof Φ is regularly varying with index (1 − α ) − so that, by Proposition 1.5.7, the composition¯ f = f ◦ Φ − is regularly varying with index α − α . The asymptotic statement about ϕ followsagain by Karamata’s theorem. (cid:3) Remark A.2. In the particular case where f ( u ) ∼ cu α , we obtainΦ( u ) ∼ c (1 − α ) u − α , Φ − ( u ) ∼ ( c (1 − α ) u ) − α and ¯ f ( u ) ∼ c − α (cid:0) (1 − α ) u (cid:1) α − α . A.2 Two concentration inequalities for martingales Lemma A.3. Let ( M n ) n ∈ Z + be a martingale for its canonical filtration ( F n ) n ∈ Z + with M = 0 .We assume that there are deterministic σ n ∈ R and M < ∞ such that almost surely • var( M n |F n − ) σ n and • M n − M n − M .Then, for any λ > and m ∈ N , P (cid:16) sup n m M n > λ + vuut m X n =1 σ n (cid:17) (cid:16) − λ P mn =1 σ n + M λ/ (cid:17) . roof. Let τ denote the first time n ∈ N for which M n > λ + p P mn =1 σ n . Then P ( M m > λ ) > m X n =1 P ( τ = n ) P (cid:16) M m − M n > − vuut m X i =1 σ i (cid:12)(cid:12)(cid:12) τ = n (cid:17) . Next, observe that var( M m − M n | τ = n ) P mi = n +1 σ i so that by Chebyshev’s inequality P (cid:16) M m − M n > − vuut m X i =1 σ i (cid:12)(cid:12)(cid:12) τ = n (cid:17) > / . On the other hand, a concentration inequality of Azuma type gives P ( M m > λ ) exp (cid:16) − λ P mn =1 σ n + M λ/ (cid:17) (see for instance Chung and Lu (2006), Theorem 2.21). Combining these estimates immediatelyproves the assertion of the lemma. 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