Universality for the distance in finite variance random graphs: Extended version Henri van den Esker∗
Remco van der Hofstad
†
Gerard Hooghiemstra∗
May 17, 2006
Abstract The asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model is generalized to a wide class of random graphs, where the degrees have finite variance. Among others, this class contains the Poissonian random graph and the generalized random graph (including the classical Erd˝ os-R´enyi graph). We prove that the graph distance grows like logν N , when the base of the logarithm equals ν = E Λ2 /E[Λ], where Λ is a positive random variable with P (Λ > x) ≤ cx1−τ , for some constant c and τ > 3. In addition, the random fluctuations around this asymptotic mean logν N are characterized and shown to be uniformly bounded. The proof of this result uses that the graph distance of all members of the class can be coupled successfully to the graph distance in the Poissonian random graph.
1
Introduction
Various papers (see e.g., [4, 6, 12, 15, 17]) study properties of random graphs with a given degree sequence. Among such properties as connectivity, cluster size and diameter, the graph distance between two uniformly chosen nodes is an important one. For two connected nodes the graph distance is defined as the minimum number of edges of a path that connects these nodes. If the nodes are not connected, then the graph distance is put equal to infinity. For the configuration model (see Section 1.4 for a definition) a distance result appeared in [12], when the distribution of the i.i.d. degrees D(C) satisfies P(D(C) > x) ≤ cx1−τ ,
(1.1)
for some constant c, all x ≥ 0, and with τ > 3. We use the superscript (C) to differentiate between models. The result in [12] states that with probability converging to 1 (whp), the typical distance between nodes in the giant component has, for ν˜ =
E[D(C) (D(C) − 1)] > 1, E[D(C) ]
(1.2)
bounded fluctuations around logν˜ N . The condition ν˜ > 1 corresponds to the supercritical case of an associated branching process. In this paper we extend the above distance result to a wide class of random graph models. Models which fall in this class are the generalized random graph (GRG), the expected degree random graph (EDRG) and the Poissonian random graph (PRG). All three models will be introduced in more detail below. ∗
Delft University of Technology, Electrical Engineering, Mathematics and Computer Science, P.O. Box 5031, 2600 GA Delft, The Netherlands. E-mail:
[email protected] and
[email protected] † Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail:
[email protected]
1
The method of proof is coupling. It is shown that the distance result holds for all models in the general class if and only if the result holds for the PRG (Section 2). In Section 4 we prove the distance result for the Poissonian random graph. This proof is parallel to that in [12] for the configuration model. In paper we included full proofs of the auxiliary lemmas contained in Section 3, since details of these proofs are different from those in [12].
1.1
Model definition
The graph models considered here are static models, meaning that the number of nodes is fixed. The graph GN has N nodes, numbered 1, 2, . . . , N . Associated with the nodes is a sequence {Λi }N i=1 of positive i.i.d. random variables, with distribution FΛ (x) = P (Λ ≤ x). We call Λi the capacity of node i. In all graphs below nodes with a large capacity will obtain a high degree, whereas nodes with small capacity have only a limited number of edges. Furthermore, we define LN = Λ1 + Λ2 + · · · + ΛN ,
(1.3)
i.e., LN is the total capacity of all nodes of the graph GN . The binary random variables {Xij }1≤i≤j≤N , are defined by setting Xij = 1, if there is a connection, i.e., one or more edges, between node i and node j in the graph GN , otherwise we set Xij = 0. If i > j, then by convention Xji = Xij . We call Xij the connection variable and pij = PN (Xij = 1) the connection probability, where PN ( · ) is the conditional distribution given the capacities {Λi }N i=1 . The graph GN obeys the following two assumptions: A1: Conditionally on the capacities, the connection variables Xij , 1 ≤ i < j ≤ N , are independent. A2: The connection probability pij , for 1 ≤ i < j ≤ N , can be written as pij = h(Λi Λj /LN ), for some function h : [0, 1] → [0, 1], satisfying h(x) − x = O x2 , for x ↓ 0. (1.4)
The current paper presents a derivation for the fluctuations of the graph distance in the graph GN with finite variance degrees, that is, we assume that for some fixed τ > 3 and some positive constant c, the capacity distribution FΛ (x) = P (Λ ≤ x) satisfies: 1 − FΛ (x) ≤ cx1−τ , for all x ≥ 0.
(1.5)
The often used condition that 1 − FΛ (x) = x1−γ L(x), for some γ > 3, with L(x) a slowly varying function is covered by (1.5), because by Potter’s Theorem [10, Lemma 2, p. 277], any slowly varying function L(x) can be bounded from above and below by an arbitrary small power of x, so that (1.5) holds for any τ , with 3 < τ < γ.
1.2
Three special cases
We give three examples of random graph models, which satisfy assumptions A1 and A2, and hence fall in the class of models considered here. The first example is the Poissonian random graph (PRG), which was introduced by Norros and Reittu in [17]. The second and third example are variants of random graph models found in the literature. The second random graph model, which we call the expected degree random graph (EDRG), is a variant of a random graph model introduced by Chung and Lu in [6, 7]. Instead of fixed weights we consider the model with i.i.d. weights {Λi }N i=1 . The third and last example is the generalized random graph (GRG), which was introduced by Britton, Deijfen and Martin-L¨ of [5]. We now define the three models and verify that they satisfy the conditions A1 and A2 .
2
• The Poissonian Random Graph model: In [17] the Poissonian random graph is intro) duced. The main feature of such a graph G(P is that, conditionally on the capacities, the N number of edges between any pair of nodes i and j is a Poisson random variable. The model in [17] is introduced as a growth model, but as a consequence of [17, Proposition 2.1], it can ) be formulated as a static model, and we will do so. Start with the graph G(P consisting of N N N nodes and capacities {Λi }i=1 . The number of edges between two different nodes i and j is (P ) given by an independent Poisson random variable Eij with random parameter Λi Λj /LN .
(1.6)
The connection variables are then Xij(P ) = 1{E (P ) >0} , so that, for 1 ≤ i ≤ j ≤ N , the connection ij
probabilities are given by Λi Λj (P ) (P ) (P ) pij = PN Xij = 1 = PN Eij > 0 = 1 − exp − = h(P ) (Λi Λj /LN ), LN where h(P ) (x) = 1 − e−x . Obviously, h(P ) (x) − x = O x2 for x ↓ 0. Since, by definition, the random variables {Xij(P ) }1≤i 3, then the expected degree of a node i is given by Λi , as N N X X (E) EN Xij = Λi Λj /LN = Λi , j=1
j=1
where we used the notation EN[ · ] as the conditional expectation under the probability measure PN ( · ).
The Erd˝os-R´enyi random graph, usually denoted by G(N, p), is a special case of the EDRG. In the graph G(N, p), an edge between a pair of nodes is present with probability p ∈ [0, 1], independently of the other edges. When p = λ/N for some constant λ > 0, then we obtain the graph G(N, λ/N ) from the EDRG by picking Λi = λ for all i, since then p(E) ij = Λi Λj /LN = λ/N = p, for all 1 ≤ i < j ≤ N .
• The Generalized Random Graph model: The GRG model is an adapted version of with N nodes as follows. The the EDRG model, see the previous example. We define G(G) N sequence of connection variables, is, conditionally on the capacities, again given by a sequence of independent Bernoulli random variables {Xij(G) }1≤i 0, to denote a Poisson random variable with parameter λ. Similarly, the expected number of offspring in the second and further generations is given by ν=
∞ X
n=1
# "∞ # " ∞ n+1 X E Λ2 1 X 1 −Λ Λ 2 ngn = E P (P oi(Λ) = n) = n(n + 1)e = E Λ . µ (n + 1)! µ µ
(1.10)
n=0
n=1
We define the graph distance or hopcount HN between two different randomly chosen nodes A1 and A2 in the graph GN as the minimum number of edges that form a path from the node A1 to node A2 where, by convention, the distance equals ∞ if the nodes A1 and A2 are not connected. Theorem 1.1 (Fluctuations of the graph distance) Fix τ > 3 in (1.5), assume that ν > 1 and that assumptions A1 and A2 are satisfied. For k ≥ 1, let σk = ⌊logν k⌋ and ak = σk − logν k. There exists random variables (Ra )a∈(−1,0] such that, as N → ∞, P (HN = σN + l | HN < ∞) = P (RaN = l) + o(1).
(1.11)
We identify the random variables (Ra )a∈(−1,0] in Theorem 1.3 below. Before doing so, we state a consequence of Theorem 1.1: Corollary 1.2 (Concentration of the graph distance) Under the assumptions of Theorem 1.1, • with probability 1 − o(1) and conditionally on HN < ∞, the random variable HN is in between (1 ± ε) logν N for any ε > 0; • conditionally on HN < ∞, the sequence of random variables HN − logν N forms a tight sequence, i.e., (1.12) lim lim sup P |HN − logν N | ≤ K HN < ∞ = 1. K→∞ N →∞
4
We use a limit result from branching process theory to identify the limiting random variables (Ra )a∈(−1,0] . It is well known, see [10, p. 244], that the process {Zl /µν l−1 }l≥1 is a martingale with uniformly bounded expectation and consequently converges almost surely to a limit W: Zl = W, a.s. l→∞ µν l−1 lim
(1.13)
Let W (1) and W (2) be two independent copies of W in (1.13), then we can identify the limit random variables (Ra )a∈(−1,0] as follows: Theorem 1.3 Under the assumptions of Theorem 1.1 and for a ∈ (−1, 0], P (Ra > j) = E exp −κν a+j W (1) W (2) |W (1) W (2) > 0 , where κ = µ(ν − 1)−1 .
1.4
Relations with the configuration model
The configuration model (CM) appeared in the context of random regular graphs as early as 1978 (see [2, 14]). Here we consider the CM as introduced in [12]. Start with an i.i.d. sequence {Di(C) }N i=1 of positive integer valued random variables , where Di(C) will denote the degree of node i. To built (C) (C) is odd is even, so if D1(C) + D2(C) + . . . + DN a graph it is mandatory that D1(C) + D2(C) + . . . + DN (C) we increase DN by one, which will have little effect. We build the graph model by attaching Di(C) stubs or half edges to node i and pair the stubs at random, so that two half edges will form one edge. In [12], the authors prove a version of Theorem 1.1-1.3 for the configuration model. The Theorems 1.1-1.3 hold verbatim for the configuration model with only two changes: 1. Replace the condition ν > 1 in Theorem 1.1 by the condition ν˜ > 1, defined in (1.2). 2. Replace the offspring distributions of the BP {Zl }l≥0 , by (a)
f˜n = P(D(C) = n),
n ≥ 1,
(b) (n + 1)f˜n+1 g˜n = P∞ , ˜ m=1 mfm
n ≥ 0.
One wonders why a result like the Theorems 1.1-1.3, holds true for the class of models introduced in Section 1.1, especially if one realizes that in the CM the degrees are independent, and the edges are not, whereas for instance in the GRG (and in the other two examples) precisely the opposite is true, i.e., in the GRG the edges are independent and the degrees are not. To understand at least at an intuitive level why the distance result holds true, we compare the configuration model with the generalized random graph. (C) By construction the degree sequence D1(C) , D2(C) , . . . , DN of the CM is an i.i.d. sequence, and (C) = dN }, the graph configuration is uniform conditionally on D = {D1(C) = d1 , D2(C) = d2 , . . . , DN over all configurations satisfying D, because the pairing is at random. Hence if we condition on both the event D and the event S = {the resulting graph has no self-loops and no multiple edges}, then the CM renders a simple graph, which is picked uniformly from all possible simple configurations with degree sequence satisfying D. Since for N → ∞ the probability of the event S converges to exp(−ν/2 − ν 2 /4) > 0 (see [3, p. 51]), it follows from [14, Theorem 9.9] that properties that hold whp in the CM also hold whp in the conditioned simple graph. Hence a property as tightness of the graph distance HN(C) in the CM is inherited by the conditioned simple graph, with the same degree sequence. This suggests that also the limiting distribution of the fluctuations of the graph 5
distance in the CM conditioned on S is the same as the one in the CM as identified in [12]. A direct proof of this claim is missing. On the other hand the GRG with given degree sequence d1 , d2 , . . . , dN is also uniform over all possible (simple) configurations. Moreover [5, Theorem 3.1] shows that the degree sequence (G) D1(G) , D2(G) , . . . , DN of the GRG is asymptotically independent with marginal distribution a mixed Poisson distribution: Z ∞ xk e−x dFΛ (x), k = 0, 1, 2 . . . , P(D(G) = k) = (1.14) k! 0 (G) as an i.i.d. sequence where FΛ is the capacity distribution. Hence starting from D1(G) , D2(G) , . . . , DN with common distribution given by (1.14), the (conditioned) CM with these degrees is close to the GRG, at least in an asymptotic sense, so that one expects that the asymptotic fluctuations of the graph distance of the CM also hold for the generalized random graph. Also note from the mixed Poisson distribution (1.14), that E Λ2 E[D(C) (D(C) − 1)] ν˜ = = , E[D(C) ] E[Λ]
which is equal to ν, according to (1.9) and (1.10). As said earlier, a proof of this intuitive reasoning is missing, and our method of proof is by coupling each random graph satisfying A1 and A2 to the Poisson random graph (PRG), and by giving a separate proof of Theorem 1.1-1.3 for the PRG. We finish this section by giving an overview of different distance results in random graphs. Let τ denote the exponent of the probability mass function of the degree distribution. In this paper and in [6, 12] the case τ > 3 is studied. Results for 2 < τ < 3 for various models appeared in [6, 13, 17, 16]. Typically in that case, the distance fluctuates around a constant times 2 log log N/| log(τ − 2)|. For 1 < τ < 2, there exists a subset of nodes with a high degree, called the core (see [9]). The core forms a complete graph and almost every node is attached to the core and, thus, the graph distance is whp at most 3.
1.5
Organization of the paper
The coupling argument that ties the fluctuations of the graph distance HN(P ) in the PRG to the fluctuations of the graph distance in random graphs satisfying assumptions A1 and A2 is treated in Section 2. In Section 4 we show that the fluctuations of the graph distance HN(P ) is given by Theorem 1.1. The derivation of the fluctuations of the graph distance HN(P ) is similar to the derivation of the fluctuations of the graph distance HN(C) in the configuration model, see [12]. The proof in [12] is more complicated than the proof presented here for the PRG model, mainly because in the latter the expansion of a given node (e.g. the nodes on a given distance) can be described by means of the so called Reittu-Norros process, a marked branching process. This branching process will be introduced in Section 3. In this paper full proofs of the auxiliary propositions and lemmas introduced in Sections 3 and 4 are presented in the appendix. These proofs were omitted in [8].
2
Coupling
In this section we denote by GN the PRG and by G′N some other random graph satisfying the assumptions A1 and A2, given in Section 1.1. We number the nodes of both GN and G′N from 1 to N and we assign the capacity Λi , for 1 ≤ i ≤ N , to node i in each graph. We denote by HN and HN′ the graph distance between two randomly chosen nodes A1 and A2 , such that A1 6= A2 , in GN and G′N , respectively. We will show that for N → ∞, P HN 6= HN′ = o(1). (2.1) 6
The above implies that whp the coupling of the graph distances is successful. Therefore, given the succesful coupling (2.1), it is sufficient to show Theorem 1.1 for the PRG.
2.1
Coupling of GN and G′N
We next describe the coupling of the connection variables of the graphs GN and G′N . A classical coupling is used, see e.g. [18]. Denote by {Xij }1≤i 0, using Lemma 2.1, such that P Sq,α ≤ N −β0 for q = 1, 2. On the event 0 S1,α0 ∩ S2,α0 , using (2.2), (2.3), (2.5) and (2.7), we bound whp the total expected number of mismatches due to a single node by EN[Ki ] =
X
EN[Kij ] =
j6=i
j6=i
=
X
CΛ2i
N CΛ2i X 2 PN (Kij = 1) ≤ 2 Λj LN j=1
SN ,2 ≤ CSS −2 Λ2i N −1 ≤ CΛ2i N −1 , N SN2 ,1
(2.8)
where C is a constant that may change from line to line. Thus, whp the total number of mismatches is bounded from above by N γ , for any γ > 0, since ! "N # N X X γ −γ c c −γ −β0 P Ki > N ≤N E ∪ S ≤ CN + O N . EN[Ki ] 1S1,α0 ∩S2,α0 + P S1,α 2,α 0 0 i=1
i=1
We see that the right hand goes to zero if N goes to infinity, which implies that whp the total number of mismatches is bounded from above by N γ for any γ > 0. Define the event AN as (N ) N \ X Ki 1{Λi >cN } = 0 = AN = Ki 1{Λi >cN } = 0 , (2.9) i=1
i=1
where cN = N ξ for each ξ > 0. Then, on the event AN , all nodes with capacity greater than cN are successfully coupled. Lemma 2.2 For each ξ > 0 there exists a constant θ > 0 such that PN (AcN ) = O N −θ .
(2.10)
Proof. Fix α0 > 0 as in Lemma 2.1. On the event S1,α0 ∩ S2,α0 , we bound PN (AcN ) using Boole’s inequality, the Markov inequality and (2.8), which yields c
PN (AN ) ≤
N X i=1
PN Ki 1{Λi >cN }
N N X C X 2 EN[Ki ] 1{Λi >cN } ≤ >0 ≤ Λi 1{Λi >cN } . N i=1
i=1
Therefore, N h i C X 2 P (AcN ∩ S1,α0 ∩ S2,α0 ) = E PN (AcN ) 1S1,α0 ∩S2,α0 ≤ E Λi 1{Λi >cN } = CE Λ2 1{Λ>cN } , N i=1
for some constant C > 0. Using integration by parts and 1 − FΛ (x) ≤ cx1−τ we have, Z ∞ ∞ w[1 − FΛ (w)]dw = O N −(τ −3)ξ , E Λ2 1{Λ>cN } = −w2 [1 − FΛ (w)] w=c + 2 N
cN
8
(2.11)
since c N w2 [1 − FΛ (w)] w=∞ ≤ c2N [1 − FΛ (cN )] = O N −(τ −3)ξ
and
Z
∞
cN
w[1 − FΛ (w)]dw ≤ c
Z
∞
cN
This yields
w2−τ dw = O N −(τ −3)ξ .
c c P (AcN ) ≤ P (AcN ∩ S1,α0 ∩ S2,α0 ) + P S1,α + P S2,α = O N − min{β0 ,(τ −3)ξ} = O N −θ 0 0
for some θ > 0.
2.2
Coupling the graph distances of GN and G′N
In this subsection we couple the graph distance of the PRG with any random graph satisfying assumptions A1 and A2. Theorem 2.3 Let GN be a PRG and let and G′N be a random graph satisfying assumption A1 and A2. Let HN and HN′ be the graph distances between two different uniformly chosen nodes A1 and A2 in, respectively, the graphs GN and G′N . Then P HN 6= HN′ = o(1). (2.12)
The above theorem implies that, whp, the coupling of the graph distances HN and HN′ is successful. In order to prove Theorem 2.3, we use the following proposition. In its statement, we consider the neighborhood shells of a uniformly chosen node A ∈ {1, 2, . . . , N }, i.e., all nodes on a fixed graph distance of node A. More precisely, ∂N0 = {A}
and
∂Nl = {1 ≤ j ≤ N : d(A, j) = l},
(2.13)
where d(i, j) denotes the graph distance between nodes i and j, i.e., the minimum number of edges in a path between the nodes i and j. Furthermore, define the set of nodes reachable in at most j steps from root A by Nl = {1 ≤ j ≤ N : d(A, j) ≤ l} =
l [
∂Nk .
(2.14)
k=0
Proposition 2.4 For N sufficiently large, l ∈ N, some constant C ′ > 0, and every b ∈ (0, 1), P HN 6= HN′ ≤ P (AcN ) + P (HN > 2l) + 2lP |Nl−1 | > N b + 2C ′ S −2 lN −(1−b) c4N . (2.15) Before giving a proof, we show that Theorem 2.3 is a consequence of Proposition 2.4. Proof of Theorem 2.3. By Lemma 2.2, we have that, for τ > 3, P (AcN ) ≤ N −θ . We fix l = lN = ⌊( 21 + η) logν N ⌋. Then, from Corollary 1.2, applied to the PRG model, we obtain that P (HN > 2l) = o(1). The third term in the right hand side of (2.15) can be bounded using the following lemma: Lemma 2.5 Let {Nl }l≥0 be the reachable sets of a uniformly chosen node A in the PRG GN . Then for η, δ ∈ (−1/2, 1/2) and all l ≤ (1/2 + η) logν N , there exists a constant β1 > 0 such that 1/2+δ P |Nl | > N = O (logν N )N − min{δ−η,β1 } . (2.16) 9
Proof.
See the proof of Lemma A.3.3.
We now prove that all terms in the right hand of (2.15) are o(1) for an appropriate choice of b. Lemma 2.5 implies that 2lP |Nl | > N b = o(1) for some appropriately chosen b > 12 . Then, provided that b < 1, we see that 2C ′ S −2 lN b−1 c4N = 2C ′ S −2 lN 4ξ+b−1 = o(1), where we substitute cN = N ξ , and picking ξ ∈ (0, (1 − b)/4). Hence, by Proposition 2.4, P (HN 6= HN′ ) = o(1), which is precisely the content of Theorem 2.3. Proof of Proposition 2.4. We use that P HN 6= HN′ ≤ P (AcN ) + P (HN > 2l) + P {HN ≤ 2l} ∩ AN ∩ {HN 6= HN′ } .
(2.17)
Let Nl(i) and Nl′(i) , for i = 1, 2, be the union of neighborhood shells of the nodes Ai in GN and G′N , respectively. Now, we use the fact that if HN ≤ 2l and if HN 6= HN′ , then Nl(1) 6= Nl′(1) and/or Nl(2) 6= Nl′(2) . By the exchangeability of the nodes, we have (2.18) P {HN ≤ 2l} ∩ AN ∩ {HN 6= HN′ } ≤ 2P {Nl 6= Nl′ } ∩ AN ,
where Nl and Nl′ are the neighborhood shells of a uniformly chosen node A in respectively GN and ′ G′N . If Nl 6= Nl′ , then there must be a k ∈ {1, . . . , l} for which Nk 6= Nk′ , but Nk−1 = Nk−1 . Thus, l X ′ P {HN ≤ 2l} ∩ AN ∩ {HN 6= HN′ } ≤ 2 P {Nk 6= Nk′ } ∩ {Nk−1 = Nk−1 } ∩ AN .
(2.19)
k=1
′ In turn, the event {Nk 6= Nk′ } ∩ {Nk−1 = Nk−1 } implies that one of the edges from ∂Nk−1 must be c , where N c miscoupled, thus Kij = 1 for some i ∈ ∂Nk−1 and j ∈ Nk−1 k−1 = {1, 2, . . . , N }\Nk−1 . The event AN implies that Λi , Λj ≤ cN . Therefore, we bound
′ } ∩ AN ≤ PN |Nk−1 | > N b PN {Nk 6= Nk′ } ∩ {Nk−1 = Nk−1 X c , Kij = 1} ∩ {|Nk−1 | ≤ N b } 1{Λi ,Λj ≤cN } . (2.20) + PN {i ∈ ∂Nk−1 , j ∈ Nk−1 i,j
c Also, under PN , since i ∈ Nk−1 and j ∈ ∂Nk−1 , the event {Kij = 1} is independent of Nk−1 and, therefore, from ∂Nk−1 as ∂Nk−1 ⊂ Nk−1 . The edge between the nodes i and j points out of Nk−1 , while Nk−1 is determined by the occupation status of edges that are between nodes in Nk−2 or pointing out of ∂Nk−2 . Thus, we can replace each term in the sum of (2.20) by c } ∩ {|Nk−1 | ≤ N b } 1{Λi ,Λj ≤cN } . (2.21) PN (Kij = 1) PN {i ∈ ∂Nk−1 , j ∈ Nk−1
Since by (2.2) and (2.7), we have
PN (Kij = 1) 1{Λi ,Λj ≤cN } = EN[Kij ] 1{Λi ,Λj ≤cN } ≤ C ′
Λ2i Λ2j 1 ≤ C ′ S −2 c4N N −2 , L2N {Λi ,Λj ≤cN }
(2.22)
we can bound the right hand side of (2.20) from above by X c } ∩ {|Nk−1 | ≤ N b } . PN {i ∈ ∂Nk−1 , j ∈ Nk−1 PN |Nk−1 | > N b + C ′ S −2 c4N N −2 i,j
Finally, we bound the sum on the right side by h i X c } ∩ {|Nk−1 | ≤ N b } ≤ N EN |∂Nk−1 |1{|Nk−1 |≤N b } ≤ N 1+b . PN {i ∈ ∂Nk−1 , j ∈ Nk−1 i,j
10
Therefore, we can bound each term in the sum of (2.19), with P replaced by PN , ′ PN {Nk 6= Nk′ } ∩ {Nk−1 = Nk−1 } ∩ AN ≤ PN |Nk−1 | > N b + C ′ S −2 c4N N −1+b .
Since, for k ≤ l, we have that PN |Nk−1 | > N b ≤ PN |Nl−1 | > N b , by summing over k = 1, . . . , l in (2.19), we arrive at PN {HN ≤ 2l} ∩ AN ∩ {HN 6= HN′ } ≤ 2lPN |Nl−1 | > N b + C ′ S −2 lc4N N −(1−b) , which in turn implies
P {HN ≤ 2l} ∩ AN ∩ {HN 6= HN′ } ≤ 2lP |Nl−1 | > N b + C ′ S −2 lc4N N −(1−b) .
Therefore, we can bound (2.17) by
P HN 6= HN′ ≤ P (AcN ) + P (HN > 2l) + 2lP |Nl−1 | > N b + C ′ S −2 lc4N N −(1−b) ,
which is precisely the claim (2.15).
3
The Poissonian random graph model
The proof of the fluctuations of the graph distance in the configuration model in [12] is done in a number of steps. One of the most important steps is the coupling of the expansion of the neighborhood shells of a node to a BP. For the PRG, we follow the same strategy as in [12], although the details differ substantially. The first step is to introduce the NR-process, which is a marked BP. The NR-process was introduced by Norros and Reittu in [17]. We can thin the NR-process in such a way that the resulting process, the NR-process, can be coupled to the expansion of the neighborhood shells of a randomly chosen node in the PRG. Finally, we introduce capacities for the NR-process and the NR-process.
3.1
The NR-process and the NR-process
The NR-process is a marked delayed BP denoted by {Z l , M l }l≥0 , where Z l denotes the number of individuals of generation l, and where the vector M l = (M l,1 , M l,2 , . . . , M l,Zl ) ∈ {1, 2, . . . , N }Zl , denotes the marks of the individuals in generation l. We now give a more precise definition of the NR-process and describe its connection with GN , the PRG. We define Z 0 = 1 and take M 0,1 uniformly from the set {1, 2, . . . , N }, corresponding to the choice of A1 , which is uniformly over all the nodes. The offspring of an individual with mark m ∈ {1, 2, . . . , N } is as follows: the total number of children has a Poisson distribution with parameter Λm , of which, for each i ∈ {1, 2, . . . , N }, a Poisson distributed number with parameter Λi Λm , LN bears mark i, independently of the other individuals. Since N X Λi Λm i=1
LN
=
N Λm X Λi = Λm , LN i=1
11
(3.1)
and sums of independent Poisson random variables are again Poissonian, we may take the number of children with different marks mutually independent. As a result of this definition, the marks of the children of an individual in {Z l , M l }l≥0 can be seen as independent realizations of a random variable M , with distribution PN (M = m) =
Λm , LN
1 ≤ m ≤ N,
(3.2)
and, consequently, EN[ΛM ] =
N X
Λm PN (ΛM = Λm ) =
N X
m=1
m=1
N 1 X 2 Λm . Λm PN (M = m) = LN
(3.3)
m=1
For the definition of the NR-process we start with a copy of the NR-process {Z l , M l }l≥0 , and reduce this process generation by generation, i.e., in the order M 0,1 , M 1,1 , . . . M 1,Z1 , M 2,1 , . . .
(3.4)
by discarding each individual and all its descendants whose mark has appeared before. The process obtained in this way is called the NR-process and is denoted by the sequence {Z l , M l }l≥0 . One of the main results of [17] is Proposition 3.1: Proposition 3.1 Let {Zl , M l }l≥0 be the NR-process and let M l be the set of marks in the l−th generation, then the sequence of sets {M l }l≥0 has the same distribution as the sequence {∂Nl }l≥0 given by (2.13). As a consequence of the previous proposition, we can couple the NR-process to the neighborhood shells of a uniformly chosen node A ∈ {1, 2, . . . , N }, i.e., all nodes on a fixed graph distance of A, see (2.13) and note that A ∼ M 0,1 . Thus, using the the above proposition, we can couple the expansion of the neighborhood shells and the NR-process in such a way that M l = ∂Nl
Z l = |∂Nl |, l ≥ 0.
and
(3.5)
Furthermore, we see that an individual with mark m, 1 ≤ m ≤ N , in the NR-process is identified with node m in the graph GN , whose capacity is given by Λm . We will now show that the offspring distribution of the BP {Z l }l≥0 converges as N → ∞ to the offspring distribution of {Zl }l≥0 , introduced in Section 1.3. The offspring distribution f (N) of Z 1 , i.e., the first generation of {Z l }l≥0 , is given by fn(N) = PN (P oi(ΛA ) = n) =
N X
m=1
N 1 X −Λm Λnm PN P oi(ΛA ) = n A = m P (A = m) = , e N n!
(3.6)
m=1
for n ≥ 0. Recall that individuals in the second and further generations have a random mark distributed as M , given by (3.2). Hence, if we denote the offspring distribution of the second and further generations by gn(N) , then we obtain (N)
gn = PN (P oi(ΛM ) = n) =
N X
m=1
=
N X
m=1
e−Λm
PN P oi(ΛM ) = n M = m PN (M = m)
N Λnm Λm 1 X −Λm Λn+1 m e = , n! LN LN n! m=1
12
(3.7)
(3.8)
for n ≥ 0. Furthermore, we can relate gn(N) and fn(N) by gn(N) =
N N (N) (n + 1)fn+1 1 X −Λm Λn+1 (n + 1) 1 X −Λm Λn+1 m m e e = = . LN n! LN /N N (n + 1)! LN /N m=1
(3.9)
m=1
Observe that {fn(N) }n≥0 is the average over Poisson probabilities, whereas (3.9) shows that {gn(N) }n≥0 (N) }n≥0 . Since {Λm }N comes from size biased sampling of {fn+1 m=1 are i.i.d. the strong law of large (N) number states that for N → ∞ the limit distributions of f and g (N) are given by N n 1 X −Λm Λnm (N) −Λ Λ e fn = →E e = fn , n ≥ 0, (3.10) N n! n! m=1
and, as a consequence, (N)
gn
(N) (n + 1)fn+1 (n + 1)fn+1 → = gn , n ≥ 0. = LN /N µ
(3.11)
Indeed, according to (1.7) and (1.8) the limit distributions are equal to the offspring distributions of the delayed BP {Zl }l≥0 introduced in Section 1.3.
3.2
Coupling the NR-process with a delayed BP
In this subsection we will introduce a coupling between the NR-process with the delayed BP {Zl }l≥0 , which is defined by (1.7) and (1.8) in Section 1.3. This coupling is used in the proof of Theorem 1.1 and 1.3 for the PRG, to express the probability distribution of HN in terms of the BP {Zl }l≥0 . Introduce the capacity of the l-th generation of the NR-process {Z l , M l }l≥0 and the NR-process {Z l , M l }l≥0 as, respectively, C l+1 =
Zl X i=1
Λ M l,i
C l+1 =
and
Zl X i=1
Λ M l,i , l ≥ 0.
Using the coupling given by (3.5), we can rewrite the capacity C l+1 as X Λi . C l+1 =
(3.12)
(3.13)
i∈∂Nl
For the proof of Theorem 1.1 and 1.3, in the case of the PRG, we need to control the difference between C l and C l for fixed l. For this we will use the following proposition: Proposition 3.2 There exist constants α2 , β2 > 0, such that for all 0 < η < α2 and all l ≤ (1/2 + η) logν N , ! l X P (3.14) (C k − C k ) > N 1/2−α2 ≤ N −β2 . k=1
Proof.
The proof is deferred to Section A.3.
In order to prove Theorem 1.1 and Theorem 1.3 we will grow two NR-processes {Z l(i) , M l(i) }l≥0 , for i = 1, 2. The root of {Z l(i) , M l(i) }l≥0 starts from a uniformly chosen node or mark Ai ∈ {1, 2, . . . , N }. These two nodes are different whp, because P (A1 = A2 ) =
1 . N
By (3.5) the NR-process can be coupled to the neighborhood expansion shells {Nl(1) }l≥0 and {Nl(2) }l≥0 . In the following lemma we compute the distribution of the number of edges between two shells with different subindeces, i.e., Nk(1) and Nl(2) . 13
Lemma 3.3 Consider the neighborhood expansion shells {Nl(1) }l≥0 and {Nl(2) }l≥0 . Then conditionally on Nk(1) and Nl(2) and given that Nk(1) ∩ Nl(2) = ∅ the number of edges between the nodes in Nk(1) and Nl(2) , for fixed positive integers k and l, is distributed as a Poisson random variable with mean (1) (2) Ck+1 C l+1 . LN
(3.15)
Proof. Conditioned on Nk(1) , Nl(2) and Nk(1) ∩ Nl(2) = ∅, the number of edges between Nk(1) and (2) Nl is given by X
(1) i∈∂Nk
X
(P ) Eij ,
(3.16)
(2) j∈∂Nl
(P ) where Eij are independent Poisson random variables with mean Λi Λj /LN , see (1.6). Sums of independent Poisson random variables are again Poissonian, thus (3.16) is a Poisson random variable with mean the expected value of (3.16): (1) (2) X X Λi Λj 1 X X C C = Λi Λj = k+1 l+1 , LN LN LN (1) (2) (1) (2)
i∈∂Nk
i∈∂Nk
j∈∂Nl
i∈∂Nl
where we have used (3.13) in the last step.
The further proof of Theorems 1.1-1.3 crucially relies on the following technical claim: Proposition 3.4 There exist constants α3 , β3 , η > 0 such that for all l ≤ (1 + 2η) logν N , as N → ∞, P
l+1 l+1 1 X X (2) (1) (1) (2) C⌈k/2⌉ − Z⌊k/2⌋ Z⌈k/2⌉ C ⌊k/2⌋ > N −α3 = O N −β3 . N k=2
Proof.
(3.17)
k=2
The proof is deferred to Section A.4.
In the next section we will use this proposition in combination with Lemma 3.3 to replace sums over capacities, which do depend on N , by sums over sizes of a BP, which do not depend on N anymore.
4
Proof of Theorem 1.1 and 1.3 for the PRG
In this section, we give the proof of Theorem 1.1 and 1.3 for the PRG model. Using the coupling result in Proposition 2.4 we obtain Theorem 1.1 and 1.3 for all random graphs satisfying the assumptions A1 and A2. As in the previous section, we denote by GN a PRG. We grow two NR-processes. Each NR-process starts from a uniformly chosen node Ai ∈ {1, 2, . . . , N }, i = 1, 2, such that A1 6= A2 , whp. Step 1: Expressing P(HN > l) in capacities. We have HN > 1 iff (if and only if) there are no edges between the nodes A1 and A2 , which is equivalent to the fact that XA1 A2 = 0. Given the capacities C 1(1) and C 1(2) , the number of edges between the nodes A1 and A2 has, according to Lemma 3.3, a Poisson distribution with mean C 1(1) C 1(2) . LN 14
(4.1)
By taking expectations P(HN > 1) = E[PN (XA1 A2
C 1(1) C 1(2) = 0)] = E exp − . LN
(4.2)
We next inspect the capacity of the first generation of Z 1(1) given by C 2(1) . Given C 2(1) and C 1(2) , that is the total capacity of the nodes in Z 1(1) and the capacity of node A2 , we again have a Poisson number of edges between node A2 and the nodes in Z 1(1) , however, this time with parameter C 2(1) C 1(2) . LN
(4.3)
In order to compute the survival probability P(HN > l) we need more notation. We write (l ,l ) (l ,l ) 1 2 QC1 2 for the conditional probabilities given {C k(1) }lk=1 and {C k(2) }lk=1 . We further write EC1 2 for (l ,l ) 1 the expectation with respect to QC1 2 . For l2 = 0, we only condition on {C k(1) }lk=1 . Lemma 3.3 implies that C k(1) C l(2) (k,l) QC (HN > k + l − 1|HN > k + l − 2) = exp − . LN Indeed, the event {HN > k + l − 2} implies that Nk(1) ∩ Nl(2) = ∅. Then from (4.1) and the above statement, (1,1) P(HN > 2) = E Q(1,1) C (HN > 1) QC (HN > 2|HN > 1) (2,1) (1,1) QC (HN > 2|HN > 1) = E Q(1,1) C (HN > 1) EC (1,1) (2,1) Q (H > 1) Q (H > 2|H > 1) = E E(1,1) N N N C C C C 2(1) C 1(2) C 1(1) C 1(2) · exp − = E exp − LN LN " ( P3 )# (1) (2) C ⌈k/2⌉ C ⌊k/2⌋ = E exp − k=2 . LN By induction we obtain as in [12, Lemma 4.1], " ( Pl+1 P(HN > l) = E exp −
(1) (2) k=2 C ⌈k/2⌉ C ⌊k/2⌋ LN
)#
.
(4.4)
Note that (4.4) is an equality, while in [12] an error needed to be taken along. Step 2: Coupling with the BP with offspring distribution {gn }. In this step we replace C l(1) and C l(2) by Zl(1) and Zl(2) . For each event B, and any two nonnegative random variables V and W , −V E e − E e−W ≤ E (e−V − e−W )1B + E (e−V − e−W )1Bc ≤ E (e−V − e−W )1B + P (B c ) .
Now take
B=
(
l+1 l+1 X 1 X (1) (2) (1) (2) Z⌈k/2⌉ Z⌊k/2⌋ − C ⌈k/2⌉ C ⌊k/2⌋ ≤ N −α3 N k=2
k=2
)
and the random variables V and W as V =
l+1 1 X (1) (2) , Z⌈k/2⌉ Z⌊k/2⌋ N
W =
k=2
l+1 1 X (1) (2) C ⌈k/2⌉ C ⌊k/2⌋ . N k=2
15
,
Then for l ≤ (1 + 2η) logν N the Proposition 3.4 implies that P(B c ) = O N−β3 , whereas on the event B we have |V − W | ≤ N −α3 . Hence, using that e−v − e−w = O v − w when v, w are small, and that e−v ≤ 1, v ≥ 0, we obtain −V E e (4.5) − E e−W ≤ O N −α3 P(B) + P(B c ) = O N −α3 + O N −β3 .
It is now clear from step 1, the above result and Lemma 2.1, where we take q = 1, that for some β > 0 and all l ≤ (1 + 2η) logν N , " ( Pl+1 (1) )# (2) k=2 Z⌈k/2⌉ Z⌊k/2⌋ P(HN > l) = E exp − + O N −β . (4.6) µN Step 3: Evaluation of the limit points. From this step on, the remainder of the proof of our main theorem is identically to the proof of Theorem 1.1 in [12]. To let the proof be self-contained, we finish the main line of the argument. Starting from (4.6) we replace l by σN + l and assume that σN + l ≤ (1 + 2η) logν N , where, as before, σN = ⌊logν N ⌋, to obtain )# " ( Pσ +l+1 (1) (2) N Z Z ⌈k/2⌉ ⌊k/2⌋ P(HN > σN + l) = E exp − k=2 + O N −β . (4.7) µN We write N = ν logν N = ν σN −aN , where we recall that aN = ⌊logν N ⌋ − logν N . Then PσN +l+1 k=2
(1) (2) Z⌈k/2⌉ Z⌊k/2⌋ = µν aN +l µN
PσN +l+1 k=2
(1) (2) Z⌈k/2⌉ Z⌊k/2⌋ . µ2 ν σN +l
In the above expression, the factor ν aN prevents proper convergence. Without the factor µν aN +l , we obtain from Appendix A4 of [12] that, with probability 1, lim
N →∞
PσN +l+1 k=2
(2) (1) Z⌊k/2⌋ Z⌈k/2⌉ W (1) W (2) = , 2 σ +l µ ν N ν−1
(4.8)
We now use the speed of convergence result of [1], which was further developed in Section 2 of [12] and which reads that there exists a positive β such that: 1 P(|W − Wk | > (log N )−α ) = O N −β , k ≤ ⌊ logν N ⌋. 2
(4.9)
for each positive α. Combining (4.7) and (4.9) we obtain that for each α > 0 and for l ≤ 2η logν N , h i P(HN > σN + l) = E exp{−κν aN +l W1 W2 } + O (log N )−α . (4.10)
From (4.10) one can finally derive as in [12], that, asymptotically as N → ∞, the probability P(HN < ∞) is equivalent to the probability q 2 = P(W1 W2 > 0), where q is the survival probability of the branching process {Zl }l≥0 , so that (4.10) induces for l ≤ 2η logν N , h i aN +l P(HN ≤ σN + l|HN < ∞) = E 1 − exp{−κν W1 W2 }|W1 W2 > 0 + o(1). (4.11)
16
A
Appendix.
A.1
Proof of Lemma 2.1
In this section we prove Lemma 2.1. For convenience we restate the lemma here as Lemma A.1.1. Recall the definition of the event Sq,α0 given in (2.4). Lemma A.1.1 For each fixed q ∈ (0, τ − 1), there exist constants α0 , β0 > 0 such that c P Sq,α ≤ N −β0 . 0 Proof.
(A.1.1)
For q < (τ − 1)/2, in which case E[Λ2q ] < ∞, we simply apply Chebychev’s inequality: c P Sq,α ≤ N 2α0 Var(SN ,q ) = N 2α0 −1 Var(Λq ) = O(N 2α0 −1 ), (A.1.2) 0
so each 0 < α0 < 12 and β0 = −1 + 2α0 will do the job. For q ∈ [(τ − 1)/2, τ − 1), the proof is more involved. Since E[Λ2q 1 ] < ∞ no longer holds, we have to cut off the large values of SN ,q . Take δ such that 21 < δ < (τ − 1)/(2q) ≤ 1, and define for r = 2qδ, the event Gr = {SN ,r ≤ N α0 }. Since r = 2qδ < τ − 1 implies E[Λr ] < ∞, the Markov inequality yields
P (Grc ) ≤ N −α0 E[SN ,r ] = N −α0 E[Λr ] = O N −α0 .
(A.1.3)
We also use Minkowski’s Inequality, [11, (2.11.24), page 30], implying that for each s ∈ [0, 1], !s N N X X Λsi . (A.1.4) ≤ Λi i=1
i=1
Applying both (A.1.3) and (A.1.4), we arrive at c c q 2 −2α0 P Sq,α ≤ P {(S − E[Λ ]) > N } ∩ G N ,q r + P (Gr ) 0 "N # !1/δ N X 2q X ≤ N 2α0 −2 E Λi 1Gr + P (Grc ) ≤ N 2α0 −2 E Λ2qδ 1Gr + P (Grc ) i i=1
i=1
i h 1/δ 1G2qδ + P (Grc ) = O N − min{−2α0 +2−(1+α0 )/δ,α0 } . = N 2α0 −2 N (1+α0 )/δ E N −α0 SN ,2qδ
Consequently, for q ∈ [(τ − 1)/2, τ − 1) we can take β0 = min{α0 , 2 − 2α0 − (1 + α0 )/δ}, and β0 > 0 provided that we choose 0 < α0 < (2δ − 1)/(2δ + 1).
A.2
Coupling of {Zi }i≥0 and {Zi }i≥0
We will couple the delayed BP {Z i }i≥0 to the delayed BP {Zi }i≥0 with law f in the first generation and law g in the second and further generations, using a classical coupling argument, see [18]. We give each individual of {Z i }i≥0 an independent indicator, which is 1 with probability ∞
q0,N =
1 X (N) fn − fn , 2 n=0
in the first generation and with probability
∞
q1,N =
1 X (N) gn − gn , 2 n=0
17
in the second and further generations. When this indicator is 0 for a certain individual, then the offspring of this individual is successfully coupled. When, on the other hand, the indicator is 1, then an error has occurred, and the coupling is not successful. In this case, the law of the offspring of {Z i }i≥0 is different from the one in {Zi }i≥0 , and we record an error. The following proposition gives bounds on the probability that the random variables q0,N and q1,N are larger than N to a certain negative power: Proposition A.2.1 For each τ > 3, there exist constants α1 > 0 and β1 > 0 such that ! ∞ X (N) −α 1 ≤ N −β1 . (n + 1) gn − gn ≥ N P
(A.2.1)
n=0
Consequently,
P max{q0,N , q1,N } ≥ N −α1 ≤ N −β1 . (A.2.2) P P Proposition A.2.1 will render a coupling between the sums lk=1 Z k and lk=1 Zk . In the proof of this proposition we need an additional lemma from analysis. Recall the definition of fn in (1.7). Lemma A.2.2 Fix τ > 3, then for s = 0, 1, 2 and δ = min{(τ − 3)/4, 1/4}, there exists a constant m0 > 0 such that for all m ≥ m0 , ∞ X
(n + 1)s fn+1 ≤ m−δ .
(A.2.3)
n=m
We defer the proof of Lemma A.2.2 to the end of this section. Before we give the proof of Proposition A.2.1, we introduce the function tn(x) =
xn −x e , x > 0. n!
(A.2.4)
Using this function we can rewrite (3.6) and (3.8) as: (N)
fn
N
N 1 X = tn (Λi ) N
and
(N)
gn
n+1X tn+1 (Λi ) . = LN
(A.2.5)
i=1
i=1
Proof of Proposition A.2.1. The proof is based on the proof of [12, Proposition 3.4]. For 0 < u < 1, v > 0 and α0 ∈ (0, 1/2) we define B = B1 ∩ B2 ∩ B3 ∩ B4 , where LN −α0 − 1 ≤ N , B1 = B1 (α0 ) = µN ( ) ∞ N 1 X X 2 −v B2 = B2 (u, v) = (n + 1) tn+1(Λi ) ≤ N , N u i=1 n=N N ( Nu ) X X 1 B3 = B3 (u, v) = (tn+1 (Λi ) − fn+1 ) ≤ N −v , (n + 1)2 N n=0 i=1 ) ( N 1 X −α0 . B4 = B4 (α0 ) = (t (Λ ) − f ) 0 i 0 ≤N N i=1
The precise values of α0 , u and v will be chosen later in the proof. The strategy of the proof is as follows. We will show that P (B c ) ≤ N −β1 , 18
(A.2.6)
for some β1 > 0, and that on the event B, ∞ X
n=0
(n + 1) gn(N) − gn = O N −α1 ,
(A.2.7)
for some α1 > 0. We start by proving (A.2.7). We bound q1,N =
∞ X
n=0
(n + 1)|gn(N) − gn | ≤
∞ X
n=0
(n + 1)|gn(N) −
Nµ Nµ gn | + (ν + 1) − 1 , LN LN
(A.2.8)
P∞ using ν = n=0 ngn . On B1 , the second term on the right-hand side of (A.2.8) is bounded by O N −α0 . The first term on the right-hand side of (A.2.8) can be rewritten by (3.9), (A.2.4), (A.2.5) and (1.8), respectively, as ∞ X
n=0
(n + 1)|gn(N) −
N ∞ X Nµ 1 X (n + 1)2 tn+1(Λi ) − fn+1 . gn | = LN LN n=0
i=1
Consequently, the first term on the right-hand side of (A.2.8) can be bounded on B1 , for N sufficiently large, by ∞ X
n=0
(n + 1)|gn(N) −
∞ N X Nµ 2 X gn | ≤ (n + 1)2 tn+1(Λi ) − fn+1 . LN µN n=0
i=1
We next split the sum over n into n < N u and n ≥ N u . On B3 , the contribution from n < N u is at most µ2 N −v , whereas we can bound the contribution from n ≥ N u on B2 by ∞ N ∞ X 2 2 X 2 X (n + 1)2 (n + 1)2 fn+1 . (tn+1(Λi ) + fn+1 ) ≤ N −v + µN µ µ u u i=1
n=N
n=N
For τ > 3, the latter term is bounded by N −uδ by Lemma A.2.2. Thus, we obtain the claim in (A.2.7) for each 0 < α1 < min {α0 , uδ, v} .
(A.2.9)
To show (A.2.6), it is sufficient to show that P (B1c ) + P (B2c ) + P (B3c ) + P (B4c ) ≤ N −β1 ,
(A.2.10)
for some β1 > 0. We will prove that P (Bic ) ≤ N −ai , where a1 , a2 , . . . , a4 depend on α0 , δ, u and v. Later we will show that we can take α0 , δ, u and vsuch that (A.2.9) is satisfied and a1 , a2 , a3 , a4 > 0. Lemma A.1.1 states that P (B1c ) = O N −β0 , thus a1 = β0 . We bound P (B2c ) by using the Markov inequality and Lemma A.2.2: P (B2c ) ≤ N v
∞ X
(n + 1)2 E[tn+1(Λ)] = N v
∞ X
n=N u
n=N u
(n + 1)2 fn+1 ≤ N v−uδ = N −a2 ,
for some constant δ > 0 and we take a2 = uδ − v and N sufficiently large. Before we bound P (B3c ) from above, we first note that # " N ! N h i X X E ( [tn+1 (Λi ) − fn+1 ])2 = Var tn+1 (Λi ) ≤ N E tn+1 (Λ)2 ≤ N, i=1
i=1
19
(A.2.11)
where we used in the last step that tn+1 (x) is a probability. In the sequence of inequalities below, which gives the bound for P (B3c ), we use the Markov inequality, Cauchy-Schwarz in the form PN u 2 PN u 2 1 PN u √ 2 x n=0 bn ) , the Jensen inequality applied to the concave function x 7→ n=0 bn ≤ ( n=0 1 and (A.2.11), respectively, to obtain u
P(B3c )
≤N
v−1
N N X X 2 (n + 1) tn+1(Λi ) − fn+1 E n=0
i=1
Nu X 1 v−1 u 2 ≤N (N + 1) E (n + 1)4 n=0
≤ 2N v+u/2−1 E
u
N X
(n + 1)4
n=0
Nu
≤ 2N v+u/2−1
X
i=1
(n + 1)4 N
n=0
N X
1/2
N X i=1
!2 1/2 tn+1(Λi ) − fn+1
!2 1/2 tn+1(Λi ) − fn+1
≤ 2N v+3u−1/2 = O N −a3 ,
for a3 = 1/2 − v − 3u. Finally, we bound the fourth term of (A.2.10) using the Chebychev inequality ! N X P (B4c ) ≤ N 2u−2 Var t0(Λi ) = N 2u−1 Var(t0(Λ)) ≤ N 2u−1 = N −u4 , i=1
where a4 = 2u − 1, because t0(Λ) ≤ 1, as it is a probability. We now verify that we can choose α0 , δ, u and v in such a way that both α1 and β1 can be taken positive. Recall that a1 = β0 ,
a2 = uδ − v,
a3 = 1/2 − 3u − v,
and
a4 = 2u − 1.
(A.2.12)
The constant δ > 0 follows from Lemma A.2.2, we pick 0 < α0 < 1/2 and use Lemma A.1.1 to find a positive β0 , then finally we choose u = 1/12 and v such that 0 < v < min{1/4, δ/12}. The reader easily verifies that these conditions imply that the constants a1 , a2 , a3 and a4 are positive, so that β1 equal to the minimum of these four quantities does the job. For α1 we can take any value that satisfies (A.2.9). This completes the proof of (A.2.9). In order to show (A.2.2) it is sufficient to show that P (q0,N ≥ N −α1 ) ≤ N −β1 , i.e., ! ∞ X (N) (n + 1) fn − fn ≥ N −α1 ≤ N −β1 , P
(A.2.13)
n=0
because we already have shown that P (q1,N ≥ N −α1 ) ≤ N −β1 . On the event B4 we have that N X (N) 1 f − f0 = (t (Λ ) − f ) ≤ N −α0 . 0 i 0 0 N i=1
Thus, using the event B, the, now proven, bound (A.2.7), LN /N < 2µ for N sufficiently large, and
20
the fact α1 ≤ α0 we have that q0,N =
∞ X
n=0
∞ X (N) (n + 1) fn+1 − fn+1 (n + 1) fn(N) − fn ≤ f0(N) − f0 + 2 n=0
∞ ∞ X X (N) LN (N) L L N N g ≤ f0 − f0 + 2 gn − µ n+1 − gn+1 + 2 N N N n=0 n=0 ∞ X LN (N) g ≤ f0(N) − f0 + 4µ − µ = O N − min{α1 ,α0 } = O N −α1 . n+1 − gn+1 + 4µ N n=0
We close this section with the proof of Lemma A.2.2. Proof of Lemma A.2.2. It is sufficient to consider only the case s = 2, because ∞ X
s
(n + 1) fn+1 ≤
n=m
∞ X
(n + 1)2 fn+1 , for s = 0, 1, 2.
n=m
Recall that we denote by P oi(λ) a Poisson random variable with mean λ. Then, using (1.7) we can bound the right side of the above equation by # # " " ∞ ∞ n−1 n+1 X X Λ Λ (1 + 1/n) ≤ 2E Λ2 P (P oi(Λ) ≥ m|Λ) = E Λ2 e−Λ (n + 1)2 e−Λ E (n + 1)! (n − 1)! n=m n=m 2 ≤ 2E Λ 1{Λ≥mw } + 2m2w E P (P oi(Λ) ≥ m|Λ) 1{Λ 0 + O N − min{η/2,β1 /2} = O N − min{η/2,β1 /2,ε} (A.3.32) ≤ P w=1
26
and, similarly,
P (D2c ) ≤ P D2c ∩
⌊(1/2+η) Xlogν N ⌋
k=0
⌊N 1/2+2η ⌋
≤ P
X
w=1
Z k < N 1/2+2η + O (logν N )N − min{η,β1 }
Yw > N 5η + O N − min{η/2,β1 /2} = O N − min{η/2,β1 /2,ε} .
(A.3.33)
To bound P(D3c ∩ D2 ∩ C), we first bound PN (D3c ∩ D2 ) on the event C. Using the Markov inequality and Boole’s inequality we have, on C, that X X Z1(d) 1D2 = N −θ EN PN (D3c ∩ D2 ) ≤ N −θ EN Z1(d) 1D2 , (A.3.34) d∈Dup⌊(1/2+η) logν N ⌋
d∈Dupk
where we, for convenience, set k = ⌊(1/2 + η) logν N ⌋. If we condition on the sequences {Λi }N i=1 and variable with {Mv }v≥0 , recall (A.3.2), then Z1(d) , for d ∈ Dupk , is an independent Poisson random P (d) N mean Λ(Md ). Thus, conditioned on the sequences {ΛP i }i=1 and {Mv }v≥0 , the sum d∈Dupk Z1 is distributed as a Poisson random variable with mean d∈Dupk Md . Which in turn can be stochastically bounded from above by a Poisson random variable with mean |Dupk |N γ on the event C, because Λ(Md ) ≤ Λ(N) ≤ N γ for all d ∈ Dupk . Since D2 and Dupk are measurable with respect to N {Mv }v≥0 , therefore the above yields X X Z1(d) 1D2 = EN EN EN P oi Z1(d) {Mv }v≥0 1D2 d∈Dupk
d∈Dupk
= ENEN P oi Z1(d) {Mv }v≥0 1D2 d∈Dupk
X
≤ EN[EN[ P oi(|Dupk |N γ )| {Mv }v≥0 ] 1D2 ] ≤ EN EN P oi N 5η+γ {Mv }v≥0 = EN P oi N 5η+γ = N 5η+γ .
(A.3.35)
Thus, we bound (A.3.34) by
X
PN (D3c ∩ D2 ) ≤ N −θ EN
d∈Dup⌊(1/2+η) logν N ⌋
Hence,
Z1(d) 1D2 ≤ N −θ+5η+γ = N −η .
P(D32 ∩ D2 ∩ C) = E[1C PN (D3c ∩ D2 )] = O N − min{η/2,β1 /2,ε}
and, finally, by (A.3.32), (A.3.33), (A.3.37) and (A.3.31),
P (Dc ) = O N − min{η/2,β1 /2,ε} ,
which proves Step 1 and therefore the proposition.
A.4
(A.3.36)
(A.3.37)
Proof of Proposition 3.4
In this section we prove Proposition 3.4, which we restate here for convenience as Proposition A.4.1. Proposition A.4.1 There exist constants α3 , β3 , η > 0 such that for all l ≤ (1 + 2η) logν N , as N → ∞, l+1 l+1 1 X X (2) (1) (1) (2) P C⌈k/2⌉ Z⌈k/2⌉ Z⌊k/2⌋ − C ⌊k/2⌋ > N −α3 = O N −β3 . N k=2
k=2
27
(A.4.1)
We start with an outline of the proof. Define T1 =
l+1 1 X (2) (1) (1) Z⌊k/2⌋ Z⌈k/2⌉ − C ⌈k/2⌉ N
and
T2 =
k=2
l+1 1 X (1) (2) (2) . C ⌈k/2⌉ Z⌊k/2⌋ − C ⌊k/2⌋ N
(A.4.2)
k=2
To show (A.4.1), it suffices to prove that for an appropriate chosen event H, 1 −α3 1 −α3 −β3 =O N and P T 2 1H > N = O N −β3 , (A.4.3) P T 1 1H > N 2 2 and that P (Hc ) = O N −ξ for some ξ > 0. We choose the event H such that on this event, for each k ≤ l ≤ (1 + 2η) logν N , (1) (2) (1) 1H = O N 1/2−u , − C ⌈k/2⌉ E Z⌈k/2⌉ 1H = O N 1/2+2η , E Z⌊k/2⌋ (A.4.4) (1) (2) (2) 1/2+2η 1/2−u E C ⌈k/2⌉ 1H = O N and E Z⌊k/2⌋ − C ⌊k/2⌋ 1H = O N , (A.4.5) for some u, α3 , η > 0 such that
α3 < u − 2η.
(A.4.6)
The claims (A.4.4), (A.4.5) and (A.4.6) implies (A.4.3). This is straightforward from the Markov inequality, the independence of Z (1) between Z (2) and the associated capacities and the inequality l ≤ (1 + 2η) logν N . We leave it to the reader to verify this. Instead of showing (A.4.4) and (A.4.5), it is sufficient to show that there exists constants α3 , η, u > 0 satisfying (A.4.6) and such that for each k ≤ l ≤ (1/2 + η) logν N , E[|Zk − C k | 1H ] = O N 1/2−u , E[Zk 1H ] = O N 1/2+2η and E[C k 1H ] = O N 1/2+2η , (A.4.7) which simplifies the notation considerably. We choose the event H as H = H1 ∩ H2 ∩ H3 ∩ H4 ∩ C, where ( l ) X (C k − C k ) ≤ N 1/2−α2 , H1 =
(A.4.8)
k=1
n o H2,k = C k − EN[ΛM ] Z k−1 < N 1/2−δ , n o H3,k = |Zk − νZk−1 | < N 1/2−δ , n o H4 = |EN[ΛM ] − ν| ≤ N −δ ,
H2 = ∩lk=1 H2,k ,
(A.4.9)
H3 = ∩lk=1 H3,k ,
(A.4.10) (A.4.11)
and the event C is given by (A.3.12). Hence, for the proof of Proposition A.4.1 it suffices to show the three claims in (A.4.7) with u−2η > 0, and that for some ξ > 0 we have that P (Hc ) = O N −ξ . For the latter statement, we use the following lemma. In this lemma we denote by dN a random variable with distribution {gn(N) }n≥0 given by (3.8). Lemma A.4.2 For every γ > 0, for {Λi }N i=1 , such that C occurs, VarN (ΛM ) = O N γ and VarN (dN ) = O N γ ,
where VarN ( · ) is the variance under PN ( · ).
28
(A.4.12)
The proof of this lemma is deferred to the end of this section. Although it seems that we can take any γ > 0, this will not be the case in the proof of Proposition A.4.1. We need that the event C happens whp, which is the case when γ > 1/(τ − 1). Proof of Proposition A.4.1 Consider the first claim given by (A.4.7). Using the triangle inequality we arrive at E[|C k − Zk | 1H ] ≤ E C k − C k 1H + E C k − EN[ΛM ] Z k−1 1H + E Z k−1 (EN[ΛM ] − ν) 1H + νE Z k−1 − Zk−1 1H + E[|νZk−1 − Zk | 1H ] , (A.4.13) The first, second and the last term on the right hand of (A.4.13), we bound by N 1/2−min {δ,α2 } using the events H1 , H2 and H3 , respectively. We bound the third term of (A.4.13) using the event H4 and (A.3.22), which gives E Z k−1 (EN[ΛM ] − ν) 1H4 ∩C ≤ N −δ E Z k−1 1C = O N 1/2+2η−δ . (A.4.14)
Finally, we need to bound the fourth term of (A.4.13). On the event C3 the following inequality holds for each k ≤ l (compare with [12, (A.1.4) and (A.1.15)]), l X EN Z k (max {ν, νN })l−k , EN Z k − Zk ≤ max {ν − αN , νN − αN }
(A.4.15)
k=1
where
αN =
∞ X
n=0
n min{gn , gn(N) }
and νN is given by (A.3.19). We bound the sum in (A.4.15), using (A.3.22), which implies l X k=1
l X EN Z k (max {ν, νN })l−k = µN νNm−1 (max {ν, νN })l−k ≤ lmax {µ, µN } (max {ν, νN })l−1 k=1
≤ l 1 + O N −α1
l
for N sufficiently large. On the event C3 we have
O N 1/2+η ≤ N 1/2+2η
∞ X n gn − gn(N) ≤ N −α1 . max {ν − αN , νN − αN } ≤
(A.4.16)
(A.4.17)
n=1
Combining (A.4.15), (A.4.16) and (A.4.17), we obtain that, on C3 , EN Z k − Zk = O N 1/2+2η−α1 . Thus, all together, the left side of (A.4.13) satisfies
E[|C k − Zk | 1H ] = O N 1/2−min {δ,α2 ,δ−2η,α1 −2η} .
(A.4.18)
This gives a bound for the right hand of (A.4.7) with u = min {δ, α2 , δ − 2η, α1 − 2η}. The second claim of (A.4.7) is evident, because k ≤ l ≤ (1/2 + η) logν N and therefore E[Zk 1H ] ≤ E[Zk ] = µν k−1 ≤ µν l−1 = O N 1/2+η . (A.4.19) Finally, the third claim of (A.4.7) follows from using in turn C k ≤ C k , Zk−1 X E[C k 1H ] ≤ E C k 1H = E Λ M k−1,v 1H . v=1
29
Now by taking conditional expectation with respect to Z i−1 and the capacities, we obtain Zk−1 Zk−1 Zk−1 X X X EN Λ M k−1,v 1H Z k−1 = EN Λ M k−1,v 1H Z k−1 ≤ EN Λ M k−1,v 1C Z k−1 v=1 v=1 v=1 = Z k−1 EN[ΛM 1C ] ≤ SSZ k−1 1C
so that (A.3.22) implies that E[C k 1H ] ≤ EEN
Zk−1
X v=1
Λ M k−1,v 1H Z k−1 ≤ SSE Z k−1 1C = O N 1/2+2η .
Thus, we have shown the claims given by (A.4.7) if we restrict η to 0 < η < min{α1 , α2 }/2 due to (A.4.18). To satisfy condition (A.4.6), we restrict η to 0 < η < min{α1 /4, α2 /2, δ/2},
(A.4.20)
and pick α3 = (u − 2η)/2. Then α3 > 0 and condition (A.4.6) is satisfied, because u − 2η < α3 = (u − 2η)/2 = min{α1 − 4η, α2 − 2η, δ − 2η}/2 > 0, for each 0 < δ < 1/2 and for each η given by (A.4.20). We finish the proof by showing that P (Hc ) = O N −ξ for some ξ > 0. We bound P (Hc ) by
P (Hc ) ≤ P (H1c ) + P (H2c ∩ C) + P (H3c ∩ C) + P (H4c ∩ C) + P (C c ) . The last term we bound by (A.3.16), which states P (C c ) = O N −ε for some ε > 0. We bound P (H1c ) using Proposition A.3.1, P (H1c ) = O N −β2 .
(A.4.21)
(A.4.22)
Next, we will bound P (H2c ∩ C) and P (H3c ∩ C) from above. We will show that by using the Chebychev inequality that c ∩ C = O N −1/2+2δ+γ+2η , for k ≤ l ≤ (1/2 + η) logν N . (A.4.23) PN H2,k
Then by Boole’s inequality we have, ! l l X [ c c c c P (H2 ) ≤ P ∩ C + P (C c ) E PN H2,k H2,k ∩ C + P (C ) ≤ ≤ lN
k=0 −1/2+2δ+γ+2η
k=0
+ P (C ) = O (logν N )N − min{1/2−2δ−γ−2η,η/2,ε} . c
(A.4.24)
Fix k, 1 ≤ k ≤ l ≤ (1/2 + η) logν N , and denote by Vv = Λ M k−1,v − EN[ΛM ] for v = 1, 2, . . . , Z k−1 . We have that EN[Vv ] = 0, and conditionally on the marks Λ1 , . . . , ΛN the sequence Z
k−1 is an i.i.d. sequence. Hence, on C and using Lemma A.4.2, {Vv }v=1 2 Zk−1 Zk−1 Zk−1 Zk−1 Zk−1 Zk−1 X X X X X X EN EN[Vv ] EN[Vw ] + Vv Z k−1 = EN[Vv Vw ] = EN Vv2 v=1 v=1 w=1 v=1 w=1,w6=v v=1
Zk−1
=
X v=1
EN Vv2 = Z k−1 VarN (ΛM ) ≤ Z k−1 N γ . 30
(A.4.25)
Therefore, using (A.3.22), on C, 2 Zk−1 Zk−1 X X VarN Vv = ENEN Vv Z k−1 ≤ EN Z k−1 N γ = O N 1/2+γ+2η . v=1 v=1 Thus, by the Chebyshev inequality, on C, Zk−1 Zk−1 X X c = PN PN H2,k Vv ≥ N 1/2−δ ≤ N 2δ−1 VarN Vv ≤ N −1/2+2δ+γ+2η . v=1 v=1
c ∩ C = O (log N )N − min{1/2−2δ−γ−2η,η/2} , when we reSimilarly, we can show that P H3,k ν
place Z k−1 by Zk−1 , set Vn = Xk−1,n − EN[dN ], where Xk−1,n is an independent copy of dN . This, yields on C, 2 ZX Zk−1 k−1 X c Vv ≥ N 1/2−δ ≤ N 2δ−1 ENEN Vv Zk−1 = PN PN H3,k v=1 v=1 = N 2δ−1 EN[Zk−1 VarN (dN )] .
Using that VarN (dN ) is constant under PN , Lemma A.3.3 and (A.4.19) we have that on C: EN[Zk−1 VarN (dN )] = VarN (dN ) EN[Zk−1 ] = O N 1/2+η+γ . The above yields, compare with (A.4.24),
P (H3c ) = O (logν N )N − min{1/2−2δ−γ−2η,η/2,ε} .
To bound P (H4c ) we will use that EN[ΛM ] = SN ,2 /SN ,1 , where SN ,q = that on S1,α0 ∩ S2,α0 , recall (2.4), that SN ,1 ≤ 1 + O N −α0 and E[Λ] which yields, together with ν = E Λ2 /E[Λ] 1 − O N −α0
≤
1 − O N −α0
ν≤
1 − O N −α0
≤
1 N
(A.4.26) q i=1 Λi ,
PN
q = 1, 2. Notice
SN ,2 ≤ 1 + O N −α0 , 2 E[Λ ]
SN ,2 ≤ 1 + O N −α0 ν. SN ,1
Thus, using Lemma 2.1, the Markov inequality and (A.4.27) we have c c P (H4c ) ≤ P (H4c ∩ S1,α0 ∩ S2,α0 ∩ C) + P S1,α ∩ C + P S ∩ C + P (C) 2,α 0 h i 0 ≤ N δ E |EN[ΛM ] − ν| 1S1,α0 ∩S2,α0 ∩C + O N − min{β0 ,ε} h i = N δ E |SN ,2 /SN ,1 − ν| 1S1,α0 ∩S2,α0 ∩C + O N − min{β0 ,ε} = O N δ−α0 + O N − min{β0 ,ε} = O N − min{α0 −δ,β0 ,ε} .
By Lemma 2.1 we can pick α0 > δ/2 and β0 > 0 for all δ < 1/2. Combining (A.4.22), (A.4.24), (A.4.26) and (A.4.28) with (A.4.21) gives P (Hc ) = O N −ξ , 31
(A.4.27)
(A.4.28)
where ξ = min{(1/2 − 2δ − γ − 2η)/2, β2 , η/2, β0 , ε, α − δ}. Remember that γ is restricted to 1 τ −1 < γ < 1/2, that δ is restricted to 0 < δ < 1/2 and η is restricted by (A.4.20). So, pick η, δ > 0 such that 2δ + η < 1/2 − γ, because we can pick δ > 0 and η > 0 arbitrary small, then ξ > 0. Proof of Lemma A.4.2 Consider the first claim of (A.4.12). The variance of a random variable is bounded from above by its second moment. Therefore, using (3.2), Lemma A.3.2 and (2.7), on C N N 1 X 3 Λ(N) X 2 VarN (ΛM ) ≤ EN Λ2M = Λm ≤ N Λm LN LN m=1
≤ Nγ
N X
1 LN
m=1
m=1
Λ2m = N γ SN ,2 SN−1,1 1C ≤ SS −1 N γ = O N γ ,
for N sufficiently large. We turn to the second claim of (A.4.12). Using (3.8) we bound (A.4.12) on C from above by VarN (dN ) 1C ≤
∞ X
n2 gn(N) 1C =
∞ X
n=1
n=1
N n2 X −Λm Λn+1 m e 1C . SN ,1 N n! m=1
We split off the term with n = 1 and use (2.7) and the fact that for n > 1, we have obtain, on C, VarN (dN ) ≤ ≤
1 SN ,1 N
N X
−Λm
e
m=1
SN ,2 2 + S SN
N X
m=1
Λ2m
+
1 SN ,1 N
N X
m=1
Λ3m
∞ X
n=2
e−Λm
n n−1
≤ 2, to
n−2 Λm n (n − 2)! n − 1
Λ3m ≤ SS −1 + 2SN ,2 S −1 Λ(N) ≤ SS −1 + 2SS −1 N γ . N
Hence, VarN (dN ) = O N γ .
Acknowledgements
The work of HvdE and RvdH was supported in part by Netherlands Organisation for Scientific Research (NWO). We thank Michel Dekking for carefully reading the manuscript and many suggestions, which resulted in a more accessible paper.
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