In this section, we show that multiple edges are negligible (Lemma 6.4.4 below) and use this result to prove Corollary 1.0.3 from the introduction, which states our main result for the case where multiple edges are ignored.
First we consider the social index of a node that is connected to the node JT by an incoming edge.
6.4.1 Lemma
i6=jT si)−1. Since the social indices are identically distributed, we thus obtain by Bayes’ Theorem
FS˜(s) = the right-hand side of (6.80) is equal to
s
6.4.2 Remark Since
S11[0,s](S1)
S1
n−1+ n−11 Pn−1 i=2 Si
≤1 almost surely and n−1S1 →0 and n−11 Pn−1
i=2 Si→E(S) almost surely by the strong law of large number as n→ ∞, Lemma 6.4.1 and Fubini’s Theorem imply that the cumulative distribution function FS˜ from Lemma 6.4.1 converges to
E(S1[0,s](S)) E(S)
as n → ∞. Note that this is just the cumulative distribution function of the so-called size-biased distribution by its definition (see e.g. (1.2.3) in [vdH16]). Thus the distribution of the social index of a nodei1 that is connected toJT at timeT by an edge that was created by i1 converges weakly to the size-biased distribution of the social index distribution. Since the size-biased distribution dominates the original one stochastically (see e.g. Section 2.3 of [vdH16]), we may conclude that the neighbours that are connected by incoming edges have asymptotically in average a higher social index thanE(S).7 Since the distribution of the social index of a neighbour that is connected by an outgoing edge is obviously just the social index distribution, the distribution of the social index of a neighbour picked uniformly at random is a mixture of the distribution given by the cumulative distribution function FSi
1 from Lemma 6.4.1 and the social index distribution. We may conclude that this distribution converges weakly to a mixture of the size biased distribution of the social index distribution and the social index distribution, which is a result that was already stated in Subsection 3.3 of [BLT11].
6.4.3 Corollary
The expected value of the social index ˜S of a node i1 that is connected to JT at time T by an edge that was created by i1 is bounded from above by
2σS2
c2E(S) + E(S2) E(S)(1−c) for any c∈(0,1).
P roof: We condition onYT = n∈ N\ {1}. By Lemma 6.4.1, the conditional expected value of the social index ˜S of a node i1 that is connected to JT at time T by an edge that was created by i1 is then equal to
∞
Z
0
s1E
s1 s1
n−1 +n−11 Pn−1 i=2 Si
PS(ds1) =E
S12
S1
n−1 +n−11 Pn−1 i=2 Si
,
and we have E
S12
S1
n−1 +n−11 Pn−1 i=2 Si
≤E
1{n−21 Pn−1
i=2Si≤E(S)(1−c)}(n−1)S1
7This is result is similar to the result that neighbours have in average a higher degree than a node picked uniformly at random (see e.g. Theorem 1.2 in [vdH16]).
+E where we use the convention 00 := 0. By Chebyshev’s inequality, we have
P which yields the desired result.
The following lemma states that multiple edges are negligible.
6.4.4 Lemma
The probability that an individual picked uniformly at random at time T has at least one multiple edge given the number of nodes is positive at timeT is of the orderO(T2e−16(λ−µ)T) as T → ∞.
Proof. LetDT denote the degree of the randomly picked nodeJT at timeT, i.e. the number of edges that are incident to the node picked uniformly at random at timeT. Letρ1 < . . . < ρDT be the birth times of these edges. Condition onJT, DT,ρ1 < . . . < ρDT and (Yt)0≤t≤T. Let Bk be the event that JT creates an outgoing edge at timeρk that is a multiple edge up to time T. Then the (conditional) probability ofBk is smaller than or equal to
DT −1 Yρk−1. Note that SDT
k=1Bk is the event that JT has at least one outgoing edge that is a multiple edge at timeT. By subadditivity, we have
P
Taking the expectation, we obtain
WritingD∞ for a random variable having the asymptotic degree distribution MixPo(M∗) with M∗ defined at the beginning of Subsection 6.3.2, we obtain by conditioning on M∗that the second moment E(D∞2 ) is equal to
2αE(S)
λ+β+µ+ 2αE((S+E(S))2)
(λ+β+µ)(λ+ 2(β+µ)) (cf. Subsection 3.3 of [BL10]).
Thus Theorem 6.3.4 and the Markov inequality imply
P(DT ≥e121(λ−µ)T|YT >0)≤E(D2∞)e−16(λ−µ)T +O(T2e−16(λ−µ)T)
For the second summand of the right-hand side of (6.81), we obtain P Thus the first summand of the right-hand side of (6.82) is smaller than or equal to
P
The second summand is smaller than or equal to µλe−(λ−µ)T by Lemma 6.3.5. By Corollary 6.3.9 and the inequality (2) in Theorem 6.14 on page 99 in [Yeh95], the first summand of (6.83) is bounded from above by where the first equality follows from Lemma 6.3.16.
We may conclude that the probability thatJT has at least one outgoing edge that is a multiple edge at timeT is of the orderO(T e−16(λ−µ)T)
Now we consider incoming edges. By conditioning onJT, (Si)i∈Nand (Yt)0≤t≤T, we obtain that the probability for the event ˜Bi(1) that node icreates an edge that connectsi toJT at time T is smaller than or equal to
E Since we showed above that the first two summands are of the order O(T e−16(λ−µ)T), the right-hand side is of the orderO(T e−13(λ−µ)T).
We condition on ˜Bi(1) now and denote the birth time of the edge (i, JT) corresponding to ˜Bi(1) by ηi. Then we have for the conditional probability of the event ˜Bi(2) that icreates another edge (i, JT) in the time interval (ηi, T]⊂(T−AJT(T), T] that survives up to time T where ˜S denotes the social index of a node connected to JT at time T by an incoming edge. By Corollary 6.4.3, the right-hand side is of the orderO(T e−13(λ−µ)T).
For simplicity, we denote the YT nodes alive at time T by 1, . . . , YT now. For the probability that JT has at least two incoming edges from the same node at time T, we then obtain by subadditivity
P
By the Markov inequality, the second summand of the right-hand side is of the order O(e−16(λ−µ)T).
For the first summand, we have e76(λ−µ)TP B˜1(1)∩B˜1(2)|YT >0
=e76(λ−µ)TP B˜1(1)|YT >0
P B˜1(2)|B˜1(1), YT >0
=e76(λ−µ)TO(T e−13(λ−µ)T)O(T e−13(λ−µ)T) =O(T2e−16(λ−µ)T).
Altogether, we obtain that the probability that JT has at least one multiple edge is of the order O(T2e−16(λ−µ)T).
Proof of Corollary 1.0.3
Recall that ˜νt denotes the distribution of the number of neighbours,νtthe degree distribution at time tand ν the asymptotic degree distribution. Lemma 6.4.4 implies thatdT V(˜νt, νt) =O(t2e−16(λ−µ)t) as t→ ∞. From Theorem 1.0.2, we know thatdT V(νt, ν) =O t2e−16(λ−µ)t
ast→ ∞. Thus the triangle inequality yields the desired result.