CONVERGENCE RATES FOR THE DEGREE DISTRIBUTION IN A
DYNAMIC NETWORK MODEL
By Fabian Kück and Dominic Schuhmacher
University of Göttingen
In the stochastic network model of Britton and Lindholm [Dynamic random networks in dynamic populations. Journal of Statistical Physics, 2010], the number of individuals evolves according to
a linear birth and death process with per-capita birth rate λ and
per-capita death rate µ < λ. A random social index is assigned to
each individual at birth, which controls the rate at which connections
to other individuals are created. Britton and Lindholm give a somewhat rough proof for the convergence of the degree distribution in
this model towards a mixed Poisson distribution. We derive a rate
for this convergence giving precise arguments. In order to do so, we
deduce the degree distribution at finite time and derive an approximation result for mixed Poisson distributions to compute an upper
bound for the total variation distance to the asymptotic degree distribution. We prove several general results about linear birth and death
processes along the way. Most notably, we derive the age distribution
of an individual picked uniformly at random at some finite time.
1. Introduction. “Network Science” is a relatively young, rapidly growing research area
dealing with complex systems with an underlying graph structure (see e.g. the recent book by
van der Hofstad [vdH15]). One of the first random graph models is the well-known Erdős-Rényi
model, which is a static model, i.e. it describes a random network at a fixed time. Although
this model shows interesting behaviour, further random graph models were needed for the description of real networks since some empirical networks have important properties that are not
represented by Erdős-Rényi Graphs. One of these is the power law property of the degree distribution. Preferential attachment models exhibit such a power law behaviour asymptotically and
were popularized by Barabási and Albert [BA99]. Peköz, Röllin and Ross [PRR13] established
convergence rates for the degree distribution in a preferential attachment model in terms of the
total variation distance. An alternative to preferential attachment models is the time-continuous
random graph model that was introduced by Britton and Lindholm [BL10] and that we consider
here.
Let us give a loop-free version of the definition of the original dynamic network model by
Britton and Lindholm; see [BL10] and [BLT11]:
We examine a finite undirected graph without loops that develops over time. The node process
(Yt )t≥0 is a linear birth and death process with initial value one. Thus each node gives birth at
MSC 2010 subject classifications: Primary 60E05; secondary 05C80, 62E20
Keywords and phrases: age distribution, birth and death process, mixed Poisson distribution, dynamic random
graph, small worlds
1
2
F. KÜCK AND D. SCHUHMACHER
constant rate λ and dies at constant rate µ independently from other nodes. The process (Yt )t≥0
and all other random variables that are defined in what follows to describe the dynamic random
graph are defined on a common underlying probability space (Ω, A, P).
We assume λ > µ, so that (Yt )t≥0 is a supercritical continuous-time Markov branching process. From standard branching process theory, we obtain that for a random variable Wt with
L(Wt ) = L(Yt e−t(λ−µ) |Yt > 0), we have
Wt → W a.s.,
λ
where W is Exp( λ−µ
) distributed (see e.g. [Har50], page 319).
We equip every node i with a positive random social index Si , where the Si are i.i.d. and
independent of all other random variables.
This allows us to define the development of the edge set. At birth every node is isolated.
During its lifetime and as long as there is at least one other node, node i generates and destroys
edges according to a birth and death process with constant birth rate αSi and per-edge death
rate β. Here α and β are positive constants. The “second” node of each newly born edge is
chosen uniformly at random from the set of all other living nodes, and all of the edge processes
(including the choices of the second nodes) are independent of each other and all other events.
In addition to the direct destruction of edges in the above process, all edges connected to a
certain node are removed when this node dies.
Remark 1.1. The only difference to the definition by Britton and Lindholm is that we do
not allow loops, because these are not present in most applications. Note that the proofs become
slightly simpler if we use the original model by Britton and Lindholm, essentially because times
where Yt = 1 need not obtain special treatment. The upper bounds remain largely the same; see
also Remark 4.2 for the pure birth case.
We refer to the distribution of the number of edges incident to a node picked uniformly
at random from all living nodes at time t given the number of nodes is positive, as degree
distribution, and denote it by νt . It is the main purpose of this paper to give a rate for the
convergence in total variation distance of νt to the asymptotic degree distribution
α
−(β+µ)A
ν = MixPo
S + E(S) 1 − e
,
β+µ
where A ∼ Exp(λ), S has the social index distribution, and A and S are independent. Here
MixPo denotes the mixed Poisson distribution. The following result is an immediate consequence
of Theorems 4.1 and 4.3 below, which are proved in this article.
Theorem 1.2.
we obtain
For the degree distribution νt in the Britton–Lindholm Model without loops,
√
1
(a) if µ = 0, then dT V (νt , ν) = O te− 2 λt as t → ∞;
1
(b) if µ > 0, then dT V (νt , ν) = O t2 e− 6 (λ−µ)t as t → ∞.
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
3
The rest of the paper is organized as follows. In Section 2 we collect some elementary facts
about linear birth and death processes. We also present the distribution of the age of a randomly
picked individual in such a process, which is an important ingredient for the proof of our main
theorem 4.3 and appears to be a substantial new result. Section 3 gives the degree distribution
at finite time by using results about the birth and death processes of the edges. In Section 4,
we derive upper bounds for the total variation distance between finite-time and asymptotic
degree distributions, treating the pure birth case and the general case separately, which leads
to Theorem 1.2 above. The appendices collect various results needed for the proof of our main
theorem 4.3. Appendix A1 derives a universal bound for the total variation distance between
two mixed Poisson distributions. In Appendix A2 we present the proof of the age distribution
result mentioned above. Appendix A3 lists a larger number of mostly technical lemmas.
This article is a condensed version of the technical report [KS15]. It assumes a certain familiarity of the reader with birth and death processes and contour processes, and omits the more
technical details of the proofs. The report [KS15] discusses these processes extensively and gives
the proofs in full detail.
2. Linear birth and death processes and the age of a randomly picked individual.
The node process (Yt )t≥0 is a linear birth and death process with birth rate λ, death rate µ < λ
and initial value one, i.e. Y0 = 1. According to (8.15) and (8.46) in [Bai64], the one-dimensional
distributions of such a process are given by the following probability mass functions:
p0 (t) = µp̃(t)
pn (t) = (1 − µp̃(t))(1 − λp̃(t))(λp̃(t))n−1 , n ≥ 1,
where
p̃(t) :=
e(λ−µ)t − 1
1 1 − e−(λ−µ)t
=
.
λ 1 − µλ e−(λ−µ)t
λe(λ−µ)t − µ
Remark 2.1. Note that p0 (t) is the probability that a linear birth and death process with
initial value one goes extinct up to time t. Due to the branching property of a linear birth and
death process, we have that p0 (t)m is the probability that a linear birth and death process with
a general initial value m goes extinct up to time t. By taking the limit t → ∞, we obtain that
the probability of eventual extinction is (µ/λ)m (see e.g. (8.59) in [Bai64]).
By elementary computations using these probability mass functions, we obtain the following
proposition (cf. (8.16), (8.17), (8.48) and (8.49) in [Bai64]).
Proposition 2.2.
We have
E(Yt ) = e(λ−µ)t and Var(Yt ) =
λ + µ 2(λ−µ)t
(e
− e(λ−µ)t ).
λ−µ
In [KS15, Section 3.1], we give furthermore expressions and bounds for E(Yt−1 | Yt > 0) and
−1/2
E(Yt
| Yt > 0) which show that these terms are essentially (up to a factor of order t) of the
1
anticipated orders e−(λ−µ)t and e− 2 (λ−µ)t , respectively, as t → ∞.
4
F. KÜCK AND D. SCHUHMACHER
For the proof of our main theorem a finer analysis is required. Denote by Bt and Dt the
numbers of births and deaths up to time t, respectively, where we set B0 = Y0 = 1 and D0 = 0.
By using a partial differential equation for the joint cumulant generating function of Bt and Yt
stated in [CM77], we obtain the following formulae for the first and second joint moments.
Proposition 2.3 (KS15, Proposition 3.6).
We have
λ (λ−µ)t
µ
e
−
λ−µ
λ−µ
λ(λ + µ) 2(λ−µ)t
2λµ (λ−µ)t
λ2
Cov(Bt , Yt ) =
e
−
te
−
e(λ−µ)t
(λ − µ)2
λ−µ
(λ − µ)2
λ2 (λ + µ) 2(λ−µ)t
4λ2 µ
2λ2 µ
λ(λ + µ) (λ−µ)t
(λ−µ)t
e
.
Var(Bt ) =
e
−
te
+
−
(λ − µ)3
(λ − µ)2
(λ − µ)3
(λ − µ)2
E(Bt ) =
µ −1
t )−1
λ
λ
Remark 2.4. Note that E(Yt ) = E(Bt ) − E(Dt ) and E(B
is the
E(Dt ) = µ = λ+µ ( λ+µ )
ratio of the probabilities of a birth and a death at each event time. Furthermore, the sum
(λ−µ)T − 2µ is the expected number of events up to time t.
E(Bt ) + E(Dt ) = λ+µ
λ−µ e
λ−µ
In the rest of this section we are mainly concerned with the age of an individual picked
uniformly at random at a fixed time T > 0 (given YT > 0). We briefly call the distribution of
this age the age distribution of (Yt )t≥0 at time T (not to be confused with the “age distribution”
treated for example in [Har02]). It is known that the age distribution converges to the Exp(λ)
distribution as T → ∞; see e.g. [HJV05]. In the pure birth case, the age distribution can be
easily derived from known results, such as Theorem 1 in [NR71].
Proposition 2.5 (from KS15, Proposition 3.9). Let µ = 0. The ages of the individuals at
time T that have been born after time zero are i.i.d. truncated exponentially distributed, more
precisely they have distribution L(Z|Z ≤ T ), where Z ∼ Exp(λ).
In the general case, the situation is far more complicated. To the best of our knowledge,
the following results are new. Our proof of Theorem 2.6 is based on a recent result by Ba,
Pardoux, and Sow [BPS12] that recovers the distribution of the random binary tree generated
by a supercritical birth and death process as the distribution of the tree inscribed under a certain
contour (or exploration) process. We give a slightly simplified proof in the appendix, referring
to [KS15, Section 3.3] for full technical details.
Theorem 2.6 (KS15, Theorem 3.12). Given YT = yT for some yT > 0, the cumulative
distribution function FyT of the age of an individual picked uniformly at random at time T is
given by
yT − 1
e−λt − e−(λ−µ)T e−µt
1 λ(1 − e−µt ) − µ(1 − e−λt )
FyT (t) =
1−
+
1{t<T } + 1{t=T }
yT
yT
λ−µ
1 − e−(λ−µ)T
for t ∈ [0, T ].
A simple computation yields then the unconditional age distribution.
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
5
Corollary 2.7 (KS15, Corollary 3.13). The cumulative distribution function F of the age
of an individual picked uniformly at random at time T is given by
(λ−µ)T
λ−µ
e−λt − e−(λ−µ)T e−µt
λe
−µ
F (t) = 1 − (λ−µ)T
1−
log
λ−µ
λe
−λ
1 − e−(λ−µ)T
(λ−µ)T
λe
−µ
λ(1 − e−µt ) − µ(1 − e−λt )
λ−µ
log
1{t<T } + 1{t=T }
+ (λ−µ)T
λ−µ
λ−µ
λe
−λ
for t ∈ [0, T ].
The next corollary states that the age distribution converges exponentially fast to the Exp(λ)
distribution in a certain sense.
Corollary 2.8 (KS15, Corollary 3.14). Let A denote the age of an individual picked uniformly at random at time T . Then there exists a random variable Z with L(Z|YT > 0) = Exp(λ)
such that
−cA
λ−µ
λ
1
λ
−cZ E e
−e
YT > 0 ≤ c + λ e(λ−µ)T − 1 + λe(λ−µ)T − λ log λ − µ + (λ − µ)T
for any c > 0.
Another random quantity we need to control for our main bound is, at any fixed time T , the
time since the last event has occurred. Let 0 = T1 < T2 < . . . be the event times of (Yt )t≥0 . Since
BT + DT is the number of events up to time T , the random variable T − TBT +DT describes the
quantity we are interested in. The following result states that the Exp((YT − 1)λ) distribution
is a stochastic upper bound given YT on {YT > 1} and follows from Theorem 2.8 above.
Theorem 2.9 (KS15, Theorem 3.19). Given YT = yT , the distribution of T − TBT +DT is
stochastically dominated by the distribution with cumulative distribution function
G(t) = 1{yT >1} (1 − e−(yT −1)λt ) + 1{yT ≤1} 1{t≥T } .
3. Degree distribution at finite time. If at least two nodes are alive, each node i spawns
edges to other nodes according to a birth and death process with constant birth rate α0 = αSi
and linear death rate with factor β. The one-dimensional distributions of this process are wellknown. For completeness a derivation can be found in [KS15, Subsections 4.1–4.2].
Proposition 3.1. Let (Zt )t≥0 be a birth and death process with constant birth rate α0 and
linear death rate β 0 = βn if the process is in state n ∈ N = {1, 2, . . .}. If the process is started
deterministically at k ∈ N, we have
Zt ∼ Po
α0
β (1
− e−βt ) ∗ Bin k, e−βt
for every t ≥ 0, where ∗ denotes convolution.
In what follows, we derive the degree distribution at finite time T based on this result.
6
F. KÜCK AND D. SCHUHMACHER
3.1. The pure birth case. We deal with the case µ = 0 first. Let the nodes be ordered by their
birth times. Let Si be the social index of node i and Ai (T ) its age at time T . For convenience,
we define A2 (T ) = 0 if YT = 1. Furthermore, given YT = yT , let the random variable JT be
uniformly distributed on {1, . . . , yT } and independent of the ages, the social indices and the rest
of the path (Yt )0≤t<T . We interpret JT as the index of a node that is randomly picked at time
T among all living nodes.
For the time being, we condition on (Yt )0≤t≤T = (yt )0≤t≤T with yT > 0, the social indices
(Sk )k∈N = (sk )k∈N and JT = jT . Let T = a1 > . . . > ayT denote the corresponding ages of the
individuals.
Firstly, we consider the number of edges created by node jT . Assuming yT > 1, the edges
created by jT form a birth and death process with constant birth rate αsjT and linear death
rate with factor β, started in zero at time T − amax(jT ,2) . By Proposition 3.1, the number of
edges alive at time T that jT has created has distribution
αsjT
−βamax(jT ,2)
(3.1)
Po
(1 − e
) .
β
We have to add to this the number of edges alive at time T that other nodes have created
but connect to jT . Consider a fixed node i ∈ {1, . . . , yT } \ {jT } and some time interval of the
form [T − al , T − al+1 ) for l ≥ i ∨ 2. The number of edges that connect i to jT and that survive
until the end of this interval, i.e. until the birth time T − al+1 of node l + 1, can be described
by a birth and death process of the above type again. The birth rate is constant and equal to
1
1
because node i creates edges at rate αsi and l−1
is the probability that an edge that is
αsi l−1
created in the interval [T − al , T − al+1 ) is connected to jT . The death rate is linear again with
factor β. By Proposition 3.1, the number of edges created by i in [T − al , T − al+1 ) that connect
αsi
to jT and survive until T − al+1 is Po (l−1)β
(1 − e−β(al −al+1 ) ) -distributed.
We can extend the time interval by one birth time, i.e. we compute the distribution of the
number of edges that i creates in [T − al , T − al+2 ), connect to jT , and survive until T − al+2
by conditioning on the number Z of edges that i creates in [T − al , T − al+1 ), connect to jT
and survive until T − al+1 . Given Z = z, by Proposition 3.1 and the Markov property, the
number of edges that i creates in [T − al , T − al+2 ), connect to jT , and survive until T − al+2
−βt(al+1 −al+2 ) ) ∗ Bin z, e−βt . We already know from above that
i
has distribution Po αs
lβ (1 − e αsi
Z ∼ Po (l−1)β
(1−e−β(al −al+1 ) ) . Thus if we do not condition on Z = z, we obtain the distribution
αsi
αsi
−β(al+1 −al+2 )
−β(al+1 −al+2 )
−β(al −al+1 )
(1 − e
)+e
(1 − e
) .
Po
lβ
(l − 1)β
(3.2)
Starting at l = max(i, jT ) and iterating the procedure that leads to (3.2) until the whole
interval [T − amax(i,jT ) , T ) is spanned, we see that the number of edges alive at time T that
connect to jT but have been created by other nodes has distribution
Po
X
yT
i=1
i6=jT
yT
yT −1 ytY
−l−1
αsi (1 − e−βayT ) X −βay X
αsi (1 − e−β(al −al+1 ) )
−β(a
−a
)
l+k
l+k+1
T
+
e
e
(yT − 1)β
(l − 1)β
i=1
i6=jT
l=i∨jT
k=1
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
7
(3.3)
X
yT
yT yT −1
α X X si
α
si
−βal
−βayT
−βal+1
) .
)+
−e
= Po
(1 − e
(e
β
yT − 1
β
l−1
i=1
i6=jT
i=1 l=i∨jT
i6=jT
Since the numbers of outgoing and incoming edges are independent, the desired degree distribution is obtained by convolving (3.1) and (3.3). Lifting the conditioning on (Yt )0≤t≤T , (Sk )k∈N
and JT , we arrive at the following result.
Theorem 3.2. For µ = 0, the degree distribution in the Britton–Lindholm model without
loops is the MixPo(ΛT ) distribution, where
ΛT =
YT
αSJT
α X
Si
(1 − e−βAmax(JT ,2) (T ) ) +
(1 − e−βAYT (T ) )
β
β
YT − 1
i=1
i6=JT
(3.4)
+
YT YX
T −1
Si
α X
(e−βAl+1 (T ) − e−βAl (T ) ).
β
l−1
i=1 l=i∨JT
i6=JT
3.2. The general case. For general µ, the degree distribution can be determined similarly to
the pure birth case.
As in Section 2, let 0 = T1 < . . . < TBT +DT be the event times up to T . We still enumerate
nodes according to their birth times. Denote by 0 = T1+ < . . . < TB+T their birth times and by
Ti− the death time of the i-th node. Finally, given the subset of {1, . . . , BT } that contains the
indices of all living nodes at time T , let JT be uniformly distributed on this set and independent of all other random variables as before. Note that SJT is (stochastically) independent of
((Yt )0≤t≤T , JT ).
We condition on (Yt )0≤t≤T = (yt )0≤t≤T with yT > 0, the social indices (Sk )k∈N = (sk )k∈N and
JT = jT again. Let bT and dT denote the corresponding number of births and deaths up to time
+
+
T , respectively. Furthermore, let 0 = t1 < t2 < . . . < tbT +dT and 0 = t+
1 < t2 < . . . < tbT be the
corresponding event times and birth times up to time T , respectively.
Since deaths of other nodes reduce the number of edges created by node jT , we cannot derive
the distribution of the number of outgoing edges at time T in the same way as for the pure birth
process. However, we can derive the total number of edges incident to jT in a similar way as the
number of incoming edges in the pure birth case. Consider a fixed node i 6= jT that is alive at
+
time T (provided there are any) and some time interval of the form [tl , tl+1 ) for tl ≥ t+
i ∨ tjT and
tl+1 ≤ T . Note that the nodes i and jT create edges with rate αsi and αsjT , respectively, and
that the probability that an edge that is created by i is connected to jT is yt 1−1 and equal to
l
the probability that an edge that is created by jT is connected to i. Thus the number of edges
created between i and jT that survive until time tl+1 can be described by a birth and death
process with constant birth rate α(si + sjT ) yt 1−1 and linear death rate with factor β like before.
l
By Proposition 3.1 we obtain that the number of edges that are created in [tl , tl+1 ) between i
8
F. KÜCK AND D. SCHUHMACHER
and jT and survive until tl+1 has distribution
α(si + sjT )
−β(tl+1 −tl )
(1 − e
)1{ytl >1} .
Po
(ytl − 1)β
Let the function r = r(yt )0≤t≤T : {1, . . . , bT } → {1, . . . , bT +dT } be defined such that t+
j = Tr(j)
for all j ∈ {1, . . . , bT }, i.e. r maps birth number to event number.
Applying the same iterative procedure as in Subsection 3.1, starting at l = r(i) ∨ r(jT ) and
+
continuing until the whole interval [t+
i ∨ tjT , T ) is spanned, we can see that the number of edges
between i and jT alive at time T has distribution
α(si + sjT )
(1 − e−β(T −tbT +dT ) )1{yT >1}
Po
(yT − 1)β
(3.5)
bT +d
T −1
X
−β(T −tbT +dT )
+e
bT +dY
T −l−1
l=r(i)∨r(jT )
e
−β(tl+k+1 −tl+k ) α(si
k=1
+ s jT )
(1 − e−β(tl+1 −tl ) )
(ytl − 1)β
1{ytl >1} .
Since the processes of edges between i and jT are mutually independent for different i, we
obtain the desired degree distribution by convolving the distribution (3.5) for i running in
{i0 ∈ {1, . . . , bT } : i0 6= jT , t−
i0 > T } and lifting the conditioning on (Yt )0≤t≤T , (Sk )k∈N and JT .
Theorem 3.3. The degree distribution in the Britton–Lindholm model without loops is the
MixPo(Λ∗T ) distribution, where Λ∗T is a random variable with L(Λ∗T ) = L(ΛT |YT > 0) and
ΛT :=
BT
α X
Si + SJT
1 − (1 − e−β(T −TBT +DT ) )1{YT >1}
β
(YT − 1) {Ti >T }
i=1
i6=JT
(3.6)
+
BT
α X
β
i=1
i6=JT
1{Ti− >T }
BT +D
XT −1
l=r(i)∨r(JT )
Si + SJT −β(T −Tl+1 )
(e
− e−β(T −Tl ) )1{YT >1} .
l
(YTl − 1)
4. Bounds on the total variation distance between the finite-time and the asymptotic degree distributions. Since in the pure birth case we obtain a much better bound with
considerably less work, we treat the cases µ = 0 and µ > 0 separately in Subsections 4.1 and 4.2,
giving the ideas of the proofs. The proofs themselves are deferred to Subsection 4.3.
4.1. The pure birth case. Let µ = 0. Fix T > 0 and let the random variable ΛT be defined
as in Theorem 3.2. Furthermore, let SJ∞ and AJ∞ be independent random variables such that
−1 (F (A (T, ω))) for all ω ∈ Ω, where F and F
SJ∞ (ω) = SJT (ω) and AJ∞ (ω) = F∞
∞ are
T
JT
T
the cumulative distribution functions of AJT (T ) and the Exp(λ) distribution, respectively. Note
that we obtain from Proposition 2.5 that L(AJT ) → Exp(λ) weakly. The random variable AJ∞
is Exp(λ) distributed since FT is continuous and hence FT (AJT (T )) is uniformly distributed
on [0, 1]. Moreover, since the exponential distribution stochastically dominates the truncated
−1 is
exponential distribution, we have F∞ (AJT (T, ω)) ≤ FT (AJT (T, ω)) for all ω ∈ Ω. Since F∞
increasing, it follows that AJT (T, ω) ≤ AJ∞ (ω) for all ω ∈ Ω.
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
9
Let S be a generic random variable that is distributed according to the distribution of the
social indices and let
M=
(4.1)
αSJ∞
αE(S)
(1 − e−βAJ∞ ) +
(1 − e−βAJ∞ ).
β
β
We know from Theorem 3.2 that MixPo(ΛT ) is the degree distribution at time T in the pure
birth case. Theorem 4.1 below implies that MixPo(ΛT ) converges at rate of just a bit slower
1
than e− 2 λT to MixPo(M) as T → ∞, which is the asymptotic degree distribution already stated
in Section 3.2 of [BL10]. Note that the theorem below is much more powerful as it gives an exact
distance bound for every finite T .
Theorem 4.1.
For T ≥
log(2)
λ ,
we have
√
32α √ − 1 λT
λ
e−λT ,
dT V (MixPo(ΛT ), MixPo(M)) ≤ √ σS T e 2 + 4αE(S) T +
β(β + λ)
λ
where σS is the standard deviation of S.
The main idea of the proof is as follows. In Appendix A1, we present a simple but crucial
theorem that allows to bound dT V (MixPo(ΛT ), MixPo(M)) by E(|ΛT − M|). In order to bound
E(|ΛT − M|) further, we use that the expected value of the second summand of the right-hand
side of (3.4) becomes small as T → ∞ since the age AYT (T ) of the youngest individual at time
T converges quickly to 0, and compare the other summands of the right-hand sides of (3.4) and
(4.1).
For the comparison of the last summands in (3.4) and (4.1), respectively, we note that the
average of the social indices becomes close to E(S) by the Law of Large Numbers. Then we use
again that the age AYT (T ) of the youngest individual at time T converges quickly to 0 and that
AJT (T ) converges quickly to AJ∞ by Proposition 2.5.
Finally, the expected absolute value of the difference between the first summand of (3.4) and
the first summand of (4.1) again becomes small since AJT (T ) converges quickly to AJ∞ .
Remark 4.2. For the original model of Britton and Lindholm with loops, we can adapt the
proof of Theorem 4.1 in such a way that the upper bound remains exactly the same.
4.2. The general case. Let SJ∞ (ω) = SJT (ω) as in the pure birth case. Furthermore, let
AJ∞ = Z, where Z is the random variable from Corollary 2.8. From the proof of this corollary
(in [KS15], proof of Corollary 3.14), we see that
−1
AJ∞ (ω) = 1{JT (ω)<YT (ω)} F∞
(F∗ (AJT (T, ω)) + 1{JT (ω)=YT (ω)} Z̃(ω),
where F∞ is the cumulative distribution function of the Exp(λ) distribution as before, Z̃ ∼ F∞
−λt −e−(λ−µ)T e−µt
independent of everything and F∗ (t) = 1 − e 1−e
(see also (A.4)). Note that, given
−(λ−µ)T
YT > 0, we have AJ∞ ∼ Exp(λ).
Recall the definition of the parameter random variable ΛT from (3.6) and let Λ∗T be a random
variable with L(Λ∗T ) = L(ΛT |YT > 0) as before. Moreover, set
(4.2)
M=
αSJ∞
αE(S)
(1 − e−(β+µ)AJ∞ ) +
(1 − e−(β+µ)AJ∞ )
β+µ
β+µ
10
F. KÜCK AND D. SCHUHMACHER
and let M∗ be a random variable with L(M∗ ) = L(M|YT > 0).
From Theorem 3.3, we know that MixPo(Λ∗T ) is the degree distribution at time T in the
general case. Theorem 4.3 below implies that MixPo(Λ∗T ) converges at a rate of just a bit over
1
e− 6 (λ−µ)T to the MixPo(M∗ ) distribution, which is the asymptotic degree distribution stated in
Section 3.2 of [BL10] as T → ∞. Note again that this theorem is much more powerful since it
gives an exact bound for the total variation distance for every finite T .
Theorem 4.3.
For T ≥
2 log(4(λ(λ−µ)−1 ))
,
λ−µ
we have
dT V (MixPo(Λ∗T ), MixPo(M∗ ))
!
√
229
27 √
5 6 λ
2
µ
2
1
E(S) +
2σS (λ − µ)T 2 e− 6 (λ−µ)T
≤α
+ + β+
+
2 λ−µ 5
2
5(λ − µ) λ + µ
10
√
3
!
1
µ µ
8
4 λ (λ + µ)
6
3
5λ
+ αβ
+
+
+
+ 6T +
+ (µT + 3) + 2 λ E(S)T 2 e−(λ−µ)T
2 4β λ
µ β (λ − µ)3
2
β
β
√
+ 3 2αλσS T 2 e−(λ−µ)T .
Note that the right-hand side is of the order
1
O(T 2 )e− 6 (λ−µ)T
as T → ∞.
The main idea of the proof of this theorem is the same as in the pure birth case: We make use of
Theorem A.1 to obtain E(|ΛT −M| |YT > 0) as an upper bound for dT V (MixPo(Λ∗T ), MixPo(M∗ ))
and establish a further bound for this expected value. In order to do so, we use that the expected
value of the first summand of the right-hand side of (3.6) converges quickly to zero since the time
T − TBT +DT since the last event before T converges quickly to 0, and compare the remaining
summand of the right-hand side of (3.6) with the right-hand side of (4.2). For this comparison,
we make vital use of the fact that the average of the social indices of the nodes living at time T
is close to E(S) by the Law of Large Numbers again and that, given Tl for some large l ∈ N, the
percentage of the nodes living at time Tl that survive up to time T is approximately e−µ(T −Tl )
by the Law of Large Numbers (see Lemma A.12 in the appendix).
A further important ingredient is that the reciprocal of the embedded jump chain, which
describes the population size after a fixed number of events, conditioned on survival is a supermartingale (see Corollary A.6 in the appendix), which makes it easy to deal with the expected
value of its maximum. Finally, we also use that the age AJT (T ) of the randomly picked individual
converges quickly to AJ∞ by Corollary 2.8.
The fact that nodes may die complicates the procedure considerably since the population size
after a fixed number of events is random in this case and additional dependencies have to be
treated (e.g. the inter-event times depend on the random population size at the previous event
time).
11
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
In order to cope with additional dependencies on the random index JT , we introduce the
random number K(T ).
Definition 4.4.
surely.
Let K(T ) := max{k : Tk ≤ 12 T }, such that we have YTK(T ) = Y 1 T almost
2
The probability that the node that is randomly picked at time T is born before the event time
TK(T )+1 decreases exponentially in T (see Lemma A.8(i) in the appendix). Thus we essentially
only have to consider the time interval [TK(T )+1 , ∞) instead of the interval [Tr(JT ) , ∞), where
Tr(JT ) is the birth time of the randomly picked individual. The choice of K(T ) makes sure that
we always have a large number of individuals in the interval [TK(T )+1 , ∞) with high probability.
4.3. Proofs.
Proof of Theorem 4.1. From Theorem A.1 follows
dT V (MixPo(ΛT ), MixPo(M)) ≤ E(|M − ΛT |),
and E(|M − ΛT |) is smaller than or equal to
α
E
SJ∞ (1 − e−βAJ∞ ) + E(S)(1 − e−βAJ∞ ) − SJT (1 − e−βAmax(JT ,2) (T ) )
β
−
YT
X
i=1
i6=JT
YT YX
T −1
X
Si
Si
−βAYT (T )
−βAl+1 (T )
−βAl (T )
)−
(1 − e
(e
−e
) YT − 1
l−1
i=1 l=i∨JT
i6=JT
YT
α X
Si
−βAYT (T ) ≤ E
(1 − e
)
β
YT − 1
i=1
i6=JT
YT YX
T −1
(Si − E(S))
α X
−βAl+1 (T )
−βAl (T ) 1{JT ≤l} (e
+ E
−e
)
β
l−1
i=1
i6=JT
+
l=i
YT YX
T −1
X
α E(S) −βAl+1 (T )
EE(S)(1 − e−βAJ∞ ) −
(e
− e−βAl (T ) ))
β
l−1
i=1 l=i∨JT
i6=J
(4.3)
T
α −βAmax(JT ,2) (T ) −βAJ∞
+ ESJ∞ (1 − e
) − SJT (1 − e
),
β
where we use the convention
0
0
:= 0.
For the first line of the right-hand side, we have
YT
X
E
i=1
i6=JT
X
YT
Si
Si
−βAYT (T ) −βAYT (T ) (1 − e
) = E E
(1 − e
)YT , JT
YT − 1
YT − 1
i=1
i6=JT
12
F. KÜCK AND D. SCHUHMACHER
X
YT
E(Si )
=E
(1 − E(e−βAYT (T ) |YT )) ,
YT − 1
(4.4)
i=1
i6=JT
where the last equality holds since, given YT , the age AYT (T ) and JT are independent.
Given YT = yT > 1, the age AyT (T ) is the minimum of yT − 1 i.i.d. truncated exponentially
distributed random variables by Proposition 2.5. Since the minimum of yT − 1 independent
Exp(λ) distributed random variables is Exp((yT − 1)λ) distributed, the distribution of AyT (T )
is stochastically dominated by the Exp((yT − 1)λ) distribution. In general, we have for random
variables X̂ and Ŷ with X̂ ≤st Ŷ that f (X̂) ≤st f (Ŷ ) and consequently E(f (X̂)) ≤ E(f (Ŷ )) for
every increasing function f . Thus the right-hand side of (4.4) is smaller than or equal to
β
1
β
1
E E(S)
≤
E(S)E
1
.
λ YT − 1 + β
λ
YT − 1 {YT >1}
λ
Note that
λT
X
∞
∞
∞
λe − λ n−1
λ X 1
λ X 1 λeλT − λ n
1
pn (T )
1
= λT
= λT
=
E
YT − 1 {YT >1}
n−1
n−1
n
λe
λeλT
λe
λeλT
n=2
n=1
n=2
∞
∞
X
1 λeλT − λ n X pn (T )
λ
1
≤ λT
=
=E
,
λT
n
n
YT
λe − λ
λe
n=1
n=1
where pn (T ) is the probability mass function from Section 2. Thus the right-hand side of (4.4)
is smaller than or equal to
β
1
E(S)E
.
λ
YT
For the second line of the right-hand side of (4.3), we obtain
YT YT −1
X X (Si − E(S))
−βAl+1 (T )
−βAl (T ) 1{JT ≤l} (e
−e
)
E
l−1
i=1
i6=JT
l=i
YT −1
X
= E
l=2
l
1 X
−βAl+1 (T )
−βAl (T ) (Si − E(S))1{JT ≤l} (e
−e
)
l−1
i=1
i6=JT
YX
T −1
= E E l=JT ∨2
l
1 X
−βAl+1 (T )
−βAl (T ) (Si − E(S))(e
−e
)
l−1
i=1
i6=JT
YT , JT
YX
l
T −1
1 X
−βAl+1 (T )
−β(Al (T )−Al+1 (T ))
≤E
E
(Si − E(S))E e
(1 − e
)|YT
l−1
l=JT ∨2
i=2
v
u l YX
T −1
X
1 u
−β(Al (T )−Al+1 (T ))
t
2
≤E
E (Si − E(S)) E(1 − e
|YT )
l−1
l=JT ∨2
(4.5)
i=1
YX
T −1
1
−β(Al (T )−Al+1 (T ))
√
≤E
σS E(1 − e
|YT ) ,
l−1
l=J ∨2
T
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
13
where the fourth line holds since the second sum in the third line is (stochastically) independent
of JT , the social indices are independent of all other random variables and, given YT , the index
JT is independent of the social indices and ages, and the second last line is obtained by applying
p
1 Pl
E|Z − E(Z)| ≤ Var(Z) to Z = l−1
i=1 Si .
Given YT and Al+1 (T ), the difference Al (T ) − Al+1 (T ) is the minimum of l − 1 i.i.d. truncated
exponentially distributed random variables by Proposition 2.5 (for l ≥ 2). Thus it is stochastically dominated by an Exp((l − 1)λ) distributed random variable Zl , where Zl can be assumed
to independent of Al+1 (T ). This implies for a > 0:
Z
T
P(Al − Al+1 ≤ a|YT , T − Al+1 = x)P(T − Al+1 ∈ dx|YT )
P(Al − Al+1 ≤ a|YT ) =
0
≥ P(Zl ≤ a|YT ).
Thus, also given YT alone, the difference Al (T ) − Al+1 (T ) is stochastically dominated by Zl . By
the same argument as for (4.4), it follows that (4.5) is smaller than or equal to
YX
YX
T −1
T −1
1
β
1
β
σS √
E
E(1{JT ≤l} |YT )
σS √
1{JT ≤l} = E
l − 1 (l − 1)λ + β
l − 1 (l − 1)λ + β
l=2
l=2
YX
T −1
1
β
l
σS √
≤E
l − 1 (l − 1)λ YT
l=2
YX
T −1
1
β
2(l − 1)
√
σS
≤E
l − 1 (l − 1)λ YT
l=2
YT −1
1
1 X
2β
√
σS E
≤
λ
YT
l−1
l=2
YT −1
2β
1 X
2
√
√
≤
σS E
λ
YT
l−2+ l−1
l=2
YT −1
√
√
2β
1 X
=
σS E
2( l − 1 − l − 2)
λ
YT
l=2
√
4β
YT − 2
=
σS E
λ
YT
4β
1
≤
σS E √
.
λ
YT
Since we arranged AJT (T ) ≤ AJ∞ , for the third line of the right-hand side of (4.3) we obtain
YT YX
T −1
X
E(S) −βAl+1 (T )
−βAJ∞
−βAl (T ) ))
EE(S)(1 − e
)−
(e
−e
l−1
i=1 l=i∨JT
i6=JT
YX
T −1
= EE(S)(1 − e−βAJ∞ ) − E(S)
l=JT ∨2
l
1 X −βAl+1 (T )
−βAl (T ) (e
−e
))
l−1
i=1
i6=JT
14
F. KÜCK AND D. SCHUHMACHER
(4.6)
YX
T −1
−βA
−βA
(T
)
−βA
(T
)
J∞
l+1
l
= EE(S)(1 − e
) − E(S)
(e
−e
))
l=JT ∨2
−βA
(T
)
−βA
(T
)
−βA
Y
max(J
,2)
J
∞ ) − E(S)(e
T
T
= EE(S)(1 − e
−e
))
−βAmax(JT ,2) (T )
−βAYT (T )
−βAJ∞
= E E(S)(1 − e
) + E(S)(e
−e
)
= E(S)E 1 − E(e−βAYT (T ) |YT ) + E(S)(E(e−βAmax(JT ,2) (T ) ) − E(e−βAJ∞ )).
By Proposition 2.5
(4.7)
E(e−βAmax(JT ,2) (T ) ) =
Z
0
T
e−βx
λ
λe−λx
dx =
−λT
1−e
1 − e−λT
Z
T
e−(β+λ)x dx =
0
λ 1 − e−(β+λ)T
,
β + λ 1 − e−λT
whence follows that (4.6) is smaller than or equal to
β 1
λ
1 − e−(β+λ)T
β
1
λ
1
E(S)E
+ E(S)
− 1 ≤ E(S)E
+ E(S)
.
λ Yt + β
β+λ
λ
YT
β + λ eλT − 1
1 − e−λT
λ
Since we arranged SJT = SJ∞ and AJT (T ) ≤ AJ∞ , for the fourth line of the right-hand side
of (4.3) we have
(4.8)
−βAmax(JT ,2) (T ) −βAJ∞
ESJ∞ (1 − e
) − SJT (1−e
) = E(S)(E(e−βAmax(JT ,2) (T ) ) − E(e−βAJ∞ )).
Note that the right-hand side of (4.8) is the same as the second summand of the right-hand side
λ
1
of (4.6), which is smaller than or equal to E(S) β+µ
(see above).
eλT −1
Altogether, we may conclude that the right-hand side of (4.3) is bounded from above by
2α
1
4α
1
2α
λ
1
.
E(S)E
+
σS E √
+
E(S)
λT
λ
YT
λ
β
β+λe −1
YT
−1/2
Using the upper bounds for E(YT−1 ) and E(YT
[KS15], we obtain the statement of the theorem.
from Proposition 3.3 and Corollary 3.4 in
Proof of Theorem 4.3. In order to simplify notation, we introduce E∗ ( · ) = E( · |YT > 0),
P∗ ( · ) = P( · |YT > 0) and MT = BT + DT .
As mentioned before, we use Theorem A.1 to obtain
dT V (MixPo(Λ∗T ), MixPo(M∗ )) ≤ E∗ |ΛT − M|.
In order to find an upper bound for E∗ |ΛT − M|, we use the triangle inequality for the absolute
value after plugging in the definitions of ΛT and M given by (3.6) and (4.2), which yields
BT
−β(T −TMT )
X
)
∗ α(1 − e
E 1{YT >1}
(Si + SJT )1{T − >T }
i
(YT − 1)β
i=1
i6=JT
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
BT
α X
(Si + SJT )1{T − >T }
i
β
+
i=1
i6=JT
M
T −1
X
(e−β(T −Tl+1 ) − e−β(T −Tl ) )
l=r(i)∨r(JT )
15
1
1
YTl − 1 {YTl >1}
α
−(β+µ)AJ∞ −
(E(S) + SJ∞ )(1 − e
).
β+µ
This is bounded from above by
B
−β(T −TMT ) X
) T
∗ α(1 − e
(4.9) E
Si 1{T − >T } 1{YT >1}
i
(YT − 1)β
i=1
i6=JT
(4.10)
∗
+E
BT
X
α(1 − e−β(T −TMT ) )
S JT
(YT − 1)β
1{Ti− >T }1{YT >1}
i=1
i6=JT
(4.11)
BT
X
+E (Si − E(S))1{T − >T }
i
β
M
T −1
X
∗ α
i=1
i6=JT
−β(T −Tl+1 )
(e
−β(T −Tl )
−e
l=r(i)∨r(JT )
1
)
1{YTl >1} YTl − 1
MX
r−1 (l)
T −1
X
α
1
+ E∗ 1{YTl >1}
(E(S) + SJT )1{T − >T } (e−β(T −Tl+1 ) − e−β(T −Tl ) )
i
β
YTl − 1
l=r(JT )
(4.12)
i=1
i6=JT
M
T −1
X
α
−(β+µ)(T −Tl+1 )
−(β+µ)(T −Tl ) (e
−e
)
(E(S) + SJT )
−
β+µ
l=r(JT )
(4.13)
M
T −1
X
∗ α
+E (E(S) + SJT )
(e−(β+µ)(T −Tl+1 ) − e−(β+µ)(T −Tl ) )
β+µ
l=r(JT )
α
−(β+µ)AJ∞ −
),
(E(S) + SJ∞ )(1 − e
β+µ
where r−1 (l) is the number of births that occur not later than the lth event for all l ∈ {1, . . . , MT },
P T
i.e. r−1 : {1, . . . , MT } → {1, . . . , BT }, l 7→ B
i=1 1{r(i)≤l} .
In the following, we deduce upper bounds for (4.9)–(4.13).
Upper bound for (4.9) and (4.10). We treat (4.9) similarly to the corresponding expression in
the pure birth case, but condition on BT , DT , JT and the information which nodes survive up
to time T , and obtain
BT
−β(T −TMT )
X
)
α
∗ α(1 − e
(4.14)
E
1{YT >1}
Si 1{T − >T } ≤ E(S)E∗ (1 − eβ(T −TMT ) ).
i
(YT − 1)β
β
i=1
i6=JT
Since we condition on JT , the same expression is also obtained for (4.10).
For T ≥
1
λ−µ
log(2), we have that the expectation E∗ (1 − e−β(T −TMT ) ) is smaller than or equal
to
(4.15)
2
λ
λ − µ + β log
+ (λ − µ)T
e−(λ−µ)T
λ
λ−µ
16
F. KÜCK AND D. SCHUHMACHER
by Lemma A.9. Plugging this expression in the right-hand side of (4.14) results in the following
upper bound for the sum of (4.9) and (4.10):
λ
4αE(S)
λ − µ + β log
+ (λ − µ)T
e−(λ−µ)T .
βλ
λ−µ
Upper bound for (4.11). Let RTl ,T be the number of nodes that are alive at time Tl and survive
up to time T . We compute
X
α BT
E (Si − E(S))1{T − >T }
i
β
M
T −1
X
∗
i=1
i6=JT
α
≤ E β
∗
α
β
≤ E∗
(4.16)
·
M
T −1
X
(e
−β(T −Tl+1 )
−β(T −Tl )
−e
RTl ,T
1
)
1{YTl >1} YTl − 1
r−1 (l)
RTl ,T − 1 1{YTl >1} X
(Si − E(S))1{T − >T } )
i
YTl − 1 RTl ,T − 1
i=1
i6=JT
(e−β(T −Tl+1 ) − e−β(T −Tl ) )
l=r(JT )
RTl ,T ≥2
1
−e
−β(T −Tl )
l=r(i)∨r(JT )
l=r(JT )
RTl ,T ≥2
M
T −1
X
(e
−β(T −Tl+1 )
RTl ,T − 1
1
YTl − 1 {YTl >1}
−1
rX
(l)
(Si − E(S))1{T − >T } i
−1
i=1
i6=JT
!!
.
(Yt )0≤t≤T , JT
Note in the second line that we can restrict the sum to RTl ,T ≥ 2 because RTl ,T ≥ 1 by l ≥ r(JT )
and a summand with RTl ,T = 1 in the first line would be 0 anyway.
Since the sum in the last line of (4.16) has exactly RTl ,T − 1 summands of the form Si − E(S)
and all other summands are zero, the right-hand side of (4.16) is smaller than or equal to
(4.17)
α
E β
∗
M
T −1
X
−β(T −Tl+1 )
(e
l=r(JT )
RTl ,T ≥2
−β(T −Tl )
−e
RTl ,T − 1
σS
.
1{YTl >1} p
)
YTl − 1
RTl ,T − 1 Since e−β(T −Tl+1 ) − e−β(T −Tl ) ≤ β(Tl+1 − Tl ) and RTl ,T ≤ YTl , the expression (4.17) is smaller
than or equal to
MX
MX
T −1 σ 1
T −1
S {YTl >1}
σ
p
q S
E∗ α
(Tl+1 − Tl ) ≤ E∗ α
YT
Y
−
1
Tl
YTl − 2 l
l=r(JT )
l=r(JT )
√
∗
≤ 2ασS T E
max
1{YTl >1} (Tl+1 − Tl )
1
p
r(JT )≤k≤MT −1
YTk
√
max
≤ 2ασS T E∗ 1{K(T )<r(JT )}
(4.18)
1
p
r(JT )≤k≤MT −1
YTk
√
1
∗
p
+ 2ασS T E 1{K(T )≥r(JT )}
max
,
r(JT )≤k≤MT −1
YTk
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
17
where K(T ) = max{k : Tk ≤ 12 T } (see Definition 4.4). We have
1
p
max
E∗ 1{K(T )<r(JT )}
r(JT )≤k≤MT −1
YTk
1
Y∞ > 0 P∗ (Y∞ > 0) + 1 · P∗ (Y∞ = 0),
p
(4.19)
≤ E 1K(T )<r(JT )}
max
r(JT )≤k≤MT −1
YTk where Y∞ := lim Yt ∈ {0, ∞}.
t→∞
By Lemma A.3, we have for the second summand of the right-hand side of (4.19)
P∗ (Y∞ = 0) =
µ −(λ−µ)T
e
.
λ
The first summand of the right-hand side of (4.19) is smaller than or equal to
1
1 1
− 16 (λ−µ)T
+E
Y∞ > 0 ≤ e
Y∞ > 0 e 3 (λ−µ)T
E max p
YTK(T )
K(T )≤k
YTk
1
λ
≤ 1 + 2 log
+ (λ − µ)T e− 6 (λ−µ)T
(4.20)
λ−µ
for T ≥
for T ≥
to
1
2
λ−µ log(2) by combining Lemma A.10, where δ = 2 and
2
λ−µ log(2), the first summand of the right-hand side of
√
(4.21)
√
1
µ
λ
−(λ−µ)T
2α σS T e
+ 2ασS 1 + 2 log
+ (λ − µ)T T e− 6 (λ−µ)T .
λ
λ−µ
By Lemma A.8(i), we obtain that for T ≥
side of (4.18) is bounded from above by
√
(4.22)
Thus for T ≥
(4.22).
γ = 16 , and Lemma A.11. Thus
(4.18) is smaller than or equal
2ασS T e
2
λ−µ
− 21 λT
1
λ−µ
log(2), the second summand of the right-hand
!
2(λ − µ)
λ
−(λ−µ)T
+
log
+ (λ − µ)T e
.
λ
λ−µ
log(2), the expression (4.11) is bounded from above by the sum of (4.21) and
Upper bound for (4.12). For (4.12), we have
E β
∗ α
M
T −1
X
l=r(JT )
−1
r (l)
X
1
1{YTl >1}
(E(S) + SJT )1{T − >T } (e−β(T −Tl+1 ) − e−β(T −Tl ) )
i
YTl − 1
i=1
i6=JT
M
T −1
X
α
−(β+µ)(T −Tl+1 )
−(β+µ)(T −Tl ) −
(E(S) + SJT )
(e
−e
)
β+µ
l=r(JT )
M
T −1
X
α(β + µ)
RTl ,T − 1
= E∗ (E(S) + SJT )
1
(e−β(T −Tl+1 ) − e−β(T −Tl ) )
β(β + µ)
YTl − 1 {YTl >1}
l=r(JT )
(4.23)
M
T −1
X
αβ
−(β+µ)(T −Tl+1 )
−(β+µ)(T −Tl ) −
(E(S) + SJT )
(e
−e
)
β(β + µ)
l=r(JT )
18
F. KÜCK AND D. SCHUHMACHER
since the second sum on the left-hand side of (4.23) has exactly RTl ,T − 1 non-zero summands.
The right-hand side of (4.23) is equal to
M
T −1 X
RT ,T − 1
α
E 1
e−β(T −Tl+1 ) (1 − e−β(Tl+1 −Tl ) )
(E(S) + SJT )
(β + µ) l
β(β + µ)
YT − 1 {YTl >1}
∗
l
l=r(JT )
(4.24)
−(β+µ)(T −Tl+1 )
− βe
−(β+µ)(Tl+1 −Tl )
(1 − e
) .
Now we use
1−e
−β(Tl+1 −Tl )
= β(Tl+1 − Tl ) −
∞
X
(−β(Tl+1 − Tl ))k
k=2
and
1−e
−(β+µ)(Tl+1 −Tl )
= (β + µ)(Tl+1 − Tl ) −
k!
∞
X
(−(β + µ)(Tl+1 − Tl ))k
k=2
k!
to obtain that (4.24) is smaller than or equal to
E MT −1
RTl ,T − 1
+ SJT ) X
−µ(T −Tl+1 )
−β(T −Tl+1 )
e
(Tl+1 − Tl )
(β + µ)β
1{YTl >1} − e
β(β + µ)
YTl − 1
∗ α(E(S)
l=r(JT )
M
∞
T −1 X
X
α
(−(β + µ)(Tl+1 − Tl ))k
∗
−(β+µ)(T −Tl+1 )
+E (E(S) + SJT )
βe
β(β + µ)
k!
l=r(JT )
k=2
∞
X
RTl ,T − 1
(−β(Tl+1 − Tl ))k −β(T −Tl+1 )
e
− (β + µ)
1
YTl − 1 {YTl >1}
k!
k=2
∞
X
RTl ,T − 1
−µ(T
−T
)
∗
l+1 (T
−
T
)
≤ E α(E(S) + SJT )
1{Tl+1 ≤T } 1{YTl >1} − e
l
l+1
YTl − 1
l=r(JT )
∗
+E
M
∞
T −1 X
X
(−(β + µ)(Tl+1 − Tl ))k
α
(E(S) + SJT )
β+µ
k!
l=r(JT ) k=2
(4.25)
∗
+E
M
∞
T −1 X
X
α
(−β(Tl+1 − Tl ))k
(E(S) + SJT )
.
β
k!
l=r(JT ) k=2
Firstly, we consider the first summand of the right-hand side of (4.25). Since the social index
SJT is independent of all other random variables appearing in (4.25), this summand is equal to
X
∞
RTl ,T − 1
∗
−µ(T
−T
)
l+1 2αE(S)E
1{Tl+1 ≤T } 1
−e
(Tl+1 − Tl ) .
YTl − 1 {YTl >1}
l=r(JT )
We derive
∗
E
X
∞
l=r(JT )
RTl ,T − 1
−µ(T −Tl+1 ) 1{Tl+1 <T } 1
−e
(Tl+1 − Tl )
YTl − 1 {YTl >1}
19
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
∗
≤E
∞
X
l=K(T )+1
+ T P∗ (K(T ) ≥ r(JT )).
(4.26)
For
RTl ,T − 1
−µ(T −Tl+1 ) 1{Tl+1 <T } 1
−e
(Tl+1 − Tl )
YTl − 1 {YTl >1}
1
λ−µ
log(2), the second summand of the right-hand side of (4.26) is smaller than or equal to
− 12 λT
Te
λ
2(λ − µ)
log
+ (λ − µ)T T e−(λ−µ)T
+
λ
λ−µ
by Lemma A.8(i). The first summand of the right-hand side of (4.26) is equal to
∗
E
∞
X
RTl ,T − 1
−µ(T −Tl+1 ) E 1{Tl+1 <T } 1{YTl >1} − e
(Tl+1 − Tl )
YTl − 1
l=K(T )+1
For the inner expectation, we have
RTl ,T − 1
−µ(T −Tl+1 ) 1{YTl >1} − e
E 1{Tl+1 <T } (Tl+1 − Tl )
YTl − 1
K(T ), YT > 0 .
K(T ), YT > 0
(4.27)
RTl ,T − 1
−µ(T
−T
)
l+1 = E 1{Tl+1 <T } 1{YTl >1} − e
(Tl+1 − Tl )
YTl − 1
K(T ), YT > 0 P(YTl > 0|K(T )) .
l
P(YT > 0|K(T ))
For the fraction, we obtain for l ≥ K(T )
P(YTK(T ) > 0|K(T ))
P(YTl > 0|K(T ))
≤
.
P(YT > 0|K(T ))
E(P(YT > 0|K(T ), YTK(T ) )|K(T ))
(4.28)
Note that, given K(T ) and YTK(T ) , we know that exactly YTK(T ) nodes are alive at time 21 T .
Thus due to the Markov property for (Yt )t≥0 and the formula for the extinction probability of a
linear birth and death process with a general initial value given in Remark 2.1, the conditional
Y
probability P(YT > 0|K(T ), YTK(T ) ) is equal to p0 ( 12 T ) TK(T ) , which implies that (4.28) is equal
to
P(YTK(T ) > 0|K(T ))
YTK(T )
E(1 − p0 ( 12 T )
|K(T ))
≤
P(YTK(T ) > 0|K(T ))
(1 −
p0 ( 12 T ))P(YTK(T )
> 0|K(T ))
1
=
λe 2 (λ−µ)T − µ
(λ − µ)e
1
(λ−µ)T
2
≤
λ
.
λ−µ
Thus the right-hand side of (4.27) is smaller than or equal to
RTl ,T − 1
−µ(T
−T
)
l+1 E E 1{Tl+1 <T } 1
−e
(Tl+1 − Tl ) YTl , Tl , K(T ) K(T ), YTl > 0
YTl − 1 {YTl >1}
λ
(4.29)
·
.
λ−µ
By the Markov property of (Yt )t≥0 , we may omit the conditioning on K(T ) if we condition on
YTl and Tl for l ≥ K(T ) + 1. Conditioning in addition on Tl+1 , we see that (4.29) is equal to
20
F. KÜCK AND D. SCHUHMACHER
RTl ,T − 1
−µ(T −Tl+1 ) E 1{Tl+1 <T } (Tl+1 − Tl )E 1{YTl >1} − e
YTl − 1
YT , Tl , Tl+1 K(T ), YT > 0
l
l
(4.30)
·
λ
.
λ−µ
By Lemma A.12, summing over l and taking the expectation yields the following upper bound
for the first summand of the right-hand side of (4.26):
X
∞
√
λ
− 12 ∗
(4.31)
E
6
E 1{Tl+1 <T } (Tl+1 − Tl )YTl K(T ), YTl > 0 .
λ−µ
l=K(T )+1
Obviously, (4.31) is equal to
∞
X
√
λ
− 12 ∗
6
E 1{Y∞ >0}
E 1{Tl+1 <T } (Tl+1 − Tl )YTl K(T ), YTl > 0
λ−µ
l=K(T )+1
∞
X
√
λ
− 21 ∗
+ 6
E 1{Y∞ =0}
E 1{Tl+1 <T } (Tl+1 − Tl )YTl K(T ), YTl > 0 .
λ−µ
l=K(T )+1
1
We may conclude that for for T ≥ λ−µ
log(2), the first summand of (4.25) is bounded from
above by
∞
X
√
λ
− 12 ∗
2αE(S)
6
E 1{Y∞ >0}
E 1{Tl+1 <T } (Tl+1 − Tl )YTl K(T ), YTl > 0
λ−µ
l=K(T )+1
∞
X
√
λ
− 21 ∗
E 1{Y∞ =0}
E 1{Tl+1 <T } (Tl+1 − Tl )YTl K(T ), YTl > 0
+ 6
λ−µ
l=K(T )+1
!
2(λ − µ)
λ
− 21 λT
−(λ−µ)T
+ Te
+
log
+ (λ − µ)T T e
(4.32)
.
λ
λ−µ
For the first outer expectation in (4.32), we have
∞
X
−1
∗
E 1{Y∞ >0}
E 1{Tl+1 <T } (Tl+1 − Tl )YTl 2
l=K(T )+1
≤ E
∞
X
−1
E 1{Tl+1 <T } (Tl+1 − Tl )YTl 2
l=K(T )+1
∞
X
−1
≤ E max YTk 2
K(T )≤k
≤
l=K(T )+1
T
−1
E max YTk 2
2
K(T )≤k
K(T ), YT > 0
l
K(T ) Y∞ > 0
1{Tl+1 <T } (Tl+1 − Tl ) Y∞ > 0
Y∞ > 0 .
2
By inequality (4.20), we obtain that for T ≥ λ−µ
log(2), this expression is smaller than or equal
to
1
T
λ
1 + 2 log
+ (λ − µ)T e− 6 (λ−µ)T .
2
λ−µ
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
For the second outer expectation in (4.32), we obtain
∞
X
−1
E 1{Y∞ =0}
E 1{Tl+1 <T } (Tl+1 − Tl )YTl 2
l=K(T )+1
21
K(T ), YT > 0 YT > 0
l
T
T µ −(λ−µ)T
≤ P(Y∞ = 0|YT > 0) =
e
,
2
2λ
where the last equality follows from Lemma A.3.
2
In conclusion, for T ≥ λ−µ
log(2), the first summand of the right-hand side of (4.25) is
bounded from above by
!
√
1
λ
λ
E(S)T
1 + 2 log
+ (λ − µ)T e− 6 (λ−µ)T + e−(λ−µ)T
6α
λ−µ
λ−µ
!
2(λ − µ)
λ
−(λ−µ)T
− 12 λT
+
.
log
+ (λ − µ)T T e
+ 2αE(S) T e
λ
λ−µ
Since the social index SJT is independent of all other random variables appearing in (4.25),
the second summand of the right-hand side of (4.25) is smaller than or equal to
MX
∞
T −1 X
2α
(−(β + µ)(Tl+1 − Tl ))k
∗
E(S)E
.
β+µ
k!
l=r(JT ) k=2
Since
by
P∞
k=2
(−x)k
k!
≤
x2
2
for x ≥ 0 (Lemma 5.18 in [KS15]), this expression is bounded from above
α
E(S)E∗
β+µ
(4.33)
MX
T −1
2
2
(β + µ) (Tl+1 − Tl )
.
l=r(JT )
2
For T ≥ ( λ−µ
log(2) ∨ 2(log(4λ)−log(λ−µ))
), Lemma A.13 implies that (4.33), which is an upper
λ+µ
bound for the second summand of the right-hand side of (4.25), is smaller than or equal to
α(β + µ)E(S)
1
µ T 2 −(λ−µ)T 60λ3 (λ + µ) 2 −(λ+µ)T
e
+
T e
+ T 2 e− 2 λT
4
λ 4
(λ − µ)
λ
2(λ − µ)
+
log
+ (λ − µ)T T 2 e−(λ−µ)T
λ
λ−µ
!
1
3 2
T
λ
+
T +
1 + 2 log
+ (λ − µ)T e− 4 (λ−µ)T .
4
2(λ + µ)
λ−µ
For the third summand of the right-hand side of (4.25), we obviously obtain the same upper
bound, but may replace the factor β + µ by β.
2
We may conclude that for T ≥ ( λ−µ
log(2) ∨ 2(log(4λ)−log(λ−µ))
), the expression (4.12) is smaller
λ+µ
than or equal to
!
√
1
λ
λ
6α
E(S)T
1 + 2 log
+ (λ − µ)T e− 6 (λ−µ)T + e−(λ−µ)T
λ−µ
λ−µ
22
F. KÜCK AND D. SCHUHMACHER
− 21 λT
+ 2αE(S) T e
+ α(2β + µ)E(S)
!
2(λ − µ)
λ
+
log
+ (λ − µ)T T e−(λ−µ)T
λ
λ−µ
1
µ T 2 −(λ−µ)T 60λ3 (λ + µ) 2 −(λ+µ)T
T e
+ T 2 e− 2 λT
e
+
4
λ 4
(λ − µ)
2(λ − µ)
λ
+
log
+ (λ − µ)T T 2 e−(λ−µ)T
λ
λ−µ
!
1
3 2
λ
T
+
1 + 2 log
+ (λ − µ)T e− 4 (λ−µ)T .
T +
4
2(λ + µ)
λ−µ
Upper bound for (4.13). Recall that we arranged SJT = SJ∞ . Since the sum in (4.13) telescopes,
this implies
M
T −1
X
α
E (E(S) + SJT )
(e−(β+µ)(T −Tl+1 ) − e−(β+µ)(T −Tl ) )
β+µ
l=r(JT )
α
−(β+µ)AJ∞ −
)
(E(S) + SJ∞ )(1 − e
β+µ
−(β+µ)(T −Tr(JT ) )
−(β+µ)(T −TMT )
−(β+µ)AJ∞
∗ α
(E(S) + SJ∞ ) e
−e
− (1 − e
) =E β+µ
(4.34)
2α
−(β+µ)(T −TMT )
∗
∗ −(β+µ)AJT (T )
−(β+µ)AJ∞ ≤
E(S) E 1 − e
+ E e
−e
,
β+µ
∗
where the last inequality holds since SJ∞ is independent of all other random variables appearing
in (4.34).
By using Lemma A.9 again, we obtain for the first conditional expectation on the right-hand
side of (4.34) the upper bound
λ
2
λ − µ + (β + µ) log
+ (λ − µ)T
e−(λ−µ)T
λ
λ−µ
1
if T ≥ λ−µ
log(2).
Corollary 2.8 implies that the second conditional expectation on the right-hand side of (4.34)
is bounded from above by
!
2λ
2(λ − µ)
λ
+
log
+ (λ − µ)T
e−(λ−µ)T
β+µ+λ
λ
λ−µ
1
if T ≥ λ−µ
log(2).
Altogether, we obtain that the right-hand side of (4.34) is smaller than or equal to
!
4α
λ−µ β+λ
λ
λ
E(S)
+
log
+ (λ − µ)T +
e−(λ−µ)T
β+µ
λ
λ
λ−µ
β+µ+λ
if T ≥
1
λ−µ
log(2).
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
23
Conclusion. Combining the upper bounds obtained for (4.9)–(4.13) and simplifying the results
as detailed on the final pages of [KS15], we obtain the desired statement.
Appendix A1: Upper bound for the total variation distance between two mixed
Poisson distributions. Theorem 2.1 in [Yan91] gives us an upper bound for the total variation
distance between two Poisson distributions. By conditioning we can generalize this result to
mixed Poisson distributions.
Theorem A.1. Let Λ and M be positive real valued random variables. Then we have for the
total variation distance between the mixed Poisson distributions MixPo(Λ) and MixPo(M):
√
√
dT V (MixPo(Λ), MixPo(M)) ≤ E(min(| Λ − M|, |Λ − M|).
Proof. Let X and Y be MixPo(Λ) and MixPo(M) distributed, respectively. Then it follows
dT V (MixPo(Λ), MixPo(M)) = sup |P(X ∈ A) − P(Y ∈ A)|
A⊂N0
= sup |E(P(X ∈ A|Λ)) − E(P(Y ∈ A|M))|
A⊂N0
≤ sup E(|P(X ∈ A|Λ) − P(Y ∈ A|M)|)
A⊂N0
≤ E sup |(P(X ∈ A|Λ)) − (P(Y ∈ A|M))|
A⊂N0
= E(dT V (L(X|Λ), L(Y |M))
√
√
≤ E(min(| Λ − M|, |Λ − M|),
where the last line follows from Theorem 2.1 in [Yan91].
Remark A.2. Note that the mixed Poisson distribution depends on the parameter random
variable only via its distribution; therefore Theorem A.1 yields
dT V (MixPo(Λ), MixPo(M)) ≤
inf
D
1
1
E(min(|Λ̃ 2 − M̃ 2 |, |Λ̃ − M̃|).
Λ̃:Λ̃=Λ
D
M̃:M̃=M
Appendix A2: The age distribution at finite time T . We give here a slightly simplified
the proof of Theorem 2.6, and refer to [KS15, Section 3.3] for a proof that includes all technical
details. The proof vitally rests on a recent result in [BPS12] that shows that the distribution of
the random binary tree generated by a supercritical birth and death process can be obtained
by “inscribing” a binary tree in the area under the following contour process (sometimes also
referred to as exploration process).
Let (Uk )k∈N0 and (Vk )k∈N0 be two mutually independent sequences of independent, identically exponentially distributed random variables with parameters µ and λ, respectively. Starting at (0, 0), alternatingly add straight lines of slope +1 and −1 whose random heights are
governed by the U - and V -variables, respectively. More precisely, for any k ∈ N0 , given that
after 2k steps of this procedure we are at position (x, y), we add a straight line from (x, y)
24
F. KÜCK AND D. SCHUHMACHER
to x + min(Uk , T − y), y + min(Uk , T − y) ; given that after 2k + 1 steps of this procedure we
are at position (x, y), we add a straight line from (x, y) to x + min(Vk , y), y − min(Vk , y) .
We stop the procedure when reaching a point (τ, 0) with τ > 0. The random function H =
(Hx )0≤x≤τ induced by this graph is what we call contour process deflected at T . We denote by
0 = W1 < . . . < W2BT +1 = τ its local extrema.
We inscribe a “left-aligned” tree under (the graph of) the contour process as follows. Note
that (W2 , HW2 ) is the leftmost local maximum of H. Then we draw a (vertical) line from (W2 , 0)
to (W2 , HW2 ). The second leftmost local maximum is (W4 , HW4 ) and the leftmost non-zero local
minimum is (W3 , HW3 ) if they exist. In this case we add a (vertical) line from (W4 , HW3 ) to
(W4 , HW4 ) and a (horizontal) line from (W4 , HW3 ) connecting the vertical line horizontally to
the rest of the tree. We continue this procedure until we have explored all non-zero local extrema;
see Figure 1.
t
HW4 = HW6 = T
HW5
age
HW2
HW3
HW1 = 0
W2 W3
W4 W5 W6
τ
x
Fig 1. The contour process deflected at T and the corresponding tree representing a linear birth and death process
up to time T
Theorem 3.1 in [BPS12] implies that the tree obtained by this procedure has the same distribution as the (left-aligned) tree obtained from a linear birth and death process with per-capita
birth rate λ and per-capita death rate µ, started with one individual. This latter tree is drawn by
picking the individual that dies at a death event uniformly at random from all living individuals
and by adjusting horizontal line lengths in the resulting tree in such a way that it can be covered
from above with an admissible realization of a contour process that has local maxima at the
leaves of the tree. Note that horizontal line lengths carry no meaning in the context of the birth
and death process.
We may then read off the ages of the individuals alive at time T as the total lengths of
vertical lines that reach T or, in terms of the contour process, as the size of those up-steps
min(Uk , T − HW2k−1 ) for k ∈ N that reach T ; see Figure 1.
We use a “double mirroring argument” for the contour process, which is explained in detail
in [KS15] (culminating in Equations (11) and (12) in [KS15]). It can be summarized as follows.
Firstly, we mirror the contour process at a vertical axis, after which the distribution of the
corresponding birth and death process is still the same. It follows that the desired ages correspond
to the down-steps min(Vk , T ) that start in T . In order to derive the distributions of these downsteps given YT = yT , we mirror the contour process at a horizontal axis and shift it to the left
such that it starts in zero. It can be shown that up to the first time zero or T is hit, this mirrored
and shifted process has the same distribution as the (deflected) contour process of a subcritical
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
25
birth and death process (Zt )t≥0 with birth rate µ and death rate λ started with one individual.
Let F∗ be the cumulative distribution function of the life time L1 of this starting individual
given the process goes extinct by time T , and let F ∗ be the cumulative distribution function of
min(L1 , T ) given the process survives until time T . We then obtain from the above that the age
distribution given YT = yT > 0 has cumulative distribution function
(A.1)
FyT (t) =
yT − 1
1 ∗
F∗ (t) +
F (t)
yT
yT
for all t ≥ 0.
By Bayes’ Theorem, F∗ has density
f∗ (t) ∝ λe−λt P(ZT = 0 | L1 = t),
(A.2)
where t < T .
Note that the birth times of the offspring of the first individual form a Poisson process with
rate µ. By conditioning on their number (up to time t) and their birth times, and by the formula
for the extinction probability from Section 2, we obtain
P(ZT = 0 | L1 = t) =
Z
∞
X
(µt)k e−µt
k=0
tk
k!
t
Zt
...
0
∞
X
1 −µt
=
e
k!
Z t
k=0
0
0
λ 1 − e(λ−µ)(T −tk )
λ 1 − e(λ−µ)(T −t1 )
.
.
.
dt1 . . . dtk
µ 1 − µλ e(λ−µ)(T −t1 )
µ 1 − µλ e(λ−µ)(T −tk )
k
e−(λ−µ)(T −u) − 1
du
λ
e−(λ−µ)(T −u) − µλ
k
∞
X
1 −µt
(λ−µ)(T −t)
(λ−µ)T
=
e
log(λe
− µ) + λt − log(λe
− µ)
k!
=
(A.3)
k=0
λe(λ−µ)(T −t)
λe(λ−µ)T
− µ (λ−µ)t
e
−µ
for t < T . Now we can easily compute the normalizing constant of the density f∗ in (A.2):
ZT
λe−λt (λe(λ−µ)(T −t) − µ)e(λ−µ)t dt =
0
ZT
(λe(λ−µ)T e−λt − µe−µt )dt
0
(λ−µ)T
=e
(1 − e−λT ) − (1 − e−µT )
= e(λ−µ)T − 1.
Thus the density f∗ is given by
f∗ (t) =
λe(λ−µ)T e−λt − µe−µt
λe−λt − µe−(λ−µ)T e−µt
=
.
e(λ−µ)T − 1
1 − e−(λ−µ)T
for t ∈ [0, T ]. By integration we obtain
(A.4)
F∗ (t) = 1 −
e−λt − e−(λ−µ)T e−µt
1 − e−(λ−µ)T
26
F. KÜCK AND D. SCHUHMACHER
for t ∈ [0, T ].
In order to prove Theorem 2.6, it remains to compute F ∗ . Note that min(L1 , T ) has a density
f˜ with respect to the measure Leb[0,T ) + δT given by
f˜(t) =
(
λe−λt
e−λT
if 0 ≤ t < T
if t = T.
Thus by Bayes’ Theorem, F ∗ has density f ∗ with respect to the measure Leb[0,T ) + δT satisfying
f ∗ (t) ∝ f˜(t)P(ZT > 0 | L1 = t)
for t ∈ [0, T ]. By (A.3) we have that
P(ZT > 0 | L1 = t) = 1 − P(ZT = 0 | L1 = t) = 1 −
λe(λ−µ)(T −t) − µ (λ−µ)t
e
λe(λ−µ)T − µ
for t ∈ [0, T ). Therefore

λe−λt µe(λ−µ)t −µ
∗
λe(λ−µ)T −µ
f (t) ∝
e−λT
if 0 ≤ t < T
if t = T.
We can compute the normalizing constant of f ∗ as
e
−λT
+
ZT
λµ
e
λe(λ−µ)T − µ
−λt
(λ−µ)t
(e
−λT
− 1)dt = e
+
λµ
ZT
λe(λ−µ)T − µ
0
(e−µt − e−λt )dt
0
= e−λT +
(A.5)
=
λ(1 −
e−µT )
− µ(1 − e−λT )
λe(λ−µ)T − µ
λ−µ
.
−µ
λe(λ−µ)T
We may conclude that f ∗ is given by
∗
f (t) =
(
−µt −e−λt
λµ e λ−µ
λe−µT −µe−λT
λ−µ
if 0 ≤ t < T
if t = T.
By integration we obtain
(A.6)
F ∗ (t) =
λ(1 − e−µt ) − µ(1 − e−λt )
1{t<T } + 1{t=T }
λ−µ
for t ∈ [0, T ]. Plugging (A.4) and (A.6) into Equation (A.1) yields the statement of Theorem 2.6.
Appendix A3: Lemmas for the proof of Theorem 4.3. Recall that we set E∗ ( · ) =
E( · |YT > 0) and P∗ ( · ) = P( · |YT > 0). We start with a number of results about the linear
birth and death process (Yt )t≥0 that could well be useful in other settings. We refer to [KS15,
Subsection 5.2] for proofs. We first compute the extinction probability given the process has
survived up to time T . We write Y∞ = lim Yt ∈ {0, ∞}.
t→∞
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
Lemma A.3 (KS15, Lemma 5.5).
we have
27
For the conditioned extinction probability given YT > 0,
P∗ (Y∞ = 0) =
µ −(λ−µ)T
e
.
λ
For the probability of YT = 1 conditioned on YT > 0, we have the following lemma.
We have for T ≥
Lemma A.4 (KS15, Lemma 5.6).
P∗ (YT = 1) ≤
1
λ−µ
log(2)
2(λ − µ)
.
λe(λ−µ)T
We next consider sub- and supermartingal properties of conditioned processes.
Lemma A.5 (KS15, Lemma 5.7).
(i) (YTk )k∈N conditioned on Y∞ = 0 is a supermartingale.
(ii) (YTk )k∈N conditioned on Y∞ > 0 is a submartingale.
It can be shown that the first part of Lemma A.5 implies the second part. Lemma A.5 (ii)
yields the following corollary (see [KS15]).
Corollary A.6 (KS15, Corollary 5.8).
gale.
)k∈N conditioned on Y∞ > 0 is a supermartin(YT−1
k
In order to cope with dependencies, we already introduced the random variable K(T ) in
Definition 4.4. For the same reason, we define the deterministic number κ(T ), which depends
on T such that the probability for more than κ(T ) events up to time T decreases exponentially
in T (see Lemma A.8(ii) below). The quantity κ(T ) is solely used in the proof of Lemma A.13
below. Essentially, we substitute the number of events BT + DT up to time T by κ(T ) since it
is difficult to treat the dependencies between the number of events up to time T and the event
times (see [KS15] for details).
Definition A.7.
3
Let κ(T ) := be 2 (λ+µ)T c.
Lemma A.8 below states essentially that, for T large, it is unlikely that the randomly picked
3
individual was born before 12 T or that more than e 2 (λ+µ)T events have occurred by time T . It
can be proved by computing E(YT−1 |YT > 0) and using Proposition 2.3 (see [KS15] for details).
Lemma A.8 (KS15, Lemma 5.10).
(i) For the probability that fewer than K(T ) events have occurred up to the birth time of the
randomly picked node JT , we have
λ−µ
λ
∗
− 21 λT
P (K(T ) ≥ r(JT )) ≤ e
+ (λ−µ)T
log
+ (λ − µ)T .
λ−µ
λe
−λ
28
F. KÜCK AND D. SCHUHMACHER
(ii) For T ≥
2(log(4λ)−log(λ−µ))
,
λ+µ
we have
P∗ (κ(T ) ≤ MT ) ≤
60λ3 (λ + µ) −(λ+µ)T
e
,
(λ − µ)4
where MT = BT + DT is the number of events up to time T as before.
The following lemma states that the time T − TMT since the last event becomes small quickly
and can be proved by using Theorem 2.9 and Lemma A.4 (see [KS15] for details).
1
Lemma A.9 (KS15, Lemma 5.11 and Lemma 5.12). For T ≥ λ−µ
log(2) and c > 0, we have
2(λ − µ) 2c
λ
−c(T −TMT )
∗
(A.7)
E (1 − e
) ≤ (λ−µ)T +
log
+ (λ − µ)T e−(λ−µ)T .
λ
λ−µ
λe
The next result follows straightforwardly from Corollary A.6 and a well-known inequality for
supermartingales.
Lemma A.10 (KS15, Lemma 5.13). For all δ, γ > 0, we have
γ
1 1
−γ(λ−µ)T
Y∞ > 0 ≤ e
+E
E
Y∞ > 0 e δ (λ−µ)T .
δ
YTK(T )
min YTk
K(T )≤k
We use the following lemma to bound the conditional expectation on the right hand side of
Lemma A.10 from above.
2
Lemma A.11 (KS15, Lemma 5.14). We have for T ≥ λ−µ
log(2)
1
1 1
λ
E
+ (λ − µ)T e− 2 (λ−µ)T .
Y∞ > 0 ≤ 2 log
YTK(T )
λ−µ
2
The lemma below allows us to control the proportion of nodes surviving up to time T of the
nodes alive at time Tl (if we reduce both numbers by one).
Lemma A.12 (KS15, Lemma 5.17). Let RTl ,T be the number of nodes that are alive at time
Tl and survive up to time T . Then for l ∈ N, we have
1
RTl ,T − 1
6 2
−µ(T −Tl+1 ) 1
.
−e
E YTl , Tl , Tl+1 ≤ YT
YTl − 1 {YTl >1}
l
For the conditional expectation of the sum of the squared inter-event times since the birth
of the randomly picked node, we have the following lemma, which can be proved by using
Lemmas A.8, A.10 and A.11 (see [KS15] for details).
Lemma A.13 (KS15, Lemma 5.19).
∗
E
MX
T −1
(Tl+1 − Tl )
l=r(JT )
2
≤
2
For T ≥ ( λ−µ
log(2) ∨
2(log(4λ)−log(λ−µ))
),
λ+µ
1
λ3 (λ + µ) −(λ+µ)T
µ T 2 −(λ−µ)T
e
+ 60T 2
e
+ T 2 e− 2 λT
4
λ 4
(λ − µ)
we have
DEGREE DISTRIBUTION IN A DYNAMIC NETWORK MODEL
29
2(λ − µ)
λ
+
log
+ (λ − µ)T T 2 e−(λ−µ)T
λ
λ−µ
1
λ
3 2
T
1 + 2 log
+ (λ − µ)T e− 4 (λ−µ)T .
+
T +
4
2(λ + µ)
λ−µ
Acknowledgement. We would like to thank Thomas Rippl for pointing out to us the connection
between birth and death and contour processes.
References.
[BA99] Albert-László Barabási and Réka Albert. Emergence of Scaling in Random Networks. Science,
286(5439):509–512, 1999.
[Bai64] Norman TJ Bailey. The elements of stochastic processes with applications to the natural sciences. Wiley,
New York, 1964.
[BL10] Tom Britton and Mathias Lindholm. Dynamic random networks in dynamic populations. Journal of
Statistical Physics, 139(3):518–535, 2010.
[BLT11] Tom Britton, Mathias Lindholm, and Tatyana Turova. A dynamic network in a dynamic population:
asymptotic properties. Journal of Applied Probability, 48(4):1163–1178, 2011.
[BPS12] Mamadou Ba, Etienne Pardoux, and Ahmadou B Sow. Binary trees, exploration processes, and an
extended Ray-Knight theorem. Journal of Applied Probability, 49(1):210–225, 2012.
[CM77] David R Cox and Hilton D Miller. The theory of stochastic processes, volume 134. CRC Press, Boca
Raton, Florida, 1977.
[HJV05] Patsy Haccou, Peter Jagers, and Vladimir A Vatutin. Branching processes: variation, growth, and
extinction of populations. Cambridge University Press, Cambridge, 2005.
[Har50] Theodore Edward Harris. Some mathematical models for branching processes. Technical report, No.
P-152, RAND Corporation, Santa Monica, 1950.
[Har02] Theodore Edward Harris. The Theory of Branching Processes. Dover Publications, Mineola, New York,
2002.
[KS15] Fabian Kück and Dominic Schuhmacher. Convergence Rates for the Degree Distribution in a Dynamic
Network Model. Technical Report, 2015. http://arxiv.org/abs/1509.04918.
[NR71] Marcel F Neuts and Sidney I Resnick. On the times of births in a linear birthprocess. Journal of the
Australian Mathematical Society, 12(04):473–475, 1971.
[PRR13] Erol A. Peköz, Adrian Röllin, and Nathan Ross. Total variation error bounds for geometric approximation. Bernoulli, 19(2):610–632, 2013.
[vdH15] Remco van der Hofstad. Random Graphs and Complex Networks. Vol. I. Lecture Notes, 2015. http:
//www.win.tue.nl/~rhofstad/NotesRGCN.pdf
[Yan91] Nikos Yannaros. Poisson approximation for random sums of Bernoulli random variables. Statistics &
Probability Letters, 11(2):161–165, 1991.
Institute for Mathematical Stochastics
University of Göttingen
Goldschmidtstraße 7
37077 Göttingen
Germany
E-mail: [email protected]
[email protected]