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# The partial duplication random graph with edge deletion

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## Abstract

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e. an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability p. In addition, every edge is deleted at constant rate, a mechanism which extends previous partial duplication models. In this model, we obtain results on the degree distribution, which shows a phase transition such that either -- if p is small enough -- the frequency of isolated vertices converges to 1, or there is a positive fraction of vertices with unbounded degree. We derive results on the degrees of the initial vertices as well as on the sub-graph of non-isolated vertices. In particular, we obtain expressions for the number of star-like subgraphs and cliques.
ALEA, Lat. Am. J. Probab. Math. Stat. 18, 325–347 (2021)
DOI: 10.30757/ALEA.v18-15
The partial duplication random graph
with edge deletion
Felix Hermann and Peter Pfaﬀelhuber
Technische Universität Berlin,
Straße des 17. Juni 136,
10623 Berlin, Germany.
URL:http://page.math.tu-berlin.de/hermann/
Albert-Ludwigs-Universität Freiburg,
Ernst-Zermelo-Straße 1,
79104 Freiburg, Germany.
URL:https://www.stochastik.uni-freiburg.de/professoren/pfaffelhuber
Abstract. We study a random graph model in continuous time. Each vertex is
partially copied with the same rate, i.e. an existing vertex is copied and every edge
leading to the copied vertex is copied with independent probability p. In addition,
every edge is deleted at constant rate, a mechanism which extends previous partial
duplication models. In this model, we obtain results on the degree distribution,
which shows a phase transition such that either if pis small enough the frequency
of isolated vertices converges to 1, or there is a positive fraction of vertices with
unbounded degree. We derive results on the degrees of the initial vertices as well
as on the sub-graph of non-isolated vertices. In particular, we obtain expressions
for the number of star-like subgraphs and cliques.
1. Introduction
Various random graph models have been studied in the last decades. Frequently,
such models try to mimic the behavior of social networks (see e.g. Cooper and Frieze,
2003 and Barabási et al.,2002) or interactions within biological networks (see e.g.
Wagner,2001,Albert,2005 and Jeong et al.,2000). For a general introduction to
random graphs see the monographs Durrett (2007), van der Hofstad (2017) and
references therein.
In this paper, we study and extend a duplication random graph model introduced
and discussed in Bhan et al. (2002), Chung et al. (2003), Pastor-Satorras et al.
Received by the editors July 8th, 2019; accepted December 10th, 2020.
2010 Mathematics Subject Classiﬁcation. 05C80 (Primary), 60K35 (Secondary).
Key words and phrases. Random graph, Degree distribution, Cliques.
325
326 F. Hermann and P. Pfaﬀelhuber
(2003), Chung et al. (2003), Bebek et al. (2006), Bebek et al. (2006), Hermann
and Pfaﬀelhuber (2016), Jordan (2018) and, more recently, in Jacquet et al. (2020)
and Frieze et al. (2020). In most applications, a vertex models a protein and
an edge denotes some form of interaction; see e.g. Pastor-Satorras et al. (2003).
Within the genome, the DNA encoding for a protein can be duplicated (which
in fact is a long evolutionary process), such that the interactions of the copied
protein are partially inherited to the copy; see Ohno (1970) for some more biological
explanations. Within the random graph model, a vertex is p-copied, i.e. a new
vertex is introduced and every edge of the parent vertex is independently copied
with the same probability p. The idea behind this is to model protein-protein
interactions, assuming that the ability to interact can be inherited from a parent
protein with a ﬁxed probability p.
In Pastor-Satorras et al. (2003), an extension of this model was suggested (but
not studied further) where edges can be randomly removed from the random graph.
Aiming for a closer look at this model, we extend the duplication random graph
from above by introducing a rate δat which each edge in the random graph is
deleted. In biological terms, this corresponds to loss of interactive abilities due to
mutation or deterioration; see e.g. Figure 3 in Wagner,2001.
While in the previous literature no rigorous limit results were shown for the model
without edge deletion, which we will call pure partial duplication model,Hermann
and Pfaﬀelhuber (2016) have determined a critical parameter p0.567143, the
unique solution of pep= 1, below which approximately all vertices are isolated.
Moreover, almost sure asymptotics and limit results for the number of k-cliques and
k-stars in the random graph as well as for the degree of a ﬁxed vertex were obtained.
Recently, Jordan (2018) has shown that for p<e1the degree distribution of the
connected component, i.e. of the subgraph of non-isolated vertices, has a limit with
tail behavior close to a power-law with exponent β > 2solving β3 + pβ2= 0
(cf. Jordan,2018, Theorem 1(c)). This ﬁnding has been complemented by Jacquet
et al. (2020) by some ﬁner asymptotics. Extending the model for adding additional
edges at random, Frieze et al. (2020) obtain results on the degree distribution and
the degree of a ﬁxed vertex. We also mention that Bienvenu et al. (2019) introduce
a similar model for speciation. However, in their model, each birth of a new vertex
is linked to removing another vertex, making the number of vertices in the network
a constant.
In this paper for the model with edge deletion, we derive results on the degree
distribution F1, F2, . . . of the full graph (see Theorem 2.4), the sub-graph of non-
isolated vertices (see Proposition 2.6) as well as the number of star-like graphs,
cliques, and degrees of initial vertices (see Theorem 2.9). The methods used to
derive these results include branching processes with disasters Z(see Section 3.1)
and piecewise deterministic jump processes X(see Section 3.2). For the former,
note that the degree distribution is closely related via P(Zt=k) = E[Fk(t)] for
all k0(see (3.2)) i.e. the expected degree distribution of the graph process
equating to the distribution of Zt. Such a connection to branching processes is
as in Jordan (2018), but now Zhas additional deaths at rate δ the rate of
edge deletion. (Note that links between random graphs and branching processes
frequently appear in the literature, e.g. van der Hofstad,2017, Section 4.2 and
Bollobás and Riordan,2009.) For the latter, such branching processes, and therefore
the degree distribution, can be studied by using piecewise-deterministic Markov
The partial duplication random graph with edge deletion 327
jump processes, a tool which we introduce in Lemma 2.3; see also Section 3.1 for the
connection to branching processes. The new connection of Xand Zis via a duality
relation (see (2.2)), which was already used in Hermann and Pfaﬀelhuber (2016),
and proved extendable in various directions. Generalizing the limit results from
Hermann and Pfaﬀelhuber,2016, Lemma 3.3 for Xto a broader class of piecewise-
deterministic Markov processes, we obtained general limit results for branching
processes with disasters in several settings in Hermann and Pfaﬀelhuber (2020). In
the present paper, we transfer these results to the degree distribution, generalizing
the results of Theorem 2.7 in Hermann and Pfaﬀelhuber (2016) to the model with
edge deletion; see Section 3. In particular, we derive a phase transition such that
if pis small, the fraction of vertices with positive degree vanishes. In this case we
ﬁnd that the sub-graph of non-isolated vertices is exponentially small, with two
possible rates depending on pand δ; see Theorem 2.4. For larger p, a positive
fraction of vertices is non-isolated, and their degree is unbounded. In Section 4, we
prove Theorem 2.9 which states limit results for binomial moments of the degree
distribution, cliques and the degree of a ﬁxed node mainly by applying martingale
theory, generalizing Theorems 2.9 and 2.14 in Hermann and Pfaﬀelhuber (2016).
2. Model and main results
After introducing the model (and its connection to a piecewise deterministic
Markov process) in Section 2.1, we give our ﬁrst main result, Theorem 2.4, on
the number of non-isolated vertices, in Section 2.2. In Section 2.3, we discuss the
(generating function of) the degree-distribution of the sub-graph of non-isolated
vertices. Theorem 2.9 on certain graph functionals is contained in Subsection 2.4.
Finally, we put our results in perspective to previous results in Subsection 2.5.
2.1. The model and a piecewise deterministic Markov process.
Deﬁnition 2.1 (Partial duplication graph process with edge deletion).Let p
[0,1],δ0and G0= (V0, E0)be a deterministic undirected graph without loops
with vertex set V0={v1, . . . , v|V0|}and non-empty edge set E0. Let P D(p, δ) =
(Gt)t0be the continuous-time graph-valued Markov process starting in G0and
evolving in the following way:
Every node vVtis p-partially duplicated (or p-copied for short) at rate
κt:= (|Vt|+ 1)/|Vt|, i.e. a new node v|Vt|+1 is added and for each wVt
with (v, w)Et,v|Vt|+1 is connected to windependently with probability
p.
Every edge in Etis removed at rate δ.
Then, P D(p, δ)is a partial duplication graph process with edge deletion with initial
graph G0,edge-retaining probability pand deletion rate δ. Within P D(p, δ), we
deﬁne the following quantities:
(1) Let Di(t) := degGt(vi)·1{i≤|Vt|} be the degree of vi, i.e. the number of its
neighbors, at time t.
(2) Let F(t) := (Fk(t))k=0,1,2,... with Fk(t) := |{1i |Vt|:Di(t) = k}|/|Vt|
for k= 0,1,2, . . . be the degree distribution at time t. Furthermore, let
F+(t) := 1 F0(t)be the proportion of vertices of positive degree.
(3) For k= 1,2, . . . let Bk(t) := P`k`
kF`(t)be the kth binomial moment of
the degree distribution.
328 F. Hermann and P. Pfaﬀelhuber
(4) For k= 1,2, . . . let Ck(t)be the number of k-cliques at time t, i.e. the
number of complete sub-graphs of size k.
Remark 2.2.(1) It is often desirable to have a discrete-time graph valued-
process where at each time nthe graph is of size n. For the P D(p, δ),
such a discretization is also possible, but quite elaborate, since the number
of edge deletions between two node additions follows a generalized negative
hypergeometric distribution heavily depending on the number of edges and
thus depending on the number of edges added with the latest node. In
what follows, we will only discuss the continuous-time version.
(2) We choose the duplication rate κt:= (|Vt|+ 1)/|Vt|in order to get a closed
recurrence relation for the degree distribution; see Lemma 3.3. Alterna-
tively, one could choose eκt:= 1, i.e. all vertices are copied at unit rate.
Since |Vt| Vet(see Lemma 4.4) and hence Rt
0κseκsds =Rt
0
1
|Vs|ds con-
verges to a ﬁnite random variable, we conjecture that the random graph
with our choice of κtbehaves the same qualitatively for t (i.e. un-
derlying the same phase transitions as in Theorems 1 and 2). However,
some limits depend on the intial graph (cf. F0in Theorem2.4(c)) and thus,
since the choice of κtinﬂuences the distribution of Gtearly on, quantitative
diﬀerences are to be expected.
(3) For kNak-star is a graph of k+1 nodes and kedges, where one particular
node, the center, is connected to each of the kother nodes. Since each node
of degree `is the center of `
kdistinct k-stars, these deliver an alternative
interpretation for the binomial moments: |Vt| · Bk(t)is equal to the total
number of distinct k-stars contained as subgraphs in Gt. Hence, the Bk(t)
as well as the Ck(t)can give an understanding of the topology of Gt. In
fact, several functionals of interest can be expressed via Bkand Ckas the
average degree in Gtequates to B1(t) = C2(t)/|Vt|while the transitivity
ratio is given by C3(t)
|Vt|B2(t).
Note that Hermann and Pfaﬀelhuber (2016) used the notation Sk(t)for the
factorial moments of the degree distribution giving Sk(t)/k! = Bk(t).
In order to formulate our results, we need an auxiliary process, which is connected
to P D(p, δ). It will appear below in Theorem 2.4.1 and in Proposition 2.6. The
proof of the following Lemma is found in Section 3.2.
Lemma 2.3 (Connection of P D(p, δ)and a piecewise-deterministic Markov pro-
cess).Let X= (Xt)t0be a Markov process on [0,1] jumping at rate 1 from Xt=x
to px, in between jumps evolving according to ˙
Xt=pXt(1Xt)δXt. Furthermore,
let
Hx(t) :=
X
k=0
(1 x)kFk(t),(2.1)
i.e. the probability generating function at 1xof the degree distribution at time t.
Then, for all t0and x[0,1], writing Ex[.] := E[.|X0=x],
E[Hx(t)] = Ex[HXt(0)] =
X
k=0
Fk(0) ·Ex[(1 Xt)k].(2.2)
The partial duplication random graph with edge deletion 329
2.2. Limits on the number of non-isolated vertices. Recall from the model with
δ= 0 that there are at least three regimes: 1) If p < 1/e (or 0> p plog 1
p),
Theorem 2.7 of Hermann and Pfaﬀelhuber (2016) shows that the frequency of iso-
lated vertices converges to 1, and Theorem 2.1 of Jordan (2018) shows that the
connected component converges to a graph with a power law distribution. 2)
If 1/e < p < p, where p0.567143 is the unique solution of pep= 1, (i.e.
pplog 1
p>0> p log 1
p), the techniques of Jordan (2018) break down (see his
Proposition 3.7), but still Theorem 2.7 of Hermann and Pfaﬀelhuber (2016) shows
that the frequency of non-isolated vertices becomes negligible. 3) If p>p(or
plog 1
p>0), the expected number of non-isolated vertices converges to a non-
trivial fraction of the whole graph. Our ﬁrst result, Theorem 2.4 below, extends
this result to the case δ0. The three cases (a), (b) and (c) of the following
Theorem have 1), 2) and 3) as special cases for δ= 0 . In Figure 2.1, we give an
illustration of all cases.
For the formulation, we need some notation: For Xas in Lemma 2.3, we deﬁne
c= exp pZ
0
E1[X2
s]
E1[Xs]ds.(2.3)
Set atbt, if at/bt
t→∞
1, as well as Q= 1, and recall that Bk(0) is the kth
binomial moment of the degree distribution of the initial graph, and in particular,
B1(0)|V0|is the initial number of edges.
Theorem 2.4 (Limit of the degree distribution).Let p(0,1) and δ0.
(a) If δpplog 1
p, then F(t)t→∞
(1,0,0, . . .)almost surely with
E[F+(t)] cet(1+δ2p),
where c(0, B1(0)) is given in (2.3).
(b) If pe1and pplog 1
pδplog 1
p, then F(t)t→∞
(1,0,0, . . .)
almost surely with
1
tlog E[F+(t)] t→∞
11
γ(1 + log γ),
where γ= log 1
p/(pδ).
(c) If p > log 1
pand δ < p log 1
p, then F(t)t→∞
(F0,0,0, . . .)almost surely,
where F0is non-deterministic and
E[F0]=11δ
p1
plog 1
p
X
k=1
Bk(0)(1)k1
k1
Y
`=1 1δ
p1p`
p` .
Remark 2.5 (Interpretations).(1) Clearly, the quantity F0is increasing in δ
and decreasing in p(and F+is decreasing in δand increasing in p). See
also the illustrations of Theorem 2.4 in Figure 2.1.
(2) For Theorem 2.4(c), we will see in the proof that the right hand side is
the hitting probability of a stochastic process, and in particular is in (0,1).
This interpretation shows that 0<E[F0]<1as long as the initial graph is
not trivial (i.e. F0(0) <1).
(3) The asymptotics given in case δpplog 1
pis more exact than the one
given for pplog 1
pδplog 1
p(in the sense that 1
tlog E[F+(t)]
1 + δ2pis a consequence of part 1 of the above Theorem). The reason is
330 F. Hermann and P. Pfaﬀelhuber
that we can give a formula for cin this case, which does not carry over to
(b); see the proof of Proposition 2.6.
(A)
p
1
p
1/e
1
δ
E[F0()] <1
E[F+()] et(1+δ2p)
E[F+()] et(1
1
γ(1+log γ))
(B)
p
1
p
1/e
1
tlog E[F+(t)]
δ= 0
δ= 1
δ= 0.5
1plog 1
pp
Figure 2.1.
Illustration of Theorem 2.4. In (A), the three cases are shown in the pδ-plane.
In (B), we draw the diﬀerent exponential rates of decrease in E[F+].
2.3. On the degree distribution of non-isolated vertices. In the case δplog 1
p,
the frequency of isolated vertices converges to 1. Hence, it is interesting to study
the (degree distribution of the) sub-graph of non-isolated vertices. In order to do
so, note that F+(t) = 1 H1(t), with Hfrom (2.1). Also note that, if at some time
ta duplication event is triggered, 1Hp(t)denotes the probability that the new
node is not isolated. Here, we aim for results on the asymptotics of the generating
function of the degree distribution of the sub-graph of non-isolated vertices,
h+
x(t) := P
k=1(1 x)kE[Fk(t)]
E[F+(t)] =E[Hx(t)H1(t)]
E[1 H1(t)] = 1 E[1 Hx(t)]
E[1 H1(t)] .
Using (2.2) and Bernoulli’s formula we compute
The partial duplication random graph with edge deletion 331
E[1 Hx(t)]
= 1
X
k=0
Fk(0)Ex[(1 Xt)k] = 1
X
`=0
Ex[X`
t](1)`X
k`
Fk(0)`
k
= 1
X
`=0
(1)`Ex[X`
t]·B`(0) =
X
`=1
B`(0)(1)`+1Ex[X`
t].(2.4)
As it turns out, we can control the right hand side as long as δ > p plog 1
p(see
Lemma 3.4). This immediately implies the following result:
Proposition 2.6 (Limit of degree distribution on the set of non-isolated vertices).
Let Xbe the process given in Lemma 2.3 and δ > p plog 1
p. Then, as t ,
E[1 Hx(t)] Ex[Xt]·B1(0),and therefore 1h+
x(t)Ex[Xt]
E1[Xt].
For the other case, plog 1
p< δ pplog 1
p, the right hand side of (2.4) is not
dominated by the Ex[Xt]-term. Here, we can rather show that (see Section 3.4)
1
tZt
0
Ex[Xk+1
s]
Ex[Xk
s]ds t→∞
(pδ)k+pk(1 + log γ)
pk
=log 1
γpk+γpk1
p =: ck(p, δ)
(2.5)
where γ= log 1
p/(pδ). However, to obtain a limit result in analogy to Proposi-
tion 2.6, we need this convergence not only to hold in the Cesaro-sense, but in the
regular sense, i.e.
Conjecture 2.7. For pe1,plog 1
p< δ pplog 1
pand k1it holds
Ex[Xk+1
t]
Ex[Xk
t]
t→∞
ck(p, δ ).(C)
With this, inserting Ex[X`
t] = Ex[Xt]Q`1
k=1
Ex[Xk+1
t]
Ex[Xk
t], (2.4) immediately provides
(a) and (b) of
Proposition 2.8. Assume that Conjecture 2.7 holds.
(a) If pe1,δ=pplog 1
p, then c1(p, δ)=0and E[1 Hx(t)] Ex[Xt]·
B1(0).
(b) If p>e1,plog 1
p< δ < p plog 1
p, then
E[1 Hx(t)] Ex[Xt]·
X
`=1
B`(0)(1)`+1c1(p, δ)· · · c`1(p, δ).
In both cases, as t ,
1h+
x(t)Ex[Xt]
E1[Xt]xexp pZ
0
E1[X2
s]
E1[Xs]Ex[X2
s]
Ex[Xs]ds.(2.6)
The last approximate equality is also shown in Section 3.4. Since the right hand
side of (2.6) is non-trivial, this proposition implies given that Conjecture 2.7
holds true that the degree distribution of the sub-graph of non-isolated vertices
converges to some non-trivial distribution.
332 F. Hermann and P. Pfaﬀelhuber
However, for a proof of Conjecture 2.7 or a closer analysis of the limits more insight
into the process Xis necessary.
2.4. Limits of some graph-functionals. We now investigate the limiting behavior of
certain functionals of the graph.
Theorem 2.9 (Binomial moments, cliques and degrees).As t , the following
statements hold almost surely:
(a) For k= 1,2, . . . ,ekBk(t)Bk(), where Bk() L1and
βk=(1 + δ2p, if δpp(1pk1)
k1,
1 + δk pk pk,otherwise.
(b) For k= 2,3, . . . ,exp(t(kpk1δk
2))Ck(t)Ck(), where Ck()
L1.
(i) If Ck(0) >0and δ < 2pk1/(k1), the convergence also holds in L1
and P(Ck()>0) >0.
(ii) Otherwise, if δ2pk1/(k1),Ck(t)=0for all tTCkfor some
ﬁnite random variable TCkand P(Ck() = 0) = 1.
(c) For i |V0|,et(pδ)Di(t)Di(), where Di() L1. Moreover,
(i) if Di(0) >0and δ < p, the convergence also holds in Lrfor all r1
and
E[Di()] = Di(0)1 + p
|V0|.(2.7)
(ii) if δp,Di(t) = 0 for all tTifor some ﬁnite random variable Ti
and P(Di() = 0) = 1.
Remark 2.10 (Interpretations).
(1) For Theorem 2.9.(a), we have β1= 1 + δ2pand for δ= 0, we have
βk= 1 pk pk,k= 1,2, . . . In all cases, we can also write βk= (1 + δ
2p)(1 + δk pk pk), which immediately shows that βkis continuous
in pand δ. In addition, for k= 2,3, . . . , we ﬁnd pp(1pk1)
k1, i.e. we can
choose δ0such that either of the two cases can in fact occur. Moreover,
βkβk1, which can be seen as follows: First, note that (1pk2)/(k2)
(1pk1)/(k1) =
(1+···+pk3)/(k2)
(1+···+pk2)/(k1) 1. So, if δpp(1pk1)
k1, both βk1and βkdo not
depend on kanyway. Then, if pp(1pk1)
k1δpp(1pk2)
k2, we have
that
βk1βk= 1 + δ2pmin(1 + δ2p, 1 + δk pk pk)0.
Finally, for pp(1pk2)
k2δ, we have
βk1βk=pδpk1+pkp(1 pk1)
k1p(1 p)pk2
=p(1 p)1 + · · · +pk2
k1pk20.
The fact that βk1βkimplies that there are much less star-like subgraphs
with k1leaves than star-like subgraphs with kleaves, k= 2,3, . . . This
can only be explained by nodes with high degree.
The partial duplication random graph with edge deletion 333
(2) Noting that B1(t)=2|V(t)| · C2(t)and 1
tlog |V(t)|t→∞
1, we see that the
results in (a) and (b) imply the same growth rate for the number of edges.
(3) Interestingly, we ﬁnd that δpimplies that all vertices of the initial graph
will eventually be isolated (i.e. have degree 0). However, the total number
of edges, denoted by C2, only dies out for δ2p. So, for p < δ < 2p, all
initial vertices become isolated, but are copied often enough such that the
number of edges is positive for all times with positive probability.
(A)
p
1
β2
δ= 0
δ= 1/2
δ=p2
δ= 1
(B)
p
1
β3
δ= 0
δ= 1/2
δ=pp(1p2)
2
δ= 1
Figure 2.2.
Illustration of Theorem 2.4(a). We display the rates of decay of E[Bk(t)] for k= 2
(A) and k= 3 (B).
2.5. Connection to previous work. Hermann and Pfaﬀelhuber (2016) analyzed the
case δ= 0. Note that Theorem 2.7.1 in that paper is extended here by considering
δ > 0as well as giving precise exponential decay rates in Theorem 2.4(a) and 1(b).
Moreover, recalling the connection Bk(t) = Sk(t)/k!mentioned in Remark 2.2.3,
Theorem 2.7.2 is also generalized here to the case δ > 0and further extended by
the almost sure convergence of each component of the degree distribution in The-
orem 2.4(c). The methods used in the proofs of Theorem 2.4 and Proposition 2.6
must be seen as extensions of tools used previously in Hermann and Pfaﬀelhuber
(2016). In particular, we found that duplication graphs with edge deletion yield a
similar connection to birth-death processes with disasters, deﬁned in the next sec-
tion. Such models can be studied using piecewiese-deterministic Markov processes;
334 F. Hermann and P. Pfaﬀelhuber
see Hermann and Pfaﬀelhuber (2020). Theorem 2.4 and Proposition 2.6 now es-
sentially follow by combining Hermann and Pfaﬀelhuber,2020, Corollaries 2.4 and
3.7 in the next section.
Theorem 2.9 in Hermann and Pfaﬀelhuber (2016) deals with cliques and k-stars
in the case δ= 0 and is extended by Theorem 2.9. More precisely, since |Vt| et,
and Hermann and Pfaﬀelhuber (2016) treats the discrete-time model, we note that
Theorem 2.9(1) of Hermann and Pfaﬀelhuber (2016) aligns with Theorem 2.9(b),
but only gives L1(rather than L2)-convergence. In Theorem 2.9(2) of Hermann
and Pfaﬀelhuber (2016), S
k, the number of k-stars in the network at time trelative
to the network size, was analyzed, which coincided with the factorial moments of
the degree distribution. There, a k-star was not deﬁned as a sub-graph of Gt, since
it depended on the order of the nodes. |Vt| · Bk(t)now gives the number of star-
like sub-graphs in the network at time tconsisting of k+ 1 nodes. Since the only
diﬀerence between S
kand Bk, as given in Theorem 2.9(a) is a factor of k!, the results
of Hermann and Pfaﬀelhuber (2016) easily apply also for Bkif δ= 0. Theorem 2.14
of Hermann and Pfaﬀelhuber (2016) treats the degrees of initial vertices and thus
can be compared to Theorem 2.9(c).
3. Proof of Theorem 1
Our analysis of the random graph P D(p, δ)is based on some main observations:
First, the expected degree distribution can be represented by a birth-death process
with binomial disasters Z, such that the distribution of Ztequals the expected
degree distribution of Gt; see (3.2). Second, asymptotics for the survival probability
of such processes were studied in Hermann and Pfaﬀelhuber (2020).
3.1. Birth-death processes with disasters and p-jump processes.
Deﬁnition 3.1. Let b > 0,d0and p[0,1]. Let Z(b, d, p)=(Zt)t0be a
continuous-time Markov process on N0that evolves as follows: Given Z0=z, the
process jumps
to z+ 1 at rate bz;
to z1at rate dz;
to a binomially distributed random variable with parameters zand pat
rate 1.
Then we call Z(b, d, p)abirth-death process subject to binomial disasters with birth-
rate b,death-rate dand survival probability p.
Remark 3.2.(1) A birth-death process with binomial disasters, Z(b, d, p)mod-
els the size of a population where each individual duplicates with rate band
dies with rate d, subjected to binomial disasters at rate 1. These disasters
are global events that kill oﬀ each individual independently of each other
with probability 1p, which generates the binomial distribution in the
third part of Deﬁnition 3.1.
(2) Hermann and Pfaﬀelhuber (2020) provides several limit results for such
branching processes with disasters. As reference for the following, let Z=
Z(b, d, p)be as above. Then, Corollary 2.7 of Hermann and Pfaﬀelhuber
(2020) states:
The partial duplication random graph with edge deletion 335
(a) If bdplog 1
p,Zgoes extinct almost surely and
lim
t→∞ 1
tlog P(Zt>0) = (1 p)(bd).
(b) If plog 1
p< b dlog 1
p,Zgoes extinct almost surely and
lim
t→∞ 1
tlog P(Zt>0) = 1 bd
log 1
p1 + log log 1
p
bd.
(c) If bd > log 1
p, then Pk(limt→∞ Zt= 0) + Pk(limt→∞ Zt=) = 1
and
Pk( lim
t→∞ Zt=) = 1d+ log 1
p
bk
X
`=1 k
`(1)`1
`1
Y
m=1 1dm + (1 pm)
bm .
By constructing a relationship between P D(p, δ)and Z(p, δ, p)in
Lemma 3.3, we are able to transfer these results to our duplication graph
processes.
Lemma 3.3. Let p(0,1),δ0and recall Fk(t)from Deﬁnition 2.1. As h0,
the entries Fkof the degree distribution yield
1
hE[Fk(t+h)Fk(t)|Gt]
=(1 + pk +δk)Fk(t) + p(k1)Fk1(t) + δ(k+ 1)Fk+1 (t)
+X
`k`
kpk(1 p)`kF`(t) + o(1).
(3.1)
Moreover, recall Z:= Z(p, δ, p)from Deﬁnition 3.1 (i.e. the binomial distribution
of the disasters has the birth rate as a parameter) and let P(Z0=k) = Fk(0) for
all kbe its initial distribution. Then, for all t0and k, it holds
P(Zt=k) = E[Fk(t)],(3.2)
i.e. the distribution of Ztequals the expected degree distribution of Gt.
Proof : Letting Φk(t) := |Vt|Fk(t), the absolute number of nodes with degree kat
time t, we obtain for h0that
1
hEk(t+h)Φk(t)|Gt]
=(pkκt+δkk(t) + p(k1)κtΦk1(t) + δ(k+ 1)Φk+1(t)
+X
`k
κtΦ`(t)`
kpk(1 p)`k+O(h),
where the ﬁrst term on the right hand side stands for the events at which a node
can lose the degree kby either obtaining a new neighbor (by one of its kneighbors
being copied, which happens with rate t, retaining at least the one relevant edge,
which has probability p) or one of its kedges being deleted, which happens at
rate δk. The second and third terms describe the corresponding gain of a node
with degree kby analogous events. Finally, the sum equals the rate of a new node
arising with degree k, which can only happen if a node of degree `kis copied
(with rate κtΦ`(t)) and the copy retains exactly kedges (which then has a binomial
probability).
336 F. Hermann and P. Pfaﬀelhuber
Now, since |Vt|only increases if a new node is added, i.e. on an event related to
κt, it follows
1
hE[Fk(t+h)Fk(t)|Gt]
=1
hEhΦk(t+h)
|Vt+h|Φk(t)
|Vt+h|Gti+1
hEhΦk(t)
|Vt+h|Φk(t)
|Vt|Gti
=κt
|Vt|+ 1pkΦk(t) + p(k1)Φk1(t) + X
`k
Φ`(t)`
kpk(1 p)`k
+δkΦk(t) + δ(k+ 1)Φk+1 (t)
|Vt|+κt|Vt|Φk(t)1
|Vt|+ 1 1
|Vt|+O(h)
=pkFk(t) + p(k1)Fk1(t) + X
`k
F`(t)`
kpk(1 p)`k
δkFk(t) + δ(k+ 1)Fk+1 (t)Fk(t) + O(h),
and (3.1) holds. Computing the Kolmogorov forwards equation for Zshows that
for k= 0,1,2, . . .
d
dt P(Zt=k) = (1 + pk +δk)P(Zt=k) + p(k1)P(Zt=k1)
+δ(k+ 1)P(Zt=k+ 1) + X
`k
P(Zt=`)`
kpk(1 p)`k,
which is the same relation as (3.1) after taking expectation and letting h0. This
shows (3.2).
3.2. Properties of the piecewise deterministic jump process X.We have seen the
connection of P D(p, δ)to a branching process with disasters in Lemma 3.3. Such
branching processes are in turn closely connected to piecewise deterministic jump
processes as in Lemma 2.3 (Hermann and Pfaﬀelhuber,2020). Hence, we can now
prove Lemma 2.3.
Proof of Lemma 2.3:Lemma 3.3 implies that E[Hx(t)] = E[(1x)Zt]. Recognizing
(Zt) = Z(p, δ, p)as a homogeneous branching process with disasters Zh
λ,q,1,p in the
sense of Hermann and Pfaﬀelhuber,2020, Deﬁnition 2.5, with death-rate λ=p+δ
and oﬀspring distribution q= (q0,0, q2,0, . . .)holding q2=p
p+δ= 1 q0, the result
follows from Lemma 4.1 in Hermann and Pfaﬀelhuber (2020).
For the process X, we now obtain a property which is needed in the proofs of
Theorem 2.4 and Proposition 2.6.
Lemma 3.4 (Moments of X).Let Xbe as in Lemma 2.3. If δ > p plog 1
p, then
Ex[Xk
t] = o(Ex[Xt]) for all k= 2,3, . . . and Ex[Xt]cet(1+δ2p), where
c:= x·exp pZ
0
Ex[X2
s]
Ex[Xs]ds(0,1).
Proof : Recall γ:= log 1
p/(pδ). Indeed, for pplog 1
p< δ pp2log 1
p, such that
γ(p1, p2), it follows from Corollary 2.4 of Hermann and Pfaﬀelhuber (2020)
that independent of x
1
tlog Ex[X2
t]
Ex[Xt]t→∞
11 + log γ
γ(1 + δ2p)=2p·c2(p, δ)>0.
The partial duplication random graph with edge deletion 337
On the other hand, if δpp2log 1
p, the same corollary gives
1
tlog Ex[X2
t]
Ex[Xt]t→∞
1+2δ2pp2(1 + δ2p)
=δp2p2(1
p1log 1
p)>0.
In either case, there is an ε > 0such that 0< r(s) := Ex[X2
s]/Ex[Xs] = O(eεs)
and it follows from (3.3)
Ex[Xt] = xexp t(1 + δ2p)pZt
0
r(s)ds,
conluding the proof.
3.3. Proof of Theorem 2.4.By (3.1) in Lemma 3.3, we get that, as h0
1
hE[F0(t+h)F0(t)|Gt] F0(t) + δF1(t) +
X
`=0
(1 p)`F`(t)
=δF1(t) +
X
`=1
(1 p)`F`(t)0.
Hence, (F0(t))tis a bounded sub-martingale and converges almost surely and in
L1. Consequently, the left hand side has to converge to 0 almost surely. Since
(1 p)`is always positive, that can only be the case if F`(t)0almost surely for
all `= 1,2, . . . , which guarantees almost sure convergence of F(t)to a vector of
the form F()=(F0,0,0, . . .)in all cases.
Let Z:= (Zt)t0:= Z(p, δ, p)be as in Deﬁnition 3.1. We note that E[F+(t)] =
P(Zt>0) by (3.2). For (a), we see from Lemma 2.3 and Lemma 3.4 that
E[F+(t)] = 1 E[H1(t)] = 1
X
k=0
Fk(0)E1[(1 Xt)k]
=
X
k=1
kFk(0)E1[Xt] + o(E1[Xt]) B1(0) ·cet(1+δ2p)
with cas in (2.3). Moreover, (b) follows directly from Corollary 2.7 in Hermann
and Pfaﬀelhuber (2020); see Remark 3.2.2. by setting b=pand d=δ. For (c), we
again use Corollary 2.7 in Hermann and Pfaﬀelhuber (2020), but use in addition
that
E[F0(t)] = P(Zt= 0) =
X
k=0
Pk(Zt= 0) ·P(Z0=k),
and Pk`Fk(0)k
`=B`(0).
3.4. Proof of claims in Subsection 2.3.It remains to show (2.5) and the last equality
in (2.6). Applying the generator of Xwe see that its moments satisfy, for k=
1,2, . . . ,
d
dt log Ex[Xk
t] = 1
Ex[Xk
t]pkEx[Xk
t]Ex[Xk
t]+(pδ)kEx[Xk
t]pkEx[Xk+1
t]
=pk+ (pδ)k1pkEx[Xk+1
t]
Ex[Xk
t](3.3)
338 F. Hermann and P. Pfaﬀelhuber
and thus, integrating, dividing by t, and using Corollary 2.4 of Hermann and
Pfaﬀelhuber (2020),
1
tZt
0
Ex[Xk+1
s]/Ex[Xk
s]ds =1
pk 1
t(log Ex[Xk
t]log(xk)) pk(pδ)k+ 1
t→∞
pδ+pk(1 + log γ)
pk =ck(p, δ),
which shows (2.5). Moreover, the last equality in (2.6) also follows from (3.3).
4. Proof of Theorem 2.9
The proof of Theorem 2.9, which is carried out in Section 4.4, will be based on the
analysis of several martingales, which are derived in Proposition 4.5 in Section 4.3.
In Section 4.2, we will analyze the total size of Gt.
4.1. Two auxiliary functions. We will need two speciﬁc functions in the sequel,
which we now analyze.
Lemma 4.1. Let p(0,1),δ0and
g:([0,)R,
x7→ 1 + δx px px.
Then, gis strictly concave and thus, x7→ g(x)/x strictly decreases. Also, the
following holds:
(1) If δp,gis strictly increasing and,
g(x)x→∞
(,if p < δ,
1,if p=δ.
(2) If plog 1
p< δ < p,
gis (strictly increasing on (0, ξ),
strictly decreasing on (ξ, )
for ξ:= log γ / log 1
pwith γ:= log 1
p/(pδ). The global maximum is g(ξ) =
11
γ(1 + log γ).
(3) If δplog 1
p,gstrictly decreases and its maximum is g(0) = 0.
Proof : All results are straight-forward to compute. First, g0(x) = δp+ log 1
p·px
for all cases. Since the right hand side strictly increases, gis strictly concave. Part
1. follows from the form of g0. For part 3., we have that g0(x)δp+ log 1
p0,
implying the result. Finally, for part 2., we have that g0(x)=0iﬀ px= log 1
p/(p
δ) = γiﬀ x= log γ / log 1
p=ξ= log γ/(γ(pδ)) and the rest follows.
Lemma 4.2. Let Γdenote the Γ-function, rR,gr(n) := Γ(n+r)
Γ(n)and n0
max{2,1r}. Then, there are 0< cr1< Cr<, such that
crnrgr(n)Crnrfor all nn0.(4.1)
Proof : First, we note that gr(n)nras n (see e.g. 6.1.46. of Stein,1970)
and hence, the result follows.
The partial duplication random graph with edge deletion 339
4.2. The size of the graph. For the asymptotics of the functionals of the random
graph in Theorem 2.9 it will be helpful to understand the asymptotics of the process
(|Vt|). Here and below, we will frequently use the following well-known lemma.
Lemma 4.3. Let X= (Xt)t0be a Markov process with complete and separable
state space (E, r), and f:ERcontinuous and bounded and such that
lim
h0
1
hE[f(Xt+h)f(Xt)|Xt=x] = λf(x), x E
for some λR, then (ef(Xt))t0is a martingale.
Proof : See Lemma 4.3.2 of Ethier and Kurtz (1986).
Lemma 4.4 (Graph size).Let gr(n) := Γ(n+r)/Γ(n). For all r > (|V0|+ 1),
the process (etrgr(|Vt|+ 1))t0is a non-negative martingale. Moreover, there is a
random variable Vsuch that the following holds:
et|Vt|t→∞
Valmost surely and in Lrfor all r1,
et/|Vt|t→∞
1/Valmost surely and in Lrfor 1r < |V0|+ 1,
Vis Γ(|V0|+ 1,1)-distributed.
Proof : Let 0< cr<1< Cr<be as in Lemma 4.2. The process V:= (|Vt|)t0
is a Markov process which jumps from vto v+ 1 at rate v+ 1. Setting gr(v) =
Γ(v+r)/Γ(v), we see that the process (gr(|Vt|+ 1))t0is well-deﬁned and non-
negative if |Vt|+1+r > 0for all t, i.e. if r > (|V0|+ 1). Then, as h0,
1
hE[gr(Vt+h)gr(Vt)|Vt=v]=(v+ 1)(gr(v+ 2) gr(v+ 1)) + o(1)
= (v+ 1)gr(v+ 1)v+1+r
v+ 1 1+o(1)
=rgr(v+ 1) + o(1)
and Lemma 4.3 implies that (etrgr(|Vt|+ 1))t0is a (non-negative) martingale
for all r > (|V0|+ 1), and therefore L1-bounded. By the martingale conver-
gence theorem, this martingale converges almost surely. Using (4.1), the martingale
(etg1(|Vt|+1))t= (et(|Vt|+1))tis Lr-bounded for every r1and therefore con-
verges in Lr. Analogously, for r=1, the martingale (etg1(|Vt|+1))t= (et/|Vt|)t
is Lr-bounded for 1r < |V0|+ 1 and converges in Lr.
Noting that (|Vt|+ 1)t0is a Yule-process starting in |V0|+ 1, we have that
|Vt|+ 1 is distributed as the sum of |V0|+ 1 independent, geometrically distributed
random variables with success probabilities et(see e.g. p. 109 of Athreya and Ney,
1972). Hence, as t , we ﬁnd that et|Vt|converges in distribution to the sum
of |V0|+ 1 independent, exponentially distributed random variables with unit rate.
This is a Γ(|V0|+ 1,1) distribution.
4.3. Some martingales. Similarly to the discrete-time pure duplication graph in
Hermann and Pfaﬀelhuber (2016) we obtain martingales for the functionals of
P D(p, δ).
Proposition 4.5 (Martingales).Fix k2.
(1) Considering the function gof Lemma 4.1, the following properties hold:
(a) If g(1) g(k),(etg(1)Bk(t))t0is a martingale that almost surely con-
verges to a limit Bk() L1.
340 F. Hermann and P. Pfaﬀelhuber
(b) If g(1) > g(k), there is a process Rk(t)such that (etg(k)(Bk(t) +
Rk(t)))t0is a positive martingale that almost surely converges to a
limit Bk() L1and etg(k)Rk(t)t→∞
0. In particular, etg(k)Bk(t)
Bk()almost surely as t .
Combining (a) and (b), we ﬁnd et(g(1)g(k))Bk(t)t→∞
Bk() L1.
(2) (et(kpk11δ(k
2))Ck(t)/|Vt|)t0is a martingale that converges almost
surely to a limit ˜
Ck(). If additionally Ck(0) >0and δ < 2pk1/(k1),
the convergence also holds in L2.
(3) Let i |V0|. Then, (et(pδ1) Di(t)/|Vt|)t0is a martingale that converges
almost surely to a limit ˜
Di() L1. Moreover,
E[et(pδ)Di(t)] = Di(0)1 + (1 et)p
|V0|(4.2)
and for r2, there is C > 0, depending only on r, p, δ such that
E[(et(pδ)Di(t))r]E[(Di(0))r] + CZt
0
es(pδ)E[(es(pδ)Di(s))r1]ds. (4.3)
Proof : 1. Since the sum in Bk(t)is almost surely ﬁnite for every kand t, it follows
for h0using equation (3.1), that
1
hE[Bk(t+h)Bk(t)|Gt]
=X
`k`
k (1 + p` +δ`)F`(t) + p(`1)F`1(t) + δ(`+ 1)F`+1 (t)
+X
m`m
`p`(1 p)m`Fm(t)!+o(1)
=Bk(t) + p(k1)Fk1(t) + X
mk
Fm(t)
m
X
`=k`
km
`p`(1 p)m`
+X
`k
F`(t) (p+δ)``
k+p``+ 1
k+δ``1
k
| {z }
=:a(`,k)
!+o(1).
Considering that n+1
mn
m=n
m1and n
m·n1
m1=n
m, we deduce
a(`, k) = p``
k1δ``1
k1= (pδ)``1
k1+p``1
k2
= (pδ)k`
k+p(k1)`
k1,
which implies that
1
hE[Bk(t+h)Bk(t)|Gt]
=Bk(t) + p(k1)Bk1(t)+(pδ)kBk(t)
+X
mk
Fm(t)
mk
X
`=0 mk
`m
kp`+k(1 p)mk`
| {z }
=(m
k)pk
+o(1)
The partial duplication random graph with edge deletion 341
=Bk(t)(pδ)k(1 pk)+p(k1)Bk1(t) + o(1)
=g(k)Bk(t) + p(k1)Bk1(t) + o(1),
recalling the function g:x7→ 1 + δx px pxfrom Lemma 4.1. In any case we
see that (etg(1)B1(t))t0is a non-negative martingale converging almost surely to
a limit B1() L1. For k= 2,3, . . . let gmin(k) := min1`kg(k)the running
minimum of g. Then, there are two cases to consider:
1. g(1) g(k): It holds by strict concavity of g(see Lemma 4.1) that in this case
g(1) = gmin(k)< g(`)for all 1< ` < k. Thus, letting
λk
m:= g(1)
g(k)
k1
Y
`=m
p`
g(`)g(1), m = 2, . . . , k
and λk
1:= 1 + 1
g(1) k
2, these coeﬃcients are well-deﬁned and positive. Considering
the linear combination Qk(t) := Pk
m=1 λk
mBm(t)we obtain
1
hE[Qk(t+h)Qk(t)|Gt]
=g(1)Bk(t) +
k1
X
m=1
Bm(t)(λk
mg(m) + λk
m+1pm) + o(1)
=g(1)Bk(t) +
k1
X
m=2
Bm(t)λk
mg(m) + pmg(m)g(1)
pm
+B1(t)g(1)λk
1+k
2+o(1)
=g(1)Qk(t) + o(1).
So now, (etg(1)Qk(t))t0is a non-negative martingale for every k. Since Bk(t)can
be represented as a linear combination of (Q`(t))1`k, also (etg(1)Bk(t))t0has to
be a non-negative martingale and thus converges to some Bk() L1.
2. g(1) > g(k): Here it holds by strict concavity of gthat g(`)> g(k) = gmin(k)
for all `= 1, . . . , k 1. Hence
λk
m:=
k1
Y
`=m
p`
g(`)g(k), m = 1, . . . , k,
are well-deﬁned and positive. We compute analogously to the ﬁrst case that, as
h0, with Qk(t) := Pk
m=1 λk
mBm,
1
hE[Qk(t+h)Qk(t)|Gt]
=g(k)Bk(t) +
k1
X
m=1
Bm(t)λk
mg(m) + pmg(m)g(k)
pm +o(1)
=g(k)Qk(t) + o(1).
Thus, (etg(k)Qk(t))t0is a non-negative martingale and has an almost sure limit
Bk() L1. Moreover, for `<k, since g(`)> g(k), we have that etg(k)Q`(t)0,
so, writing Rk(t) = Qk(t)Bk(t), we see that Rk(t) = Pk1
`=1 µk
`Q`(t)for some
µk
1, . . . , µk
k1and etg(k)Rk(t)0and etg(k)Bk(t)Bk()follows.
2. For the cliques ﬁx t0and let Nk(v)for every node vVtdenote the number
342 F. Hermann and P. Pfaﬀelhuber
of k-cliques that node is part of. Then, PvVtNk(v) = kCk(t). Analogously deﬁne
Mk(e)as the number of cliques that the edge eEtis contained in, such that
PeEtMk(e) = k
2Ck(t). Also, let ˜
Ck(t) := Ck(t)/|Vt|. Note that for a new k-
clique to arise, a node vinside of such a clique has to be copied. Then, every of the
Nk(v)cliques vis part of has a chance of pk1that the copy obtains the k1edges
it needs to form a new k-clique. Also, whenever an edge eis deleted, all Mk(e)
k-cliques are destroyed. We deduce
1
hE[˜
Ck(t+h)˜
Ck(t)|Gt]
=|Vt|+ 1
|Vt|X
vVtCk(t) + pk1Nk(v)
|Vt|+ 1 ˜
Ck(t)δX
eEt
Mk(e)
|Vt|+o(1)
=X
vVt˜
Ck(t) + pk1Nk(v)
|Vt||Vt|+ 1
|Vt|˜
Ck(t)δ
|Vt|k
2Ck(t) + o(1)
=˜
Ck(t)kpk11δk
2
| {z }
=:qk
+o(1).(4.4)
This shows that (etqk·˜
Ck(t))t0is a non-negative martingale and hence converges
almost surely to an integrable random variable ˜
Ck().
It remains to show the L2-convergence of the martingale (etqk˜
Ck(t))t0for
qk+ 1 >0, i.e. δ < 2pk1/(k1). This will be done by considering the number of
(unordered) pairs of k-cliques, CCk(t) := Ck(t)
2= (Ck(t)2Ck(t))/2, and verifying
that the process given by g
CC k(t) := et·2qkC Ck(t)
|Vt|(|Vt|−1) is L1-bounded, which implies
L2-boundedness of the martingale (etqk·˜
Ck(t))t0and concludes the proof.
Let us denote by Ck,`(t)the number of (unordered) pairs of k-cliques which have
exactly `shared vertices. Since the overlap of such a pair (i.e. the sub-graph both
cliques have in common) is an `-clique with `
2edges, the number of edges making
up the pair equals 2k
2`
2. Hence, arguing as in the proof of Theorem 2.9 in
Hermann and Pfaﬀelhuber (2016), considering that (i) one new such pair arises if
one of the 2(k`)non-shared vertices is fully copied (probability pk1), and (ii) one
new pair arises if one of the `shared vertices is fully copied (probability p2k`1),
by taking the copied node instead of the original one; in addition, there are two
new pairs of k-cliques, one original and one copied, which share `1vertices, and
(iii) if one of the `shared vertices is chosen, but only one of the two cliques is fully
copied (probability 2pk1(1 pk`)) one new pair of k-cliques arises, which shares
`1vertices. In addition, such a pair will be destroyed if one of its edges is deleted,
hence we deduce for `k2
1
hE[Ck,`(t+h)Ck,`(t)|Gt]
= (|Vt|+ 1) ·2(k`)pk1+`p2k`1
|Vt|Ck,`(t) + 2(`+ 1)pk1
|Vt|Ck,`+1(t)
δ·2k
2`
2·Ck,`(t) + o(1),
which implies for e
Ck,`(t) := et·2qkCk,` (t)
|Vt|(|Vt|−1) , that
1
hE[e
Ck,`(t+h)e
Ck,`(t)|Gt]
The partial duplication random graph with edge deletion 343
=2qke
Ck,`(t) + et·2qk(|Vt|+ 1)
· (2(k`)pk1+`p2k`1)Ck,`(t) + 2(`+ 1)pk1Ck,`+1(t)
|Vt| · (|Vt|+ 1)|Vt|
+Ck,`(t)·1
(|Vt|+ 1)|Vt|1
|Vt|(|Vt| 1)!
2δk
2e
Ck,`(t) + δ`
2e
Ck,`(t) + o(1)
=e
Ck,`(t) 2qk2δk
2+δ`
2+ (2(k`)pk1+`p2k`1)·|Vt| 1
|Vt|2!
+e
Ck,`+1(t)·2(`+ 1)pk1·|Vt| 1
|Vt|+o(1)
e
Ck,`(t)2`pk1+`p2k`1+δ`
2+e
Ck,`+1(t)·2(`+ 1)pk1+o(1)
=`e
Ck,`(t)pk1(2 pk`)δ
2(`1)+ 2(`+ 1)pk1e
Ck,`+1(t) + o(1).
Analogously, for `=k1, additional pairs arise if a clique with kvertices is
completely copied (probability pk1), so
1
hE[e
Ck,k1(t+h)e
Ck,k1(t)|Gt]
(k1) e
Ck,k1(t)pk1(2 p)δ
2(k2)
+ 2kpk1et·2qk·Ck(t)
|Vt|(|Vt| 1) +o(1).
Also, letting b
Ck(t) := et·2qkCk(t)/(|Vt|(|Vt| 1)) and combining the calculation
above with the one in (4.4), it follows that
1
hE[b
Ck(t+h)b
Ck(t)|Gt]
=2qkb
Ck(t) + et·2qk kpk1Ck(t)
|Vt|(|Vt| 1) ·|Vt| 1
|Vt|+Ck(t)
|Vt|Ck(t)|Vt|+ 1
|Vt|(|Vt| 1)!
δk
2b
Ck(t) + o(1)
b
Ck(t)2qk+kpk12δk
2+o(1)
=kpk1δk
2b
Ck(t) = kb
Ck(t)pk1(2 p0)δ
2(k1)+o(1).
Now, since
δ < 2pk1
k1= min
2mkn2pk1(2 pkm)
m1o,
the coeﬃcients given by
λ`:=
`
Y
m=1
2pk1
pk1(2 pkm)δ
2(m1)
344 F. Hermann and P. Pfaﬀelhuber
for 1`kare well-deﬁned and positive and we obtain for the linear combination
Rk(t) := e
Ck,0(t) +
k1
X
`=1
λ`e
Ck,`(t) + λkb
Ck(t)
that
lim
h0
1
hE[Rk(t+h)Rk(t)|Gt]
k1
X
`=1 e
Ck,`(t)·`λ`pk1(2 pk`)δ
2(`1)+λ`12pk1
+b
Ck(t)·kλkpk1(2 pkk)δ
2(k1)+λk12pk1= 0.
Thus, (Rk(t)) is a non-negative super-martingale, L1-bounded and, since λmin :=
min({1} {λ`; 1 `k})>0and g
CC k(t)Rk(t)min , the proof of 2. is
complete.
3. For the degree Di(t), we set ˜
Di(t) = Di(t))/|Vt|and compute, as h0,
1
hE[˜
Di(t+h)˜
Di(t)|Gt]
= (|Vt|+ 1) pDi(t)
|Vt|Di(t)+1
|Vt|+ 1 Di(t)
|Vt|+1pDi(t)
|Vt| Di(t)
|Vt|+ 1 Di(t)
|Vt|!
+δDi(t)Di(t)1
|Vt|Di(t)
|Vt|+o(1)
=Di(t)
|Vt| p|Vt| Di(t)
|Vt|1pDi(t)
|Vt|δ!+o(1)
=˜
Di(t)(pδ1) + o(1).
Lemma 4.3 shows that (et(pδ1)Di(t)/|Vt|)t0is a non-negative martingale, and
hence converges almost surely. Furthermore, we write with gr(n) := Γ(n+r)/Γ(n)
1
hE[gr(Di(t+h)gr(Di(t)) |Gt]
= (|Vt|+ 1)pDi(t)
|Vt|(gr(Di(t) + 1) gr(Di(t)))
+δDi(t)(gr(Di(t)1) gr(Di(t))) + o(1)
=gr(Di(t))pr |Vt|+ 1
|Vt|+δDi(t)Di(t)1
Di(t) + r11+o(1)
=gr(Di(t))r(pδ) + gr(Di(t))rp|Vt|+ 1
|Vt|1δDi(t)
Di(t) + r11
+o(1)
=gr(Di(t))r(pδ) + gr(Di(t))rp1
|Vt|+δ(r1) 1
Di(t) + r1(4.5)
+o(1).
The partial duplication random graph with edge deletion 345
Letting h0, this gives (4.2) for r= 1, since
E[et(pδ)Di(t)] = Di(0) + Zt
0
pE[es(pδ)Di(s)/|Vs|]ds
=Di(0) + Zt
0
pesDi(0)/|V0|ds,
where in the last step we used the martingale (et(pδ1)Di(t)/|Vt|)t0. Moreover,
since gr(Di(t))
Di(t)+r1=gr1(Di(t)), (4.5) gives for some C > 0, depending on r, p, δ
d
dtE[etr(pδ)gr(Di(t))] C·E[etr(pδ)gr1(Di(t))],
and (4.3) follows with Lemma 4.2 and integration.
4.4. Proof of Theorem 2.9.(a). Recalling the function gfrom Lemma 4.1, we note
βk= (1 + δ2p)1+(δp)kpk=g(1) g(k).
Hence, Lemma 4.5.1 shows that ekBk(t)is non-negative and converges to some
Bk() L1. So, (a) follows.
(b) We combine Lemma 4.5.2 (recall the random variable ˜
Ck()) with the almost
sure convergence etVt
t→∞
Vfrom Lemma 4.4. In all cases, we have that
exp tkpk1δk
2Ck(t)
= exp tkpk11δk
2Ck(t)/|Vt| · et|Vt|
t→∞
˜
Ck()·V=: Ck(),
(4.6)
where V>0almost surely.
If δ2pk1/(k1), it is kpk1δk
20and the convergence can only hold if
Ck(t)t→∞
0almost surely. Since Ck(t)N0, the ﬁrst hitting time TCkof 0 has to
be ﬁnite. On the other hand, for δ < 2pk1/(k1), combining the L2-convergences
in Lemma 4.5.2 and Lemma 4.4 we obtain that the convergence in (4.6) also holds
in L1. Since (et(kpk11δ(k
2))Ck(t)/|Vt|)is an L2-convergent and thus uniformly
integrable martingale, P(Ck()>0) = P(˜
Ck()>0) >0.
(c) Finally, ﬁx i {1,...,|V0|}. Again, we combine Lemma 4.5.3 (recall the random
variable ˜
Di()) with the almost sure convergence etVt
t→∞
Vfrom Lemma 4.4.
In all cases, we have that
et(pδ)Di(t) = et(pδ1) Di(t)
|Vt|et|Vt|t→∞
˜
Di()·V=: Di().(4.7)
If δ < p, we ﬁnd by (4.2) that (et(pδ)Di(t))t0is L1-bounded. Then, inductively
using (4.3) shows that (et(pδ)Di(t))t0is Lr-bounded for all r1. In particular,
this implies that the convergence in (4.7) also holds in Lrfor all r1. This gives
convergence of ﬁrst moments, and (2.7) follows by taking t in (4.2).
If δp, the almost sure convergence in (4.7) implies, since Di(t)N0, that
Di() = 0, so there must be a ﬁnite hitting time Tiof 0.