Power-law decay of the degree-sequence probabilities of multiple random graphs with application to graph isomorphism

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by  Jefferson Elbert Simões

POWER-LAW DECAY OF THE DEGREE-SEQUENCE
PROBABILITIES OF TWO RANDOM GRAPHS WITH
APPLICATION TO GRAPH ISOMORPHISM
JEFFERSON ELBERT SIMÕES
6
1
0
2
DANIEL R. FIGUEIREDO

y
a
M
VALMIR C. BARBOSA

5
Systems Engineering and Computer Science Program, COPPE

] Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
R
P
. Abstract. We consider events over the probability space generated by the
h
degreesequencesoftwoindependentErdős-Rényirandomgraphs,andconsider
t
a an approximation probability space where such degree sequences are deemed
m tobesequences ofi.i.d.random variables. Weshowthat, foranysequence of
events with probabilities asymptotically smaller than some power law in the
[
approximationmodel,thesameupperboundalsoholdsintheoriginalmodel.

3 We accomplish this by extending an approximation framework proposed in
v a seminal paper by McKay and Wormald. Finally, as an example, we apply
8 the developed framework to bound the probability of havingan isomorphism
7 betweentwoindependent random graphs.
4
2
0
1. 1. Introduction
0
The Erdős-Rényi random graph model, also known as the G(n,p) model [7][9]
6
1 is the most traditional probabilistic model for graphs. Introduced in 1959, in this
: model, a graph over n vertices is randomly generated by adding edges indepen-
v
dently between each vertex pair with probability p(n). Despite its inability to
i
X modelreal-worldnetworks,its simplicityandtheconsequentanalyticaltractability
r have allowedthorough theoretical analysis [5] and applications such as percolation
a
models [15] and graph theory via the probabilistic method [1].
One of the toughest challenges in understanding the overall structure of the
G(n,p)randomgraphisobtainingaprecisecharacterizationofitsdegreesequence.
The main reasonfor this is that, eventhoughthe degrees ofany twospecific nodes
are only mildly correlated (due to the possible edge between them), it is still a
nontrivial task to compose these correlations into a manageable joint distribution
for the degrees.
Mostresultsonthismatteraddressthedistributionofthet-thlargestdegree,for
some t(n) generally bounded. More recently, though, a framework has been set by
McKay and Wormald [13] for approximating the degree sequence by a sequence of
independent random variables, with tight bounds on the error of the probabilities
E-mail addresses: {elbert,daniel}@land.ufrj.br, [email protected].
ThisworkhasbeenpartiallyfundedbyCAPES,CNPq,andFAPERJBBPgrants.
1
2 J.E.SIMÕES,D.R.FIGUEIREDO,ANDV.C.BARBOSA
of events estimated by this approximation. This framework has been successfully
applied in several contexts: for instance, Kostochka and West [11] use it to ana-
lyze the middle degree asymptotics of random graphs, which relates to Chvátal’s
condition for Hamiltonian graphs, and Skerman [16] applies a similar technique to
analyze degrees in a random bipartite graph model.
In this paper, we consider the problems of comparing the degree sequences of
two random graphs, and of approximating these degree sequences by correspond-
ing sequences of independent random variables. Our main result (Theorem 3.2)
directly relates power-law decaying probabilities in the two models: any event se-
quencethathasprobabilityo(n−a)intheapproximationmodelalsohasprobability
o(n−a)intheoriginaldegreesequencemodel. Toachievethis,weextendtheframe-
workin[13]toestablisharelationshipbetweenthedegreesequencesofbothgraphs
and the corresponding independent sequences through a series of intermediate ap-
proximations. Thestepwiseerrorbounds,formallyestablishedbyTheorem 3.1,lay
down a roadmap for handling asymptotic probabilities of properties that compare
the structures of a pair of graphs.
As an example, we apply Theorem 3.2 to the problem of graph isomorphism.
Not only is this problem an interesting theoretical problem in its own right, but it
alsohasimplicationsinpracticalproblemssuchasnetworkprivacyandanonymiza-
tion [14] and computer vision [8]. In particular, we show that, for a certain range
of model parameters, two random graphs will not be isomorphic with probability
1 o(n−1/2).
−
This paper is structured as follows: in section 2 we review the degree sequence
approximation framework, detailing its steps and stating the main results used.
We then proceed to extending the framework to two independent random graphs,
providingcorrespondingstatementsandproofsinsection3. Oursampleapplication
will be presented in section 4, where we apply the framework to the problem of
isomorphism, after which we conclude with some final remarks in section 5.
2. Related work
McKay and Wormald [13] have previously formalized, under quite loose con-
straints, the very intuitive result that the degree sequence of a G(n,p) random
graph is similar to a sequence of independent random variables, each having dis-
tribution Bin(n 1,p). This result takes the form of a number of theorems and
−
lemmas, each performing one of four steps in the approximation process that is
detailed in this section. Notation will be kept as similar as possible to the original
work [13].
For some fixed n N, take the set I = 0,...,n 1 n equipped with the
n
∈ { − }
discrete σ-algebra as our measurable space. Let d = (d ,...,d ) be some element
1 n
in this space. Also, let p=p(n) (0,1), and denote N = n and q =1 p.
∈ 2 −
In the binomial model , d is distributed as a sequence of n independent
n,p
B (cid:0) (cid:1)
Bin(n 1,p) random variables. This can be achieved by evaluating d under the
probab−ility measure P = Bin(n 1,p)⊗n. We would like to assert that this
Bn,p −
model is similar to the degree sequence of a G(n,p) random graph. We call this
the degree sequence model ( ), and denote by P the probability measure
Dn,p Dn,p
under which d has this distribution. Note that the sum of degrees in any graph
is necessarily even, which means d will take, with probability 1, values on the set
E = d I : M(d) is even (where M = M(d) = d is the sum of the
n n 1
{ ∈ } k k
components of d).
Theapproximationprocessrequiresthreeadditionalmodels(withcorresponding
probability measures) that will perform a transition from the binomial model to
the degreesequence model,with twoofthemmaking dacquirepropertiesfromthe
DEGREE-SEQUENCE PROBABILITIES OF TWO RANDOM GRAPHS 3
degree sequence model that are not present in the binomial model, and the third
one acting as a technical middleman. The first model is the even-sum binomial
model ( ). It ensures that d indeed takes values in E with probability 1. To
n,p n
E
ensure minimum distortion between probability of elements of E , this model is
n
simply set to be the restriction of the binomial model to the set E .1 Then, the
n
weighted even-sum binomial model ( ′ )ensuresthestrongerpropertythatM has
En,p
the same distribution as it does under the degree sequence model (namely, that
M/2isdistributedasBin(N,p)). Toinsertaslittleinterferenceaspossibleintothe
relative probabilities of any two points in E , the probabilities of all points E are
n n
rescaled (or reweighted) uniformly on each set S = d E : M(d) = m , to
m n
{ ∈ }
make these sets have the desired probability.
Toperformthebridgebetween ′ and ,theyhaveintroducedtheintegrated
En,p En,p
model , which is essentially a “noisy” version of the even-sum model . The
n,p n,p
I E
model isobtainedfrom byswitchingfromafixedparameterptoarandom
n,p n,p
I E
parameter p′ that quickly concentrates around p. More specifically, p′ must be
distributedasatruncatednormalvariable,withexpectedvalue p,variancepq/2N,
and restricted to the unit interval.
We can informally summarize the approximation scheme as follows:
P P P P P
Bn,p ≈ En,p ≈ In,p ≈ En′,p ≈ Dn,p
Now, for these approximations to work, it is necessary for p(n) to lie in a “good
behavior range”, in which case p=p(n) is said to be acceptable. The last approxi-
mation, in particular, is hard to tighten in general, so the necessary conditions for
this approximationto work are brought into the definition of acceptable function:
Definition 2.1. Let λ=λ(d)=M(d)/2N and γ =γ (d)=(n 1)−2 n (d
2 2 − i=1 i−
M(d))2. A function p=p(n) is acceptable if the following conditions hold:
P
(1) pqN =ω(n)logn;
(2) there is a set R (n) E and a real function δ(n)=o(1) such that:
p n
⊂
(a) P (R (n)),P (R (n))=1 n−ω(n);
Dn,p p En,p p −
(b) for every d R (n), there is some δ such that δ δ(n) and
p d d
∈ | |≤
PDn,p(d) =exp 1 1 γ22 exp δ .
PEn′,p(d) (cid:26)4(cid:18) − λ2(1−λ)2(cid:19)(cid:27)· { d}
ThesecondconditioninthisdefinitionrequiresasetR (n)toexistinoursample
p
space E , with very large probability in and (the probability of its com-
n n,p n,p
D E
plement in both models vanishes faster than any standard exponential), in which
the models and ′ uniformly agree to a ratio that approaches 1. This con-
Dn,p En,p
dition is required for the proofs to be carried out, though it has been conjectured
by McKay and Wormald that condition 1 in the definition is sufficient for p(n) to
be acceptable — to the best of our knowledge, this conjecture is still open. For
our purposes, they have identified an interesting regime for p(n) in which these
conditions hold [13]:
Theorem 2.2. p(n) is acceptable whenever ω(n)logn/n2 pq o(n−1/2).
≤ ≤
The execution of this approximationscheme has been broken down into a num-
ber of pieces with various levels of complexity, so to fit different possibilities of
applications. In our particular case, we would like to ensure that this scheme is
well-suitedforapproximatingprobabilitiesthatvanishfasterthanpowerlawsinn.
For this purpose, we extract the following results from [13], condensed in a single
theorem.
1That is, the corresponding probability measure is the measure for the binomial model con-
ditionaltotheevent En,evaluatedonlyontheevents En.
4 J.E.SIMÕES,D.R.FIGUEIREDO,ANDV.C.BARBOSA
Theorem 2.3. Write φ(x;µ,σ2) the density function of the normal distribution,
and V = 1φ(x;p,pq/2N)dx. Then the following statements hold:
n,p 0
(1) For any event A E ,
R n ⊆ n
2P (A )
P (A )= Bn,p n ;
En,p n 1+(q p)2N
−
(2) For any event A E ,
n n
⊆
1 1
P (A )= φ(x;p,pq/2N)P (A )dx;
In,p n V En,x n
n,p Z0
(3) If pqN and y =y(n)=o(√6pqN), then
→∞
1+ y 3
PIn,p(d)=PEn′,p(d) 1+O √pq|N|
(cid:18) (cid:18) (cid:19)(cid:19)
uniformly over d E : M(d) 2Np 2y√Npq ;
n
{ ∈ | − |≤ }
(4) If ω(n)logn/n2 pq o(n−1/2), then there are sets R (n),R′(n) E
≤ ≤ p p ⊆ n
and a real function δ(n)=o(1) such that:
(a) P (R (n)),P (R′(n))=1 n−ω(n);
Dn,p p Dn,p p −
(b) in R′(n), γ =λ(1 λ)(1+o(1));
p 2 −
(c) for every d R (n), there is some δ such that δ δ(n) and
p d d
∈ | |≤
PDn,p(d) =exp 1 1 γ22 exp δ .
PEn′,p(d) (cid:26)4(cid:18) − λ2(1−λ)2(cid:19)(cid:27)· { d}
Proof. All referenced results used in this proof have been extracted from [13], to
which we refer the reader for notation and statements. Statement 1 is a particular
caseofcorollary4.3takingf =I theindicatorfunctionoftheeventA ,simplified
An n
bytheorem4.2andtheobservationthat,sincef =0inI E ,f =f˜. Statement2
n n
is a rewriting of lemma 2.4, consequence of the construc\tion of P from P
In,p En,p
and an application of the law of total probability — we note that, for x [0,1],
∈
φ(x;p,pq/2N)/V is the density function of the randomparameter p′ used in the
n,p
construction. Statement 3 simply restates theorem 3.6. Statement 4 comes from
the definition of acceptability and corollary3.5, noting that the hypothesis implies
p(n) is acceptable. (cid:3)
Theorem 2.3 suffices for us to ensure suitability of the scheme to our purposes:
Theorem 2.4. Let A be a sequence of events in E . If p satisfies ω(logn/n)
n n
p o(n−1/2), then for any fixed a > 0, P (A ) = o(n−a) implies P (A ) =≤
≤ Bn,p n Dn,p n
o(n−a).
Eachstepintheproofofthistheoremisasimplifiedversionofthecorresponding
stepinthe proofofTheorem 3.2,whichconsiderstwoindependentrandomgraphs,
to be presented in the next section. For brevity, we will explicitly provide proof
only for the latter theorem.
It is worth noting that, in the abstract of their paper, McKay and Wormald
state that their techniques can be used to determine highly accurate asymptotics
for probabilities that, in , are greater than any (fixed) power law. While this
n,p
D
statement, andparticularlythe meaning of“highly accurate”,has been posedin an
informal fashion, it can be understood, in light of their approximation scheme, to
mean the following formal statement:
If p(n) is acceptable, then for any event A E , if P (A ) = ω(n−k)
n ⊆ n Dn,p n
for some fixed k, then there are functions α(n) = θ(1) and ε(n) = n−ω(n)
such that P (A )=P (A ) α(n)+ε(n).
Dn,p n Bn,p n ·
DEGREE-SEQUENCE PROBABILITIES OF TWO RANDOM GRAPHS 5
Ourresultcould,inprinciple,bederivedasacorollaryofthisfactsinceitasserts
that, if p(n) is acceptable, P (A ) = Ω(n−k) implies P (A ) = Ω(n−k),
Dn,p n Bn,p n
which is the contrapositive of our result. However,since this fact has neither been
formallystatednorproven,wechoosetoproveourresultdirectlybyusingthesame
fundamental techniques (i.e., individual pieces of the approximation scheme) that
would be required to prove the aforementioned fact.
3. Results
Inseveraldomains,wecanidentifyproblemsthatcanbereducedtounderstand-
ingwhetherthestructureoftwogivengraphsaresimilar. Inthiswork,weconsider
thesituationwherethesetwographsaregiveninstancesofthe G(n,p)model,with
the same size but possibly with different values of p — that is, two randomgraphs
G(n,p) and G(n,p′). We also assume that these instances are independent.
Naturally, we would like to compare the degree sequences of these two graphs,
as such comparisoncan be used as a proxy for more complicated properties. Intu-
itively, it would seem trivial that, since the two degree sequences are independent
of each other and can both be individually approximated by i.i.d. sequences with
small errors on the corresponding probabilities of events, the joint approximation
of both degree sequences should similarly yield a small error as well. However, we
find it essential that this extension of the single-graph case be obtained formally.
As we seeinwhatfollows,eventhoughsuchextensionis indeedpossible,achieving
it is far from trivial.
Before we proceed, let us introduce some notation. For p,p′ (0,1), denote by
P theprobabilitymeasureP P overI2 —andsim∈ilarlyformeasures
Bn,p,p′ Bn,p⊗ Bn,p′ n
in other models, over E2. Our goal is to perform the following approximation
n
scheme:
P P P P P
Bn,p,p′ ≈ En,p,p′ ≈ In,p,p′ ≈ En′,p,p′ ≈ Dn,p,p′
Let us stress that P = P P is the distribution of the degree
Dn,p,p′ Dn,p ⊗ Dn,p′
sequence of two random graphs G(n,p) and G(n,p′) independent of each other,
and P is the corresponding approximation by two independent sequences of
Bn,p,p′
i.i.d. random variables.
We will extend our notation further and write q′ =1 p′, and denote by d d′
− ×
someelementofI2. Wewillalsowriteλ=λ(d d′)=M(d)/2N,λ′ =λ′(d d′)=
M(d′)/2N, γ = γn (d d′) = (n 1)−2 n (×d M(d))2 and γ′ = γ′(d ×d′) =
(n 1)−2 n2 (d′2 M×(d′))2. − i=1 i− 2 2 ×
− i=1 i− P
In the following, we will prove that an extended version of Theorem 2.3 holds
P
for these models:
Theorem 3.1. Let φ(x;µ,σ2) and V as in Theorem 2.3. Then the following
n,p
statements hold:
(1) For any event A E2,
n ⊆ n
4P (A )
P (A )= Bn,p,p′ n ;
En,p,p′ n [1+(q p)2N][1+(q′ p′)2N]
− −
(2) For any event A E2,
n ⊆ n
1 1 1 pq p′q′
P (A )= φ(x;p, )φ(x′;p′, )P (A )dxdx′;
In,p,p′ n Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′ n
(3) If pqN,p′q′N and y =y(n) is both o(√6pqN) and o(√6p′q′N), then
→∞
1+ y 3 1+ y 3
PIn,p,p′(d×d′)=PEn′,p,p′(d×d′) 1+O √pq|N| +O √p′q|′N|
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
6 J.E.SIMÕES,D.R.FIGUEIREDO,ANDV.C.BARBOSA
uniformly over d d′ E2 : M(d) 2Np 2y√Npq, M(d′) 2Np′
{ × ∈ n | − |≤ | − |≤
2y√Np′q′ ;
}
(4) If ω(n)logn/n2 ≤ pq,p′q′ ≤ o(n−1/2), then there are sets Sp,p′(n) ⊆ En2
and S′ (n) E2 and a real function ε(n)=o(1) such that:
p,p′ ⊆ n
(a) PDn,p,p′(Sp,p′(n)),PDn,p,p′(Sp′,p′(n))=1−n−ω(n);
(b) in S′ (n), γ =λ(1 λ)(1+o(1)) and γ′ =λ′(1 λ′)(1+o(1));
p,p′ 2 − 2 −
(c) for every d d′ Sp,p′(n), thereis some εd×d′ such that εd×d′ ε(n)
× ∈ | |≤
and
PPDEn′n,,pp,,pp′′((dd××dd′′)) =exp(cid:26)14(cid:18)2− λ2(1γ−22 λ)2 − (λ′)2((γ12′−)2λ′)2(cid:19)(cid:27)·exp{εd×d′}.
Proof. Statement 1: Let be the family of subsets of E2 for which the
F n
statement’s equality holds. We will provethat (i) containsallrectangles
F
(i.e.,eventsofthe formB B′,withB, B′ E )and(ii) is aλ-system.
n
× ⊂ F
This is enough since, by Dynkin’s theorem, must contain the σ-algebra
F
generated by the rectangles, which is the discrete σ-algebra over E2.
n
Forthefirstclaim,foranyrectangleA=B B′inE2,byTheorem 2.3(1),
× n
we have that
P (A)=P P (B B′)
En,p,p′ En,p ⊗ En,p′ ×
=P (B)P (B′)
En,p En,p′
= 2PBn,p(B) 2PBn,p′(B′)
1+(q p)2N · 1+(q′ p′)2N
− −
4P (B)P (B′)
= Bn,p Bn,p′
[1+(q p)2N][1+(q′ p′)2N]
− −
4P (A)
= Bn,p,p′ .
[1+(q p)2N][1+(q′ p′)2N]
− −
Therefore, contains all rectangles.
F
For the secondclaim,note that containsE2, since itis a rectangle;
F n F
is closed by complements, since for any A , it holds that
∈F
P (E2 A)
En,p,p′ n\
=P (E2) P (A)
En,p,p′ n − En,p,p′
4P (E2) 4P (A)
= Bn,p,p′ n Bn,p,p′
[1+(q p)2N][1+(q′ p′)2N] − [1+(q p)2N][1+(q′ p′)2N]
− − − −
4P (E2 A)
= Bn,p,p′ n\
[1+(q p)2N][1+(q′ p′)2N]
− −
DEGREE-SEQUENCE PROBABILITIES OF TWO RANDOM GRAPHS 7
and E2 A ; and is also closed by disjoint enumerable unions, since
n\ ∈F F
for any sequence B ,B ,... in , if B ,B ,... are disjoint, then
1 2 1 2
F
∞ ∞
P B = P (B )
En,p,p′ i! En,p,p′ i
i=1 i=1
] X
∞ 4P (B )
= Bn,p,p′ i
[1+(q p)2N][1+(q′ p′)2N]
i=1 − −
X
4 ∞ P (B )
= i=1 Bn,p,p′ i
[1+(q p)2N][1+(q′ p′)2N]
−P −
4P ( ∞ B )
= Bn,p,p′ ⊎i=1 i
[1+(q p)2N][1+(q′ p′)2N]
− −
and ∞ B . Since fits the three requirements, by definition, is a
⊎i=1 i ∈F F F
λ-system.
Statement 2: We willusethe samestrategyasinstatement1. Let be the
G
family of subsets of E2 for which the statement’s equality is true. First,
n
takeanarbitraryrectangleA=B B′ inE2. UsingTheorem 2.3(2)yields
× n
P (A)=P P (B B′)
In,p,p′ In,p ⊗ In,p′ ×
=P (B)P (B′)
In,p In,p′
1 1
= φ(x;p,pq/2N)P (B)dx
V En,x
n,p Z0
1 1
φ(x;p′,p′q′/2N)P (B′)dx′
· Vn,p′ Z0 En,x′
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P (B)P (B′)dxdx′
Vn,pVn,p′ Z0 Z0 2N 2N En,p En,p′
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P (B)dxdx′,
Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′
which means contains all rectangles.
G
Secondly, satisfiesthethreerequirementsofthedefinitionofλ-systems:
G
it containsE2, since it is a rectangle;it is closedunder complements, since
n
8 J.E.SIMÕES,D.R.FIGUEIREDO,ANDV.C.BARBOSA
for any A ,
∈G
P (E2 A)=P (E2) P (A)
In,p,p′ n\ In,p,p′ n − In,p,p′
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P (E2)dxdx′
Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′ n
1 1 1 pq p′q′
φ(x;p, )φ(x′;p′, )P (A)dxdx′
− Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P (E2)
Vn,pVn,p′ Z0 Z0 (cid:20) 2N 2N En,x,x′ n
pq p′q′
φ(x;p, )φ(x′;p′, )P (A) dxdx′
− 2N 2N En,x,x′
(cid:21)
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )
Vn,pVn,p′ Z0 Z0 2N 2N
[P (E2) P (A)]dxdx′
· En,x,x′ n − En,x,x′
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P (E2 A)dxdx′
Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′ n\
and E2 A ; and is also closed by disjoint enumerable unions, since
n\ ∈G F
for any sequence B ,B ,... in , if B ,B ,... are disjoint, then
1 2 1 2
G
∞
P B
In,p,p′ i!
i=1
]
∞
= P (B )
In,p,p′ i
i=1
X
∞ 1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P (B )dxdx′
i=1 Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′ i
X
1 1 1 pq p′q′ ∞
= φ(x;p, )φ(x′;p′, ) P (B ) dxdx′
Vn,pVn,p′ Z0 Z0 2N 2N "i=1 En,x,x′ i #
X
1 1 1 pq p′q′
= φ(x;p, )φ(x′;p′, )P ( ∞ B )dxdx′
Vn,pVn,p′ Z0 Z0 2N 2N En,x,x′ ⊎i=1 i
and ∞ B . Since is a λ-system and contains all rectangles, by
⊎i=1 i ∈ G G
Dynkin’s theorem, it must also contain the σ-algebra generated by the
rectangles, which is the discrete σ-algebra over E .
n
Statement 3: Take d d′ E2 satisfying both M(d) 2Np 2y√Npq
× ∈ n | − | ≤
and M(d′) 2Np′ 2y√Np′q′. Then,usingTheorem 2.3(3)wecanwrite
| − |≤
P (d d′)
In,p,p′ ×
=P (d) P (d′)
In,p · In,p
1+ y 3 1+ y 3
=PEn′,p(d) 1+O √pq|N| PEn′,p′(d′) 1+O √p′q|′N| .
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
NotethattheinequalityfromTheorem 2.3(3)wasappliedtwice,for
n,p
I
andfor n,p′. Since bothinequalitiesareuniformintheirrespectivesets—
I
d E : M(d) 2Np 2y√Npq and d′ E : M(d′) 2Np′
n n
{ ∈ | − |≤ } { ∈ | − | ≤
2y√Np′q′ —, the resulting inequality is uniform in the set d d′
} { × ∈
DEGREE-SEQUENCE PROBABILITIES OF TWO RANDOM GRAPHS 9
E2 : M(d) 2Np 2y√Npq, M(d′) 2Np′ 2y√Np′q′ . Algebraic
n | − | ≤ | − | ≤ }
manipulations yield
1+ y 3 1+ y′ 3
PIn,p,p′(d×d′)=PEn′,p(d)PEn′,p′(d′) 1+O √pq|N| +O √p′|q′N|
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)
1+ y 3 1+ y′ 3
+O | | O | |
√pqN √p′q′N
(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
1+ y 3 1+ y 3
=PEn′,p,p′(d×d′) 1+O √pq|N| +O √p′q|′N| ,
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
since y = o(√6pqN) (which implies (1+ y 3)/√pqN = o(1)). This proves
| |
the result.
Statement 4: This proof will follow by construction. Under the stated as-
sumptions for p, there exist sets R (n),R′(n) E and a real function
p p ⊆ n
δ(n)satisfyingtheconditionsofTheorem 2.3(4),withδ foreachd R (n)
d p
∈
in condition (b). Applying the same reasoning for p′, there are also sets
Rp′(n),Rp′′(n) ⊆ En and real function δ′(n) satisfying these same condi-
tions, this time with δd′′ for eachd′ ∈Rp′(n) in condition (b) (δ and δ′ will
not necessarily be equal for d=d′). Note that δ(n),δ′(n) must be positive
real functions.
Now, take:
Sp,p′(n)=Rp(n) Rp′(n),
×
Sp′,p′(n)=Rp′(n)×Rp′′(n),
ε(n)=δ(n)+δ′(n).
We will show the desired results hold for Sp,p′, Sp′,p′ and ε, using prop-
erties of Rp,Rp′,Rp′,Rp′′,δ, and δ′ thoroughly in the next steps:
4a. Note that
PDn,p,p′(Sp,p′(n))=PDn,p(Rp(n))·PDn,p′(Rp′(n))
=(1 n−ω(n))(1 n−ω(n))
− −
=1 n−ω(n),
−
and similarly for P (S′ (n)).
Dn,p,p′ p,p′
4b. Note that, for any d d′ S′ , it holds that d R′(n) and d′
× ∈ p,p′ ∈ p ∈
R′ (n). The former implies γ = λ(1 λ)(1 +o(1)) and the latter
p′ 2 −
implies γ′ =λ′(1 λ′)(1+o(1)).
2 −
4c. By construction, for any d d′ Sp,p′(n), it holds that d Rp(n)
× ∈ ∈
and d′ ∈ Rp′(n). Therefore, taking εd×d′ = δd + δd′′, it holds that
|εd×d′|≤|δd|+|δd′′|≤δ(n)+δ′(n)=ε(n), and
P (d d′) P (d) P (d′)
Dn,p,p′ × = Dn,p · Dn,p′
PEn′,p,p′(d×d′) PEn′,p(d)·PEn′,p′(d′)
1 γ2
=exp 1 2 exp δ
4 − λ2(1 λ)2 · { d}
(cid:26) (cid:18) − (cid:19)(cid:27)
1 (γ′)2
·exp 4 1− (λ′)2(12 λ′)2 ·exp{δd′}
(cid:26) (cid:18) − (cid:19)(cid:27)
1 γ2 (γ′)2
=exp 4 2− λ2(1 2 λ)2 − (λ′)2(12 λ′)2 ·exp{δd+δd′}
(cid:26) (cid:18) − − (cid:19)(cid:27)
1 γ2 (γ′)2
=exp 4 2− λ2(1 2 λ)2 − (λ′)2(12 λ′)2 ·exp{εd×d′}.
(cid:26) (cid:18) − − (cid:19)(cid:27)
10 J.E.SIMÕES,D.R.FIGUEIREDO,ANDV.C.BARBOSA
(cid:3)
Using the above stepwise approximation through the models, we can derive a
general-purpose rule for vanishing probabilities of events involving independent
G(n,p) random graphs, similar to the one stated in Theorem 2.4.
Theorem 3.2. Let A be a sequence of events in E2. If p,p′ satisfy ω(logn/n)
p,p′ o(n−1/2), thennP (A )=o(n−a) impliesnP (A )=o(n−a) for an≤y
≤ Bn,p,p′ n Dn,p,p′ n
fixed a>0.
Proof. Before anything, we note that our hypotheses imply that ω(1/n) p,p′
≤ ≤
o(1), which we will use several times along the proof. Let a > 0 be fixed, and
assume P (A )=o(n−a).
Bn,p,p′ n
Inagreementwiththeapproximationschemepreviouslypresented,wewillprove
our assertion in four steps, each addressing one of the following statements:
(1) P (A )=o(n−a) implies P (A )=o(n−a);
Bn,p,p′ n En,p,p′ n
(2) P (A )=o(n−a) implies P (A )=o(n−a);
En,p,p′ n In,p,p′ n
(3) PIn,p,p′(An)=o(n−a) implies PEn′,p,p′(An)=o(n−a);
(4) PEn′,p,p′(An)=o(n−a) implies PDn,p,p′(An)=o(n−a).
Step 1: Assume P (A )=o(n−a). Theorem 3.1(1) states that
Bn,p,p′ n
4P (A )
P (A )= Bn,p,p′ n .
En,p,p′ n [1+(q p)2N][1+(q′ p′)2N]
− −
Since p = ω(1/n), it follows that 2Np and (q p)2N = (1
→ ∞ − −
2Np/2N)2N 0. By an analogous argument, (q′ p′)2N 0. Thus,
P (A ) →4P (A ) and, since P (A )−= o(n−a→), it follows
En,p,p′ n ∼ Bn,p,p′ n Bn,p,p′ n
that P (A )=o(n−a).
En,p,p′ n
Step 2: Assume P (A )=o(n−a). We turn to the expressionthat links
En,p,p′ n
n,p,p′ to n,p,p′, presented in Theorem 3.1(2).
E I
The normalization constant Vn,pVn,p′ is the probability that two inde-
pendent random variables N(p,pq/2N) and N(p′,p′q′/2N) assume values
in [0,1]. Standardizing these random variables and denoting by Q() the
·
Q-function (tail distribution of a standard normal random variable), we
have
p q
V =Q Q
n,p
− pq/2N!− pq/2N!
p2Np p2Nq
=Q Q
−s q !− s p !
1,
→
where the limit comes from the facts that 2Np/q = ω(1) whenever p =
ω(1/n) and 2Nq/p = ω(1) whenever p = o(1). The same limit applies to
Vn,p′ by the same argument. Thus 1/Vn,pVn,p′ =θ(1).
For the integral, we will split the domain of integration into several
rectanglesanddeal with them separately. To simplify our notation, we de-
note our integrand by g(x,x′) = φ(x;p, pq)φ(x′;p′,p′q′)P (A ). Pick
2N 2N En,x,x′ n
some constant c > a, and let δ = δ(n) = c qlogn/Np and δ′ = δ′(n) =
c q′logn/Np′. Note that np = ω(logn) implies δ = c pqlogn/Np2 =
p
o(pqlogn/log2n) = o(p). Similarly, np′ = ω(logn) implies δ′ = o(p′).
p p