Counting triangles, tunable clustering and the small-world property in random key graphs (Extended version)

0.33 MB·

by  Osman Yağan

1
Counting triangles, tunable clustering and the
small-world property in random key graphs
(Extended version)
Osman Yag˘an, Member, IEEEand Armand M. Makowski, Fellow, IEEE
Abstract—RandomkeygraphswereintroducedtostudyvariouspropertiesoftheEschenauer-Gligorkeypredistributionschemefor
wirelesssensornetworks(WSNs).Recentlythisclassofrandomgraphshasreceivedmuchattentionincontextsasdiverseas
7 recommendersystems,socialnetworkmodeling,andclusteringandclassificationanalysis.Thispaperisdevotedtoanalyzingvarious
1 propertiesofrandomkeygraphs.Inparticular,weestablishazero-onelawforthetheexistenceoftrianglesinrandomkeygraphs,and
0 identifythecorrespondingcriticalscaling.Thiszero-onelawexhibitssignificantdifferenceswiththecorrespondingresultin
2 Erdo˝s-Re´nyi(ER)graphs.Wealsocomputetheclusteringcoefficientofrandomkeygraphs,andcompareittothatofERgraphsinthe
manynoderegimewhentheirexpectedaveragedegreesareasymptoticallyequivalent.Fortheparameterrangeofpracticalrelevance
n
inbothwirelesssensornetworkandsocialnetworkapplications,randomkeygraphsareshowntobemuchmoreclusteredthanthe
a
correspondingERgraphs.WealsoexplorethesuitabilityofrandomkeygraphsassmallworldmodelsinthesenseofWattsand
J
Strogatz.
3
1
IndexTerms—Randomkeygraphs;existenceoftriangles;clusteringcoefficient;wirelesssensornetworks;socialnetworks.

] ✦
I
S
. 1 INTRODUCTION
s
c
[ Random key graphs are random graphs that belong to municationrangeofeachother,thensecurecommunication

the class of random intersection graphs [26]; they are also betweentwonodes requires only thattheir keyrings share
1
calleduniformrandomintersectiongraphsbysomeauthors at least one key. The resulting notion of adjacency defines
v
8 [3],[11],[12].Theyhaveappearedrecentlyinapplicationar- the class of random key graphs; see Section 2 for precise
5 easasdiverseasepidemicsinsocialnetworks[2],clustering definitions.
7 analysis [11], [12], collaborative filtering in recommender Much efforts have recently been devoted to developing
3 systems [18], and random key predistribution for wireless zero-one laws for the property of connectivity in random
0
sensornetworks(WSNs)[9].Inthislastcontext,randomkey key graphs. A key motivation can be found in the need to
.
1 graphsnaturallyoccurinthestudyofarandomkeypredis- obtain conditions under which the scheme of Eschenauer
0 tribution scheme introduced byEschenauer and Gligor [9]: and Gligor guarantees secure connectivity with high prob-
7
Beforedeployment,eachsensorinaWSNisindependently ability in large networks [32]. An interesting feature of
1
assignedKdistinctcryptographickeyswhichareselectedat this work lies in the following fact: Although random key
:
v randomfromalargepoolofP keys.TheseKkeysconstitute graphs are not stochastically equivalent to the classical
i the key ring of the sensor node and are inserted into its Erdo˝s-Re´nyi graphs [8], it is possible to formally transfer
X
memory module. Two sensor nodes can then establish a well-known zero-one laws for connectivity in Erdo˝s-Re´nyi
r
a secure edge between them if they are within transmission graphs to random key graphs by asymptotically matching
range of each other and if their key rings have at least one theiredgeprobabilities.Thisapproach,whichwasinitiated
key in common; see [9] for implementation details. If we by Eschenauer and Gligor in their original analysis [9],
assumefullvisibility,namelythatnodes areallwithincom- has now been validated rigorously; see the papers [3], [7],
[24],[28],[33],[37]forrecentdevelopments.Rybarczyk[24]
This work was supported in part by NSF Grants CCF-0729093 and CCF- has shown that this transfer from Erdo˝s-Re´nyi graphs also
1617934. Part of the work was conducted during Fall 2014 while A.M.
worksforanumberofissuesrelatedtothegiantcomponent
MakowskiwasaVisitingProfessorwiththe DepartmentofStatisticsofthe
Hebrew University of Jerusalem with the support of a fellowship from the anditsdiameter.
LadyDavisTrust.Theauthorsalsothankacolleague(whowishestoremain In view of these developments, it is natural to wonder
anonymous) for suggestions that lead to a shorter proof of the one law in
whether this (formal) transfer technique applies to other
Theorem3.4.
graph properties. In particular, in the literature on random
Partsofthematerialwerepresentedinthe47thAnnualAllertonConference
onCommunication,ControlandComputing,Monticello(IL),September2009, graphs there is long standing interest [4], [8], [15], [16],
andintheFirstWorkshoponApplicationsofGraphTheoryinWirelessAd [23], [26] in the containment of certain (small) subgraphs,
hocNetworksandSensorNetworks(GRAPH-HOC2009),Chennai(India),
the simplest one being the triangle. This particular case is
December2009.
O. Yag˘an is with the Department of Electrical and Computer Engineering alsoofsomepracticalrelevance:Thenumberoftrianglesin
andCyLab,CarnegieMellonUniversity,Pittsburgh,PA15213USA(e-mail: a graph is closely related to its clustering coefficient, and
[email protected]). for random key graphs this has implications on network
A.M.MakowskiiswiththeDepartmentofElectricalandComputerEngineer-
resiliencyundertheEGscheme(e.g.,see[7])andalsoonits
ing,andtheInstituteforSystemsResearch,UniversityofMaryland,College
Park,MD20742USA(e-mail:[email protected]). applicability and relevance in different domains including
2
socialnetworks–moreonthatlater. the class of random key graphs in Section 2, with various
With these in mind, in the present paper we study the definitionsfortheclusteringcoefficientpresentedinSection
triangle containment problem in random key graphs. In 2.2.InSection2.3weevaluatethefirstandsecondmoments
particular, we establish a zero-one law for the existence of ofthenumberoftrianglesinrandomkeygraphs.Ourmain
triangles and identify the corresponding critical scaling. By results are presented in Section 3: A zero-one law concern-
the help of this result (and its proof), we conclude that in ing the containment of triangles in random key graphs is
the many node regime, the expected number of triangles discussed in Section 3.2 while its clustering coefficient is
in random key graphs is always at least as large as the computedinSection3.3.Relevantdefinitionsandfactscon-
corresponding quantity in asymptotically matched Erdo˝s- cerning Erdo˝s-Re´nyi graphs are given in Section 4. Section
Re´nyigraphs.Fortheparameterrangeofpracticalrelevance 5 and Section 6 are devoted to comparing random key
in WSNs, we show that this expected number of triangles graphs and Erdo˝s-Re´nyi graphs in terms of their number
can be orders of magnitude larger in random key graphs of trianglesandclusteringcoefficients, respectively.Section
than in Erdo˝s-Re´nyi graphs, confirming the observations 7 and Section 8 discuss the implications of our results on
madeearlierviasimulationsbyDiPietroetal.[7]. utilizingrandomkeygraphinthecontextofWSNandsocial
Theseresultsshowthattransferringresultsfrom Erdo˝s- network applications, respectively. The proofs of the main
Re´nyigraphstorandomkeygraphsbymatchingtheiredge results of the paper are available in Section 10, while some
probabilities is not a valid approach in general, and can technicalresultsareestablishedinSection9.
be quite misleading in the context of WSNs. In particu- A word on the notation and conventions in use: Un-
lar, our results indicate that the asymptotic equivalence of less specified otherwise, all limiting statements, including
random key graphs and Erdo˝s-Re´nyi graphs (in the sense asymptotic equivalences, are understood with n going to
discussed in [26]) is possible only when the size of key infinity. The random variables (rvs) under consideration
rings is comparable to the network size, a case not very are all defined on the same probability triple (Ω,F,P);
realistic in WSNs due to the severe constraints imposed its construction is standard and omitted in the interest
on the memory and computational capabilities of sensors. of brevity. Probabilistic statements are made with respect
This points to the inadequacy of Erdo˝s-Re´nyi graphs to to this probability measure P, and we denote the corre-
capture some key properties of the EG scheme in realistic spondingexpectationoperatorbyE.Wedenotealmostsure
WSN implementations, and reinforces the call for a direct convergence (under P) by a.s. The indicator function of an
investigationofrandomkeygraphs. eventE isdenotedby1[E].ForanydiscretesetS wewrite
The number (and fraction) of triangles in a network is |S|foritscardinality.Wedenotealmostsureconvergenceby
closely related to its clustering coefficient, a metric known a.s.
to have a significant impact on the dynamics of many
interestingprocessesthattakeplaceonthenetwork;e.g.,the
2 MODEL AND DEFINITIONS
diffusion of information and epidemic diseases [10], [20],
[21], [38], [34], the propagation of influence [13], [39], and 2.1 Randomkeygraphs
cascadingfailures[14].Withthisinmind,wealsostudythe Pick positive integers K and P such that K ≤ P, and fix
clustering coefficient of random key graphs and compare n = 3,4,.... We shallgroup the integers P and K intothe
it with that of an Erdo˝s-Re´nyi graph. We observe that the orderedpairθ ≡(K,P)inordertolightenthenotation.
clusteringcoefficientofarandomkeygraphisneversmaller The model of interesthere is parametrizedby the num-
than the clustering coefficient of the corresponding Erdo˝s- ber n of nodes, the size P of the key pool and the size
Re´nyi graph with identical expected average degree. For K of each key ring. For each node i = 1,...,n, let K (θ)
i
the parameter range that is relevant for large scale social denote the random set of K distinct keys assignedtonode
networks (as well as WSNs), we show that random key i. Thus, under the convention that the P keys are labeled
graphs are in fact much more clustered than Erdo˝s-Re´nyi 1,...,P, the random set K (θ) is a subset of {1,...,P}
i
graphs when expected average degrees are asymptotically with |K (θ)| = K. The rvs K (θ),...,K (θ) are assumed
i 1 n
equivalent. Recalling the fact that random key graphs also tobei.i.d.,eachofwhichisuniformlydistributedwith
have a small diameter [24], [31], we then conclude that
−1
randomkeygraphsaresmall-worldsinthesenseintroduced P
P[K (θ)=S]= , i=1,...,n (1)
byWattsandStrogatz[27].Thisreinforcesthepossibilityof i K!
using random key graphs in a wide range of applications
includingsocialnetworkmodeling. for any subset S of {1,...,P} with |S| = K. This corre-
Inlinewithresultscurrentlyavailableforotherclassesof spondstoselectingkeysrandomlyandwithoutreplacement
graphs, e.g., Erdo˝s-Re´nyi graphs [15, Chap. 3] and random fromthekeypool.
geometric graphs [23, Chap. 3], it would be interesting Distinct nodes i,j = 1,...,n are said to be adjacent if
to consider the containment problem for small subgraphs theyshareatleastonekeyintheirkeyrings,namely
otherthantrianglesinthecontextofrandomkeygraphs.To
K (θ)∩K (θ)6=∅, (2)
i j
the best of our knowledge, this issue has not been consid-
ered in the literature. Future work mayalsoconsider other inwhichcaseanundirectededgeisassignedbetweennodes
properties of random key graphs that might be relevant i and j. The adjacency constraints (2) define an undirected
in various applications; e.g., Hamiltonicity, spectral radius, random graph on the vertex set {1,...,n}, hereafter de-
percolation,etc. notedK(n;θ).Werefertothisrandomgraphastherandom
Thepaperisorganizedasfollows:Weformallyintroduce keygraph.
3
Itiseasytocheckthat naturaltoconsiderthe averageofthe localclusteringcoeffi-
cientC (G)overthegraphG,i.e.,
P[K (θ)∩K (θ)=∅]=q(θ) (3) Avg
i j
1
C (G)= C (G) (8)
with Avg |V′| i
0 if P <2K iX∈V′
where V′ = {i ∈ V : d ≥ 2}. This last quantity, while
q(θ)= (P−K) (4) natural,isoftenreplacedbiythe globalclusteringcoefficient
 (KKP) if 2K ≤P. definedasthe“fractionoftransitivetriples”overthewhole
The probability p(θ)of edge occurrence between any two graphG,namely,
nodesisthereforegivenby T (G)
C⋆(G)= i∈V i (9)
1 d (d −1)
p(θ)=1−q(θ). (5) 2 Pi∈V i i
IfP < 2K thereexistsanedge betweenanypairof nodes, pCr⋆o(vGid)e=d0oti∈hVerdwii(sdei. − 1)P> 0. It is convenient to set
andK(n;θ)coincideswiththecompletegraphonthevertex P
In the context of random graphs, related (but simpler)
set{1,...,n}.Whileitisalwaysthecasethat0≤q(θ)<1,
definitions are possible when the edge assignment rvs are
itisplainfrom(4)thatq(θ)>0ifandonlyif2K ≤P.
exchangeable(asisthecasefortherandomgraphsofinterest
Theexpression(4)isaconsequenceofthegeneralfact here). Recall that an undirected random graph G defined
P−|S| over the set of nodes {1,...,n} is characterized by the
P[S∩Ki(θ)=∅]= KP , i=1,...,n (6) {0,1}-valued edge rvs {ξij, i,j = 1,...,n} with the in-
(cid:0) K (cid:1) terpretation that ξij = 1 (resp. ξij = 0) if there is an edge
(resp. no edge) between nodes i and j. As we consider
validforanysubsetS of{1,..(cid:0).,P(cid:1)}with|S|≤P −K.
graphswhichareundirectedwithnoself-loops,weimpose
Weclosebyintroducingtheevents
theconditions
E (θ)=[K (θ)∩K (θ)6=∅], i,j =1,...,n
ij i j ξ =ξ and ξ =0, i,j =1,...,n.
ij ji ii
whoseindicatorfunctions A case of great interest arises when the rvs {ξ , 1 ≤
ij
ξ (θ)=1[K (θ)∩K (θ)6=∅], i,j =1,...,n i < j ≤ n} form a family of exchangeable rvs [1]. In
ij i j
that setting, a popular approach (e.g., see [6]) is to define
aretheedgervsdefiningtherandomkeygraphK(n;θ).For the clustering coefficient of the random graph G as the
each i = 1,...,n, it is a simple matter to check with the conditionalprobability
help of (6) that the events {Eij(θ), j 6= i,j = 1,...,n} C(G)=P[E |E ∩E ] (10)
12 13 23
are mutually independent, or equivalently, that the rvs
{ξij(θ), j 6=i,j =1,...,n}formacollectionofi.i.d.rvs. wherewehaveusedthenotation
E =[ξ =1], i,j =1,...,n.
ij ij
2.2 Clusteringcoefficient
For the random graphs considered here, we show that the
Many networks encountered in practice exhibit high clus- quantity (10) provides a good approximation to the global
tering (or transitivity) in that the neighbors of a node are clustering coefficient defined at (9), when n is large; see
likely to be neighbors to each other [25] – Your friends are Theorem 3.7 and Theorem 4.1. It is for this reason that we
likelytobefriends!Clusteringpropertiesareknowntohave usethesimplerdefinition(10)forstudyingclusteringinthe
a significant impact on the dynamics of many interesting remainderofthispaper.
processesthattakeplaceonanetwork,e.g.,thediffusionof
information and epidemic diseases [10], [20], [21], [38], the
2.3 Countingtriangles
propagation of influence [13], [39], and cascading failures
[14]. With this in mind we shall investigate clustering in Pick positive integers K and P such that K ≤ P, and fix
randomkeygraphsundervariousparameterregimes. n = 3,4,... For distinct i,j,k = 1,...,n, we define the
Aformaldefinitionofclusteringisgivennext.Consider indicatorfunction
Van.uFonrdieraecchteidingrVap,hleGt Twi(iGth)ndoensoeltfe-ltohoepnsuomnbthereovfedrtiesxtinsectt χijk(θ)=1 Naotdreiasnig,ljeainndKk(nf;oθr)m . (11)
triangles in G that contain vertex i. The local clustering (cid:20) (cid:21)
coefficientofnodeiisgivenby The number of distinct triangles in K(n;θ) is then simply
givenby
Ci(G)= 12dTii((dGi−)1) ifdi ≥2 (7) Tn(θ)=Xn1≤i<j<k≤nχijk(θ). (12)
 0 otherwise Of particular interest is the event that there exists at least
onetriangleinK(n;θ),namely[T (θ)>0]=[T (θ)=0]c.
n n
wheredi isthedegreeofnodeiinG. One of our main results is a zero-one law for the exis-
There are, however, several possible definitions for a tence of triangles in random key graphs. These results will
graph-wide notion of clustering [22]: Inspired by (7), it is be established by the method of first and second moments
4
appliedtothe count variables(12), e.g., see [4, p. 2], [15, p. 3.1 Twoasymptoticequivalences
55].Theyarestatedintermsofthequantity
The two asymptotic equivalence results (under such scal-
K3 K2 3 θ =(K,P) ings)presentednextwillproveusefulinanumberofplaces.
τ(θ)= + , (13) They provide easy asymptotic expressions for the edge
P2 P K,P =1,2,...
(cid:18) (cid:19) probabilityandfortheprobabilityofatriangle,respectively,
As we shall see soon in Proposition 3.2, this quantity inlargerandomkeygraphs.Thefirstone,alreadyobtained
givesthe asymptoticprobabilityofatriangleinrandom key in[33],isgivenhereforeasyreference.
graphs,whentheparametersK andP aresuitablyscaled. Lemma3.1.ForanyscalingP,K :N →N ,wehave
0 0
Key to much of the discussion carried out in this pa-
per are the first two moments of the count variables (12). lim q(θ )=1 ifandonlyif lim Kn2 =0, (21)
n
The first moment, computed next, will be conveniently n→∞ n→∞ Pn
expressedwiththehelpofthequantityβ(θ)givenby
andundereitherconditionat(21),theasymptoticequiv-
β(θ)=(1−q(θ))3+q(θ)3−q(θ)r(θ) (14) alence
K2
1−q(θ )∼ n (22)
withr(θ)definedby n P
n
0 if P <3K holds.
r(θ)= (P−2K) (15) The next result shows that under certain conditions the
 (KP) if 3K ≤P. quantity (13) behaves asymptoticallylike (14) (which gives
K the probability that three nodes form a triangle in random
Notethatr(θ)correspondstotheprobability(6)when|S|= keygraphs).
2K. Proposition 3.2.ForanyscalingP,K : N → N satisfying
0 0
Proposition2.1.Fixn=3,4,....ForpositiveintegersKand (21),wehavetheasymptoticequivalence
P suchthatK ≤P,wehave
β(θ )∼τ(θ ). (23)
n n
E[χ (θ)]=β(θ) (16)
123
withβ(θ)definedat(14),sothat
AproofofProposition3.2isgiveninSection9.3.Inwords,
n this result shows that under (21) the probability of three
E[T (θ)]= β(θ). (17)
n 3! verticesformingatriangleinrandomkeygraphsisasymp-
toticallyequivalentto
A proof of Proposition 2.1 is given in Section 9.1. We see K3 K2 3
τ(θ )= n + n .
from (16) that the quantity β(θ) gives the probability that n P2 P
three distinct vertices form a triangle in K(n;θ). For future n (cid:18) n(cid:19)
reference,wenotethat
3.2 Zero-onelawsfortheexistenceoftriangles
r(θ)≤q(θ)2 (18) The zero-law, which is given first, is established in Section
10.1.
bydirectinspection,whence
Theorem3.3.ForanyscalingP,K :N →N ,thezero-law
0 0
β(θ)≥(1−q(θ))3 >0. (19)
lim P[T (θ )>0]=0
n n
The second moment of the count variables (12) is com- n→∞
puted next; it will play a crucial role in the proofs of both holdsunderthecondition
Theorem3.4andTheorem3.7thatareforthcoming.
lim n3τ(θ )=0. (24)
n
Proposition 2.2. ForpositiveintegersK andP suchthat n→∞
K ≤P,wehave
n−3 n−3 The one-lawgivennext assumesa more involvedform;
E T (θ)2 =E[T (θ)]+ 3 +3 2 (E[T (θ)])2
n n n n n itsproofisgiveninSection10.2.
(cid:2) (cid:3) (cid:0) 3 (cid:1) (cid:0) 3 (cid:1)! Theorem 3.4.ForanyscalingP,K : N0 → N0forwhichthe
+3(n−3) n(cid:0)·(cid:1)E[χ123(cid:0)(θ(cid:1))χ124(θ)] (20) limitlimn→∞q(θn)=q⋆exists,theone-law
3!
lim P[T (θ )>0]=1
n n
foralln=3,4,... n→∞
holdsifeither0≤q⋆ <1,orifq⋆ =1andtheadditional
TheproofofProposition2.2isavailableinSection9.2.
condition
lim n3τ(θ )=∞ (25)
n
3 MAIN RESULTS n→∞
holds.
Forsimplicityofexpositionwerefertoanypairoffunctions
P,K : N → N as a scaling (for random key graphs) To facilitate an upcoming comparison with analogous
0 0
provided the natural condition K ≤ P holds for all resultsinERgraphs,wecombineTheorem3.3andTheorem
n n
n=3,4,.... 3.4intoasinglesymmetricstatement.
5
Theorem 3.5. ForanyscalingP,K : N → N forwhich K P 1−q(θ) CK(θ) Cbn⋆(θ)
0 0
lim q(θ )exists,wehave 4 103 0.0159 0.2590 0.2587
n→∞ n 8 5×103 0.0127 0.1348 0.1349
0 if limn→∞n3τ(θn)=0 16 2×104 0.0127 0.0737 0.0736
lim P[T (θ )>0]= 20 4×104 0.0100 0.0590 0.0590
n→∞ n n  1 if limn→∞n3τ(θn)=∞. 2342 110055 00..00015072 00..00446098 00..00446088
40 5×105 0.0032 0.0280 0.0280
By Lemma 3.1 the condition limn→∞n3τ(θn) = 0 64 106 0.0041 0.0196 0.0196
implies lim q(θ ) = 1, hence he limit lim q(θ )
n→∞ n n→∞ n
TABLE1
necessarilyexistswithq⋆ =1.
Clusteringcoefficientswithfixedθforrandomkeygraphs
3.3 Clusteringinrandomkeygraphs
Inaccordancewithdefinition(10),theclusteringcoefficient 4 FACTS CONCERNING ERDO˝S-RE´NYI GRAPHS
oftherandomkeygraphK(n;θ)isdefinedby
A little later in this paper, we shall compare random key
C (θ)=P[E (θ)|E (θ)∩E (θ)]. (26)
K 12 13 23 graphstorelatedErdo˝s-Re´nyi(ER)graphs[8],butfirstsome
Aclosedformexpressionforthisquantityisgivennext. notation: For each n = 2,3,... and each p in [0,1], let
Proposition3.6.ForpositiveintegersK,PsuchthatK ≤P, G(n;p) denote the ER graph on the vertex set {1,...,n}
wehave with edge probability p. The ER graph G(n;p) is char-
β(θ) n(n−1)
acterized by the fact that the possible undirected
CK(θ)= (1−q(θ))2 (27) edgesbetweenthennodesareinde2pendentlyassignedwith
withβ(θ)givenby(14). probabilityp.Thus,ifinanalogywithearliernotation,with
distinct i,j = 1,...,n, we denote by E (p) the event that
ij
there is an (undirected) edge between nodes i and j in
Proof. ThedefinitionsofCK(θ)andχ123(θ)yield G(n;p), then the events {E (p), 1 ≤ i < j ≤ n} are
ij
P[E12(θ)∩E13(θ)∩E23(θ)] E[χ123(θ)] mutually independent, each of probability p. For ease of
C (θ)= = (28)
K P[E13(θ)∩E23(θ)] (1−q(θ))2 expositionitwillalwaysbeunderstoodthatEij(p)=Eji(p)
fordistincti,j =1,...,n.
sincetheeventsE (θ)andE (θ)areindependent,with
13 23
Random key graphs are not stochastically equivalent to
P[E (θ)∩E (θ)]
13 23 ER graphs even when their edge probabilities are matched
=P[K (θ)∩K (θ)6=∅,K (θ)∩K (θ)6=∅] exactly: As graph-valued rvs, the random graphs G(n;p)
1 3 2 3
=(1−q(θ))2 (29) andK(n;θ)havedifferentdistributionsevenundertheexact
matchingcondition
byvirtueof(6)(andcommentsfollowingit).Theconclusion
(27)isimmediateuponsubstituting(16)into(28). p=1−q(θ)=p(θ). (31)
See [29] for a discussion of (dis)similarities. Under (31) the
For random key graphs there is strong consistency be- random graphs G(n;p) and K(n;θ) are said to be exactly
tweenthedefinitions(9)and(26)ofclusteringcoefficient. matched.
Theorem 3.7. ForpositiveintegersK,P suchthatK ≤ P, Inanalogywith(12)letTn(p)denotethenumberofdis-
wehave tincttrianglesinG(n;p).Undertheenforcedindependence,
wenotethat
lim C⋆(K(n;θ))=C (θ) a.s. (30)
K
n→∞ n
E[T (p)]= τ⋆(p), n=3,4,... (32)
n
3!
A proof of Theorem 3.7 is given in Section 10.3. To
the best of our knowledge, Theorem 3.7 is the first rigor- with
ous result in the literature that shows that the conditional τ⋆(p)=p3, 0≤p≤1.
probabilitydefinition(10)ofclusteringcoefficientconverges
The edge assignment rvs being exchangeable in ER
asymptotically almost surely to the empirical clustering
graphs, we can again define the clustering coefficient in
coefficient measure of (9). For instance, Deijfen and Kets
G(n;p)accordingto(10)bysetting
indicated [6], for another class of random graphs, that the
twodefinitionsshouldbecloselyrelated,butthatarigorous C (p)=P[E (p)|E (p)∩E (p)]. (33)
ER 12 13 23
proofwouldneedsignificantadditionalwork.
Simulationresults givenin TableI illustratethe conver- Bymutualindependenceoftheedgervsitfollowsthat
gence(30)forseveralrealisticparametervalues.Thenumer- P[E (p)∩E (p)∩E (p)]
ical values of CK(θ) are obtained directly from the expres- CER(p)= 1P2[E (p)1∩3 E (p)2]3 =p. (34)
sions(26).ThequantityC⋆(θ)standsfortheclusteringcoef- 13 23
n
ficientofK(n;θ), calculatedthrough(9)andaveragedover Here as well, strong consistency holds between the two
1000 realizations; the nubmber of nodes is set to n = 1000 notionsofclustering(9)and(26).
inallsimulations.Thedatasupportthevalidityof(30),and Theorem4.1.Foreverypin(0,1),wehave
confirm the claim that for large networks the quantity (9)
capturesessentiallythesamestructuralinformationas(26). lim C⋆(G(n;p))=CER(p) a.s. (35)
n→∞
6
K P CK(θ) Cbn⋆(θ) CER(p) Cbn⋆(p) by virtue of (18). Consequently, the expected number of
4 103 0.2590 0.2587 0.0159 0.0159 trianglesinarandomkeygraphisalwaysatleastaslargeas
8 5×103 0.1348 0.1349 0.0127 0.0128
16 2×104 0.0737 0.0736 0.0127 0.0128 thecorrespondingquantityinanERgraphexactlymatched
20 4×104 0.0590 0.0590 0.0100 0.0100 toit.ThiswasalreadysuggestedbyDiPietroetal.[7]with
24 105 0.0469 0.0468 0.0057 0.0057 thehelpoflimitedsimulations.
32 105 0.0408 0.0408 0.0102 0.0102 An analogous result is available when the scalings are
40 5×105 0.0280 0.0280 0.0032 0.0031
64 106 0.0196 0.0196 0.0041 0.0041 onlyasymptoticallymatched.
Corollary5.1.ConsiderascalingK,P :N →N satisfying
0 0
TABLE2 (21),andascalingp : N → [0,1].Undertheasymptotic
Clusteringcoefficientswithfixedθandp=1−q(θ)computedvia(4) 0
matchingcondition(36),wehavetheequivalence
E[T (θ )] P
n n n
∼1+ . (39)
E[T (p )] K3
This result can be established by arguments similar to the n n n
onesprovidedintheproofofProposition3.7;seeAppendix
A for details. Table II expands on Table I given earlier in In other words, for large n the expected number of
that we now compare the clustering coefficients of exactly trianglesinrandomkeygraphsisalwaysatleastaslargeas
matched random key graphs and ER graphs for the pa- the corresponding quantity in asymptotically matched ER
rameter values used in Table I. The quantities for random graphs – In fact, if the ratio P /K3 is large, the number
n n
key graphs are as before. The numerical values of CER(p) of triangles in random key graphs can be several orders
areobtaineddirectlyfrom the expressions(33). HereC⋆(p) of magnitude larger than that of ER graphs. In Sections 7
n
stands for the clustering coefficient of G(n;p). It is calcu- and8thisissueisexploredinthecontextofwirelesssensor
lated through (9) and averaged over 1000 realizationsb.The networksandsocialnetworks,respectively.
number of nodes is still set to n = 1000 in all simulations.
Againthedatasupporttheclaimthatforlargenetworksthe Proof. Replacingθbyθ andpbyp accordingtothegiven
n n
definition (9) captures essentially the same information as scalingsintheexpression(38),weget
thequantity(33).
E[T (θ )] β(θ )
Anymappingp : N0 →[0,1]willbecalledascalingfor n n = n , n=3,4,...
E[T (p )] τ⋆(p )
ER graphs. In order to meaningfully compare the asymp- n n n
totic regime of random key graphs with that of ER graphs Under(21),Proposition3.2yields
undertheirrespectivescalings,weshallsaythatthescaling
p : N0 → [0,1] (for ER graphs) is asymptotically matched to E[Tn(θn)] ∼ τ(θn) (40)
thescalingP,K :N0 →N0 (forrandomkeygraphs)if E[Tn(pn)] τ⋆(pn)
with
p ∼p(θ )=1−q(θ ). (36)
n n n
τ(θ ) 1 K3 1 K2 3
Sometimes, when (36) holds, we shall also say that the n = · n + · n , n=3,4,...
random graphs G(n;pn) and K(n;θn) are asymptotically τ⋆(pn) p3n (cid:18)Pn2(cid:19) p3n (cid:18)Pn(cid:19)
matched. Under condition (21), by Lemma 3.1 the asymp- Withthehelpof(37),weconclude
toticmatchingcondition(36)amountsto
τ(θ ) P
n n
K2 ∼1+ (41)
pn ∼ n. (37) τ⋆(pn) Kn3
P
n
andtheequivalence(39)followsfrom(40).
Condition(31)(resp.(36))isequivalenttorequiringthat
the expected degrees in K(n;θ)and G(n;p)(resp. K(n;θ )
n
andG(n;p ))coincide(resp.areasymptoticallyequivalent). From(41)itfollowsthatundertheasymptoticmatching
n
condition (36) (together with (21)), triangles will start ap-
pearing earlier in the evolution of a random key graph as
5 COMPARING THE NUMBER OF TRIANGLES IN
comparedtoanER graph(asymptotically)matchedtoit. It
RANDOM KEY GRAPHS AND ER GRAPHS
should also be clear from (41) that the larger the quantity
Fixpin(0,1],andpositiveintegersK andP suchthatK ≤ P /K3,themorepronouncedwillsuchdifferencebe.
n n
P.From(17)and(32)itisplainthat We close this section by comparing Theorem 3.5 with
E[T (θ)] β(θ) its analog for ER graphs. Fix n = 3,4,... and p in [0,1].
n
= , n=3,4,... (38) Consider the event that there exists at least one triangle in
E[Tn(p)] τ⋆(p) G(n;p), i.e., [T (p) > 0]. The following zero-one law for
n
Undertheexactmatchingcondition(31),withp(θ)given triangle containment in ER graphsis wellknown [4, Chap.
by(5),thislastexpressionyields 4],[15,Thm.3.4,p.56].
E[T (θ)] β(θ) q(θ)2−r(θ) Theorem5.2.Foranyscalingp:N0 →[0,1],wehave
n
= =1+ ·q(θ)
E[Tn(p(θ))] τ⋆(p(θ)) (1−q(θ))3 0 if limn→∞n3τ⋆(pn)=0
lim P[T (p )>0]=
foreachn=3,4,...,whence n→∞ n n  1 if limn→∞n3τ⋆(pn)=∞.
E[T (p(θ))]≤E[T (θ)], n=3,4,...
n n

7
This result, which is also established by the method followsfromtheeasilycheckedfactthat2P3−4P2−P+3<
of first and second moments, is easily understood once 2(2P −3)3 forallP =2,3,....
we recall (32). As we compare Theorem 3.5 with Theorem The cases K = 1 and K = 2 may not be interesting
5.2, we note a direct analogy since the terms τ(θ ) and fromtheperspectiveofenvisionedmodelingapplicationsof
n
τ⋆(p )correspondtothe(asymptotic)probabilitythatthree randomkeygraphs.However,thediscussionalreadyshows
n
arbitrary nodes form a triangle in random key graphs and thattheparametersofthecorrespondingrandomkeygraph
ERgraphs,respectively. can be selected (e.g., by taking P very large in these two
cases)sothatithasamuchlargerclusteringcoefficientthan
the ER graph exactly matched to it. Additional limited nu-
6 COMPARING THE CLUSTERING COEFFICIENTS
mericalevidence alongtheselines isalsoavailableinTable
OF RANDOM KEY GRAPHS AND ER GRAPHS II discussedearlier. Infact, for anygivenK we see thatthe
Fixpin(0,1],andpositiveintegersK andP suchthatK ≤ linearbehaviorfound in(45)and(46)holds asymptotically
P.Combining(27)and(34)weget forlargeP.
C (θ) β(θ) Corollary6.1.ForeachpositiveintegerK,itholdsthat
K
= . (42)
CER(p) p(1−q(θ))2 CK(θ) ∼1+ P (P →∞). (47)
C (p(θ)) K3
Undertheexactmatchingcondition(31)wefind ER
C (θ) β(θ)
K
= Thus, exactly matched random key graphs and ER
C (p(θ)) (1−q(θ)))3
ER graphs will have vastly different clustering coefficients
q(θ)2−r(θ) whenP islarge.ThiswillbeespeciallysoforWSNswhere
= 1+ ·q(θ) (43)
(1−q(θ))3 thesizeofthekeypoolP intheEschenauer-Gligorscheme
is expected to be in the range 217 − 220 (with K much
aswerecall(5).Thus,
smaller)[9].
C (p(θ))≤C (θ) (44)
ER K
Proof. FixpositiveintegersK andP suchthat2K ≤P.We
by virtue of (18) – The clustering coefficient of a random
canrewrite(43)as
keygraphisatleastaslargeasthatoftheERgraphexactly
matchedtoit. C (θ) q(θ) 3 r(θ)
K
=1+ · 1− . (48)
Severalconclusions can be extracted from these expres- C (p(θ)) 1−q(θ) q(θ)2
ER (cid:18) (cid:19) (cid:18) (cid:19)
sions: Equality in (44) holds only when P < 2K, i.e., from
With K fixed and P getting large, we see from Lemma
(4)weget
3.1that1−q(θ)∼ K2 andq(θ)∼1(P →∞),sothat
P
C (θ)
K =1 ifK ≤P <2K q(θ) 3 P 3
CER(p(θ)) ∼ (P →∞).
1−q(θ) K2
since then q(θ) = r(θ) = 0. If 2K ≤ P < 3K, then 0 < (cid:18) (cid:19) (cid:18) (cid:19)
q(θ)<1butr(θ)=0,whence The arguments given in the proof of Proposition 3.2 to
establish(69)canalsobeusedtoestablish
C (θ) q(θ) 3
K =1+ >1. r(θ) K3
C (p(θ)) 1−q(θ) 1− ∼ (P →∞). (49)
ER (cid:18) (cid:19) q(θ)2 P2
Understandingthecase3K ≤P ismorechallengingdue
Collectingweconcludetothevalidityof(47).
to a lack of simple expressions. Therefore, before dealing
with the case of an arbitrary positive integer K, we first
consider a couple of special cases as a way to explore the Next we compare the clustering coefficients of asymp-
relative ranges possibly exhibited by the clustering coeffi- toticallymatchedrandom keygraphsandERgraphswhen
cients. For K = 1 it is a simple matter to check from (43) theparametersθandparescaledwithn.
that Corollary6.2.ConsiderascalingK,P :N →N satisfying
C (1,P) 0 0
K =P (45) (21)andascalingp : N0 → [0,1].Undertheasymptotic
C (p(θ))
ER matchingcondition(36),wehavetheequivalence
foreachP = 2,3,.... ForK = 2 uninterestingcalculations
C (θ ) P
K n n
showthat ∼1+ . (50)
C (p ) K3
C (2,P) P 2P3−4P2−P +3 ER n n
K
= ·
CER(p(θ)) 2 (2P −3)3 Proof. Aswereplaceθbyθ andpbyp accordingtothese
n n
foreachP =6,7,...,whence scalingsintheexpression(42),weget
P C (2,P) C (θ ) β(θ )
< K <P (46) K n = n , n=3,4,... (51)
8 C (p(θ)) C (p ) p (1−q(θ ))2
ER ER n n n
on that range. This upper bound is seen to hold by noting Notethat
that 4(2P3−4P2−P +3) = (2P −3)3+(P −1)(20P − C (θ ) β(θ ) τ(θ )
K n n n
38)+1 > (2P −3)3 for allP = 2,3,.... The lower bound C (p ) ∼ (1−q(θ ))3 ∼ (1−q(θ ))3. (52)
ER n n n
8
The first equivalence is a consequence of (36) while the notpracticalsincerequiringK ≫ n .Furthermore,they
n logn
second equivalence follows by Proposition 3.2 under (21). will also result in high node degrees, and this in turn will
With (37) being still valid here, we easily conclude (50) by decrease network resiliency against node capture attacks. It
thesameargumentsastheonesusedtoobtain(39). wasproposedbyDiPietroetal.[7,Thm.5.3]thatresiliency
inlargeWSNsagainstnode captureattackscanbeensured
byselectingK and P suchthat Kn ∼ 1. Under(55) this
Under(21)and(36),weconcludethat n n Pn n
additionalrequirementthenleadstoK ∼c·logn,whence
n
C (θ ) K3 P ∼c·nlogn,and(39)nowimplies
lim K n =1 if lim n =∞, (53) n
n→∞CER(pn) n→∞ Pn E[Tn(θn)] n
lim = lim 1+ =∞. (58)
and C (θ ) K3 n→∞E[Tn(pn)] n→∞(cid:18) (c·logn)2(cid:19)
lim K n =∞ if lim n =0. (54) Therefore, for such realistic WSN implementations the ex-
n→∞CER(pn) n→∞ Pn pected number of triangles in the induced random key
Thus, asymptotically matched random key graphs and ER graphswillbeordersofmagnitudelargerthaninERgraphs.
graphscaninprinciplehavevastlydifferentclusteringcoef- Concerningtheclusteringcoefficients,weseethatunder
ficients.We explore thispossibilityinthe next twosections the condition (55), (53) can hold only if the key ring size
where the implications of the main results are discussed K is much larger than n/logn. As already discussed, this
in the context of wireless sensor networks and of social condition can not be satisfied in a practical WSN scenario
networksbasedoncommoninterestrelationships. due to storage limitations at the sensor nodes and security
constraints. In fact, we see from (39) and (58) that, in a
realistic WSN, the condition (54) is always in effect and
7 WIRELESS SENSOR NETWORKS
the clustering coefficient of the random key graph is much
Random key graphs were originally introduced to model
largerthanthatoftheasymptoticallymatchedERgraph.
the random key pre-distribution scheme proposed by Es-
chenauer and Gligor [9] in the context of WSNs. When
8 SOCIAL NETWORKS – CAN RANDOM KEY
the WSN comprises n nodes, it is natural to select the
parametersK andP inorderfortheinducedrandomkey GRAPHS BE SMALL WORLDS?
n n
graphtobeconnected.However,thereisatradeoffbetween Withanobviouschangeinterminology,randomkeygraphs
connectivity and security [7], requiring that Kn2 be kept as canbe usedto model certaintypesof social networks, e.g.,
closeaspossibletothe criticalscaling logn foPrnconnectivity see [2], [37]: Instead of viewing {1,...,P} as a collection
n of cryptographic keys randomly assigned to the nodes of
(but above it); see the papers [3], [7], [24], [28], [33]. The
a WSN according to the Eschenauer-Gligor scheme, we
desiredregimeneartheboundarycanbeachievedbytaking
can think of it as a list of “interests,” e.g., hobbies, books,
Kn2 ∼c· logn (55) movies, sports, etc., which are pursued by the members
P n of a social group. In that reformulation, the i.i.d. random
n
sets K (θ),...,K (θ) appearing in the definition of the
withc>1butclosetoone. 1 n
random key graph K(n;θ) can now be interpreted as the
Now,considerthesituationwheretherandomkeygraph
K(n;θ )ismatchedasymptoticallytotheERrandomgraph interestsassignedtotheindividualmembersofthatgroup.1
G(n;pn) under the asymptotic matching condition (36). It The random key graph K(n;θ) then naturally describes a
n
common-interest relationship between community members
followsfrom(39)that
sincetwoindividualsarenowadjacentinK(n;θ)whenthey
E[T (θ )] P
n n n haveatleastoneinterestincommon.
∼1 ifandonlyif =o(1) (56)
E[Tn(pn)] Kn3 InparalleltothediscussiongiveninSection7forWSNs,
weexploretheparameterrangeslikelytoappearinpractice
under the condition (21). This last condition obviously oc-
for random key graphs modeling social networks. We do
curs when (55) holds, in which case the condition at (56)
so with an eye towards understanding the behavior of the
amountstotaking
expressionappearingat(39).
1 logn
We begin with the observation that most real-world
=o(1) c· .
Kn (cid:18) n (cid:19) social networks are known to be sparse in the sense that
the expected number of edges per node appears to remain
Thus,undertheconnectivitycondition(55)itholdsthat
(nearly) constant as the size of the network increases. In
E[T (θ )] K
n n ∼1 ifandonlyif lim n =∞. (57) the case of random key graphs, the expected degree of a
E[Tn(pn)] n→∞n/logn nodeisgivenby(n−1)(1−q(θn))andsparsityamountsto
1−q(θ )∼ c forsomec>0,orequivalently,to
The expected number of triangles in random key graphs n n
is then of the same order as the corresponding quantity K2 c
in asymptotically matched ER graphs with E[T (θ )] ∼ n ∼ (59)
n n P n
E[T (p )] ∼ c3 (logn)3 – This is a direct consequence of n
n n 6
(32), (37) and (55). This conclusion holds regardless of the 1.Here we assume that each individual has exactly K interests
valueofcin(55). drawnfromthelist{1,...,P}.Morerealisticmodelscanbeobtained
through more complex randomization mechanisms as in thework of
However,giventhelimitedmemoryandcomputational
Godehardtetal.ongeneralrandomintersectiongraphs[11],[12]and
powerofthesensornodes,keyringsizessatisfying(57)are asintheworkofYag˘anoninhomogeneousrandomkeygraphs[35].
9
byvirtueofLemma3.1,whence mayindeedbeconsideredgoodcandidatemodelsforsmall
worlds!
P n
n
∼ . (60)
K3 cK
n n 9 PROOFS OF THE PRELIMINARY RESULTS
Inviewof Corollary5.1andCorollary6.2,inthe sparse
In Sections 9.1 and 9.2, we fix positive integers K and P
regime, random keygraphswill havemanymore triangles
suchthatK ≤P,andn=3,4,....
and will be muchmore clustered(by orders of magnitude)
thantheasymptoticallymatchedERgraphsunless
9.1 AproofofProposition2.1
n
limsup <∞. (61) As exchangeability yields (17), we need only show the
K
n→∞ n validity of (16). We make repeated use of the fact that for
Thisconditionisequivalentto anypairofeventsE andF inF,wehave
Kn =Ω(n), (62) P[E∩F]=P[E]−P[E∩Fc]. (63)
andisevenmorestringentthanthecorrespondingcondition Thus,byrepeatedapplicationof(63)wefind
(57)derivedforWSNapplications.Moreimportantly,under E[χ (θ)]
thecondition(59)wehaveP ∼c−1nK2,requiring 123
n n K (θ)∩K (θ)6=∅,K (θ)∩K (θ)6=∅,
= P 1 2 1 3
P =Ω(n3) K (θ)∩K (θ)6=∅
n 2 3
(cid:20) (cid:21)
= P[K (θ)∩K (θ)6=∅,K (θ)∩K (θ)6=∅]
if(62)isalsoenforced.Thus,under(59)and(62)generating 1 2 1 3
the random key graph will require each of the n nodes to −P K1(θ)∩K2(θ)6=∅,K1(θ)∩K3(θ)6=∅,
choose Kn = Ω(n) objects from a universe of size Pn = (cid:20) K2(θ)∩K3(θ)=∅ (cid:21)
Ω(n3). The computational complexity of this task quickly = P[K (θ)∩K (θ)6=∅,K (θ)∩K (θ)6=∅]
1 2 1 3
becomesprohibitivelyhighasthenumberofindividualsin −P[K (θ)∩K (θ)6=∅,K (θ)∩K (θ)=∅]
1 2 2 3
thesocialnetworkbecomeslarge.Yet,wewouldexpectthe
K (θ)∩K (θ)6=∅,K (θ)∩K (θ)=∅,
realistic values for the number K of interests of a single +P 1 2 1 3 .
n K (θ)∩K (θ)=∅
individual to be much smaller than the network size n, in (cid:20) 2 3 (cid:21)
sharpcontrastwith(62).Inotherwords,thecondition(62)is Byindependence,withthehelpof(6),wereadilyobtainthe
naturallyeliminatedinrealisticapplicationsofrandomkey expressions
graphstosuchsocialnetworks–Theresultingrandomkey P[K (θ)∩K (θ)6=∅,K (θ)∩K (θ)6=∅]=(1−q(θ))2
1 2 1 3
graphswillnaturallyhaveveryhighclusteringandcontain
verylargenumberoftriangleswhenusedforsocialnetwork and
modeling. P[K (θ)∩K (θ)6=∅,K (θ)∩K (θ)=∅]
1 2 2 3
Since random key graphs can be highly clustered, a
= (1−q(θ))q(θ).
natural question arises as to their suitability for modeling
the small world effect. This notion is linked to a well-known Next,asweuse(63)onemoretime,weget
series of experiments conducted by Milgram [19] in the
K (θ)∩K (θ)6=∅,K (θ)∩K (θ)=∅,
latesixties.The results,commonlyknownassixdegreesof P 1 2 1 3
K (θ)∩K (θ)=∅
2 3
separation,suggestthatthesocialnetworkofpeopleinthe (cid:20) (cid:21)
= P[K (θ)∩K (θ)=∅,K (θ)∩K (θ)=∅]
UnitedStatesissmallinthesensethatpathlengthsbetween 1 3 2 3
pairsofindividualsareshort.AsawaytocaptureMilgram’s −P K1(θ)∩K2(θ)=∅,K1(θ)∩K3(θ)=∅ .
experiments,WattsandStrogatz[27]introducedsmallworld K2(θ)∩K3(θ)=∅
(cid:20) (cid:21)
network models that are highly clustered and yet have a Again, by independence, with the help of (6) we conclude
smallaveragepathlength.More precisely,arandom graph that
is considered tobe a smallworld if its average pathlength
P[K (θ)∩K (θ)=∅,K (θ)∩K (θ)=∅]=q(θ)2
is of the same order as that of an ER graph with the same 1 3 2 3
expectedaveragedegree,butwithamuchlargerclustering and
coefficient.
K (θ)∩K (θ)=∅,K (θ)∩K (θ)=∅
P 1 2 1 3
Theresultsofthis paperalreadyshowthatrandom key
K (θ)∩K (θ)=∅
2 3
graphs can satisfy the high clustering coefficient require- (cid:20) (cid:21)
K (θ)∩K (θ)=∅,
ment of a small world. Under (55), Rybarczyk [24] has = P 1 2
K (θ)∩(K (θ)∪K (θ))=∅
shownthat (cid:20) 3 1 2 (cid:21)
logn = q(θ)r(θ)
diam[K(n;θ )]∼
n
loglogn
since |K (θ) ∪ K (θ)| = 2K when K (θ) ∩ K (θ) = ∅.
1 2 1 2
with high probability where K(n;θn) is the largest con- Collectingthesefactswefind
nectedcomponentofK(n;θ ).Thissuggeststhatthediam-
n E[χ (θ)]
eter, hence the average path length, in random key graphs 123
issmallaswasthecasewithERgraphs[5].Wealsonote[31, = (1−q(θ))2−(1−q(θ))q(θ)+q(θ)2−q(θ)r(θ)
Corollary5.2]thatrandomkeygraphshaveverysmall(e.g.,
andtheconclusion(16)followsbyelementaryalgebra.
≤ 2) diameter under certain parameter ranges (e.g., with
P = O(nδ) with 0 < δ < 1). Thus, random key graphs
n 2
10
9.2 AproofofProposition2.2 As Lemma 3.1 already implies q(θ )3 ∼ 1 and
n
Byexchangeabilityandthebinarynatureofthervsinvolved (1−q(θn))3 ∼ KPnn2 3,theasymptoticequivalenceβ(θn)∼
wereadilyobtain τ(θn)willbees(cid:16)tablis(cid:17)hedifweshowthat
E Tn(θ)2 1− r(θn) ∼ Kn3. (69)
q(θ )2 P2
(cid:2)= E[T(cid:3) (θ)]+ n 3 n−3 E[χ (θ)χ (θ)] n n
n 3! 2! 1 ! 123 124 This is an easy consequence of the fact that all terms in-
volvedarenon-negative.
n 3 n−3
+ E[χ (θ)χ (θ)] To establish (69) we proceed as follows: With positive
123 145
3! 1! 2 ! integersK,P suchthat3K ≤P,wenotethat
n n−3 r(θ)
+ E[χ (θ)χ (θ)]. (64)
3! 3 ! 123 456 q(θ)2
(P −2K)! 2 (P −2K)! P!
Under the enforced independence assumptions the rvs = · ·
(P −K)! (P −3K)! (P −K)!
χ123(θ) and χ456(θ) are independent and identically dis- (cid:18) (cid:19)
(P −2K)!(P −2K)! P!(P −2K)!
tributed.Asaresult, = ·
(P −K)!(P −3K)! (P −K)!(P −K)!
E[χ (θ)χ (θ)]=E[χ (θ)]E[χ (θ)]=β(θ)2,
123 456 123 456 K−1 P −2K−ℓ K−1 P −ℓ
= ·
andusingtherelation(17)yields P −K−ℓ P −K−ℓ
ℓ=0 (cid:18) (cid:19) ℓ=0 (cid:18) (cid:19)
Y Y
n n−3 n−3 K−1 K K−1 K
E[χ (θ)χ (θ)]= 3 (E[T (θ)])2. (65) = 1− · 1+
3! 3 ! 123 456 (cid:0) n3 (cid:1) n ℓY=0 (cid:18) P −K−ℓ(cid:19) ℓY=0 (cid:18) P −K−ℓ(cid:19)
(cid:0) (cid:1) K−1 K 2
Ontheotherhand,withthehelpof(6)wereadilycheck = 1−
that the indicator rvs χ123(θ) and χ145(θ) are independent ℓ=0 (cid:18)P −K−ℓ(cid:19) !
andidenticallydistributedconditionallyonK (θ)with Y
1 upongroupingfactorsappropriately.Elementarybounding
P[χ (θ)=1|K (θ)]=P[χ (θ)=1]=β(θ). (66) argumentsnowyieldthetwobounds
123 1 123
Asasimilarstatementappliestoχ (θ), we conclude that K 2 K r(θ)
145 1− 1− ≤1−
the rvs χ123(θ) and χ145(θ) are (unconditionally) indepen- (cid:18)P −K(cid:19) ! q(θ)2
dentandidenticallydistributedwith
and
E[χ (θ)χ (θ)]=E[χ (θ)]E[χ (θ)]=β(θ)2. r(θ) K 2 K
123 145 123 145 1− ≤1− 1− .
Againbyvirtueof(17),thislastobservationyields q(θ)2 (cid:18)P −2K(cid:19) !
Picka scalingP,K : N → N satisfyingthe equivalent
0 0
n 3 n−3 conditions (21) and consider n sufficiently large in N so
E[χ (θ)χ (θ)] 0
3! 1! 2 ! 123 145 that3Kn <Pn.Onthatrange,wereplaceθbyθninthelast
chain of inequalities according to this scaling. A standard
n−3
= 3 2 ·(E[T (θ)])2. (67) sandwich argument will yield the desired equivalence (69)
n n
(cid:0) 3 (cid:1) ifweshowthat
Substituting (65) and (67(cid:0)) (cid:1)into (64) establishes Proposition K 2 Kn K3
1− 1− n ∼ n, c=1,2. (70)
2.2. (cid:18)Pn−cKn(cid:19) ! Pn2
Todosoweproceedasfollows:Fixc=1,2.With
K
n
A (c)= , n=1,2,...
9.3 AproofofProposition3.2 n P −cK
(cid:18) n n(cid:19)
Since1 ≤ K ≤ K 2 foralln = 1,2,...,thecondition(21) standardcalculusyields
n n
impliesboth
K 2 Kn
n
1 K 1− 1−
nl→im∞Pn =0 and nl→im∞ Pnn =0. (68) = 1−(cid:18)1Pn−−AcK(cn)2(cid:19)K!n
n
Therefore, lim P = ∞, and for any c > 0, we have
n→∞ n 1
cKn <PnforallnsufficientlylargeinN0(dependentonc). = KnA(cid:0)n(c)2 1(cid:1)−An(c)2t Kn−1dt (71)
Thus, we have 3Kn < Pn for all n sufficiently large in N0. Z0
Onthatrangewecanusetheexpression(14)towrite ontheappropriaterange.The(cid:0)asymptotice(cid:1)quivalences
r(θ ) K 2 K 2
β(θn)=(1−q(θn))3+q(θn)3 1− q(θn)2 . An(c)2 = P −ncK ∼ Pn (72)
(cid:18) n (cid:19) (cid:18) n n(cid:19) (cid:18) n(cid:19)