Table Of ContentAchieving SK Capacity in the Source Model:
When Must All Terminals Talk?
Manuj Mukherjee† Navin Kashyap† Yogesh Sankarasubramaniam‡
Abstract—In thispaper,we addresstheproblemof character- of interest to us is the following: characterize the instances
izing the instances of the multiterminal source model of Csisza´r of the multiterminal source model in which omnivocality is
and Narayan in which communication from all terminals is
necessary for maximal-rate SK generation. In this paper, we
4 needed for establishing a secret key of maximum rate. We give
report partial progress made towards such a characterization.
1 an information-theoreticsufficientcondition for identifyingsuch
0 instances. We believe that our sufficient condition is in fact an The paper is organized as follows. After establishing the
2 exact characterization, but we are only able to prove this in the required notation and background in Section II, we give, in
caseofthethree-terminalsourcemodel.Wealsogivearelatively SectionIII, asufficientconditionunderwhichomnivocalityis
n
simple criterion for determining whether or not our condition
a necessary for achieving SK capacity in a source model with
holds for a given multiterminal source model.
J m ≥ 3 terminals. This condition is satisfied, for example, in
0 I. INTRODUCTION the case of the complete graph pairwise independentnetwork
2 (PIN) model of Nitinawarat and Narayan [7]. We conjecture
We are concerned with the multiterminal source model of
that our sufficient condition is also necessary, but at present,
] Csisza´r and Narayan [3], which can be briefly described as
T we can only prove this in the m = 3 case. Finally, in
follows. There are a certain number, m ≥ 2, of terminals,
I SectionIV,wegivea usefulcriterionforcheckingwhetheror
. each of which observes a distinct component of a source of
s not our condition holds for a given source model.
c correlated randomness. The terminals must agree on a shared
[ SK by communicating over a noiseless public channel. This II. PRELIMINARIES
key must be protected from a passive eavesdropper having
1
Throughout,weuseNtodenotethesetofpositiveintegers.
v access to the public communication. The SK capacity, which
In the multiterminal source model [3], a set of m ≥ 2
7 is the supremum of the rates of SKs that can be generated,
3 has been characterized in various ways [2], [3], [7]. What is terminals, denoted by [m] , {1,2,...,m}, has access to a
0 source (Xn,Xn,...,Xn), n∈N, where Xn denotesn i.i.d.
less well-understood is the nature of public communication 1 2 m i
5 copies of a random variable (rv) X taking values in a finite
that is needed to achieve SK capacity in this model. In a i
.
1 companionpaper[6],wegavealowerboundontheminimum setXi.ThervsX1,X2,...,Xm areingeneralcorrelated,and
0 foreachi∈[m],theithterminalobservesonlythecomponent
rate of communication required to generate a maximal-rate
4 Xn. For any subset A ⊆ [m], we will use X to denote the
1 (i.e., capacity-achieving)SK, building upon the prior work of i A
collection of rvs (X : i ∈ A), and p to denote their joint
: Tyagi[9]onthetwo-terminalmodel.Inthispaper,weaddress i XA
v probability mass function.
arelatedquestion:whenmustallmterminalsnecessarilyhave
i
X to communicate in order to generate a maximal-rate SK? The terminals communicate through a noiseless public
channel, any communicationsent throughwhich is accessible
r It is well known that, in order to generate a maximal-rate
a to all terminals and to potential eavesdroppers as well. The
SKinthetwo-terminalmodel(m=2),itissufficientforonly
terminals communicate in a round-robin fashion, following
oneterminaltocommunicate[1],[5],[3].Allthisterminalhas
thecyclicorder(1,2,...,m).Anytransmissionsentbytheith
to do is convey its local observations to the other terminal at
terminalisadeterministicfunctionofXn andalltheprevious
the least possible rate of communication required to do so. i
communication.Formally,avalidcommunicationisafinitely-
Thus, when m = 2, it is never necessary for both terminals
supported random vector F = (F ,F ,...,F ), r ∈ N, with
to communicate to generate a capacity-achieving SK. Even 1 2 r
F denoting a communication sent by the terminal i ∈ [m]
when m > 2, there are examples wherein not all terminals j
withi≡j(mod m),andH(F |F ,...,F ,Xn)=0.The
need to communicate — see remark following Theorem 1 in j 1 j−1 i
rate of the communicationis taken to be 1 log |F|, where F
[3].However,aswewillshowinthispaper,thereareplentyof n 2
is the finite set on which F is supported. Terminal i∈[m] is
otherexampleswhereallterminalsmustcommunicateinorder
said to be silent if F = 0 (with probability 1) for all j ≡ i
toachieveSKcapacity.Wecointheterm“omnivocality”tode- j
(mod m). An omnivocal communication is one in which no
scribethestate whenallterminalscommunicate.Theproblem
terminal is silent.
†M. Mukherjee and N. Kashyap are with the Department of Electrical Given an ǫ>0, we say that an rv U is ǫ-recoverable from
Communication Engineering, Indian Institute of Science, Bangalore. Email: an rv V if there exists a function g of V such that Pr[U =
{manuj,nkashyap}@ece.iisc.ernet.in. g(V)]≥1−ǫ.
‡Email:[email protected]
Definition 1. For any ǫ > 0, an ǫ-SK for [m] is an rv K = allowed to communicate, while terminals in [m] \ T must
K(n)(Xn ), for some n ∈ N, such that there exists a valid remainsilent.Thus,wearerestrictedtovalidcommunications
[m]
communication F with the following properties: Finwhichtheterminalsin[m]\T aresilent,butwhichallow
(i) I(K;F)≤ǫ; and all m terminalsto agree upona weak SK. In other words, we
(ii) K is ǫ-recoverable from (Xn,F) for each i∈[m]. only consider weak ǫ-SKs for [m] that are obtainable through
i
The rate of this ǫ-SK is given by 1H(K). valid communicationsF in which all terminals in [m]\T are
n
silent. Thesupremumof ratesachievableby suchSKs will be
A real number R ≥ 0 is an achievable SK rate if for
denoted by C([m]kT).
any ǫ > 0, there exists an ǫ-SK of rate greater than R−ǫ.
The SK capacity C([m]) is defined as the supremum of all Theorem1 ([4],Theorem6). C([m]kT)=H(XT)−RT(min),
achievable SK rates. The SK capacity can be expressedas [2, where R(min) = min R , the rate region R being the
Theorem 1.1] (see also [3, Eq. (26)]) T R∈RT i∈T i T
set of all points R=(RX,i∈T) such that
i
1
C([m])=I(X[m]),mPin|P|−1D pX[m] k pXA! Ri ≥H(XA∩T|XAc) ∀A([m], A∩T 6=∅.
AY∈P (1) i∈XA∩T
the minimumbeing taken overall partitionsP of [m], of size
Note that if C([m]) > C([m]kT) for all T ⊂ [m] of size
|P| ≥ 2. The quantity D(·k·) denotes relative entropy, and
|T| = m−1, then omnivocality is necessary for achieving
for a partition P = {A ,...,A }, the notation p
1 k A∈P XA SK capacity. Thus, our approach for showing that omnivocal
represents the product p ×···×p . Note that when
m=2, the quantity I(XXA)1 defined inX(1A)ksimplyQreduces to communicationisneededincertaincasesisto useTheorem1
[m]
the mutual information I(X ;X ). Thus, I(X ) should be to prove that C([m]) > C([m]kT) for all (m − 1)-subsets
viewed as a multiparty exten1sion2of mutual inf[omr]mation. T ⊂ [m]. For this, we will need a lower bound on RT(min)
when |T| = m−1. To prove this bound, we use a simpler
Before proceeding further, a couple of clarifications con-
characterization (than that given in Theorem 1) of the rate
cerning Definition 1 are needed. We have adopted the notion
region R when |T|=m−1.
of strong secrecy (property (i) in the definition), as opposed T
to weak secrecy, which only requires 1I(K;F) ≤ ǫ. All the Lemma 2. Let T = [m]\{u} for some u ∈ [m]. The rate
n
results proved in this paper would hold just as well under region R is the set of all points (R , i∈T) such that
T i
either type of secrecy. In particular, our main result shows
R ≥H(X |X ) ∀B (T,B 6=∅, (2)
that omnivocal communication is necessary for achieving i B T\B
SK capacity if a certain condition on the singleton partition iX∈B
S is satisfied. Our proof of this result relies only on the
and R ≥H(X |X ).
i T u
expression for SK capacity given in (1), which remains the
i∈T
same under both forms of secrecy [3], and on a theorem of X
Proof:ObservethatR isdefinedbyconstraintsonsums
GohariandAnantharam[4],whichis statedandprovedunder T
of the form R for non-empty subsets B ⊆ T. When
the weak secrecy notion. Thus, our proof in fact shows that i∈B i
B =T, the constraint is simply R ≥H(X |X ).
omnivocal communication is necessary even under a weak P i∈T i T u
Now, consider any non-empty B ( T. From Theorem 1,
secrecy requirement on SKs. P
we see that constraints on R arise as constraints on
i∈B i
A second clarification concerning Definition 1 is that,
R in two ways: when A = B and when A = B ∪
usually, the definition of an ǫ-SK includes an additional {ui}∈.AT∩hTus,iwe have two conPstraints on R :
requirement that K be almost uniformly distributed over its P i∈B i
alphabet K, i.e., H(K) ≥ log|K|−ǫ [3]. However, this can R ≥H(X |X P),
i B [m]\B
always be dropped without affecting SK capacity — see e.g., i∈B
X
the discussion on p. 3976 in [4].
obtained when A=B, and
Asmentionedabove,wemakeuseofaresultofGohariand
R ≥H(X |X ),
Anantharam[4,Theorem6]insomeofourproofs.Tostatethis i B T\B
result,we explicitlydefinea weakǫ-SK for[m] to beanrvK iX∈B
as in Definition 1, except that the strong secrecy condition (i) obtained when A = B ∪{u}. The latter constraint is clearly
is replaced by the weak secrecy condition, 1I(K;F) ≤ ǫ. stronger, so we can safely discard the former.
n
Then, R≥0 is an achievable weak-SK rate if for any ǫ>0, We can now prove the desired lower bound on R(min).
T
there exists a weak ǫ-SK of rate greater than R − ǫ. It is
Lemma 3. Let m ≥ 3 be given. For T ⊂ [m] with |T| =
known that the supremum of achievable weak-SK rates is the
m−1, we have
sameastheSKcapacitygivenby(1).TheGohari-Anantharam
1
resultconcernsachievableweak-SKratesundertheadditional R(min) ≥ H(X |X ).
assumption that some fixed subset of terminals remains silent T m−2 T\{j} j
j∈T
X
throughout. Let T ⊆ [m] be such that terminals in T are
Proof:ConsideranyT ⊂[m]with|T|=m−1.Foreach The pairwise independent network (PIN) model of Niti-
j ∈ T, let B = T \{j}. Now, let (R ,i ∈ T) be any point nawarat and Narayan [7] is defined on an underlying graph
j i
in R . Applying (2) with B =B , we get G = (V,E) with V = [m]. For n ∈ N, let G(n) be the
T j
multigraph (V,E(n)), where E(n) is the multiset of edges
R ≥H(X |X ),
i T\{j} j formed by taking n copies of each edge of G. Associated
iX∈Bj with each edge e∈E(n) is a Bernoulli(1/2) rv ξe; the rvs ξe
for each j ∈T. Summing over all j ∈T, we obtain associated with distinct edges in E(n) are independent. With
this, the rvs Xn, i ∈ [m], are defined as Xn = (ξ : e ∈
Ri ≥ H(XT\{j}|Xj). (3) E(n) and e is inicident on i). When G = K i, the coemplete
m
jX∈TiX∈Bj jX∈T graphon m vertices, we have the complete graphPIN model.
Exchanging the order of summation in the double sum on We show in the next section (Corollary 7.2) that for the
the left-hand side (LHS) above, we have R = complete graph PIN model, the singleton partition S is the
Puit∈tiTng thji∈sBbiaRcki i=nto (3i∈),Tw(me g−et2)Ri =P(mj∈−T P2)i∈Bij∈TiRi. duinaitqeulye fmoilnloiwmsizferromforTIh(eXor[mem]).4T.he result below then imme-
P P P P
1 Corollary 4.1. In the PIN model defined on the complete
R ≥ H(X |X ).
i m−2 T\{j} j graph Km, m ≥ 3, omnivocal communication is necessary
i∈T j∈T
X X for achieving C(X ).
[m]
Sincethisholdsforany(R ,i∈T)∈R ,thelemmafollows.
i T
In conjunction with Theorem 6 in [6], we now have the
following picture for a capacity-achieving communication in
III. OMNIVOCAL COMMUNICATION the complete graph PIN model: the communication must be
omnivocal, and if it is constrained to be a linear function
AspointedoutintheIntroduction,inthesourcemodelwith
of the observations Xn , then it must have rate at least
two terminals, omnivocality is never necessary for generating [m]
a maximal-rate SK. However, the situation is different when m(m−2)/2. It should be noted that the capacity-achieving
there are three or more terminals. In this section, we give a communicationintheproofof[7,Theorem1]isanomnivocal,
sufficientconditionfor omnivocalitybeingneededfor achiev- linear communication of rate m(m−2)/2.
ing SK capacity when there are m≥3 terminals, and give an For the proof of Theorem 4, we need some convenient
example where the sufficient condition is met. The sufficient notation. For T ⊂ [m], |T| = m − 1, define ∆ (S) ,
T
conditionalsoturnsouttobenecessarywhenthereareexactly 1 [ H(X )−H(X )].
m−2 i∈T i T
three terminals.
LemmPa 5. For m≥3 terminals, if the singleton partition S
Tostateourresults,weneedafewdefinitions.Thepartition
is the uniqueminimizer forI(X ), then ∆ (S)<∆(S) for
{{1},{2},...,{m}}consistingofm singletoncellswillplay [m] T
all T ⊂[m] with |T|=m−1.
aspecialroleinourresults;wecallthisthesingletonpartition
and denote it by S. For any partition P of [m] with |P|≥2, Proof: For any u ∈ [m], consider T = [m]\{u}. Using
define ∆(S)= 1 [ m H(X )−H(X )] and the definition of
m−1 i=1 i [m]
∆ (S) above, it is easy to verify the identity
∆(P), 1 H(X )−H(X ) . (4) T P
|P|−1 A [m] m−1∆(S)=∆ (S)+ 1 I(X ;X ).
"A∈P # m−2 T m−2 u T
X
Equivalently, Re-arranging the above, we obtain
1 ∆T(S)−∆(S)= m1−2[∆(S)−I(Xu;XT)]
∆(P)= D p k p ,
|P|−1 X[m] XA = 1 [∆(S)−∆(P)], (5)
A∈P ! m−2
Y
the notation being as in (1). Thus, C([m]) = I(X ) = where P is the 2-cell partition {{u},T} of [m]. By assump-
[m]
min ∆(P). In all that follows, we say that the singleton tion, the expression in (5) is strictly negative.
P
partition S is a minimizer for I(X ) if ∆(S) = I(X ), With this, we are ready to prove Theorem 4.
[m] [m]
andthatS istheuniqueminimizerforI(X )iftheminimum Proof of Theorem 4: We will show that C([m]) >
[m]
in (1) is uniquely achieved by S, i.e., ∆(S) < ∆(P) for all C([m]kT) for any T ⊂ [m] with |T| = m−1. First, note
partitions P of [m], P 6=S, with |P|≥2. that since S is, by assumption, a minimizer for I(X ), we
[m]
We can now state the main result of this section. have C([m]) = I(X ) = ∆(S). Next, by Theorem 1 and
[m]
Lemma 3, we have
Theorem4. Form≥3terminals,ifS istheuniqueminimizer
for I(X[m]), then omnivocal communication is necessary for C([m]kT)≤H(XT)− m1−2 H(XT\{i}|Xi),
achieving the SK capacity C([m]).
i∈T
X
Before provingthe theorem,we give an examplewhere the = 1 (m−2)H(X )− [H(X )−H(X )]
m−2 T T i
condition of the theorem is met.
(cid:20) i∈T (cid:21)
X
=∆ (S). a communication in which terminals 2 and 3 are both silent.
T
Therefore, C([m]kT) ≤ ∆ (S) < ∆(S) = C([m]), the Now, consider Case II, in which we obviously have
T
second inequality coming from Lemma 5. C([3]) = I(X ;X ). The idea here is to show that a
{1,2} 3
valid communication of rate H(X |X ) exists in which
For the three-terminal source model, it turns out that {1,2} 3
terminal 3 is silent, and which allows ǫ-recoverability of
the unique minimizer condition in Theorem 4 is also
(Xn,Xn) at all three terminals. Given this, an application
necessary for the conclusion of the theorem to hold. 1 2
of [3, Lemma B.3] shows that an SK rate of H(X )−
Note that when m = 3, (1) reduces to C(X ) = {1,2}
[3] H(X |X )=I(X ;X )isachievable.Thus,thereisa
min{I(X ;X ),I(X ;X ),I(X ;X ),∆(S)}; so {1,2} 3 {1,2} 3
{1,2} 3 {1,3} 2 {2,3} 1 C([3])-achievingcommunicationinwhichterminal3issilent.
the unique minimizer condition is equivalent to
To show that the desired communication exists, we argue
∆(S)<min{I(X{1,2};X3),I(X{1,3};X2),I(X{2,3};X1)}. as follows. For i=1,2, let Ri be the rate at which terminali
communicates. A standard random binning argument shows
Theorem 6. In the three-terminal source model, omnivocal
that an achievable (R ,R ) region, with terminal 3 silent,
1 2
communication is necessary for achieving SK capacity iff the
for a communication intended to allow ǫ-recoverability of
singleton partition S is the unique minimizer for I(X[m]). (Xn,Xn) at all terminals is given by
1 2
Proof: The “if” part is by Theorem 4.
R ≥H(X |X ), R ≥H(X |X ),
1 1 2 2 2 1
For the “only if” part, suppose that ∆(S) ≥ (9)
R +R ≥H(X |X ).
min{I(X ;X ),I(X ;X ),I(X ;X )}. Then, 1 2 {1,2} 3
{1,2} 3 {1,3} 2 {2,3} 1
∆(S) is either (a) greater than or equal to at least two of the Now, using the assumption in Case II that ∆(S) ≥
three terms in the minimum, or (b) greater than or equal to I(X ;X ), we will prove that the inequality
{1,2} 3
exactly one term. Up to symmetry, it suffices to distinguish
H(X |X )+H(X |X )≤H(X |X ) (10)
between two cases: 1 2 2 1 {1,2} 3
Case I: ∆(S)≥max{I(X{1,2};X3),I(X{1,3};X2)} holds.Itwouldthenfollowfrom(9)thatthereexistachievable
Case II: min{I(X{1,3};X2),I(X{2,3};X1)} > ∆(S) ≥ rate pairs (R1,R2) with R1 + R2 = H(X{1,2}|X3), thus
I(X{1,2};X3) completing the proof for Case II.
In each case, we demonstrate a capacity-achieving communi- So,letusprove(10).Wehave∆(S)= 1[H(X )+H(X )+
2 1 2
cation in which at least one terminal remains silent. H(X ) − H(X )] and I(X ;X ) = H(X ) +
3 [3] {1,2} 3 {1,2}
H(X )−H(X ). Using these expressions in the inequality
We deal with Case I first. Observe that 3 [3]
∆(S) = 1 3 H(X )−H(X ) can also be written ∆(S)≥I(X{1,2};X3), and re-arranging terms, we obtain
2 i=1 i [3]
a∆s(S12)[I≥(X1I;(XXhP2) +;XI(X),{1u,p2o};nXs3o)m].eTirhe-uosr,gatnhiezataiossnu,mypietilodns 12[H(X1)+H(X2)−2H(X{1,2})]≥ 12[H(X3)−H(X[3])],
{1,2} 3
I(X ;X )≥I(X ;X ), i.e., which is equivalent to (10). This completes the proof of the
1 2 {1,2} 3
theorem.
I(X ;X )≥I(X ;X )+I(X ;X |X ). (6)
1 2 1 3 2 3 1 We in fact conjecture that the result of Theorem 6 should
Similarly, using the identity ∆(S) = 1[I(X ;X ) + extend to more than three terminals as well.
2 1 3
I(X ;X )] in the assumption ∆(S)≥I(X ;X ), we
{1,3} 2 {1,3} 2 Conjecture 1. In the multiterminalsource modelwith m≥3
obtain I(X ;X )≥I(X ;X ), i.e.,
1 3 {1,3} 2 terminals, omnivocal communication is necessary for achiev-
I(X ;X )≥I(X ;X )+I(X ;X |X ). (7) ing SK capacity iff the singleton partition is the unique
1 3 1 2 1 3 2
minimizer for I(X ).
[m]
The equalities in (6) and (7) can simultaneously hold iff
At this point, we do not have a systematic approach for
I(X ;X )=I(X ;X ) and
1 2 1 3 (8) proving the “only if” part of the conjecture for m≥4.
I(X ;X |X )=I(X ;X |X )=0.
1 3 2 2 3 1
IV. SINGLETONPARTITIONS
From (8), it is not hard to deduce that the quantities
I(X ;X ), I(X ;X ), I(X ;X ) and ∆(S) are The condition that the singleton partition be a unique
{1,2} 3 {1,3} 2 {2,3} 1
all equal to I(X ;X ). In particular, C(X )=I(X ;X ). minimizer for I(X ) plays a key role in the results of
1 2 [3] 1 2 [m]
From the first equality in (8), we also have H(X |X ) = Section III. Thus, it would be very useful to have a way of
1 2
H(X |X ). Now, it can be shown by a standard random checkingwhetherthisconditionholdsforagivensourceX ,
1 3 [m]
binning argument that there exists a communication from m ≥ 3. The brute force method of comparing ∆(S) with
terminal 1 of rate H(X |X ) = H(X |X ) such that Xn ∆(P) for all partitions P with at least two parts requires
1 2 1 3 1
is ǫ-recoverable at both terminals 2 and 3. It then follows an enormous amount of computation. Indeed, the number
from the “balanced coloring lemma” [3, Lemma B.3] that an of partitions of an m-element set is the mth Bell number,
SK rate of H(X1)−H(X1|X2) = I(X1;X2) is achievable. Bm, an asymptoticestimate forwhich is (logw)1/2wm−wew,
Thus, the SK capacity, C([3])=I(X ;X ), is achievable by where w = m [1 + o(1)] is the solution to the equation
1 2 logm
m = wlog(w+1) [8, Example 5.4]. The proposition below graph K (as defined in Section III) are not exchangeable
m
bringsdownthenumberofcomparisonsrequiredforverifying when m≥3, but they are isentropic.
the unique minimizer condition to a “mere” 2m−m−2.
Corollary 7.1. If X ,X ,...,X , m ≥ 3, are isentropic
For any non-empty subset B = {b ,b ,...,b } of [m] 1 2 m
1 2 |B| rvs, then S is a minimizer for I(X ).
with |B| < m, define P , {Bc,{b },{b },...,{b }} to [m]
B 1 2 |B|
bethepartitionof[m]consistingof|B|+1cells,ofwhich|B| Proof:ForapartitionP of[m]with|P|≥2,letusdefine
cells are singletons comprising the elements of B. Note that 1
if |B| =m−1, then PB =S. δ(P) , |P|−1 H(XAc|XA) = H(X[m])−∆(P).
A∈P
X
Proposition 7. For m ≥ 3, let Ω = {B ⊂ [m] : 1 ≤ |B|≤
By virtue of Proposition 7(a), we need to show that δ(P )≤
m−2}. The singleton partition S is B
δ(S) for all B ∈Ω.
(a) a minimizer for I(X ) iff ∆(S)≤∆(P ) ∀B ∈Ω;
(b) the unique minimiz[emr]for I(X ) iff ∆B(S) < ∆(P ) For isentropic rvs, the quantity H(XB|XBc), for any
[m] B B ⊆ [m], depends only on |B|. Hence, defining g(k) ,
∀B ∈Ω.
H(X |X ) for 1 ≤ k ≤ m, we can write δ(P ) =
[k] [m]\[k] B
Proof: We prove (b); for (a), we simply have to replace 1 g(|B|)+g(m−1) and δ(S) = m g(m−1). Thus, we
|B| m−1
the ‘>’ in (11) below with a ‘≥’. havetoshowthat g(|B|) ≤ g(m−1) forallB ∈Ω.Thisfollows
|B| m−1
The “only if” part is obvious. For the “if” part, suppose
from the fact that for isentropic rvs, the function g(k)/k is
that ∆(S)<∆(PB) for all B ⊂[m] with 1≤|B| ≤m−2. non-decreasing in k — see Appendix A.
Consider any partition P of [m], P 6= S, with |P| ≥ 2. We
OursecondapplicationofProposition7istothePINmodel.
wish to show that ∆(P)>∆(S).
Recall from Section III that this model is defined on an
The following identity can be obtained from the definition underlying graph G = ([m],E). From the way that the rvs
in (4) by some re-grouping of terms: X , i ∈ [m], are defined, it is not difficult to verify that for
i
|Ac|∆(P )=(|P|−1)[∆(P)+(m−1)∆(S)]. any partition P of [m] with |P|≥2, we have
Ac
A∈P |E(P)|
X ∆(P)= ,
Thus, we have |P|−1
1 where|E(P)|denotesthenumberofedgese={i,j}∈E such
∆(P)= |Ac|∆(P )−(m−1)∆(S)
|P|−1 Ac that i and j are in different cells of P. This, in conjunction
A∈P
X with Proposition 7, gives us a relatively simple criterion for
1
> |Ac|∆(S)−(m−1)∆(S) (11) verifying whether S is a (unique) minimizer for I(X ). As
|P|−1 [m]
A∈P anillustration,weapplythistothecompletegraphPINmodel.
X
=m∆(S)−(m−1)∆(S) = ∆(S). (12)
Corollary7.2. ForthePINmodelonthecompletegraphK ,
m
The inequality in (11) is due to the fact that at least one m ≥ 3, the singleton partition S is the unique minimizer for
A ∈ P is not a singleton cell, so that P 6= S, and I(X ).
Ac [m]
hence, ∆(P ) > ∆(S) by assumption. To verify the first
Ac Proof: It is easy to see that for any non-emptyB ([m],
equality in (12), observe that A∈P|Ac|= A∈P i∈/A1= |E(P )|= m − |Bc| = 1|B|(2m−|B|−1). Hence,
m 1=m(|P|−1). B 2 2 2
i=1 A∈P:i∈/A P P P
(cid:0) (cid:1)|E((cid:0)PB)(cid:1)| 2m−|B|−1 m
PNexPt, we apply the above proposition to some interesting ∆(PB)= = ≥ =∆(S),
|B| 2 2
special cases.
with equality iff |B| = m−1, i.e., P = S. The result now
Random variables X ,X ,...,X , m ≥ 2, are called B
1 2 m
follows from Proposition 7(b).
isentropic if H(X ) = H(X ) for any pair of non-empty
A B
subsetsA,B ⊆[m] havingthe same cardinality.Equivalently,
APPENDIX A
X ,...,X are isentropic if, for all non-empty A ⊆ [m],
1 m
the entropyH(XA) dependsonlyon|A|. One obviousconse- Here, we prove that for isentropic rvs X1,...,Xm, the
quenceofthisdefinitionisthatfordisjointnon-emptysubsets function 1H(X |X ), defined for 1 ≤ k ≤ m, is non-
k [k] [m]\[k]
A,B ⊂[m],theconditionalentropyH(X |X )onlydepends decreasing in k. Define g(k) = H(X |X ). We show
A B [k] [m]\[k]
on |A| and |B|. that the difference kg(k +1)−(k +1)g(k) is always non-
Clearly, i.i.d. rvs are isentropic. More generally, ex- negative, from which the result follows.
changeable rvs are isentropic — rvs X1,X2,...,Xm are We have g(k + 1) = H(X[m]) − H(X{k+2,...,m}) and
exchangeable if for all permutations σ of [m], the joint g(k) = H(X[m]) − H(X{k+1,...,m}) = g(k + 1) −
distribution of (X1,X2,...,Xm) is the same as that of H(Xk+1|X{k+2,...,m}). Thus,
(X ,X ,...,X ). However, isentropic rvs need
σ(1) σ(2) σ(m) kg(k+1)−(k+1)g(k)
not be exchangeable. It may be verified that the rvs
=(k+1)H(X |X )−g(k+1).
X ,X ,...,X in the PIN model defined on the complete k+1 {k+2,...,m}
1 2 m
WeneedtoshowthattheRHSoftheequalityisnon-negative.
This is straightforward:
g(k+1)=H(X |X )
[k+1] {k+2,...,m}
k+1
≤ H(X |X )
i {k+2,...,m}
i=1
X
=(k+1)H(X |X ),
k+1 {k+2,...,m}
since, for 1 ≤ i ≤ k + 1, H(X |X ) =
i {k+2,...,m}
H(X |X ) by isentropy.
k+1 {k+2,...,m}
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