On mixing times for stratified walks on the d-cube

On Mixing Times for Stratified Walkson the d-Cube

Nancy Garcia,1 Jose Luis Palacios2

1Universidade Estadual de Campinas, Campinas, Brazil2Universidad Simon Bolıvar, Caracas, Venezuela

Received 21 March 2000; revised 19 March 2001; accepted 5 February 2002Published online xx Month 2002 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/rsa.10043

ABSTRACT: Using the electric and coupling approaches, we derive a series of results concerningthe mixing times for the stratified random walk on the d-cube, inspired in the results of Chung andGraham (1997) [Random Struct. Alg., 11, 199–122]. © 2002 Wiley Periodicals, Inc. Random Struct.Alg., 20: 540–552, 2002

Keywords: effective resistance; coupling; birth and death chains

1. INTRODUCTION

The stratified random walk (SRW) on the d-cube Qd is a Markov chain X � {X(n), n �0} whose state space is the set of vertices of the d-cube and whose transition probabilitiesare defined by: Given a set of nonzero probabilities p � ( p0, p1, . . . , pd�1), from anyvertex with k 1’s, the process moves either to any neighboring vertex with k � 1 1’s with

probability pkd ; or to any neighboring vertex with k � 1 1’s with probability pk � 1

d ; or toitself with the remaining probability. The simple random walk on the d-cube correspondsto the choice pk � 1 for all k.

Vaguely speaking, the mixing time of a Markov chain is the time it takes the chain tohave its distribution close to the stationary distribution under some measure of closeness.Chung and Graham studied the SRW on the d-cube in [5], mainly with algebraic methods,and found bounds for the mixing times under total variation and relative pointwisedistances. Here we use nonalgebraic methods, the electric and coupling approaches, in

Correspondence to: N. Lopes Garcia; e-mail: [email protected].© 2002 Wiley Periodicals, Inc.

540

order to study the same SRW and get exact results for maximal commute times andbounds for cover times and mixing times under total variation distance. We take advantageof the fact that there are several inequalities linking mixing times, hitting times, commutetimes, cover times and any definition of mixing time with any other under any measure ofcloseness (see Aldous and Fill [2] and Lovasz and Winkler [8]).

Notice first that the transition matrix for X � {X(n), n � 0}, the SRW on the d-cube,is doubly stochastic and therefore its stationary distribution is uniform. A bit of notation:we shall denote by � and � the probability measure and the expectation in the probabilityspace given by the Markov chain. The transition matrix of the chain is denoted by P and�x(X(t) � �) � �(X(t) � ��X(0) � x) is the distribution of the chain at time t given thatthe chain initiated at state x and �x its corresponding expectation. Formally speaking, inorder to assess the rate of convergence of the SRW on the cube Qd to the uniformstationary distribution �, we will bound the mixing time � defined as

� � min�t : d�t�� 1/2e, for all t� � t�,

where

d�t� � maxx�Qd

��x�X�t� � � � � ��,

and ��1 � �2� is the variation distance between probability distributions �1 and �2, one ofwhose alternative definitions is (see Aldous and Fill [2]), Chap. 2):

��1 � �2� � min ��V1 V2�,

where the minimum is taken over random pairs (V1, V2) such that Vm has distribution �m,m � 1, 2.

The bound for the mixing time is achieved using a coupling argument that goes asfollows: Let {X(t), t � 0} and {Y(t), t � 0} be two versions of the SRW on Qd suchthat X(0) � x and Y(0) �. Then

��x�X�t� � � � � �� X�t� Y�t��. (1.1)

In this case, a coupling between the processes X and Y is a bivariate process such thateach of the process X and Y has as marginal distribution the distribution of SWR and suchthat once the bivariate process enters the diagonal, it stays there forever. If we denote by

Tx � inf�t; X�t� � Y�t��

the coupling time, i.e., the hitting time of the diagonal, then (1.1) translates as

��x�X�t� � � � � �� Tx � t�, (1.2)

and, therefore,

d�t� � maxx�Qd

��Tx � t�. (1.3)

ON MIXING TIMES FOR STRATIFIED WALKS ON THE d-CUBE 541

If we can find a coupling such that � Tx � O( f(d)), for all x � Qd and for a certainfunction f of the dimension d, then we will also have � � O( f(d)). Indeed, if we take t �2ef(d), then (1.3) and Markov’s inequality imply that d(t) � 1/ 2e and the definition of� implies � � O( f(d)).

The main contribution of this work is to study the order of magnitude of � for SRWrestricted to three cases:

(a) Simple random walk ( pk � 1):

� � O�d log d�.

[This result is well-known, and it is put here only for completeness and to showthat our approach yields the right order of magnitude in simpler cases.]

(b) Aldous cube ( pk � k/(d � 1)):

� � O�d log d�.

[This is the same bound obtained by [5].](c) Slower walks ( pk � ((k � 1)/(d � 1)), 1):

� � O�d�.

[We improve the rate of the mixing time provided in [5] by a factor of log d.]

In order to bound the coupling time Tx we will relate it to commute times of a birth anddeath chain and these will be obtained through the connection between electrical networksand Markov chains described in Section 2. A specific coupling leading to the good boundsdescribed above is the object of Section 3.

2. THE ELECTRIC APPROACH

In this work we are considering the following well-known connection between electricalnetworks and Markov chains: On a connected undirected graph G � (V, E) such that theedge between vertices i and j is given a resistance rij (or equivalently, a conductanceCij � 1/rij), we can define the random walk on G as the Markov chain X � {X(n)}n�0

that from its current vertex v jumps to the neighboring vertex w with probability pvw �Cvw/C(v), where C(v) � ¥w:wv Cvw, and w v means that w is a neighbor of v. Itis acceptable to consider a conductance Czz from a vertex z to itself, giving rise to atransition probability from z to itself. Some notation: �a Tb denote the expected value,starting from the vertex a of the hitting time Tb of the vertex b; �a C denote the expectedvalue, starting from the vertex a, of the cover time C (C is the number of jumps neededto visit all the states in V); Rab is the effective resistance, as computed by means of Ohm’slaw, between vertices a and b.

A Markov chain is reversible if �iP(i, j) � �jP( j, i) for all i, j, where {��} is thestationary distribution and P(�, �) are the transition probabilities. Such a reversible Markovchain can be described as a random walk on a graph if we define conductances thus:

Cij � �iP�i, j�. (2.1)

542 GARCIA AND PALACIOS

We will be interested in finding a closed form expression for the commute time �0

Td � �d T0 between the origin, denoted by 0, and its opposite vertex, denoted by d.As pointed before, the transition matrix for X � {X(n), n � 0}, the SRW on the

d-cube, is doubly stochastic, and therefore its stationary distribution is uniform. If we nowcollapse all vertices in the cube with the same number of 1’s into a single vertex, and welook at the SRW on this collapsed graph, we obtain a new reversible Markov chain S �{S(n), n � 0}, a birth-and-death chain in fact, on the state space {0, 1, . . . , d}, withtransition probabilities

P�k, k � 1� �d � k

dpk, (2.2)

P�k, k � 1� �k

dpk�1, (2.3)

P�k, k� � 1 � P�k, k � 1� � P�k, k � 1�. (2.4)

It is plain to see that the stationary distribution of this new chain is the Binomial withparameters d and 1

2 . It is also clear that the commute time between vertices 0 and d is thesame for both X and S. For the latter we use the electric machinery described above,namely, we think of a linear electric circuit from 0 to d with conductances given by (2.1)

for 0 � i � d, j � i � 1, i, i � 1, and where �i � (id) 1

2d .

Theorem 2.1. For the collapsed chain S, the maximal commute time is given by

�0 Td � �d T0 � R0d � 2d �k�0

d�1 1

pk� kd�1�

. (2.5)

Proof. It is well known (at least since Chandra et al. proved it in [4]) that

�a Tb � �b Ta � Rab �z

C� z�, (2.6)

where Rab is the effective resistance between vertices a and b.If this formula is applied to a reversible chain whose conductances are given as in (2.1),

then it is clear that

C� z� � �z,

and therefore the summation in (2.6) equals 1. We get then this compact formula for thecommute time:

�a Tb � �a Tb � Rab, (2.7)

where the effective resistance is computed with the individual resistors having resistances


rij �1

Cij�

1

�iP�i, j�.

In our particular case of the collapsed chain, because it is a linear circuit, the effectiveresistance R0d equals the sum of all the individual resistances ri,i�1, so that (2.7) yields

�0 Td � �d T0 � R0d � 2d �k�0

d�1 1

pk� kd�1�

. (2.8)

�

Because of the particular nature of the chain under consideration, it is clear that �0

Td � �d T0 equals the maximal commute time (�* in the terminology of Aldous [2])between any two vertices.

(i) For simple random walk, formula (2.5) is simplified by taking all pk � 1. Thisparticular formula was obtained in [8] with a more direct argument, and it wasargued there that

�k�0

d�1 1

� kd�1�

� 2 � o�1�.

It should be clear that in this case, �0 Td � �d T0 is also the maximal commutetime for the X process. An application of Matthews’ result (see [9], Theorem 2.6),linking maximum and minimum expected hitting times with expected cover times,yields immediately that the expected cover time is �v C � �(�V�log�V�), whichis the asymptotic value of the lower bound for cover times of walks on a graphG � (V, E) (see [6]). Thus we could say this SRW is a “rapidly covered” walk.

(ii) The so-called Aldous cube (see [5]) corresponds to the choice pk � kd�1 . This

walk takes place in the “punctured cube” that excludes the origin. Formula (2.5)thus must exclude k � 0 in this case, for which we still get a closed-formexpression for the commute time between vertex d, all of whose coordinates are1, and vertex s which consists of the collapse of all vertices with a single 1:

�s Td � �d Ts � 2d �k�1

d�1 1

�k�1d�2�

. (2.9)

The same argument used in (i) tells us that the summation in (2.9) equals 2 �o(1), and, once again, Matthews’ result tells us that the walk on the Aldous cubehas a cover time of order �V�log�V�.

(iii) The choice pk � 1� kd � 1�

would be in the terminology of [5] a “slow walk”: Thecommute time is seen to be exactly equal to �V�log2�V� and thus the expected covertime is O(�V�log2�V�). In general, the SRW will be rapidly covered if and only if

�k�0

d�1 1

pk� kd�1�

� c � o�1�,


for some constant c.

Remarks.

(i) It should be noted that the maximal commute time of the S process computedabove in cases (ii) and (iii) does not necessarily coincide with the maximalcommute time for the X process.

(ii) A formula as compact as (2.5) could be easily obtained through the commute timeformula (2.7). It does not seem that it could be obtained that easily, by just addingthe individual hitting times �i Ti�1. (A procedure that is done, for instance, in [5],[10], [11], and in the next section).

3. THE COUPLING APPROACH

In this section we are going to construct a coupling that gives the rate of convergencestated in the Introduction. Our strategy will be to split Tx as Tx � Tx

1 � Tx2, where Tx

1 isa coupling time for the birth-and-death process S, and Tx

2 is another coupling time for thewhole process X, once the bivariate S process enters the diagonal, and we will bound thevalues of � Tx

1 and � Tx2.

3.1. Bounds for Tx1

For any x, y � Qd define

s�x� � �i�1

d

xi, (3.1)

d�x, y� � �i�1

d

�xi � yi�. (3.2)

Define also for the birth-and-death process S(t) � s(X(t)) its own mixing time:

��S� � inf�t; dS�t� � 1/2e�,

where dS(t) � maxi��i(S(t) � �) � �S��, and �S is the stationary distribution of S.Notice that s(Y(0)) �S since Y(0) �.

Now we will prove that �(S) � O( fS(d)), for a certain function fS of the expectedhitting times of the “central states,” and that this bound implies an analogous bound for� Tx

1. Indeed, as shown by Aldous [1], we can bound �(S) by a more convenient stoppingtime

��S� � K2�1�3�, (3.3)


where �1(3) � min� maxi minUi

�i Ui and the innermost minimum is taken over stoppingtimes Ui such that �i(S(Ui) � �) � ��. In particular,

�1�3� � min

b

maxi

minUi

b

�i Uib (3.4)

� minb

maxi

�i Tb (3.5)

� max��0 Td/2, �d Td/2� (3.6)

where the innermost minimum in (3.4) is taken over stopping times Uib such that

�i(S(Ui) � b) � 1. Expression (3.6) follows from (3.5) since we are dealing with birthand death chains. Therefore, combining (3.3) and (3.6), we have

��S� � K2max��0 Td/2, �d Td/2� :� fS�d�. (3.7)

In general, a coupling implies an inequality like (1.2); for example, we could use thenatural coupling that lets both chains evolve independently until they meet. Once theymeet, they evolve together. However, the inequality becomes an equality for a certainmaximal (non-Markovian) coupling, described by Griffeath [7]. Let Tx

1 be the couplingtime for the maximal coupling between s(X(t)) and s(Y(t)) such that

��x�S�X�t�� S�� Tx1 � t�.

Then

dS�t� � maxx�Qd

��x�S�X�t�� S�� maxx�Qd

��Tx1 � t�.

By the definition of �(S) it is clear that P(Tx1 �(S)) �

12e . Moreover, by the “submul-

tiplicativity” property (see Aldous and Fill [2], Chap. 2)

d�s � t� � 2d�s�d�t�, s, t � 0, (3.8)

we have that

P�Tx1 � k��S��

1

2ek , k � 1. (3.9)

Thus

� Tx1 � �

k�1

�

��Tx11��k � 1��S� Tx

1 � k��S��


� ��S� �k�1

�

k��k � 1��S� Tx1 � k��S��

� ��S� �k�1

�

k��k � 1��S� Tx1�

� ��S��1 � �k�2

�

k1

2ek�1� .

Since the series on the right-hand side converges, we have

� Tx1 � O� fS�d��.

3.2. Bounds for Tx2

Once the bivariate S process hits the diagonal,

D � ��x, y� � Qd � Qd; �i�1

d

xi � �i�1

d

yi�, (3.10)

we devise one obvious coupling that forces the bivariate X process to stay in D and suchthat the distance defined in (3.2) between the marginal processes does not decrease. Inwords: We select one coordinate at random; if the marginal processes coincide in thatcoordinate, we allow them to evolve together; otherwise we select another coordinate inorder to force two new coincidences. Formally, for each (X(t), Y(t)) � D, let I1, I2, andI3 be the partition of {0, 1, . . . , d} such that

I1 � �i; Xi�t� � Yi�t��,

I2 � �i; Xi�t� � 0, Yi�t� � 1�,

I3 � �i; Xi�t� � 1, Yi�t� � 0�.

Given (X(t), Y(t)) � D, choose i u.a.r. from {0, 1, . . . , d}.

(a) If i � I1:1. If Xi(t) � 1, then make Xi(t � 1) � Yi(t � 1) � 0 with probability ps(X(t))�1;

otherwise Xi(t � 1) � Yi(t � 1) � 1.2. If Xi(t) � 0, then make Xi(t � 1) � Yi(t � 1) � 1 with probability ps(X(t));

otherwise Xi(t � 1) � Yi(t � 1) � 0.(b) If i � I2:

1. Select j � I3 u.a.r.;2. Make Xi(t � 1) � Yj(t � 1) � 1 with probability ps(X(t)); otherwise Xi(t � 1) �

Yj(t � 1) � 0.(c) If i � I3:


1. Select j � I2 u.a.r.;2. Make Xi(t � 1) � Yj(t � 1) � 0 with probability ps(X(t))�1; otherwise Xi(t �

1) � Yj(t � 1) � 1.

Then, it is easy to check that (X(t � 1), Y(t � 1)) � D and d(X(t � 1), Y(t � 1)) �d(X(t), Y(t)). Moreover, noticing that �I2� � �I3� � d(X(t), Y(t))/ 2, we have

��d�X�t � 1�, Y�t � 1�� d�X�t�, Y�t�� 2 � X�t�, Y�t��

�d�X�t�, Y�t��

2d�ps�X�t�� ps�X�t��1� (3.11)

��d�X�t � 1�, Y�t � 1�� d�X�t�, Y�t�� X�t�, Y�t��

� 1 �d�X�t�, Y�t��

2d�ps�X�t�� ps�X�t��1�. (3.12)

In this case, it is straightforward to compute

m�i, s�X�t�� i � � d�X�t � 1�, Y�t � 1�� d�X�t�, Y�t�� i, X�t�, Y�t��

�i

d�ps�X�t�� ps�X�t��1�. (3.13)

Let Tx2 be the coupling time for the second coupling just described. That is, let Tx

2 �inf{t Tx

1 : d(X(t), Y(t)) � 0}. Then, as a consequence of the optional samplingtheorem for martingales we have the following comparison lemma (cf. Aldous and Fill[2], Chap. 2).

Lemma 3.1.

� Tx2 � d�X�Tx

1�, Y�Tx1�� L, �X�Tx

1�, Y�Tx1�� D, s�X�Tx

1�� s� � �i�1

L d

i�ps � ps�1�(3.14)

for all s � 0, 1, . . . , d.

Proof. Define (X�(t), Y�(t)) � (X(t � Tx1), Y(t � Tx

1) for all t � 0. Define Z(t) �d(X�(t), Y�(t)) and F t � �(X�(t), Y�(t)). Then, it follows from (3.13) that

m�i, s� � i � E Z�1� � Z�0� � i, s�X��0�� s�. (3.15)

Also, for all s � {1, . . . , d � 1}, 0 � m(1, s) � m(2, s) � . . . � m(d, s). Fix s �{1, . . . , d � 1} and write

h�i� � �j�1

i 1

m�i, s�(3.16)


and extend h by linear interpolation for all real 0 � x � d. Then h is concave and forall i � 1

� h�Z�1�� Z�0� � i, s�X��0�� s� � h�i � m�i, s��

� h�i� � m�i, s�h��i�

� h�i� � 1,

where the first inequality follows from the concavity of h and h� is the first derivative ofh. Now, defining h such that

h�i� � 1 � �j

� h�Z�1�� Z�0� � i, s�X��0�� s�h� j� � h�i� (3.17)

and

M�t� � t � h�Z�t�� u�0

t�1

h�Z�u�� (3.18)

we have that M is an F t-martingale, and, applying the optional sampling theorem to thestopping time T0 � inf{t; Z(t) � 0}, we have

� M�T0� � Z�0� � i, s�X��0�� s� � � M�0� � Z�0� � i, s�X��0�� s� � h�i�. (3.19)

Noticing that M(T0) � T0 and T0 � Tx2, we obtain the desired result. �

Since s(X(t)) is distributed as �S, we can write


1�, Y�Tx1�� L, �X�Tx

1�, Y�Tx1�� D� � �

s�0

d

�S�s� �i�1

L d

i�ps � ps�1�:� g�d�.

(3.20)

Putting the pieces together, we have found a coupling time Tx for the whole processsuch that

� Tx � fS�d� � g�d�.

The task now is to find explicit bounds for fS(d) and g(d) for particular workable cases.To avoid unnecessary complications, we will assume d � 2m, and compute only the

hitting times for the S process of the type E0 Tm. Hitting times in birth-and-deathprocesses assume the following closed-form (see [11] for an electrical derivation):

�k Tk�1 �1

�kP�k, k � 1� �i�0

k

�i, 0 � k � d � 1,


and in our case this expression turns into

�k Tk�1 �¥i�0

k � id�

� kd�1�pk

.

Therefore,

�0 Tm � �k�0

m�1¥i�0

k � i2m�

� k2m�1�pk

. (3.21)

(i) In case all pk � 1, we have the simple random walk on the cube, and it turns outthere is an even more compact expression of (3.21), namely,

�0 Tm � �k�0

m�1¥i�0

k � i2m�

� k2m�1�

� m �k�0

m�1 1

2k � 1. (3.22)

as was proved in [3], and the right-hand side of (3.22) equals m[H(2m) � 12 H(m)], where

H(n) � 1 � 12 � . . . � 1

n , allowing us to conclude immediately that in this case �0 Tm ��0 Td/ 2 � d

4 log d � d4 log 2.

Also, we have


1�, Y�Tx1�� L, �X�Tx

1�, Y�Tx1�� D� �

d

2p �i�1

L 1

i�

d

2pO�log L�. (3.23)

Thus in this case both fS(d) and g(d), and a fortiori � Tx and �, are O(d log d).(ii) For the Aldous cube, pk � k

d�1 , and (3.21) becomes (recall this cube excludes theorigin):

�1 Tm � �k�1

m�1¥i�0

k � i2m�

� k�12m�2�

� �k�0

m�2¥i�0

k � i2m�

� k2m�2�

� �k�0

m�2 �k�12m �

� k2m�2�

. (3.24)

After some algebra, it can be shown that the second summand in (3.24) equals (2m �1)[H(2m � 1) � 1/m], and the first summand can be bounded by twice the expressionin (3.22), on account of the fact that ( k

2m�1) � 2( k2m�2), for 0 � k � m � 1. Therefore,

we can write

�1 Td/2 �3

2d log d � smaller terms,

thus improving by a factor of 12 the computation of the same hitting time in [5].

Also, we have



1�, Y�Tx1�� L, �X�Tx

1�, Y�Tx1�� D, s�X�Tx

1�� s� � �i�1

L d�d � 1�

i�2s � 1�.

(3.25)

Thus, in this case


1�, Y�Tx1�� d, �X�Tx

1�, Y�Tx1�� D�

� �s�1

d

�S�s� �i�1

d d�d � 1�

i�2s � 1�

� ��d

3 � �i�1

d d�d � 1�

i� �1 � ��

d

3 �� i�1

d d�d � 1�

i�2d/3 � 1�

� e�d/9d�d � 1�log d � �1 � e�d/9�d log d. (3.26)

And so � � O(d log d) also in this case.(iii) Slower walks. Consider the case when the probability pk grows exponentially in

k, more specifically

pk � � k � 1

n � 1�

(3.27)

with 1. In this case, it seems that (3.21) is useless to get a closed expression for �0

Td/ 2. However, Graham and Chung [5] provide the following bound:

�i Td/2 � c0��d, for all d � d0��, 0 � i � d (3.28)

where c0() and d0() are constants depending only on . Moreover, (3.20) becomes

g�d� � �s�0

d

�S�s� �i�1

d d�d � 1�

i��s � 1� � s�

� d�d � 1� �s�0

d�S�s�

�s � 1� � s �i�1

d 1

i

� d�d � 1�log d �s�0

d�S�s�

�s � 1� � s

� d�d � 1�log d �s�0

d�S�s�

�s � 1�

� d�d � 1�log d� 1

�1 � X�� , (3.29)

where X is a binomial (d, 12) random variable. Except for � 1, the exact value of the


expectation in (3.30) does not seem easy to work out. We can find, however, a simpleupper bound (Jensen’s inequality gives a lower bound of the same order) as follows:

� 1

�1 � X�� 1

�1 � X� 1 X�d/3�� 1

�1 � X� 1 X�d/3�� X d/3� �

1

�1 � d/3� � X � d/3�

� ��d/3� �1

�1 � d/3� � O�1/d�. (3.30)

Combining (3.30) with (3.29) we get that g(d) � O(d log d). This fact together with(3.29) allows us to conclude that � � O(d) in this case, thus improving on the rate ofthe mixing time provided in [5] by a factor of log d.

ACKNOWLEDGMENTS

This paper was initiated when both authors were visiting scholars at the StatisticsDepartment of the University of California at Berkeley. The authors wish to thank theirhosts, especially David Aldous with whom they shared many fruitful discussions. Thiswork was partially supported by FAPESP 99/00260-3.

REFERENCES

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[6] U. Feige, A tight lower bound on the cover time for random walks on graphs, Random StructAlg 6 (1995), 433–438.

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[8] L. Lovasz and P. Winkler, “Mixing times,” DIMACS Series in Discrete Mathematics andTheoretical Computer Science, 41, American Mathematical Society, Providence, RI, 1998.

[9] P. C. Matthews, Covering problems for Brownian motion on spheres, Ann Probab 16 (1988),189–199.

[10] J. L. Palacios, Another look at the Ehrenfest urn via electric networks, Adv Appl Probab 26(1994), 820–824.

[11] J. L. Palacios and P. Tetali, A note on expected hitting times for birth and death chains, StatProbab Lett 30 (1996), 119–125.


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On mixing times for stratified walks on the d-cube