Expected Hitting and Cover Times of Random Walks on Some Special Graphs

Josk Luis Palacios* Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07 7 02

ABSTRACT

We give general bounds (and in some cases exact values) for the expected hitting and cover times of the simple random walk on some special undirected connected graphs using symmetry and properties of electrical networks. In particular we give easy proofs for an N - l H N - , lower bound and an N2 upper bound for the cover time of symmetric graphs and for the fact that the cover time of the unit cube is O(NlogN). We give a counterexample to a conjecture of Friedland about a general bound for hitting times. Using the electric approach, we provide some general upper and lower bounds for the expected cover times in terms of the diameter of the graph. These bounds are tight in many instances, particularly when the graph is a tree. 0 1994 John Wiley & Sons, Inc.

Key Words: hitting time, cover time, effective resistance

1. INTRODUCTION

A simple random walk on a finite connected undirected graph, G = (V, E ) , is the Markov chain X,,, n L 0, that from its current vertex u jumps to one of the A(u) neighboring vertices with uniform probability. Throughout the paper, we shall assume for the vertex set V that IVI = N . The hitting time T , of the vertex u is the minimum number of steps the random walk takes to reach the vertex: T , =

in€{ n 2 0: X,, = u } . The cover time C is the minimum number of steps the random walk takes to visit all vertices in the graph, i.e., C = max, T,. The commute time between vertices i and j is E,T, + EjTi . (See [l] for historical details).

We give here bounds (and in some cases exact values) for the expected hitting

* Present address: Departamento de Matemiticas, CESMA, Apartado 89,000, Caracas, Venezuela.

Random Structures and Algorithms, Vol. 5 , No. 1 (1994) 0 1994 John Wiley & Sons, Inc. CCC 1042-9832/94/010173-10

173

174 PALACIOS

and cover times of some special graphs, as well as some other general bounds, using elementary techniques. Our results improve and/or simplify considerably the proofs of previously known results. In Section 2 we apply the electric approach in order to easily compute some cover times. In Section 3 we state an important result due to Matthews. In Section 4 we find through a symmetry argument exact values and upper and lower bounds for hitting and cover times on symmetric graphs, In Section 5 we apply the foregoing ideas to the lollipop graph and the d-dimensional unit cube; in particular, we provide a counterexample to a conjecture of Friedland [lo] and give a simple argument showing that the cover time of the unit cube is O(N log N ) . Finally in Section 6 we give general bounds for cover times in terms of the diameter of the graph. The bounds are tight in many instances, particularly when the graph is a tree.

2. THE ELECTRIC APPROACH

There is a strong connection between random walks on graphs and electric networks, spelled out beautifully in the monograph of Doyle and Snell [9] (see also references therein). We will use the following fact of this electric analog involving the “escape probability” P,(TZ > T,) and the effective resistance R,, between a and b when every edge is considered to be a unit resistor:

1

Here d(a) is the degree of a and T,’ = inf{n > 0: X , = a } . Using renewal theory (see [6] and [18], for instance) or some other probabilis-

tic argument [13], one can prove the following equation for the commute time between a and b:

From (1) and ( 2 ) we get then

E,T, + E,T, = 21EIRa, , (3)

If we can assure further that E,T, = E,T,, for instance, under some symmetry assumption, then (3) simplifies to En Tb = /El R,, .

This simple formula involving the effective resistance between a and b (derived directly by Chandra et al. [7]) enables one to compute the commute time (and hitting time in the presence of symmetry) not only of a single vertex but also of a set of vertices because one can short together those vertices (they have “zero voltage,” see [9], p. 53) into a single vertex. Moreover, one can short together all vertices sharing the same potential, a fact that simplifies things considerably.

The examples that follow use the above ideas and the fact that the effective resistance of a set of resistors is: (i) the sum of the individual resistances in case the resistors are in series and (ii) the inverse of the sum of the inverse individual resistances in case the resistors are in parallel.

Examples in Sections 2.1 and 2.2 are solved in [20] using difference equations.

COVER TIMES OF RANDOM WALKS 175

2.1. The Cycle Graph

Assume j vertices have already been visited and the current vertex is a as shown in Figure l(a). To find the hitting time of the next new vertex, we short those remaining N - j vertices into a single vertex b and get the graph shown in Figure

-1- By symmetry E,Tb = EbTa, and for this graph (El = 1 + j and R,, - 1 + 1 / ; - &, so that ( 3 ) yields

and therefore

N N - 1

; = 1

2.2. The Chain Graph

Starting from an arbitrary j , in order to cover all vertices, the walk must (i) first hit either 1 or N and (ii) hit the opposite vertex. The expected time to do ( i ) is computed via the graph in Figure 2(b) so that

l a l b

Fig. 1. (0) Visited; (0) not visited.

2 a

Fig. 2.

2 b

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b

Fig. 3. (0) Visited; (0) not visited.

and the expected time to do (ii) is given via Figure 2(a), yielding the value I EIR,, = ( N - so that

E ~ C = ( N - j ) ( j - I) + ( N - 112

2.3. The Star with m k-Long Arms

Starting from the center, in order to cover the whole tree the walk must cover all leaves. Given that m - j leaves have been visited, in order to visit a new leaf, the walk must return to the center c and visit one among j leaves. Thus we need the commute time between the center and a group of j leaves, which are shorted into the single vertex b , as shown in Figure 3:

Thus starting from the center, the time to visit all leaves and return to the starting point is

‘7l 1 j = l I

2mk’ 2 7 = 2k2m log m .

and the cover time is (subtracting one return to the center)

“ 1 E,C= (Zm c 7 - I) k2

j = 1 I

3. MATTHEWS’ RESULT

A fundamental result linking the cover time and the maximum and minimum hitting times was obtained by Matthews [12]. We shall use here the version presented in [IS], to where we refer the reader interested in a concise proof of the result:

Theorem 3.1. V = { u, , . . . , u N } the following inequalities hold:

For a simple random walk on a connected graph with vertex set

min min EiTj 5 E,,Cf HN- , max max EiTj , H N - l 2 5 1 5 N l s i 5 N 2 5 j s N 1 I i S N

COVER TIMES OF RANDOM WALKS 177

where EiTj is the expected time to reach u j starting from ui and H N - , is the ( N - 1)th harmonic number.

This result is the key to find easy-to-obtain bounds for the expected cover time as we shall show in the next sections.

4. SYMMETRY

The results of this section are taken from [17]. It is not difficult to prove (see [15] and [19]) using elementary facts that the

following equality holds for the expected hitting times of a random walk on a connected graph:

c E,Tj = 2 ) E ) ( N - 1) i , j : ( i , if€€

(4)

Notice that there are 21EI summands on the left-hand side of (4), since both E,T, and E,T, appear in the summation for every (i, j ) E E. [In fact, (4) is saying that the sum of all commute times for adjacent vertices is 2(EI(N - l ) . ] Now, if we can guarantee that all terms on the left-hand side of (4) are equal, then their common value must be N - 1. We can guarantee that fact if all the edges of the graph are identifiable with one another via a suitable relabeling of the vertices, which is precisely the concept of symmetry in graph theory. We give a brief review of the concept as presented in [ 5 ] , to which we refer the reader for more details.

An automorphism of a graph G is an isomorphism of G with itself. Thus each automorphism LY of G is a permutation on the vertex set V , which preserves adjacency. Two vertices i and j of G are similar if for some automorphism LY of G, a(i) = j . Two edges el = ( i , , j , ) and e2 = ( i2 , j 2 ) are called similar if there is an automorphism LY of G such that a ( { i l , j , } ) = { i 2 , j 2 } . A graph is vertex-symmetric if every pair of vertices are similar; it is edge-symmetric if every pair of edges are similar; and its is symmetric if it is both vertex-symmetric and edge-symmetric.

Numerous familiar graphs are symmetric, such as the complete graph K,, the complete bipartite graph K N , N , the cycle graph, the unit cube (see Section 5.2), and all distance-transitive graphs. For all these, due to the simple observation above, we can say that E,T, = N - 1 if ( i , j ) E E. This implies, for example, that, for the complete graph K N , EST, = N - 1 for every pair i, j , so that the maximum and minimum hitting times are both N - 1, and therefore by Matthews’ result E,C = ( N - l)HNL1 for all u E V. (This is the well-known formula for the coupon collector’s problem.) More generally, we can state the following:

Proposition 4.1. For a symmetric graph we have

( i ) E,Tj = N - 1 if ( i , j ) E E , ( i i ) min, E,C 2 ( N - l ) H N - , ,

(iii) maxi,j EITl 5 3N(N - l ) i d , where d is the common degree of all vertices, ( i v ) if d = O ( N ) , then E,C = O ( N log N ) for all u E V, ( v ) max, E,C 5 2 ( ~ - I)*.

PALACIOS 178

Proof. proved noting that

(i) and (ii) follow from the comments above and Theorem 3.1. (iii) is

E,Tj 5 ( N - 1)6(i, j ) , ( 5 )

where 6(i, j ) is the distance between vertices i and j . If we define the diameter of the graph as A=max,,j6(i, j ) , it is well known that for a regular graph the following inequality holds:

and thus inserting (6) into (5) yields (iii). Now (ii) and (iii) together with the hypothesis d = O ( N ) imply (iv). To prove (v), we consider a spanning tree of the given graph with vertices u = u,,, u l , . . . , u N - l , and a sequence of vertices u = w,,, wl, . . . , w M , with M s 2(N - 1) describing a certain way to traverse all vertices in the spanning tree. Clearly E,C is bounded by the time to visit all vertices in the order w l , w2,. . . , w M , i.e., by the sum Ew1Tw2+ Ew2Tw3 + .. . E w M - I T W y . But this sum contains at most 2(N - 1) summands, each of which equals N - 1, and we are done.

Comments. In [8], they prove (i) and (ii) for distance-regular graphs with a more involved argument. Although distance-regularity places no restrictions on the automorphism group of G, most distance-regular graphs are also distance- transitive and therefore symmetric; for instance, all cubic distance-regular graphs but one are distance-transitive (see [5] for more details). On the other hand, not all symmetric graphs are distance-regular (see, for example, the graph shown in Fig. 8.8, p. 167 in [ 5 ] ) , and therefore our results extend those in [8].

Since all symmetric graphs are regular, (v) is a weaker version of Kahn et al.’s result (see [ll]), although their constant is 4 and ours is 2. Moreover, it seems plausible that every symmetric graph contains a Hamiltonian path, and, if that is the case, a revision of our proof shows that the constant can be reduced to 1. The order N2, however, is optimal as Section 2.1 shows. Incidentally, Kahn et al.’s [I l l proof is based on an erroneous formula for W ( E ) , the sum of all commute times for adjacent vertices, whose right expression is formula (4) in this paper; the error stems from Corollary 1 in [11], which is incorrect and should read instead:

2 EiTj = 2)EI - di . j : ( i , j )€E

(7 )

Their bound, however, seems to be correct, because all they need in order to prove it is the fact that W ( E ) 5 2 / V / IEl.

5. TWO EXAMPLES

We apply the results of the previous sections to two well-known examples.

COVER TIMES OF RANDOM WALKS 179

5.1. The lollipop Graph

The m-lollipop graph on N vertices is a complete graph on m vertices one of which-call it 6-is attached to a chain graph on N - m vertices. If c is the endpoint of the chain part and a is any vertex in the complete part with a # b, then it is clear that

E, T, = E, T , + E, T, ,

and from (3) we also have

E,T, + E,T, 2(N - m)’ + ( N - m)m(m - 1 ) . (9)

But since E,T, = ( N - m)’, as we saw in example 2.2, we insert this into (9) to get

E, T, = ( N - m)(N + m2 - 2m) . (10)

Finally, since E,T, = m - 1 as seen in Section 4, we insert this result and (10) into (8) to get

E,T, = m - 1 + ( N - m ) ( N + m2 - 2 m ) . (11)

It is worthwhile to compare the simple derivation of (11) above with the brute force approach as in [14]. The maximum of this expression occurs roughly for m = 2N/3, for which the value of the hitting time is 4N3/27 - N 2 / 9 . This example shows that hitting times can be cubic in N , and indeed, as Brightwell and Winkler show in [4], this is the maximum possible value of a hitting time over all graphs on N vertices. Similarly one can show for the m-lollipop graph that the commute time is [2(N - m) + m(m - l ) ] (2/m f N - m), an expression that also is maxi- mized roughly when m = 2N/3 (see [13]), although the maxima for the one-way hitting time and the commute time do not necessarily occur for the same m. Finally, the 3-lollipop graph on 8 vertices provides a counterexample to a conjecture proposed by Friedland (see [lo], where he proves the conjecture for the special case of indecomposable graphs), stating that, for an undirected connected graph, EiTj does not exceed IEI(N- 1). For the 3-lollipop on 8 vertices ( E ( ( N - 1) = 56, whereas E,T, = 57, as given by formula (11).

5.2. The &Dimensional Unit Cube

The vertex set of the d-dimensional unit cube is the set of all d-tuples of 0’s and 1’s. Two vertices are neighbors if they differ in only one coordinate. As mentioned in the previous section, this is a symmetric graph for which E,Tj = N - 1 in case (i, j ) E E , so the value of the left-hand side bound in Theorem 3.1 is

In order to find the maximum value of the hitting time between two vertices, we

180 PALACIOS

Fig. 4.

notice that if we start at any vertex that we relabel as the “origin” (0, . . . , O), the vertex it takes longest to hit is ( 1 , . . . , 1). Now if we apply a unit voltage between these two vertices so that the voltage at (0 , . . . , 0) is 1 and the voltage at (1, . . . , 1) is 0, then all vertices having the same number of 1’s share the same voltage and can be shorted. Figure 4 shows the effect of doing this on the 3-D cube.

In general, what we obtain is a new graph with d + 1 vertices, where the kth new vertex consists of the shorting of all vertices in the unit cube with k 1’s. Since every vertex in the unit cube with k 1’s is connected to d - k vertices with ( k + 1 ) l’s, there are (d - k) (;I) = d( dil ) resistors between vertex k and k + 1 in the new graph, 0 5 k 5 d - 1. Then formula (3) and symmetry give us

d - 1 A

1 E ( O ,..., 0) T ( 1 ,.._. 1 ) = EOTd = IEIROd = d2d-’ zo d(“,’)

=(2+o(1))2d- ’ = ( l + o ( l ) ) N .

Therefore, the right-hand side bound in Theorem 3.1 is H N - , ( l + o(l))N, and this together with (12) implies that E,C = O(N log N ) for any vertex u in the unit cube.

6. BOUNDS FOR COVER TIMES IN TERMS OF THE DIAMETER

Using the electrical approach [in fact, the proof hinges on formula (3) and Theorem 3.11 Chandra et al. [7] proved very useful bounds in terms of R, the electrical resistance of the graph, defined as the maximum effective resistance between any pair of vertices:

Theorem 6.1. vertices. Then

Let R = maxi,j R, be the electrical resistance of a graph on N

I E I R 9 C , 9 ( 2 + o( 1))l E I R log N . (13)

When applied to these inequalities, Raleigh’s monotonicity law [9] yields nice general bounds in terms of the diameter A of the graph. Indeed, according to the monotonicity law, the effective resistance of any part of the graph is decreased (resp. increased) whenever the resistance of any edge of the graph is decreased

COVER TIMES OF RANDOM WALKS 181

(resp. increased). In particular, the effective resistance can only decrease when we add more edges to the original graph. It is clear then that R 5 A, since the whole graph can be considered as built up from the subgraph that corresponds to the diameter. In the case of a tree, the equality R = A is attained because in a tree there is only one possible path between vertices a and b, so that the effective resistance R,, is the sum of the unit resistances (connected in series) in that unique path.

These observations applied to Theorem 6 . 1 yield:

Theorem 6.2. For any graph the following inequality holds:

C, I (2 + o( 1))l E \ \ A log N .

For any tree the following inequalities hold:

( N - 1)A 5 C, 5 (2 + O( 1 ) ) ( N - 1)A log N . (15)

Comments. These results were arrived at in [16] using a more involved argu- ment; they are similar to those of Zuckerman’s [21] except that his constant e is improved to 2.

For trees, the bounds in (15) are tight in many instances: the upper bound is (4 + o(l))(N - 1)k log N for the star discussed in Section 2.3, and

for the b-ary tree of height m. Both these results are tight within a constant factor of 2 (see [2]). Also, the lower bound is tight for the chain graph (Section 2.2).

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