Journal of Theoretical Probability, Vol. 5, No. 3, 1992

Expected Cover Times of Random Walks on Symmetric Graphs

Jos~ Luis Palaeios l

Received July 5, 1991

We give very simple proofs for an (N- 1)HN_I lower bound and a n N 2 upper bound for the expected cover time of symmetric graphs.

KEY WORDS: Cover time; symmetric graph.

1. I N T R O D U C T I O N

A simple random walk on a finite connected undirected graph, G = (V, E) is the Markov chain X., n i> 0, that from its current vertex v jumps to one of the d v neighboring vertices with uniform probability. The hitting time Tv of the vertex v is the minimum number of steps the random walk takes to reach that vertex: T~ = inf{n ~> 0: X~ = v}. The cover time C is the minimum number of steps the random walk takes to visit all verticesSn the graph, i.e., C=max~ T~. (See Ref. 1 for historical details.) Unless otherwise stated we take I V[ = N.

We give here very simple proofs for an ( N - 1 )Hu_ i lower bound and a n N 2 upper bound for the expected cover time of symmetric graphs. The lower bound extends similar results of Devroye and Sbihi (Ref. 5); the upper bound is a special case of a result proved by Kahn, Linial, Nisan, Saks (Ref. 6).

The tools used are a fundamental result linking the cover time and the maximum and minimum hitting times, obtained by Matthews in Ref. 7, and a closed form formula for the sum of all hitting times EiTj for ( i , j ) e E proved in Ref. 8. These results are given as Lemmas l and 2, respectively, in the next section.

Department of Mathematics, New Jersey Institute of Technology, Newark, New Jersey 07102.

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0894-9840/92/0700-0597506.50/0 ~C@ 1992 Plenum Publishing Corporation

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2. B O U N D S FOR COVER TIMES OF S Y M M E T R I C GRAPHS

Lemma 1. For a simple random walk on a connected graph with vertex set V= {Vl,...,VN} the following inequalities hold:

HN_ 1 min rain EiTj~Ev, C<~HN_ l max max Ei l ) (2.1) 2<~j<~N l<~i<~N 2<.j<~N l<~i<~N

where EiTj is the expected time to reach vj starting from v~ and HN_ 1 is the ( N - 1 ) t h harmonic number.

Lemma 2. following holds:

For a simple random walk on the graph G = (V, E) the

EiTj= 21EI (N-1) (2.2) i , j : ( i , j ) ~ E

Notice that there are 21E] summands in the left-hand side of (2.2), since both E~Tj and EjTe appear in the summation for every (i, j )~ E. Now if we can guarantee that all terms in the left-hand side of (2.2) 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 Ref. 3, 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 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 ~ of G, ~( i )= j . Two edges (il,jl) and (i2, J2 ) are called similar if there is an automorphism a of G such that ~({il, j , } ) = {i2, J2}- A graph is vertex-symmetric if every pair of vertices are similar; it is edge-symmetric if every pair of edges are similar; and it is symmetric if it is both vertex-symmetric and edge-symmetric.

Numerous familiar graphs are symmetric, such as the complete graph KN, the complete bipartite graph KN, N, the cycle graph, the unit cube, and all distance-transitive graphs. For all these, due to the simple observation above we can say that EiTj= N - 1 if (i, j ) ~ E . This implies, for example, that for the complete graph KN, E~Tj = N - 1 for every pair i, j, so the maximum and minimum hitting times are both N - 1 and therefore Eq. (2.1) applied to this example yields E~C= (N-1)HN_~ for all ve V. More generally, we can state the following:

Theorem 1. For a symmetric graph we have

(i) EiTj=N- 1 if (i , j)eE (ii) minEvC>~(N-1)HN_I

v

Expected Cover Times of Random Walks 599

3 N ( N - t ) (iii) max Ei Tj ~< - -

;,j d

where d is the common degree of all vertices.

(iv) If d = O(N) then E~C= O(Nlog N) for all v ~ V

(v) maxE~C<~2(N-1) 2 v

Proof (i) and (ii) follow from the comments above and Eq. (2.1). (iii) is proved noting that

EiTi~< ( N - 1) 6(i,j) (2,3)

where 6(i, j) is the distance between vertices i and j. If we define the diameter of the graph as A =maxi, y6(i,j), it is well known that for a regular graph the following inequality holds:

3N d < ~ - (2.4)

d

and thus inserting (2.4) into (2.3) 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 {V=Vo, vl . . . . . l ) U _ l } , and a sequence of vertices {v = wo, wl,..., wM}, with M~< 2 ( N - 1) describing a certain way to traverse all vertices in the spanning tree. Clearly EvC is bounded by the time to visit all vertices in the order Wo, wl ..... WM, i.e., by the sum Ewo Tw, + Ew, T,,~ + ... + EwM_, Twg. But this sun contains at most 2 ( N - 1 ) summands, each of which equals N - 1 , and we are done. []

Comments. In Ref. 5, (i) and (ii) are proved, with a more involved argument, for distance-regular graphs. 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 Ref. 3 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 Ref. 3), and therefore our results extend those in Ref. 5.

In Ref. 4 it is shown that there is a sharp threshold for the cover time of d-regular graphs at d=N/2: For d>~N/2 the cover time is O(Nlog N), whereas for d<~ N / 2 - 1 there are examples for which the cover time is O(N); our (iv) above shows that, if there is such a threshold for symmetric graphs, it must be considerably smaller.

Since all symmetric graphs are regular, (v) is a weaker version of Kahn et al.'s result (see Ref. 6), although their constant is 4 and ours is 2. Moreover, it seems plausible that every symmetric graph contains a

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Hamil tonian path, and if that is the case a revision of our proof shows that the constant can be reduced to 1. The order N 2, however, is optimal since the cover time for the cycle graph on N vertices is (2w). Our simple proof of (v), based on the fact that EiT~ is linear in N for an edge (i, j), cannot be extended to regular graphs as the following examples shows: Let G be the graph obtained from a K2N by deleting edges (vi, vi+l), 1 ~ i < ~ 2 N - 3 and adding a new vertex v and new edges (vi, v), 1 ~< i<<,2N-2; take a copy G' of G and let H be the union of G and G' plus the edge (v~ v') (v' is the copy of v); then (v, v') is a cut-edge of the 2 N - 1-regular graph H so that (see Ref. 2) EvT~,= IEI =o(Igl2). Incidentally, K a h n et al.'s p roof is based on an erroneous formula for W(E), whose right expression is Eq. (2.2) above; the error stems from Corol lary 1 in Ref. 6, which is also false and should read instead:

E E i T j = 2 1 E I - d i (2.5) j : ( i , j ) ~ E

Their bound holds, however, because all they need in their a rgument is the fact that W(E) <. 2[VI IEI.

A final comment : We have seen that the graph being symmetric implies that all hitting times EiTj with ( i , j ) e E are equal to N - 1 . Conversely, from Eq. (2.5) we see that if E, T j = N - 1 for ( i , j ) e E then di = 2 IE[/N for all i, so at least we can assert that the graph is regular.

A C K N O W L E D G M E N T

Thanks are due to Professor S. Northshield for useful discussions.

R E F E R E N C E S

1. Aldous, D. J. (1989). An introduction of covering problems for random walks on graphs. J. Theor. Prob. 2, 87-89.

2. AMiunas, R., Karp, R. M., Lipton, R. J., Lovasz, L., and Rackoff, C. (1979). Random walks, universal traversal sequences, and the complexity of maze problems. In 20th Annual Symposium on Foundations of Computer Science. Association for Computer Machinery, San Juan, PR, pp. 218-223.

3. Buckley, F., and Harary, F. (1990). Distance in Graphs. Addison-Wesley, New York. 4. Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R., and Tiwari, P. (1989) . The

electrical resistance of a graph captures its commute and cover times. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing. Association for Computer Machinery, Seattle, WA, pp. 574-586.

5. Devroye, L., and Sbihi, A. (1990). Random walks on highly symmetric graphs. 9". Theor. Prob. 3, 497-514.

6. Kahn, J. D., Linial, N., Nisan, N., and Saks, M. E. (1989). On the cover time of random walks on graphs, or. Theor. Prob. 2, 121-128.

7. Matthews, P. (1988). Covering problems for Markov chains. Ann. Prob. 16, 12t5-1228. 8. Palacios, J. L. (1990). On a result of Aleliunas et al. concerning random walks on graphs.

Prob. Eng. Info. Sci. 4, 489-492. Printed in Belgium Verantwoordelijke uitgever: Hubert Van Maele Altenastraat 20- B-8310 St.-Kruis