SUNY Plattsburgh
Digital Commons @ SUNY Plattsburgh
Mathematics Faculty Publications
Mathematics
11-1995
On the Commute Time of Random Walks on
Graphs
Sam Northshield
SUNY Plattsburgh
José Luis Palacios
Follow this and additional works at: http://digitalcommons.plattsburgh.edu/mathematics_facpubs
Recommended Citation
Northshield, S., & Palacios, J. (1995). On the Commute Time of Random Walks on Graphs. Brazilian Journal of Probability and
Statistics, 9 (2), 169-175.
This Article is brought to you for free and open access by the Mathematics at Digital Commons @ SUNY Plattsburgh. It has been accepted for inclusion
in Mathematics Faculty Publications by an authorized administrator of Digital Commons @ SUNY Plattsburgh.
Reviata Brasileira de PTobabilUade e Estatistica (1995), 2, pp.169-175.
©Associaqao Brasilcira de Estati'stica
ON THE COMMUTE TIME OF RANDOM WALKS ON
GRAPHS
Sam Northshield
Department of Mathematics
State University of New York at Plattsburgh, USA
Jose Luis Palacios
Departamento de Matemdticas
Universidad Simon Bolivar
Caracas, Venezuela
Summary
Given a simple random walk on an undirected connected graph, the commute
time of the vertices x and y is defined as C(x,y) = ExTy + EyTx. We give a new
proof, based on the optional sampling theorem for martingales, of the formula
C(x,y) = wyWy x) i" terms of the escape probability e ( y , x ) (the probability
that once the random walk leaves x, it hits y before it returns to x) and the
stationary distribution TT(-). We use this formula for C(x,y) to show that the
maximum commute time among all barbell-type graphs in N vertices is attained
for the lollipop graph and its value is O(~j-).
Key words: Commute time; cover time; escape probability; lollipop graph.
1
Introduction
A simple random walk {^fn}n>o on a finite connected undirected graph,
G = (V, E) is the Markov chain that from its current vertex v j u m p s to
one of the d(v) neighboring vertices with uniform probability. We define
the hitting time of the vertex y as Ty = min{A; > 0 : Xk = J/}, and the
commute time of the vertices x and y as C(x,y) = ExTy + EyTx, whore /•,',
denotes "expected value given that XQ = zn.
In their seminal paper, Aleliunas et al. (1979) established that if x and
y are adjacent, then C(x,y) < 1\E\ with equality if and only if jry is a.
cut edge of G. They also noted that this fact implies by i n d u c t i o n t h a t
C(x,y) < 2\E\d(x, y), where d(x,y) is the distance 1 between x ;uid y. In
Palacios (1990), there is a. simple proof of the fact that the sum of the
108
Northshield and Palacios
170
commute times over all adjacent vertices equals 2\E\(N — 1). A formula
for C(x,y) that improves the inequalities in Aleliunas et al. (1979) and is
expressed in terms of the effective resistance of the graph, can be found
for instance in Chandra et al. (1989) or Palacios (1993); this formula
can also be expressed in terms of the escape probability, either by direct
proof (see Burdzy, 1990) or as a consequence of interpreting the escape
probability as the effective conductance of the graph, when thought of as
an electric network (see Doyle and Snell, 1984). In this note we use a
discrete version of Ito's formula and the optional sampling theorem for
martingales in order to give a new proof of the formula for C(x,y), x and
y not necessarily adjacent, in terms of the escape probability of x and y
and the stationary distribution of the walk. The method of proof may be
valuable in providing new insights into the analysis of random walks on
graphs. This formula enables us to compute exact values for C(x,y) in
many instances. In particular, we use it to find that the expected cover
time of the random walk on 2"n is (™), and to show that the maximum
commute time over all barbell-type graphs on N vertices is attained for
the lollipop graph composed of a clique of ^ vertices attached to a tail of
y vertices; the value of this maximum is O(^-), agreeing with a result of
Brightwell and Winkler (1990) concerning hitting times.
On the commute time of random walks on graphs
171
Theorem 2.1 (Ito's formula) For any real function f we can write
f(Xn) - f ( X 0 ) = Mn
k=o
where Mn is a centered martingale.
Proof: Let fk = <r(X0,Xi, . . .,Xk). We observe that:
f(Xn)-f(X0)
=
A-=0
k=0
The first term is a martingale and the second term is
k-O
2
The formulas
First we introduce some notation: Xk denotes the Ar-th step of the walk,
and P = {P(x,y)}xyev its transition matrix; P defines a linear operator
on the set of real-valued functions:
because
E(f(Xk+1)\fk)
- f(Xk) = EXk(f(Xk+l))
Theorem 2.2 For x,
- f(Xk] = Af(Xk).
y: C(z,y) = ——-,--.
ir(y)c(y,x)
Proof: Let fy(x) = ExTy. If x ^ y, then
and so does the Laplacian A = P — I (I is the identity); Tx = m'm{k > 0 :
X/c = x} is the hitting time of x, and T* = m\n{k > 0 : Xk = x}. Note
the main difference between Tx and T+:
ExTy = EXT+ if x ^ t/ v whereas 0 = EyTy < EyT+ = ^y, where
n(x) — ^fej is the stationary distribution of the walk.
We also define Nff(x) = 23£=o l-{r}(^t); finally we define the escape
probability of x and y as
Hence
Also
1
E,TV) = 1 + Pfy(y)
where the last equation holds since fy(y)
i.e., the probability that once the walk leaves x, it hits y before it returns
to x.
A/„(*) = - I |
- 0. Thus
= 1 + A /„
a
Northshield and Palacios
172
On the commute time of random walks on graphs
By Ito's formula, for any a, fy(Xa] = f y ( X 0 ) + Ma - a +
j . If a is
a stopping time, then Ma is a centered random variable (optional sampling
theorem) and we have
173
Since TT = 1, an application of Theorem 2.2 yields
ExTy + EyTx = 2d(x,y)(n - d ( x , y ) ) .
Part (a) follows from symmetry.
Let (Tk be the first time that a total of k + 1 vertices are hit. (Set
<T_I = 0 ) . E(o~k -CTfc_i)= E0T, where T is the hitting time of the set
{/:, n - 1}. This equals the expected time the walk on Zjt+i started at 0
needs to hit k. By part (a), this number is k. Hence
Letting a = Tx, we get
ExTy
=
—EyTx
But No(y) has a geometric distribution with parameter e(y, x) and
= ety x\i finishing the proof.
O
Corollary 2.1 If xy G G then C(x,y) < 2\E\ with equality if and only if
xy is a cut edge of G.
n-l
n-l
k=0
k=Q
For the next application we need the following:
Lemma 3.1 Select two distinct vertices a and b in the complete graph Kn,
n arbitrary. Consider a new vertex c and attach the edge ac to Kn. Then
the simple random walk on this graph satisfies
Proof: We know that TT(J/) = j M (d(y) 1S ^e degree of y). Also, e(y,x) >
[d(y)]~l, with equality if and only if xy is a cut edge of G.
O
3
regardless of n and any additional graphs that may be attached to the above
configuration by means of edges cd or db (d being a new vertex not in the
above configuration}.
The applications
We give first an elementary application: the well known expected "cover
time" for the simple random walk on a cycle graph.
Corollary 3.1 If G = Zn, the cycle graph on n vertices, a is the cover
time of G (i.e., the time needed for the walk to visit all vertices of G) and
d(x,y) is the distance between x and y, then
Proof: Conditioning on the first step:
e(a,b)
=
X
lra
(a)
=
d(x,y}(n-d(x,y));
2
Proof: Given the simple random walk on Z and 0 < j < fc, Pj(T0 >
£ (this is the classic ruin problem). Using this fact and conditioning on
the first step we get
u i
2 [d(x,y)
n-d(x,y)\'
~2
n
I
2'
Consider now two complete graphs Km and /Ov-m-fc attached by a
linear path of length k + 1, with 1 < m < N — 1 and 0 < k < N — m— 1.
This is a graph on N vertices that we will call the (N, m,&)-barbell. To
simplify matters we will assume from now on that 3 divides TV.
Theorem 3.1 The maximum commute time for the simple random walk
among all(N,m,k)-barbells occurs for the lollipop graph, or(N, ^, ^- 1 )barbell. The value of this maximum is O(^af-), and if 3 divides N the exact
value i.9^(f + I) 2 .
Northshield and Palacios
174
On the commute time of random walks on graphs
Proof: Call a and b respectively the vertices where A'm and AOv-m-fc are
attached to the path. Select vertices x 6 A'm and y £ /Ov-m-fc with x ^ a
and y ^ b. Then
C(b,y}.
(3.1)
Since all the edges in the linear path are cut edges, C(a,b) = 2\E\(k +1).
By the lemma e(a, x) = e(b,y] = \, so that C(x,a) = 4-4 and C(b,y) =
175
Theorem 3.1 agrees with a previous result in Brightwell and Winkler
(1990): there it is shown that the maximum hitting time among all connected graphs occurs in the lollipop graph with a ^ clique and its value is
0(^27~)- Our theorem 3.1 strongly supports the conjecture that the same
is true for the commute time, and to the best of our knowledge, it is the
only known maximal result concerning commute times (see Aldous, 1993).
(Received April, 1995. Revised August, 1995.)
C(y, b) = Jjg. Also, the number of edges is | E\ = (?) + ( yv "™~*) + * + 1.
Inserting these values into (3.1) we obtain
C(x,y]
References
= [TV2 - N + 2 - 2m(N - m - k) + k(k - 27V + 3)]
(3.2)
Aldous, D.J. (1993). Reversible Markov chains and random walks on
graphs. Book preprint.
In order to find the maximum value of (3.2) we restrict our attention to
those summands which might yield an O(N3) term; therefore we consider
the continuous variable function
Aleliunas, R., Karp, R.M., Lipton, R.J, Lovasz, L., Rackoff, C. (1979).
Random walks, universal traversal sequences and the complexity of
maze problems. In: 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 218-223.
x [* + , + - **-*>
- m - Jb)J
F(m,k] = [TV2 - 2m(N - m - k) + k(k - 2N)]k
Brightwell, G., Winkler, P. (1990). Maximum hitting time for random
walks on graphs. Random Structures and Algorithms, 1, 263-276.
with domain the triangle
l<m<N-l,
Burdzy, K., Lawler, G. (1990). Rigorous inequalities for random walks.
Journal of Physics A: Mathematical and General, 23, L23-L28.
Q<k<N-m-l.
Inside the triangle it is an exercise in Calculus to show that the only candidates for critical points are found along the line m = —f-- On this
line the maximum is found at k = m = y (this is the symmetric barbell)
and its value is 0(^-). Along the vertical axis m = 1 the one- variable
function F(l,k) has a maximum at k — y (this is a lollipop graph) with
value O(^f-). Along the horizontal axis k = 0, F(m,0) = 0, so that no
O(N3) terms are found here. Finally, on the line k = N -m-1 we find a
maximum at m = ^ (another lollipop graph) with value 0(^-}.
It is not difficult to find by checking in formula (3.2), once the search
is constrained to these lollipop graphs, that the maximum commute time
among all barbell-type graphs is attained at the last candidate above: the
lollipop graph with a clique of ^ vertices, and its exact value is found
replacing m = ^ and k = y - 1 in (3.2) yielding
D
Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R., Tiwari, P.
(1989). The electrical resistance of a graph captures its commute and
cover times. In: Proceedings of the First Annual ACM Symposium
on Theory of Computing, Seattle, Washington, 574-586.
Doyle, P.G., Snell, J.L. (1984). Random walks and electrical networks.
The Mathematical Association of America, Washington, D.C.
Palacios, J.L. (1990). On a result of Aleliunas et al. concerning random
walks on graphs. Probability in the Engineering and Informational
Sciences, 4, 489-492.
Palacios, J.L. (1993). Fluctuation theory for the Ehrenfest urn via electrical networks. Advances in Applied Probability, 25, 472-476.