Foundations of Data Science

on this undirected graph. Since there are n d states, the cover time is at least n d and thus
exponentially dependent on d. It is possible to show (Exercise 5.20) that Φ is Ω(1/dn).
Since all π i are equal to n −d , the mixing time is O(d 3 n 2 ln n/ε 2 ), which is polynomially
bounded in n and d.
The d-dimensional lattice is related to the Metropolis-Hastings algorithm and Gibbs
sampling although in those constructions there is a nonuniform probability distribution at
the vertices. However, the d-dimension lattice case suggests why the Metropolis-Hastings
and Gibbs sampling constructions might converge fast.
A clique
Consider an n vertex clique with a self loop at each vertex. For each edge, p xy = 1 n
and thus for each vertex, π x = 1 . Let S be a subset of the vertices. Then
n
and
This gives a ε−mixing time of
∑
∑
x∈S
(x,y)∈(S, ¯S)
Φ(S) =
O
π x = |S|
n .
π x p xy = 1 n 2 |S||S|
∑
(x,y)∈(S, ¯S) π xp xy
∑
x∈S π ∑
x x∈ ¯S π x
= 1.
( )
ln
1 ( )
π min ln n
= O .
Φ 2 ɛ 3 ɛ 3
A connected undirected graph
Next consider a random walk on a connected n vertex undirected graph where at each
vertex all edges are equally likely. The stationary probability of a vertex equals the degree
of the vertex divided by the sum of degrees. The sum of the vertex degrees is at most n 2
and thus, the steady state probability of each vertex is at least 1 . Since the degree of a
n 2
vertex is at most n, the probability of each edge at a vertex is at least 1 . For any S, the
n
total conductance of edges out of S is greater than or equal to
1 1
n 2 n = 1 n . 3
Thus, Φ is at least 1 . Since π
n 3 min ≥ 1 , ln 1
n 2 π min
O(n 6 ln n/ε 3 ).
158
= O(ln n). Thus, the mixing time is