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We are interested in studying random walks in graphs by using the matrix
P
. Examples of real life phenomena that can be modeled by random walks
include the orderings of a deck of cards or state graphs. A graph is ergodic
if there exists a unique stationary distribution, Π, such that
lim
s
→∞
fP
s
(
v
) = Π(
v
)
where
f
is the initial probability distribution among the vertices.
There are two necessary conditions for the random walks to be Markov
chains. These conditions also end up being suﬃcient. The graph must be
irreducible, meaning that for all pairs of vertices
u
and
v
, there exists an in
teger
s
such that
P
s
(
u,v
)
>
0. This means that there exists a walk between
any two vertices. The graph must also be aperiodic. This means that for all
pairs of vertices
u
and
v
,
gcd
{
s
:
P
s
(
u,v
)
>
0
}
= 1. This means that the
lengths of all the walks from
u
to
v
have relatively prime length. For undi
rected graphs, irreducibility is equivalent to connectivity, and aperiodicity is
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 Fall '08
 aterras
 Math

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