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Unformatted text preview: is described by giving
two things: (i) a set of vertices V (which we suppose is a ﬁnite set) and (ii) an
adjacency matrix A(u, v ), which is 1 if there is an edge connecting u and v and
0 otherwise. By convention we set A(v, v ) = 0 for all v 2 V .
3
3 4 @ @ @
3 3
4 3
5 3 3 @ @
@
3
3 The degree of a vertex u is equal to the number of neighbors it has. In symbols,
X
d(u) =
A(u, v )
v since each neighbor of u contributes 1 to the sum. To help explain the concept,
we have indicated the degrees on our example. We write the degree this way to
make it clear that
(⇤) p(u, v ) = A(u, v )
d(u) deﬁnes a transition probability. In words, if Xn = u, we jump to a randomly
chosen neighbor of u at time n + 1.
It is immediate from (⇤) that if c is a positive constant then ⇡ (u) = cd(u)
satisﬁes the detailed balance condition:
⇡ (u)p(u, v ) = cA(u, v ) = cA(v, u) = ⇡ (v )p(u, v )
P
Thus, if we take c = 1/ u d(u), we have a stationary probability distribution.
In the example c = 1/40. 35 1.6. SPECIAL EXAMPLES
For a concrete example, consider Example 1.34. Rand...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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