Download Distribution Over Nodes

Mathematics for Computer Science
MIT 6.042J/18.062J
Stationary
Distributions
Albert R Meyer,
December 11, 2013
stationary.1
Distribution Over Nodes
1/4
1/4
1/4
1/4
B
1/2
1/2
1/4
1
1/3
0
1/4
1/4
1/4
1
2/3
00
(pB, pO, pG)
Suppose you start at B: (1, 0, 0)
1/2
1/2
What are p’B, p’O, p’G after 1 step?
Albert R Meyer,
December 11, 2013
stationary.2
Distribution Over Nodes
1/4
1/4
1/4
1/4
B
1/4
1/2
1
1/2
1/2
1/2
1/3
0
1/4
1/4
1/4
1
2/3
00
Dist after 1 step: (p’B,p’O,p’G),
only get places from B,
so
Albert R Meyer,
December 11, 2013
stationary.3
Distribution Over Nodes
1/4
B
1/2
1/2
1/4
1/4
1
1/3
2/3
1/4
Dist after 1 step:
Dist after 2 steps: (p’’B, p’’O, p’’G)
Albert R Meyer,
December 11, 2013
stationary.6
Distribution Over Nodes
1/4
B
1/2
1/2
1/4
1/4
1
1/3
2/3
1/4
Dist after 1 step:
p’’O = Pr[B to O|at B]•p’B
+ Pr[O to O|at O]•p’o
+ Pr[G to O|at G]•p’G
Albert R Meyer,
December 11, 2013
stationary.7
Distribution Over Nodes
1/4
B
1/2
1/2
1/4
1/4
1
1/3
2/3
1/4
Dist after 1 step:
1/4
p’’O = Pr[B to
O|at B]•p’1/2
B
1/4
+ Pr{O to
1/3
O|at O}•p’
o
1/4
+ Pr{G to0O|at G}•p’
G
Albert R Meyer,
December 11, 2013
= 5/24
stationary.8
Distribution Over Nodes
1/4
B
1/2
5/24
1/4
1/2
1
1/3
2/3
7/24
distribution after 2 steps:
’’
(p B,
’’
p O,
Albert R Meyer,
’’
p G)
December 11, 2013
stationary.9
Distribution Over Nodes
1/4
B
1/2
pt
B
ptO
1/4
1
1/3
2/3
pt G
distribution after t steps?
Albert R Meyer,
December 11, 2013
stationary.10
Linear Algebra
the edge probability matrix
December 11, 2013
stationary.11
Linear Algebra
the edge probability matrix
December 11, 2013
stationary.12
Linear Algebra
the edge probability matrix
December 11, 2013
stationary.13
Linear Algebra
the edge probability matrix
December 11, 2013
stationary.14
Linear Algebra
December 11, 2013
stationary.15
Distribution Over Nodes
ptO
1/4
B
1/2
pt
B
1/4
1
1/3
2/3
pt G
distribution after t steps?
…and as t ?
Albert R Meyer,
December 11, 2013
stationary.16
Stationary Distribution
1/4
B
1/2
pB
pO
1/4
1
1/3
2/3
pG
distribution (pB, pO, pG) is stationary
if next-step distribution is the same.
What is a stationary dist. here?
Albert R Meyer,
December 11, 2013
stationary.17
Finding Stationary Dist.
1/4
1/2
pB
pO
1/4
1
1/3
2/3
pG
pB = pB’ = (1/2)pB + 1pG
pO = pO’ = (1/4)pB + (1/3)pO
pG = pG’ = (1/4)pB + (2/3)pO
pB + pO + pG = 1
Albert R Meyer,
December 11, 2013
stationary.18
Finding Stationary Dist.
1/4
1/2
pB
pO
1/4
1/3
2/3
pG
solving for
(pB, pO, pG):
Albert R Meyer,
December 11, 2013
stationary.19
Linear Algebra
Find stationary dist vector s
by solving:
sM = s
|s | = 1
December 11, 2013
stationary.20
Stationary Difficulties
1
0
p
10
does not converge to
stable distribution
q
uncountably many
stable distributions
December 11, 2013
stationary.21
Questions on Stationary Dist
Yes
stationary dist exist? (if graph finite)
• Is it unique?
Sometimes
• Does a random walk
approach it from any Sometimes
starting distribution?
– How quickly?
Varies
•
Albert R Meyer,
December 11, 2013
stationary.22