Post on 03-Jul-2020
UpdatingMarkovChains
Carl MeyerAmy Langville
Department of MathematicsNorth Carolina State UniversityRaleigh, NC
A. A. Markov Anniversary Meeting
June 13, 2006
IntroAssumptions
Very large irreducible chain
— m = O(109) number of states
— Qm×m old transition matrix
— φT = (φ1, φ2, . . ., φm) old stationary distribution
IntroAssumptions
Very large irreducible chain
— m = O(109) number of states
— Qm×m old transition matrix
— φT = (φ1, φ2, . . ., φm) old stationary distribution
Updates
Change some transition probabilities
Add or delete some states
— Pn×n new transition matrix (irreducible)
— πT = (π1, π2, . . ., πn) new distribution (unknown)
IntroAssumptions
Very large irreducible chain
— m = O(109) number of states
— Qm×m old transition matrix
— φT = (φ1, φ2, . . ., φm) old stationary distribution
Updates
Change some transition probabilities
Add or delete some states
— Pn×n new transition matrix (irreducible)
— πT = (π1, π2, . . ., πn) new distribution (unknown)
Aim
Use φT to compute πT
Exact (Theoretical) UpdatingPerturbation Formula
P = Q − E =⇒ πT = φT − εT εT = φTEZ(I + EZ)−1
Z ={
Fundamental Matrix
(I − Q)#(group inverse)
Exact (Theoretical) UpdatingPerturbation Formula
P = Q − E =⇒ πT = φT − εT εT = φTEZ(I + EZ)−1
Z ={
Fundamental Matrix
(I − Q)#(group inverse)Requires m = n (No states added or deleted)
Exact (Theoretical) UpdatingPerturbation Formula
P = Q − E =⇒ πT = φT − εT εT = φTEZ(I + EZ)−1
Z ={
Fundamental Matrix
(I − Q)#(group inverse)Requires m = n (No states added or deleted)
One Row At A Time (Sherman–Morrison)
pTi = qT
i − δTi =⇒ πT = φT − εT
Exact (Theoretical) UpdatingPerturbation Formula
P = Q − E =⇒ πT = φT − εT εT = φTEZ(I + EZ)−1
Z ={
Fundamental Matrix
(I − Q)#(group inverse)Requires m = n (No states added or deleted)
One Row At A Time (Sherman–Morrison)
pTi = qT
i − δTi =⇒ πT = φT − εT
εT =(
φi
1 + δTi A#
∗i
)δTA# A# = (I − Q)#
Exact (Theoretical) UpdatingPerturbation Formula
P = Q − E =⇒ πT = φT − εT εT = φTEZ(I + EZ)−1
Z ={
Fundamental Matrix
(I − Q)#(group inverse)Requires m = n (No states added or deleted)
One Row At A Time (Sherman–Morrison)
pTi = qT
i − δTi =⇒ πT = φT − εT
εT =(
φi
1 + δTi A#
∗i
)δTA# A# = (I − Q)#
(I − P)# = A# + eεT[A# − γI
]− A#
∗iεT
φiγ = εTA#
∗iφi
Exact (Theoretical) UpdatingPerturbation Formula
P = Q − E =⇒ πT = φT − εT εT = φTEZ(I + EZ)−1
Z ={
Fundamental Matrix
(I − Q)#(group inverse)Requires m = n (No states added or deleted)
One Row At A Time (Sherman–Morrison)
pTi = qT
i − δTi =⇒ πT = φT − εT
εT =(
φi
1 + δTi A#
∗i
)δTA# A# = (I − Q)#
(I − P)# = A# + eεT[A# − γI
]− A#
∗iεT
φiγ = εTA#
∗iφi
Not Practical For Large Problems
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
— Requires φT ≈ πT (generally means P ≈ Q)
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
— Requires φT ≈ πT (generally means P ≈ Q)
— Asymptotic rate of convergence: R = − log10 |λ2|Need about 1/R iterations to eventually gain oneadditional significant digit of accuracy
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
— Requires φT ≈ πT (generally means P ≈ Q)
— Asymptotic rate of convergence: R = − log10 |λ2|Need about 1/R iterations to eventually gain oneadditional significant digit of accuracy
Example:
� Suppose digit1(φT ) = digit1(πT )
� Want 12 digits of accuracy
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
— Requires φT ≈ πT (generally means P ≈ Q)
— Asymptotic rate of convergence: R = − log10 |λ2|Need about 1/R iterations to eventually gain oneadditional significant digit of accuracy
Example:
� Suppose digit1(φT ) = digit1(πT )
� Want 12 digits of accuracy
� RPM — about 11/R iterations
� Start from scratch — about 12/R iterations
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
— Requires φT ≈ πT (generally means P ≈ Q)
— Asymptotic rate of convergence: R = − log10 |λ2|Need about 1/R iterations to eventually gain oneadditional significant digit of accuracy
Example:
� Suppose digit1(φT ) = digit1(πT )
� Want 12 digits of accuracy
� RPM — about 11/R iterations
� Start from scratch — about 12/R iterations
� Only 8.3% reduction
Restarted Power MethodxT
j+1 = xTj P with xT
0 = φT(assume aperiodic)
— Requires m = n (no states added or deleted)
— Requires φT ≈ πT (generally means P ≈ Q)
— Asymptotic rate of convergence: R = − log10 |λ2|Need about 1/R iterations to eventually gain oneadditional significant digit of accuracy
Example:
� Suppose digit1(φT ) = digit1(πT )
� Want 12 digits of accuracy
� RPM — about 11/R iterations
� Start from scratch — about 12/R iterations
� Only 8.3% reduction
A Little Better, But Not Great
CensoringPartition (not necessarily NCD)
Pn×n =
⎛⎜⎜⎜⎝
G1 G2. . . Gk
G1 P11 P12. . . P1k
G2 P21 P22. . . P2k...
......
. . ....
Gk Pk1 Pk2. . . Pkk
⎞⎟⎟⎟⎠
CensoringPartition (not necessarily NCD)
Pn×n =
⎛⎜⎜⎜⎝
G1 G2. . . Gk
G1 P11 P12. . . P1k
G2 P21 P22. . . P2k...
......
. . ....
Gk Pk1 Pk2. . . Pkk
⎞⎟⎟⎟⎠
Censored Chains
Records state of process only when chain visits states in Gi.
Visits to states outside of Gi are ignored.
CensoringPartition (not necessarily NCD)
Pn×n =
⎛⎜⎜⎜⎝
G1 G2. . . Gk
G1 P11 P12. . . P1k
G2 P21 P22. . . P2k...
......
. . ....
Gk Pk1 Pk2. . . Pkk
⎞⎟⎟⎟⎠
Censored Chains
Records state of process only when chain visits states in Gi.
Visits to states outside of Gi are ignored.
Censored Transition Matrices
Ci = Pii + Pi�(I − P�i )−1P�i Stochastic Complements
Aggregation
Censored Distributions
sTi Ci = sT
i
Aggregation Matrix
Ak×k =
⎡⎣ sT
1P11e . . . sT1P1ke
.... . .
...sT
k Pk1e . . . sTk Pkke
⎤⎦ e =
⎡⎣ 1...
1
⎤⎦
Aggregation
Censored Distributions
sTi Ci = sT
i
Aggregation Matrix
Ak×k =
⎡⎣ sT
1P11e . . . sT1P1ke
.... . .
...sT
k Pk1e . . . sTk Pkke
⎤⎦ e =
⎡⎣ 1...
1
⎤⎦
Aggregated Distribution
αTA = αT αT = (α1, α2, . . ., αk)
Aggregation
Censored Distributions
sTi Ci = sT
i
Aggregation Matrix
Ak×k =
⎡⎣ sT
1P11e . . . sT1P1ke
.... . .
...sT
k Pk1e . . . sTk Pkke
⎤⎦ e =
⎡⎣ 1...
1
⎤⎦
Aggregated Distribution
αTA = αT αT = (α1, α2, . . ., αk)
Aggregation Theorem
πT = (πT1 |πT
2 | . . . |πTk ) = (α1sT
1 |α2sT2 | . . . |α2sT
2 )
PartitioningIntuition
Update relatively small number of states in large sparse chain
— Effects are primarily local
— Most stationary probabilities not significantly affected.
PartitioningIntuition
Update relatively small number of states in large sparse chain
— Effects are primarily local
— Most stationary probabilities not significantly affected.
State Space Partition
S = G ∪ G G ={
states likely to be most affectednewly added states
PartitioningIntuition
Update relatively small number of states in large sparse chain
— Effects are primarily local
— Most stationary probabilities not significantly affected.
State Space Partition
S = G ∪ G G ={
states likely to be most affectednewly added states
— g = |G| << |G|
PartitioningIntuition
Update relatively small number of states in large sparse chain
— Effects are primarily local
— Most stationary probabilities not significantly affected.
State Space Partition
S = G ∪ G G ={
states likely to be most affectednewly added states
— g = |G| << |G|
Induced Matrix Partition
Pn×n =
( G GG P11 P12
G P21 P22
)=
⎡⎢⎢⎢⎣
p11. . . p1g P1�
......
...pg1
. . . pgg Pg�
P�1. . . P�g P22
⎤⎥⎥⎥⎦
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Old Distribution (reorded)
φT = (φ1, φ2, . . . |φT)
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Old Distribution (reorded)
φT = (φ1, φ2, . . . |φT)
The Assumption
φT ≈ πT
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Old Distribution (reorded)
φT = (φ1, φ2, . . . |φT)
The Assumption
φT ≈ πT =⇒ φ
T ≈ αg+1sTg+1
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Old Distribution (reorded)
φT = (φ1, φ2, . . . |φT)
The Assumption
φT ≈ πT =⇒ φ
T ≈ αg+1sTg+1 =⇒ sT
g+1 ≈ φT
φT
e
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Old Distribution (reorded)
φT = (φ1, φ2, . . . |φT)
The Assumption
φT ≈ πT =⇒ φ
T ≈ αg+1sTg+1 =⇒ sT
g+1 ≈ φT
φT
e
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
−−−−−→ αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT =(π1, . . ., πg |πT
)= (α1, . . ., αg, |αg+1sT
g+1)
Old Distribution (reorded)
φT = (φ1, φ2, . . . |φT)
The Assumption
φT ≈ πT =⇒ φ
T ≈ αg+1sTg+1 =⇒ sT
g+1 ≈ φT
φT
e
Specialized AggregationCensored Transition Matrices
C1 = . . . = Cg = [1]1×1 Cg+1 = P22 + P21(I − P11)−1P12
Censored DistributionssT
1 = . . . = sTg = 1 sT
g+1 = sTg+1Cg+1
Aggregation Matrix Aggregated Distribution
A =[
P11 P12esT
g+1P21 1 − sTg+1P21e
](g+1)×(g+1)
−−−−−→ αT =(α1, . . ., αg, αg+1
)Aggregation Theorem
πT = (α1, . . ., αg, |αg+1sTg+1)
Old Distribution (reorded) Updated Distribution
φT = (φ1, φ2, . . . |φT)
The Assumption
φT ≈ πT =⇒ φ
T ≈ αg+1sTg+1 =⇒ sT
g+1 ≈ φT
φT
e
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
Approximate Censored Distribution
sT ← φT/(φ
Te)
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
Approximate Censored Distribution
sT ← φT/(φ
Te)
Form Approximate Aggregation Matrix
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
Approximate Censored Distribution
sT ← φT/(φ
Te)
Form Approximate Aggregation Matrix
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
Compute Approximate Aggregated DistributionαT ←
(α1, . . ., αg, αg+1
)
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
Approximate Censored Distribution
sT ← φT/(φ
Te)
Form Approximate Aggregation Matrix
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
Compute Approximate Aggregated DistributionαT ←
(α1, . . ., αg, αg+1
)Approximate Updated Distribution
πT ← (α1, . . ., αg, |αg+1sT )
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
Approximate Censored Distribution
sT ← φT/(φ
Te)
Form Approximate Aggregation Matrix
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
Compute Approximate Aggregated DistributionαT ←
(α1, . . ., αg, αg+1
)Approximate Updated Distribution
πT ← (α1, . . ., αg, |αg+1sT )
Iterate ?
SummaryReorder & Partition Updated State Space
S = G ∪ G G = {New + Most affected} G = {Less affected}
Use Less Affected Components From Old Distribution
φT ← components from φT corresponding to states in G
Approximate Censored Distribution
sT ← φT/(φ
Te)
Form Approximate Aggregation Matrix
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
Compute Approximate Aggregated DistributionαT ←
(α1, . . ., αg, αg+1
)Approximate Updated Distribution
πT ← (α1, . . ., αg, |αg+1sT )
Iterate ? No! — At A Fixed Point
Iterative AggregationMove Off Of Fixed Point With Power Step
sT ← φT/(φ
Te)
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
αT ←(α1, . . ., αg, αg+1
)πT ← (α1, . . ., αg, |αg+1sT )ψT ← πTP (Also makes progress toward convergence when aperiodic)
If ‖ψT − χT‖ < τ then quit — else
sT ←− ψT/(ψTe)
−−→
Iterative AggregationMove Off Of Fixed Point With Power Step
sT ← φT/(φ
Te)
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
αT ←(α1, . . ., αg, αg+1
)πT ← (α1, . . ., αg, |αg+1sT )ψT ← πTP (Also makes progress toward convergence when aperiodic)
If ‖ψT − χT‖ < τ then quit — else
sT ←− ψT/(ψTe)
−−→
TheoremIf C = P22 + P21(I − P11)−1P12 is aperiodic, then convergent forall partitions S = G ∪ G
Iterative AggregationMove Off Of Fixed Point With Power Step
sT ← φT/(φ
Te)
A ←[
P11 P12esTP21 1 − sTP21e
](g+1)×(g+1)
αT ←(α1, . . ., αg, αg+1
)πT ← (α1, . . ., αg, |αg+1sT )ψT ← πTP (Also makes progress toward convergence when aperiodic)
If ‖ψT − χT‖ < τ then quit — else
sT ←− ψT/(ψTe)
−−→
TheoremIf C = P22 + P21(I − P11)−1P12 is aperiodic, then convergent forall partitions S = G ∪ G
— |λ2(C)| determines rate of convergence
Google’s PageRankRandom Walk On WWW Link Structure
Hij ={
1/(total # outlinks from page Pi) if Pi → Pj,
0 otherwise
Google’s PageRankRandom Walk On WWW Link Structure
Hij ={
1/(total # outlinks from page Pi) if Pi → Pj,
0 otherwise
Google Matrix
P = α(H + E) + (1 − α)F
— (H + E) & F are stochastic rank (E) = rank (F) = 1
— 0 < α < 1
Google’s PageRankRandom Walk On WWW Link Structure
Hij ={
1/(total # outlinks from page Pi) if Pi → Pj,
0 otherwise
Google Matrix
P = α(H + E) + (1 − α)F
— (H + E) & F are stochastic rank (E) = rank (F) = 1
— 0 < α < 1
— PageRank = πT
Google’s PageRankRandom Walk On WWW Link Structure
Hij ={
1/(total # outlinks from page Pi) if Pi → Pj,
0 otherwise
Google Matrix
P = α(H + E) + (1 − α)F
— (H + E) & F are stochastic rank (E) = rank (F) = 1
— 0 < α < 1
— PageRank = πT
Power Law Distribution
If ordered by magnitude π(1) ≥ π(2) ≥ . . . ≥ π(n), then
— π(i) ≈ αi−k for k ≈ 2.109[Donato, Laura, Leonardi, 2002]
[Pandurangan, Raghavan,& Upfal, 2004]
— Relatively few large states (i.e., important sites)
“L” Curves
0 200 400 6000
0.005
0.01
0.015
0.02
0.025
0.03
0.035censorship
0 200 400 6000
0.01
0.02
0.03
0.04movies
0 200 400 6000
0.01
0.02
0.03
0.04
0.05mathworks
0 1000 2000 0 5000 100000
0.005
0.01
0.015
0.02
0.025abortion
0 1000 2000 30000
0.005
0.01
0.015
0.02
0.025
0
0.002
0.004
0.008
0.006
0.01genetics CA
Experiments
The Updates
# Nodes Added = 3
# Nodes Removed = 50
# Links Added = 10 (Different values have little effect on results)
# Links Removed = 20
Stopping Criterion
1-norm of residual < 10−10
Movies
Power Method Iterative Aggregation
Iterations Time
17 .40
|G| Iterations Time
5 12 .3910 12 .3715 11 .3620 11 .3525 11 .3150 9 .31100 9 .33200 8 .35300 7 .39400 6 .47
nodes = 451 links = 713
Censorship
Power Method Iterative Aggregation
Iterations Time
38 1.40
|G| Iterations Time
5 38 1.6810 38 1.6615 38 1.5620 20 1.0625 20 1.0550 10 .69100 8 .55200 6 .53300 6 .65400 5 .70
nodes = 562 links = 736
MathWorks
Power Method Iterative Aggregation
Iterations Time
54 1.25
|G| Iterations Time
5 53 1.1810 52 1.2915 52 1.2320 42 1.0525 20 1.1350 18 .70100 16 .70200 13 .70300 11 .83400 10 1.01
nodes = 517 links = 13,531
Abortion
Power Method Iterative Aggregation
Iterations Time
106 37.08
|G| Iterations Time
5 109 38.5610 105 36.0215 107 38.0520 107 38.4525 97 34.8150 53 18.80100 13 5.18250 12 5.62500 6 5.21750 5 10.221000 5 14.61
nodes = 1,693 links = 4,325
Genetics
Power Method Iterative Aggregation
Iterations Time
92 91.78
|G| Iterations Time
5 91 88.2210 92 92.1220 71 72.5350 25 25.42100 19 20.72250 13 14.97500 7 11.141000 5 17.761500 5 31.84
nodes = 2,952 links = 6,485
California
Power Method Iterative Aggregation
Iterations Time
176 5.85
|G| Iterations Time
500 19 1.121000 15 .921250 20 1.041500 14 .902000 13 1.175000 6 1.25
nodes = 9,664 links = 16,150
“L” Curves
0 200 400 6000
0.005
0.01
0.015
0.02
0.025
0.03
0.035censorship
0 200 400 6000
0.01
0.02
0.03
0.04movies
0 200 400 6000
0.01
0.02
0.03
0.04
0.05mathworks
0 1000 2000 0 5000 100000
0.005
0.01
0.015
0.02
0.025abortion
0 1000 2000 30000
0.005
0.01
0.015
0.02
0.025
0
0.002
0.004
0.008
0.006
0.01genetics CA
“L” Curves
0 200 400 6000
0.005
0.01
0.015
0.02
0.025
0.03
0.035censorship
1000 200
50400 600
0
0.01
0.02
0.03
0.04movies
500 200 400 600
0
0.01
0.02
0.03
0.04
0.05mathworks
2500 1000 2000
0
0.005
0.01
0.015
0.02
0.025abortion
2500 1000 2000 3000
0
0.005
0.01
0.015
0.02
0.025genetics
20000 5000 10000
0
0.002
0.004
0.008
0.006
0.01CA
Quadratic Extrapolationnodes = 10,000 links = 101,118 [Kamvar, Haveliwala, Manning, Golub, 2003]
0 20 40 60 80 100 120 140 160 180 12
10 10
8 8
6 6
4 4
2 2
0
iteratioiteration
lolog 1010
res
idua
l r
esid
ual
power methopower methodpower method with quad. extrap.power method with quad. extrap.
Quadratic Extrapolationnodes = 10,000 links = 101,118 [Kamvar, Haveliwala, Manning, Golub, 2003]
0 20 40 60 80 100 120 140 160 180 12
10 10
8 8
6 6
4 4
2 2
0
iteratioiteration
lolog 1010
res
idua
l r
esid
ual
power methopower methodpower method with quad. extrap.power method with quad. extrap.i A D A D
Quadratic Extrapolationnodes = 10,000 links = 101,118 [Kamvar, Haveliwala, Manning, Golub, 2003]
0 20 40 60 80 100 120 140 160 180 12
10 10
8 8
6 6
4 4
2 2
0
iteratioiteration
lolog 1010
res
idua
l r
esid
ual
power methopower methodpower method with quad. extrap.power method with quad. extrap.i A D A Di A D A D with quad. extrap. with quad. extrap.
Timings
Iterations Time (sec) |G|
Power 162 9.69
Power+Quad 81 5.93
IAD 21 2.22 2000
IAD+Quad 16 1.85 2000
nodes = 10,000 links = 101,118
Conclusion
Iterative aggregation shows
promise for updating
Markov chains
Especially for those having
power law distributions