random walks on dynamic graphs: mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6....
TRANSCRIPT
![Page 1: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/1.jpg)
Random walks on dynamic graphs:Mixing times, hitting times, and return probabilities
Thomas Sauerwald and Luca Zanettito appear in ICALP’19, full version arXiv:1903.01342
7 May 2019
Random Walk on a Dynamic Graph Sequence
The random walk stays with probability 1/2 at the current location.
Lazy Random Walks
t = 1
t = 3
t = 2
t = 4
Intro 6
![Page 2: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/2.jpg)
Outline
Intro
Random Walks on Sequences of Connected Graphs
Random Walks on Sequences of (Possibly) Disconnected Graphs
Conclusion
Intro 2
![Page 3: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/3.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 4: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/4.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 5: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/5.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 6: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/6.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 7: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/7.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 8: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/8.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 9: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/9.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 10: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/10.jpg)
Random Walks and Markov Chains
A class of Markov chains where a particle is moving on the vertices of a graph:
start from some specified vertexat each step, jump to a randomly chosen neighbor
Random Walks on Graphs
Intro 3
![Page 11: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/11.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ
⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 12: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/12.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:
For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ
⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 13: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/13.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ
⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 14: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/14.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ
⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 15: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/15.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ
⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 16: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/16.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 17: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/17.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 18: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/18.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 19: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/19.jpg)
Hitting Times (and Cover Times) on Static Graphs
Let thit (u, v) be the expected time for a random walk to go from u to v
Let thit (G) := maxu,v thit (u, v) be the hitting time of the graph G
Let tcov (G) the expected time to visit all vertices in G
Hitting and Cover Times
Some Classical Results:For any graph, thit (G) ≤ tcov (G) ≤ thit ·O(log n)[Matthews, Annals of Prob.’88]
For any graph, thit (G) ≤ tcov (G) ≤ 2|E |(|V | − 1) = O(n3)[Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79]
For any graph, thit (G) ≤ tcov (G) ≤ 16 |E||V |δ⇒ thit (G) = O(n2) if G regular.
[Kahn, Linial, Nisan and Saks, J. Theoretical Prob.’88]
For any graph, thit (G) ≤ ( 427 + o(1)) · n3
[Brightwell and Winkler, RSA’90]
For any graph, tcov (G) ≤ ( 427 + o(1)) · n3
[Feige, RSA’95]
Intro 4
![Page 20: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/20.jpg)
Motivation: Dynamic Graphs
Many prevalent networks are dynamically changing.
a.k.a. as evolving, temporal or time-varying graph
Wireless/Mobile Networks
Big data scenario
Genome sequences for many species are available: each megabytes to gigabytes in size.
There are about 1 billion monthly active users in Facebook.
There are 5 billion global mobile phone users.
100 hours of videos uploaded per minute
Social Networks
(Distributed) Algorithms
710
310
Particle Processes
Intro 5
![Page 21: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/21.jpg)
Motivation: Dynamic Graphs
Many prevalent networks are dynamically changing.
a.k.a. as evolving, temporal or time-varying graph
Wireless/Mobile Networks
Big data scenario
Genome sequences for many species are available: each megabytes to gigabytes in size.
There are about 1 billion monthly active users in Facebook.
There are 5 billion global mobile phone users.
100 hours of videos uploaded per minute
Social Networks
(Distributed) Algorithms
710
310
Particle Processes
Intro 5
![Page 22: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/22.jpg)
Motivation: Dynamic Graphs
Many prevalent networks are dynamically changing.
a.k.a. as evolving, temporal or time-varying graph
Wireless/Mobile Networks
Big data scenario
Genome sequences for many species are available: each megabytes to gigabytes in size.
There are about 1 billion monthly active users in Facebook.
There are 5 billion global mobile phone users.
100 hours of videos uploaded per minute
Social Networks
(Distributed) Algorithms
710
310
Particle Processes
Intro 5
![Page 23: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/23.jpg)
Motivation: Dynamic Graphs
Many prevalent networks are dynamically changing.
a.k.a. as evolving, temporal or time-varying graph
Wireless/Mobile Networks
Big data scenario
Genome sequences for many species are available: each megabytes to gigabytes in size.
There are about 1 billion monthly active users in Facebook.
There are 5 billion global mobile phone users.
100 hours of videos uploaded per minute
Social Networks
(Distributed) Algorithms
710
310
Particle Processes
Intro 5
![Page 24: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/24.jpg)
Motivation: Dynamic Graphs
Many prevalent networks are dynamically changing.
a.k.a. as evolving, temporal or time-varying graph
Wireless/Mobile Networks
Big data scenario
Genome sequences for many species are available: each megabytes to gigabytes in size.
There are about 1 billion monthly active users in Facebook.
There are 5 billion global mobile phone users.
100 hours of videos uploaded per minute
Social Networks
(Distributed) Algorithms
710
310
Particle Processes
Intro 5
![Page 25: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/25.jpg)
Motivation: Dynamic Graphs
Many prevalent networks are dynamically changing.
a.k.a. as evolving, temporal or time-varying graph
Wireless/Mobile Networks
Big data scenario
Genome sequences for many species are available: each megabytes to gigabytes in size.
There are about 1 billion monthly active users in Facebook.
There are 5 billion global mobile phone users.
100 hours of videos uploaded per minute
Social Networks
(Distributed) Algorithms
710
310
Particle Processes
Intro 5
![Page 26: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/26.jpg)
Random Walk on a Dynamic Graph Sequence
The random walk stays with probability 1/2 at the current location.
Lazy Random Walks
t = 1
t = 3
t = 2
t = 4
Intro 6
![Page 27: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/27.jpg)
Random Walk on a Dynamic Graph Sequence
The random walk stays with probability 1/2 at the current location.
Lazy Random Walks
t = 1
t = 3
t = 2
t = 4
Intro 6
![Page 28: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/28.jpg)
Random Walk on a Dynamic Graph Sequence
The random walk stays with probability 1/2 at the current location.
Lazy Random Walks
t = 1
t = 3
t = 2
t = 4
Intro 6
![Page 29: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/29.jpg)
Random Walk on a Dynamic Graph Sequence
The random walk stays with probability 1/2 at the current location.
Lazy Random Walks
t = 1
t = 3
t = 2
t = 4
Intro 6
![Page 30: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/30.jpg)
Random Walk on a Dynamic Graph Sequence
The random walk stays with probability 1/2 at the current location.
Lazy Random Walks
t = 1
t = 3
t = 2
t = 4
Intro 6
![Page 31: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/31.jpg)
Outline
Intro
Random Walks on Sequences of Connected Graphs
Random Walks on Sequences of (Possibly) Disconnected Graphs
Conclusion
Random Walks on Sequences of Connected Graphs 7
![Page 32: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/32.jpg)
Agenda of this Talk
We are interested in studying the following quantities on a sequence of dynamicgraphs G = (G1,G2, . . .) on a fixed set vertices:
Mixing time Number of steps needed for the distribution of the walk tobecome ε-close to the stationary distribution
Hitting times Expected number of steps to go from u to v thit (u, v)
For static connected graphs:
regular case O(n2) mixing and hitting times
general case O(n3) mixing and hitting times
For dynamic connected graphs:
If π(t) changes over time, in general, we don’t have mixing
Can we at least say something about hitting times?
Random Walks on Sequences of Connected Graphs 8
![Page 33: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/33.jpg)
Agenda of this Talk
We are interested in studying the following quantities on a sequence of dynamicgraphs G = (G1,G2, . . .) on a fixed set vertices:
Mixing time Number of steps needed for the distribution of the walk tobecome ε-close to the stationary distribution
Hitting times Expected number of steps to go from u to v thit (u, v)
For static connected graphs:
regular case O(n2) mixing and hitting times
general case O(n3) mixing and hitting times
For dynamic connected graphs:
If π(t) changes over time, in general, we don’t have mixing
Can we at least say something about hitting times?
Random Walks on Sequences of Connected Graphs 8
![Page 34: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/34.jpg)
Agenda of this Talk
We are interested in studying the following quantities on a sequence of dynamicgraphs G = (G1,G2, . . .) on a fixed set vertices:
Mixing time Number of steps needed for the distribution of the walk tobecome ε-close to the stationary distribution
Hitting times Expected number of steps to go from u to v thit (u, v)
For static connected graphs:
regular case O(n2) mixing and hitting times
general case O(n3) mixing and hitting times
For dynamic connected graphs:
If π(t) changes over time, in general, we don’t have mixing
Can we at least say something about hitting times?
Random Walks on Sequences of Connected Graphs 8
![Page 35: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/35.jpg)
Agenda of this Talk
We are interested in studying the following quantities on a sequence of dynamicgraphs G = (G1,G2, . . .) on a fixed set vertices:
Mixing time Number of steps needed for the distribution of the walk tobecome ε-close to the stationary distribution
Hitting times Expected number of steps to go from u to v thit (u, v)
For static connected graphs:
regular case O(n2) mixing and hitting times
general case O(n3) mixing and hitting times
For dynamic connected graphs:
If π(t) changes over time, in general, we don’t have mixing
Can we at least say something about hitting times?
Random Walks on Sequences of Connected Graphs 8
![Page 36: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/36.jpg)
Agenda of this Talk
We are interested in studying the following quantities on a sequence of dynamicgraphs G = (G1,G2, . . .) on a fixed set vertices:
Mixing time Number of steps needed for the distribution of the walk tobecome ε-close to the stationary distribution
Hitting times Expected number of steps to go from u to v thit (u, v)
For static connected graphs:
regular case O(n2) mixing and hitting times
general case O(n3) mixing and hitting times
For dynamic connected graphs:
If π(t) changes over time, in general, we don’t have mixing
Can we at least say something about hitting times?
Random Walks on Sequences of Connected Graphs 8
![Page 37: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/37.jpg)
Agenda of this Talk
We are interested in studying the following quantities on a sequence of dynamicgraphs G = (G1,G2, . . .) on a fixed set vertices:
Mixing time Number of steps needed for the distribution of the walk tobecome ε-close to the stationary distribution
Hitting times Expected number of steps to go from u to v thit (u, v)
For static connected graphs:
regular case O(n2) mixing and hitting times
general case O(n3) mixing and hitting times
For dynamic connected graphs:
If π(t) changes over time, in general, we don’t have mixing
Can we at least say something about hitting times?
Random Walks on Sequences of Connected Graphs 8
![Page 38: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/38.jpg)
Related Work: A Dichotomy for dynamic graphs
1. If π(t) changes over time,
hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),
mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
Random Walks on Sequences of Connected Graphs 9
![Page 39: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/39.jpg)
Related Work: A Dichotomy for dynamic graphs
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),
mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
Random Walks on Sequences of Connected Graphs 9
![Page 40: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/40.jpg)
Related Work: A Dichotomy for dynamic graphs
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),
mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
Random Walks on Sequences of Connected Graphs 9
![Page 41: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/41.jpg)
Related Work: A Dichotomy for dynamic graphs
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
Random Walks on Sequences of Connected Graphs 9
![Page 42: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/42.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 43: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/43.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 44: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/44.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 45: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/45.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 46: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/46.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 47: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/47.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 48: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/48.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 49: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/49.jpg)
Hitting Times can be bad! (The Sisyphus Graph)
n − 2
n1
2
3
4 n − 3n − 1
t = 1
n − 3
nn − 1
1
2
3 n − 4n − 2
t = 2
n − 4
nn − 2
n − 1
1
2 n − 5n − 3
t = 3
n − 5
nn − 3
n − 2
n − 1
1 n − 6n − 4
t = 4
Random Walks on Sequences of Connected Graphs 10
![Page 50: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/50.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,
mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,
mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 51: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/51.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,
mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,
mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 52: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/52.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,
mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,
mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 53: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/53.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,
mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 54: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/54.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,
mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 55: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/55.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 56: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/56.jpg)
Our Results
1. If π(t) changes over time,hitting (and covering) can take exponential time
this holds even if π(t) changes slowly
2. If all graphs are connected and regular (⇒ π(t) is always uniform),mixing in O(n2 log(n)) steps
hitting and covering in O(n3 log2(n)) steps
Avin, Koucky, and Lotker (ICALP’08, RSA’18)
1. If all graphs are connected and regular,mixing and hitting in O(n2) steps (optimal)
2. More generally, if π(t) = π for any t ,mixing in O(n3) steps (optimal)hitting in O(n3 log(n)) steps (nearly optimal)
Our Results
How can we derive these results?
Random Walks on Sequences of Connected Graphs 11
![Page 57: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/57.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 58: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/58.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 59: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/59.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 60: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/60.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 61: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/61.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 62: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/62.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |
Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 63: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/63.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 64: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/64.jpg)
Classical Proof (Spanning Tree Approach)
For any static graph G, tcov (G) ≤ 2(n − 1)|E |).Aleliunas, Karp, Lipton, Lovász and Rackoff, FOCS’79
Proof:
Take a spanning tree T in G
Consider a traversal that goesthrough every edge in T twice
For any connected vertices i, j ,thit (i, j) + thit (j, i) = 2|E |Thus,
tcov (G) ≤∑
(i,j)∈E(T )
thit (i, j) + thit (j, i)
≤ 2(n − 1) · |E |.
1 2
3 4
5
6
7
8
Random Walks on Sequences of Connected Graphs 12
![Page 65: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/65.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1)
≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 66: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/66.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1)
≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 67: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/67.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.
Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1)
≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 68: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/68.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1)
≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 69: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/69.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1) ≤D−1∑i=0
2|E |
= 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 70: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/70.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1) ≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 71: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/71.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1) ≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 72: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/72.jpg)
Classical Proof (Refinement based on Shortest Path)
For any static graph with diameter D, thit (G) ≤ 2|E | · D.(cf. Aldous, Fill’02)
Proof:
Fix two vertices s, t , and consider a shortest path P = (u0 = s, u1, . . . , ul = t)
As before thit (ui , ui+1) ≤ 2|E |.Thus,
thit (s, t) ≤D−1∑i=0
thit (ui , ui+1) ≤D−1∑i=0
2|E | = 2|E |D
This proves not only a bound of O(n3) for any graph, but also O(n2) for regular graphs.
Both proofs crucially rely on a static spanning tree or static shortest path!
Random Walks on Sequences of Connected Graphs 13
![Page 73: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/73.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 74: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/74.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 75: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/75.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 76: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/76.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 77: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/77.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 78: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/78.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 79: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/79.jpg)
Return Times on Dynamic Graphs
A fundamental fact of the return times is that:
thit (u, u) =1
π(u).
Is this true for dynamic graphs?
No!
11
121
2
3
4
5
67
8
9
10
t = 0
11
127
2
3
4
5
61
8
9
10
t = 1
11
127
2
3
4
5
61
8
9
10
t = 2
Many combinatorial and probabilistic arguments seem to fail,but what about the t-step probabilities (and return probabilities)?
Random Walks on Sequences of Connected Graphs 14
![Page 80: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/80.jpg)
Diffusion of a Random Walk on a Static Cycle
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Step: 0
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 81: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/81.jpg)
Diffusion of a Random Walk on a Static Cycle
0.500
0.250
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.250
Step: 1
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 82: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/82.jpg)
Diffusion of a Random Walk on a Static Cycle
0.375
0.250
0.062
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.062
0.250
Step: 2
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 83: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/83.jpg)
Diffusion of a Random Walk on a Static Cycle
0.312
0.234
0.094
0.016
0.000
0.000
0.000
0.000
0.000
0.016
0.094
0.234
Step: 3
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 84: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/84.jpg)
Diffusion of a Random Walk on a Static Cycle
0.273
0.219
0.109
0.031
0.004
0.000
0.000
0.000
0.004
0.031
0.109
0.219
Step: 4
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 85: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/85.jpg)
Diffusion of a Random Walk on a Static Cycle
0.246
0.205
0.117
0.044
0.010
0.001
0.000
0.001
0.010
0.044
0.117
0.205
Step: 5
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 86: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/86.jpg)
Diffusion of a Random Walk on a Static Cycle
0.226
0.193
0.121
0.054
0.016
0.003
0.000
0.003
0.016
0.054
0.121
0.193
Step: 6
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 87: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/87.jpg)
Diffusion of a Random Walk on a Static Cycle
0.209
0.183
0.122
0.061
0.022
0.006
0.002
0.006
0.022
0.061
0.122
0.183
Step: 7
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 88: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/88.jpg)
Diffusion of a Random Walk on a Static Cycle
0.196
0.175
0.122
0.067
0.028
0.009
0.004
0.009
0.028
0.067
0.122
0.175
Step: 8
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 89: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/89.jpg)
Diffusion of a Random Walk on a Static Cycle
0.185
0.167
0.121
0.071
0.033
0.012
0.006
0.012
0.033
0.071
0.121
0.167
Step: 9
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 90: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/90.jpg)
Diffusion of a Random Walk on a Static Cycle
0.176
0.160
0.120
0.074
0.037
0.016
0.009
0.016
0.037
0.074
0.120
0.160
Step: 10
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 91: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/91.jpg)
Diffusion of a Random Walk on a Static Cycle
0.168
0.154
0.119
0.076
0.041
0.020
0.013
0.020
0.041
0.076
0.119
0.154
Step: 11
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 92: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/92.jpg)
Diffusion of a Random Walk on a Static Cycle
0.161
0.149
0.117
0.078
0.044
0.023
0.016
0.023
0.044
0.078
0.117
0.149
Step: 12
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 93: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/93.jpg)
Diffusion of a Random Walk on a Static Cycle
0.155
0.144
0.115
0.079
0.048
0.027
0.020
0.027
0.048
0.079
0.115
0.144
Step: 13
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 94: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/94.jpg)
Diffusion of a Random Walk on a Static Cycle
0.149
0.139
0.113
0.080
0.050
0.030
0.023
0.030
0.050
0.080
0.113
0.139
Step: 14
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 95: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/95.jpg)
Diffusion of a Random Walk on a Static Cycle
0.144
0.135
0.112
0.081
0.053
0.033
0.027
0.033
0.053
0.081
0.112
0.135
Step: 15
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 96: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/96.jpg)
Diffusion of a Random Walk on a Static Cycle
0.140
0.132
0.110
0.082
0.055
0.037
0.030
0.037
0.055
0.082
0.110
0.132
Step: 16
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 97: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/97.jpg)
Diffusion of a Random Walk on a Static Cycle
0.136
0.128
0.108
0.082
0.057
0.040
0.033
0.040
0.057
0.082
0.108
0.128
Step: 17
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 98: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/98.jpg)
Diffusion of a Random Walk on a Static Cycle
0.132
0.125
0.107
0.082
0.059
0.042
0.036
0.042
0.059
0.082
0.107
0.125
Step: 18
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 99: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/99.jpg)
Diffusion of a Random Walk on a Static Cycle
0.129
0.122
0.105
0.083
0.061
0.045
0.039
0.045
0.061
0.083
0.105
0.122
Step: 19
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 100: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/100.jpg)
Diffusion of a Random Walk on a Static Cycle
0.126
0.120
0.104
0.083
0.062
0.048
0.042
0.048
0.062
0.083
0.104
0.120
Step: 20
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 101: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/101.jpg)
Diffusion of a Random Walk on a Static Cycle
0.123
0.117
0.103
0.083
0.064
0.050
0.045
0.050
0.064
0.083
0.103
0.117
Step: 21
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 102: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/102.jpg)
Diffusion of a Random Walk on a Static Cycle
0.120
0.115
0.101
0.083
0.065
0.052
0.047
0.052
0.065
0.083
0.101
0.115
Step: 22
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 103: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/103.jpg)
Diffusion of a Random Walk on a Static Cycle
0.117
0.113
0.100
0.083
0.066
0.054
0.050
0.054
0.066
0.083
0.100
0.113
Step: 23
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 104: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/104.jpg)
Diffusion of a Random Walk on a Static Cycle
0.115
0.111
0.099
0.083
0.067
0.056
0.052
0.056
0.067
0.083
0.099
0.111
Step: 24
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 105: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/105.jpg)
Diffusion of a Random Walk on a Static Cycle
0.113
0.109
0.098
0.083
0.069
0.058
0.054
0.058
0.069
0.083
0.098
0.109
Step: 25
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 106: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/106.jpg)
Diffusion of a Random Walk on a Static Cycle
0.111
0.107
0.097
0.083
0.070
0.060
0.056
0.060
0.070
0.083
0.097
0.107
Step: 26
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 107: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/107.jpg)
Diffusion of a Random Walk on a Static Cycle
0.109
0.106
0.096
0.083
0.070
0.061
0.058
0.061
0.070
0.083
0.096
0.106
Step: 27
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 108: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/108.jpg)
Diffusion of a Random Walk on a Static Cycle
0.107
0.104
0.095
0.083
0.071
0.063
0.059
0.063
0.071
0.083
0.095
0.104
Step: 28
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 109: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/109.jpg)
Diffusion of a Random Walk on a Static Cycle
0.106
0.103
0.094
0.083
0.072
0.064
0.061
0.064
0.072
0.083
0.094
0.103
Step: 29
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 110: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/110.jpg)
Diffusion of a Random Walk on a Static Cycle
0.104
0.101
0.094
0.083
0.073
0.065
0.063
0.065
0.073
0.083
0.094
0.101
Step: 30
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 111: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/111.jpg)
Diffusion of a Random Walk on a Static Cycle
0.103
0.100
0.093
0.083
0.074
0.067
0.064
0.067
0.074
0.083
0.093
0.100
Step: 31
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 112: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/112.jpg)
Diffusion of a Random Walk on a Static Cycle
0.101
0.099
0.092
0.083
0.074
0.068
0.065
0.068
0.074
0.083
0.092
0.099
Step: 32
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 113: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/113.jpg)
Diffusion of a Random Walk on a Static Cycle
0.100
0.098
0.092
0.083
0.075
0.069
0.066
0.069
0.075
0.083
0.092
0.098
Step: 33
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 114: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/114.jpg)
Diffusion of a Random Walk on a Static Cycle
0.099
0.097
0.091
0.083
0.075
0.070
0.068
0.070
0.075
0.083
0.091
0.097
Step: 34
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 115: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/115.jpg)
Diffusion of a Random Walk on a Static Cycle
0.098
0.096
0.091
0.083
0.076
0.071
0.069
0.071
0.076
0.083
0.091
0.096
Step: 35
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 116: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/116.jpg)
Diffusion of a Random Walk on a Static Cycle
0.097
0.095
0.090
0.083
0.076
0.071
0.070
0.071
0.076
0.083
0.090
0.095
Step: 36
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 117: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/117.jpg)
Diffusion of a Random Walk on a Static Cycle
0.096
0.094
0.090
0.083
0.077
0.072
0.071
0.072
0.077
0.083
0.090
0.094
Step: 37
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 118: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/118.jpg)
Diffusion of a Random Walk on a Static Cycle
0.095
0.094
0.089
0.083
0.077
0.073
0.071
0.073
0.077
0.083
0.089
0.094
Step: 38
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 119: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/119.jpg)
Diffusion of a Random Walk on a Static Cycle
0.094
0.093
0.089
0.083
0.078
0.074
0.072
0.074
0.078
0.083
0.089
0.093
Step: 39
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 120: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/120.jpg)
Diffusion of a Random Walk on a Static Cycle
0.094
0.092
0.089
0.083
0.078
0.074
0.073
0.074
0.078
0.083
0.089
0.092
Step: 40
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 121: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/121.jpg)
Diffusion of a Random Walk on a Static Cycle
0.093
0.092
0.088
0.083
0.078
0.075
0.074
0.075
0.078
0.083
0.088
0.092
Step: 41
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 122: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/122.jpg)
Diffusion of a Random Walk on a Static Cycle
0.092
0.091
0.088
0.083
0.079
0.075
0.074
0.075
0.079
0.083
0.088
0.091
Step: 42
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 123: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/123.jpg)
Diffusion of a Random Walk on a Static Cycle
0.092
0.091
0.088
0.083
0.079
0.076
0.075
0.076
0.079
0.083
0.088
0.091
Step: 43
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 124: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/124.jpg)
Diffusion of a Random Walk on a Static Cycle
0.091
0.090
0.087
0.083
0.079
0.077
0.075
0.077
0.079
0.083
0.087
0.090
Step: 44
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 125: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/125.jpg)
Diffusion of a Random Walk on a Static Cycle
0.091
0.090
0.087
0.083
0.080
0.077
0.076
0.077
0.080
0.083
0.087
0.090
Step: 45
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 126: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/126.jpg)
Diffusion of a Random Walk on a Static Cycle
0.090
0.089
0.087
0.083
0.080
0.077
0.076
0.077
0.080
0.083
0.087
0.089
Step: 46
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 127: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/127.jpg)
Diffusion of a Random Walk on a Static Cycle
0.090
0.089
0.087
0.083
0.080
0.078
0.077
0.078
0.080
0.083
0.087
0.089
Step: 47
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 128: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/128.jpg)
Diffusion of a Random Walk on a Static Cycle
0.089
0.089
0.086
0.083
0.080
0.078
0.077
0.078
0.080
0.083
0.086
0.089
Step: 48
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 129: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/129.jpg)
Diffusion of a Random Walk on a Static Cycle
0.089
0.088
0.086
0.083
0.081
0.079
0.078
0.079
0.081
0.083
0.086
0.088
Step: 49
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 130: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/130.jpg)
Diffusion of a Random Walk on a Static Cycle
0.089
0.088
0.086
0.083
0.081
0.079
0.078
0.079
0.081
0.083
0.086
0.088
Step: 50
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 131: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/131.jpg)
Diffusion of a Random Walk on a Static Cycle
As long as the probability mass is concentrated on a small set of vertices,substantial progress in the `2-norm
More precisely, ‖ptu,. − 1
n‖22 ∼ 1/
√t
This property only requires each graph Gt to be connected (& regular) at each time
Random Walks on Sequences of Connected Graphs 15
![Page 132: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/132.jpg)
Mixing in Dynamic Graphs: Definition
Sequence of graphs G = G(t)∞t=1 on V with transition matrices P(t)∞t=1
πP(t) = π for any t
tmix (G) = min
t
∣∣∣∣∣∣∑y∈V
(P [0,t](x , y)− 1
n
)2
≤ 110n
∀ x ∈ V
.
`2-mixing time
extends to non-regular in a natural way
Random Walks on Sequences of Connected Graphs 16
![Page 133: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/133.jpg)
Mixing in Dynamic Graphs: Definition
Sequence of graphs G = G(t)∞t=1 on V with transition matrices P(t)∞t=1
πP(t) = π for any t
tmix (G) = min
t
∣∣∣∣∣∣∑y∈V
(P [0,t](x , y)− 1
n
)2
≤ 110n
∀ x ∈ V
.
`2-mixing time
extends to non-regular in a natural way
Random Walks on Sequences of Connected Graphs 16
![Page 134: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/134.jpg)
Mixing in Dynamic Graphs: Definition
Sequence of graphs G = G(t)∞t=1 on V with transition matrices P(t)∞t=1
πP(t) = π for any t
tmix (G) = min
t
∣∣∣∣∣∣∑y∈V
(P [0,t](x , y)− 1
n
)2
≤ 110n
∀ x ∈ V
.
`2-mixing time
extends to non-regular in a natural way
Random Walks on Sequences of Connected Graphs 16
![Page 135: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/135.jpg)
A Bound on the `2-Decrease
Let P be the transition matrix of a random walk on a connected, regular graphG = (V ,E). Then for any probability distribution σ,
∑u,v∈V
(σ(u)− σ(v))2 · Pu,v &∑u∈V
(σ(u)− 1
n
)2
.
Key Lemma
Proof Sketch:
As long as ‖σ − 1n‖
22 is large⇒ σ is concentrated on a small set of vertices
⇒∃ short path between x? = argmaxx σ(x) and y s.t. σ(y) σ(x?)
⇒ Let ` be the length of such path. Then,∑u,v∈V
(σ(u)− σ(v))2Pu,v ≥(σ(x?)− σ(y))2
2`is large
Random Walks on Sequences of Connected Graphs 17
![Page 136: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/136.jpg)
A Bound on the `2-Decrease
Let P be the transition matrix of a random walk on a connected, regular graphG = (V ,E). Then for any probability distribution σ,
∑u,v∈V
(σ(u)− σ(v))2 · Pu,v &∑u∈V
(σ(u)− 1
n
)2
.
Key Lemma
Proof Sketch:
As long as ‖σ − 1n‖
22 is large⇒ σ is concentrated on a small set of vertices
⇒∃ short path between x? = argmaxx σ(x) and y s.t. σ(y) σ(x?)
⇒ Let ` be the length of such path. Then,∑u,v∈V
(σ(u)− σ(v))2Pu,v ≥(σ(x?)− σ(y))2
2`is large
Random Walks on Sequences of Connected Graphs 17
![Page 137: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/137.jpg)
A Bound on the `2-Decrease
Let P be the transition matrix of a random walk on a connected, regular graphG = (V ,E). Then for any probability distribution σ,
∑u,v∈V
(σ(u)− σ(v))2 · Pu,v &∑u∈V
(σ(u)− 1
n
)2
.
Key Lemma
Proof Sketch:
As long as ‖σ − 1n‖
22 is large⇒ σ is concentrated on a small set of vertices
⇒∃ short path between x? = argmaxx σ(x) and y s.t. σ(y) σ(x?)
⇒ Let ` be the length of such path. Then,∑u,v∈V
(σ(u)− σ(v))2Pu,v ≥(σ(x?)− σ(y))2
2`is large
Random Walks on Sequences of Connected Graphs 17
![Page 138: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/138.jpg)
A Bound on the `2-Decrease
Let P be the transition matrix of a random walk on a connected, regular graphG = (V ,E). Then for any probability distribution σ,
∑u,v∈V
(σ(u)− σ(v))2 · Pu,v &∑u∈V
(σ(u)− 1
n
)2
.
Key Lemma
Proof Sketch:
As long as ‖σ − 1n‖
22 is large⇒ σ is concentrated on a small set of vertices
⇒∃ short path between x? = argmaxx σ(x) and y s.t. σ(y) σ(x?)
⇒ Let ` be the length of such path. Then,∑u,v∈V
(σ(u)− σ(v))2Pu,v ≥(σ(x?)− σ(y))2
2`is large
Random Walks on Sequences of Connected Graphs 17
![Page 139: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/139.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 140: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/140.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 141: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/141.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 142: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/142.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 143: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/143.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 144: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/144.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 145: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/145.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 146: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/146.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 147: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/147.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 148: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/148.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 149: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/149.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 150: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/150.jpg)
Main Result (covering also non-regular graphs)
Let G be a sequence of connected graphs of n vertices with unique stationarydistribution π. Moreover, denote with π∗ = minx π(x). Then:
tmix (G) = O(n/π∗)
thit (G) = O(n log n/π∗).
If all graphs in G are regular, thit (G) = O(n2).
Theorem
To prove the bound on mixing:
Key Lemma⇒ if variance is ε, after O(n/(π∗ε)) steps it is less than ε/2
Hence after O(n/π∗) steps, variance will be small constant⇒ walk mixed
To prove the bound on hitting:
first obtain a refined bound on the variance decrease at each step
relate t-step probabilities to the decrease in variance of the walk
use probabilistic arguments to relate t-step probabilities to hitting times
What if the graphs in the sequence have good expansion?
If every graph G is a regular expander, tmix (G) = O(log n) and thit (G) = O(n)
Refinement of Theorem⇒ thit (G) = O(n) if the isoperimetric dimension ofeach (bounded-degree) graph in G is 2 + ε
solves a conjecture by Aldous and Fill, which was proved by Benjamini andKozma (Combinatorica’05) for static graphs
Random Walks on Sequences of Connected Graphs 18
![Page 151: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/151.jpg)
Outline
Intro
Random Walks on Sequences of Connected Graphs
Random Walks on Sequences of (Possibly) Disconnected Graphs
Conclusion
Random Walks on Sequences of (Possibly) Disconnected Graphs 19
![Page 152: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/152.jpg)
What happens when the connectivity properties of the graphchange over time?
Random Walks on Sequences of (Possibly) Disconnected Graphs 20
![Page 153: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/153.jpg)
How to bound mixing when connectivity is intermittent
In static graphs, the eigenvalues of the individual transition matrices give agood bound on mixing:
11− λ . tmix (G) .
log(n)
1− λ
This is not necessarily true for dynamic graphs:
Odd t
1− λ(P(t)) = 0
Even t
1− λ(P(t)) = 0
Random Walks on Sequences of (Possibly) Disconnected Graphs 21
![Page 154: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/154.jpg)
How to bound mixing when connectivity is intermittent
In static graphs, the eigenvalues of the individual transition matrices give agood bound on mixing:
11− λ . tmix (G) .
log(n)
1− λThis is not necessarily true for dynamic graphs:
Odd t
1− λ(P(t)) = 0
Even t
1− λ(P(t)) = 0
Random Walks on Sequences of (Possibly) Disconnected Graphs 21
![Page 155: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/155.jpg)
How to bound mixing when connectivity is intermittent
In static graphs, the eigenvalues of the individual transition matrices give agood bound on mixing:
11− λ . tmix (G) .
log(n)
1− λThis is not necessarily true for dynamic graphs:
Odd t
1− λ(P(t)) = 0
Even t
1− λ(P(t)) = 0
Random Walks on Sequences of (Possibly) Disconnected Graphs 21
![Page 156: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/156.jpg)
How to bound mixing when connectivity is intermittent
In static graphs, the eigenvalues of the individual transition matrices give agood bound on mixing:
11− λ . tmix (G) .
log(n)
1− λThis is not necessarily true for dynamic graphs:
Odd t
1− λ(P(t)) = 0
Even t
1− λ(P(t)) = 0
Random Walks on Sequences of (Possibly) Disconnected Graphs 21
![Page 157: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/157.jpg)
Average transition probabilities
Odd t : 1− λ(P(t)) = 0
Even t : 1− λ(P(t)) = 0
Average transition probabilities P
1− λ(P) = Ω(1)
Random Walks on Sequences of (Possibly) Disconnected Graphs 22
![Page 158: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/158.jpg)
Average transition probabilities
Odd t : 1− λ(P(t)) = 0
Even t : 1− λ(P(t)) = 0
Average transition probabilities P
1− λ(P) = Ω(1)
Random Walks on Sequences of (Possibly) Disconnected Graphs 22
![Page 159: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/159.jpg)
Mixing based on average connectivity properties
Consider a sequence G with transition matrices P(t)∞t=1 such that
1. πP(t) = π for any t
2. there exists a time window T ≥ 1 such that, for any i ≥ 0, P[i·T+1,(i+1)·T ]
is ergodic with spectral gap greater or equal than 1− λThen, tmix (G) = O(T 2 log(1/π∗)/(1− λ))
Theorem
Suppose that for any time window I = [i · T + 1, (i + 1) · T ] and any subsetof vertices A ⊆ V there exists i ∈ I such that ΦP(i)(A) ≥ φ. Then,
tmix (G) = O(T 3 log(1/π∗)/φ2)
Corollary
Since thit (G) = O(tmix (G)/π∗), does polynomial mixing time imply polynomialhitting times?
NO! When the graphs are disconnected, π∗ can be exponentially smallWhy? We can simulate a random walk on a directed graph:
Random Walks on Sequences of (Possibly) Disconnected Graphs 23
![Page 160: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/160.jpg)
Mixing based on average connectivity properties
Consider a sequence G with transition matrices P(t)∞t=1 such that
1. πP(t) = π for any t
2. there exists a time window T ≥ 1 such that, for any i ≥ 0, P[i·T+1,(i+1)·T ]
is ergodic with spectral gap greater or equal than 1− λThen, tmix (G) = O(T 2 log(1/π∗)/(1− λ))
Theorem
Suppose that for any time window I = [i · T + 1, (i + 1) · T ] and any subsetof vertices A ⊆ V there exists i ∈ I such that ΦP(i)(A) ≥ φ. Then,
tmix (G) = O(T 3 log(1/π∗)/φ2)
Corollary
Since thit (G) = O(tmix (G)/π∗), does polynomial mixing time imply polynomialhitting times?
NO! When the graphs are disconnected, π∗ can be exponentially smallWhy? We can simulate a random walk on a directed graph:
Random Walks on Sequences of (Possibly) Disconnected Graphs 23
![Page 161: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/161.jpg)
Mixing based on average connectivity properties
Consider a sequence G with transition matrices P(t)∞t=1 such that
1. πP(t) = π for any t
2. there exists a time window T ≥ 1 such that, for any i ≥ 0, P[i·T+1,(i+1)·T ]
is ergodic with spectral gap greater or equal than 1− λThen, tmix (G) = O(T 2 log(1/π∗)/(1− λ))
Theorem
Suppose that for any time window I = [i · T + 1, (i + 1) · T ] and any subsetof vertices A ⊆ V there exists i ∈ I such that ΦP(i)(A) ≥ φ. Then,
tmix (G) = O(T 3 log(1/π∗)/φ2)
Corollary
Since thit (G) = O(tmix (G)/π∗), does polynomial mixing time imply polynomialhitting times?
NO! When the graphs are disconnected, π∗ can be exponentially smallWhy? We can simulate a random walk on a directed graph:
Random Walks on Sequences of (Possibly) Disconnected Graphs 23
![Page 162: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/162.jpg)
Mixing based on average connectivity properties
Consider a sequence G with transition matrices P(t)∞t=1 such that
1. πP(t) = π for any t
2. there exists a time window T ≥ 1 such that, for any i ≥ 0, P[i·T+1,(i+1)·T ]
is ergodic with spectral gap greater or equal than 1− λThen, tmix (G) = O(T 2 log(1/π∗)/(1− λ))
Theorem
Suppose that for any time window I = [i · T + 1, (i + 1) · T ] and any subsetof vertices A ⊆ V there exists i ∈ I such that ΦP(i)(A) ≥ φ. Then,
tmix (G) = O(T 3 log(1/π∗)/φ2)
Corollary
Since thit (G) = O(tmix (G)/π∗), does polynomial mixing time imply polynomialhitting times?
NO! When the graphs are disconnected, π∗ can be exponentially small
Why? We can simulate a random walk on a directed graph:
Random Walks on Sequences of (Possibly) Disconnected Graphs 23
![Page 163: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/163.jpg)
Mixing based on average connectivity properties
Consider a sequence G with transition matrices P(t)∞t=1 such that
1. πP(t) = π for any t
2. there exists a time window T ≥ 1 such that, for any i ≥ 0, P[i·T+1,(i+1)·T ]
is ergodic with spectral gap greater or equal than 1− λThen, tmix (G) = O(T 2 log(1/π∗)/(1− λ))
Theorem
Suppose that for any time window I = [i · T + 1, (i + 1) · T ] and any subsetof vertices A ⊆ V there exists i ∈ I such that ΦP(i)(A) ≥ φ. Then,
tmix (G) = O(T 3 log(1/π∗)/φ2)
Corollary
Since thit (G) = O(tmix (G)/π∗), does polynomial mixing time imply polynomialhitting times?
NO! When the graphs are disconnected, π∗ can be exponentially smallWhy? We can simulate a random walk on a directed graph:
Random Walks on Sequences of (Possibly) Disconnected Graphs 23
![Page 164: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/164.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 165: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/165.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1
t = 2t = 3t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 166: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/166.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1
t = 2
t = 3t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 167: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/167.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2
t = 3
t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 168: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/168.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3
t = 4
t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 169: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/169.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4
t = 5
t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 170: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/170.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5
t = 6
t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 171: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/171.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6
t = 7
t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 172: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/172.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7
t = 8
Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 173: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/173.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 174: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/174.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1
t = 2t = 3t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 175: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/175.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1
t = 2
t = 3t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 176: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/176.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2
t = 3
t = 4t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 177: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/177.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3
t = 4
t = 5t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 178: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/178.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4
t = 5
t = 6t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 179: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/179.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5
t = 6
t = 7t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 180: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/180.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6
t = 7
t = 8Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 181: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/181.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7
t = 8
Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 182: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/182.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8
Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 183: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/183.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8
Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 184: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/184.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8
Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 185: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/185.jpg)
Simulating a Directed Graph using Dynamic Graphs
1/4 1/4
1/8 1/8
1/16 1/16
1/32 1/32
1/64 1/64
t = 1t = 2t = 3t = 4t = 5t = 6t = 7t = 8
Random Walk Behaviour:
Since the stationary distribution is exponentially small for thevertices at the bottom, hitting time is exponential in n
However, average transition matrix P can be easily made ergodic(add same cycle of n − 2 matrices in reverse order)
⇒mixing time polynomial in n by our theorem!
Random Walks on Sequences of (Possibly) Disconnected Graphs 24
![Page 186: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/186.jpg)
Outline
Intro
Random Walks on Sequences of Connected Graphs
Random Walks on Sequences of (Possibly) Disconnected Graphs
Conclusion
Conclusion 25
![Page 187: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/187.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 188: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/188.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 189: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/189.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 190: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/190.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 191: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/191.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 192: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/192.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 193: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/193.jpg)
Conclusions and further work
We have exhibited a dichotomy for random walks on dynamic graphs:
If stationary distribution does not change over time, behaviour is comparableto static graphs
otherwise, they lose many nice properties associated with random walks onstatic graphs (even when the changes in the stationary distribution are small,e.g., all graphs are bounded-degree)
Bad counter-examples often simulate random walks on directed graphs.
Is there a more profound link between dynamic graphs and directed graphs?
Here we have only considered worst-case changes.
Can our methods be applied to settings where the graph changes randomly?
Conclusion 26
![Page 194: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/194.jpg)
The End
Conclusion 27
![Page 195: Random walks on dynamic graphs: Mixing times, hitting times, …tms41/dynamic.pdf · 2020. 6. 3. · Randomwalksondynamicgraphs: Mixingtimes,hittingtimes,andreturnprobabilities Thomas](https://reader034.vdocument.in/reader034/viewer/2022052103/603d46b42a665e3b9d67927f/html5/thumbnails/195.jpg)
The End
Conclusion 27