Back to Random Walks on Graphs

1. Back to Random Walks on Graphs Random walk on a graph: Stationary distribution:

2. Back to Random Walks on Graphs Random walk on a graph: Stationary distribution:

3. Detailed balance condition Detailed balance condition: An ergodic Markov chain satisfying the detailed balance condition is called reversible.

4. Hitting, Commute, Cover Time Given is a MC M=(,P). Hitting time from u to v, for u,v2: hu,v(M) = expected number of steps of M started from u until first reach v

5. Hitting, Commute, Cover Time Given is a MC M=(,P). Hitting time from u to v, for u,v2: hu,v(M) = expected number of steps of M started from u until first reach v Commute time between u and v, for u,v2: Cu,v(M) = expected number of steps of M started from u to reach v and get back to u

6. Hitting, Commute, Cover Time Given is a MC M=(,P). Hitting time from u to v, for u,v2: hu,v(M) = expected number of steps of M started from u until first reach v Commute time between u and v, for u,v2: Cu,v(M) = expected number of steps of M started from u to reach v and get back to u Cover time: Cu(M) = expected number of steps of M started from u until every state in  has been visited at least once C(M) = maxu2 Cu(M)

7. Hitting Time of a Random Walk on a Graph Given is a graph G.

8. Electrical Networks Resistive electrical network: Resistive electrical network: Resistive electrical network: Rectangles: branch resistance Injecting a current of 1 ampere into b: a 1 1 b c 2

9. Electrical Networks Resistive electrical network: Resistive electrical network: Resistive electrical network: Goal: find voltages at every node such that: Kirhoff’s Law: sum of the currents in = sum of the currents out Ohm’s Law: voltage difference across resistance = product of the current and the resistance a 1 1 b c 2

10. Electrical Networks Resistive electrical network: Resistive electrical network: Resistive electrical network: Effective resistance between two nodes u,v: Ru,v = voltage difference when one ampere is injected into u and removed from v Example: effective resistance vs branch resistance between b,c a 1 1 b c 2

11. Electrical Networks and the Commute Time Resistive electrical network: Resistive electrical network: Resistive electrical network: Effective resistance between two nodes u,v: Ru,v = voltage difference when one ampere is injected into u and removed from v Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v, where branch resistance = 1 on every edge. a 1 1 b c 2

12. Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v where branch resistance = 1 for every edge.

13. Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v where branch resistance = 1 for every edge.

14. Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v where branch resistance = 1 for every edge.

15. Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v where branch resistance = 1 for every edge.

16. Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v where branch resistance = 1 for every edge.

17. Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,v where branch resistance = 1 for every edge. Corollary: For any u,v: Cu,v· n3.

18. Electrical Networks and the Cover Time Thm: For a random walk on a graph G with m edges and n vertices: C(G) · 2m(n-1).

19. Electrical Networks and the Cover Time Thm: For a random walk on a graph G with m edges and n vertices: mR(G) · C(G) · 2e3mR(G)ln n + n, where R(G) = maxu,vRu,v.