Back to Random Walks on Graphs Random walk on a graph: Stationary - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

Electrical Networks Resistive electrical network: Resistive electrical network: Resistive electrical network: a 1 1 2 b c 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

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

Electrical Networks and the Commute Time Resistive electrical network: Resistive electrical network: Resistive electrical network: a 1 1 2 b c Effective resistance between two nodes u,v: R u,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: C u,v = 2mR u,v , where branch resistance = 1 on every edge.

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

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

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

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

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

Electrical Networks and the Commute Time Thm: For a random walk on a graph G with m edges: C u,v = 2mR u,v where branch resistance = 1 for every edge. Corollary: For any u,v: C u,v ≤ n 3 .

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).

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) ≤ 2e 3 mR(G)ln n + n, where R(G) = max u,v R u,v .