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∈: hu,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∈: hu,v(M) = expected number of steps of M started from u until first reach v Commute time between u and v, for u,v∈: Cu,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∈: hu,v(M) = expected number of steps of M started from u until first reach v Commute time between u and v, for u,v∈: 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) = maxu∈ Cu(M)

Hitting Time of a Random Walk on a Graph

Given is a graph G.

Resistive electrical network: Resistive electrical network:

Electrical Networks

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

Resistive electrical network: Resistive electrical network:

Electrical Networks

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 c 1 1 2 b

Resistive electrical network: Resistive electrical network:

Electrical Networks

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 c 1 1 2 b

Resistive electrical network: Resistive electrical network:

Electrical Networks and the Commute Time

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 c 1 1 2 b

Thm: For a random walk on a graph G with m edges: Cu,v = 2mRu,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: Cu,v = 2mRu,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: Cu,v = 2mRu,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: Cu,v = 2mRu,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: Cu,v = 2mRu,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: Cu,v = 2mRu,v where branch resistance = 1 for every edge. Corollary: For any u,v: Cu,v ≤ n3.

Electrical Networks and the Commute 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) ≤ 2e3mR(G)ln n + n, where R(G) = maxu,vRu,v.

Electrical Networks and the Cover Time