Back to Random Walks on Graphs Random walk
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: 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 2 : 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 2 : 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 2 : 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 2 : 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 2 : 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 2 Cu(M)
Hitting Time of a Random Walk on a Graph Given is a graph G.
Electrical Networks Resistive electrical network: 1 b a 2 Rectangles: branch resistance Injecting a current of 1 ampere into b: 1 c
Electrical Networks Resistive electrical network: 1 b a 1 2 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: 1 b a 1 2 c 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
Electrical Networks and the Commute Time Resistive electrical network: 1 b a 1 2 c 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 = 2 m. Ru, 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: Cu, v = 2 m. Ru, 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 = 2 m. Ru, 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 = 2 m. Ru, 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 = 2 m. Ru, 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 = 2 m. Ru, 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 = 2 m. Ru, v where branch resistance = 1 for every edge. Corollary: For any u, v: Cu, 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) · 2 m(n-1).
Electrical Networks and the Cover Time Thm: For a random walk on a graph G with m edges and n vertices: m. R(G) · C(G) · 2 e 3 m. R(G)ln n + n, where R(G) = maxu, v. Ru, v.
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