1 1/3 2 1/3 2/3 3 0 1 0 π0)=（经，2，） P 1/3 0 2/3 1/3 1/3 1/3 π四0=4(0,1,0)+(3,0，)+4(3,3，)
1/3 1/3 1/3 1/3 2/3 1 1 2 3 P = ⇤ 010 1/302/3 1/3 1/3 1/3 ⇥ (0) = ( 1 ⌅ 4 , 1 2 , 1 4 ) (1) = 1 4 (0, 1, 0) + 1 2 ( 1 3 , 0, 2 3 ) + 1 4 ( 1 3 , 1 3 , 1 3 )
Random Walks fair +1 random walk:flipping a fair coin,the state is the difference between heads and tails; random walk on a graph; card shuffling:random walk in a state space of permutations; random walk on q-coloring of a graph;
Random Walks • fair ±1 random walk: flipping a fair coin, the state is the difference between heads and tails; • random walk on a graph; • card shuffling: random walk in a state space of permutations; • random walk on q-coloring of a graph;
Convergence 0 1 0 1/3 P= 1/3 0 2/3 3 /3 1/3 1/3 1/3 0.2500 0.3750 0.3750 P20≈ 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 V distribution m, πP20≈(8，)
Convergence P = ⇤ 010 1/302/3 1/3 1/3 1/3 ⇥ ⌅ 1/3 1/3 1/3 1/3 2/3 1 1 2 3 P20 ⇤ 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 0.2500 0.3750 0.3750 ⇥ ⌅ distribution , P20 ( 1 4 , 3 8 , 3 8 )
Stationary Distribution Markoy chain=(,P) stationary distribution πP=π (fixed point) Perron-Frobenius Theorem: stochastic matrix P:P1 =1 1 is also a left eigenvalue of P (eigenvalue of pT) ●the left eigenvectorπP=πis nonnegative stationary distribution always exists
Stationary Distribution • stationary distribution π: • Perron-Frobenius Theorem: • stochastic matrix P: • 1 is also a left eigenvalue of P (eigenvalue of PT) • the left eigenvector is nonnegative • stationary distribution always exists Markov chain M = (⌦, P) ⇡P = ⇡ P1 = 1 ⇡P = ⇡ (fixed point)
Perron-Frobenius Perron-Frobenius Theorem: A:a nonnegative nxn matrix with spectral radius (A) (A)>0 is an eigenvalue of A; there is a nonnegative (left and right)eigenvector associated with o(A); if further A is irreducible,then: there is a positive (left and right)eigenvector associated with o(A)that is of multiplicity 1; for stochastic matrix A the spectral radius o(A)=1
Perron-Frobenius • A : a nonnegative n×n matrix with spectral radius ρ(A) • ρ(A) > 0 is an eigenvalue of A; • there is a nonnegative (left and right) eigenvector associated with ρ(A); • if further A is irreducible, then: • there is a positive (left and right) eigenvector associated with ρ(A) that is of multiplicity 1; • for stochastic matrix A the spectral radius ρ(A)=1. Perron-Frobenius Theorem: