Electric network for non-reversible Markov chains joint with Aron - PowerPoint PPT Presentation

Reversible Irreversible Properties Electric network for non-reversible Markov chains joint with ´ Aron Folly M´ arton Bal´ azs University of Bristol School and Workshop on Random Interacting Systems Bath, June 2014.

Reversible Irreversible Properties Reversible chains and resistors Thomson, Dirichlet principles Monotonicity, transience, recurrence Irreversible chains and electric networks The part From network to chain From chain to network Properties Effective resistance What works Nonmonotonicity Dirichlet principle

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Reversible chains and resistors Irreducible Markov chain: on Ω , a � = b , x ∈ Ω , h x : = P x { τ a < τ b } ( τ is the hitting time ) is harmonic: h x = P xy h y , h a = 1 , h b = 0 . � y y a x b

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Reversible chains and resistors Irreducible Markov chain: on Ω , a � = b , x ∈ Ω , h x : = P x { τ a < τ b } ( τ is the hitting time ) is harmonic: h x = P xy h y , h a = 1 , h b = 0 . � y y a x b y a x b R xy R yb 1 V R ax 0 V i Electric resistor network: the voltage u is harmonic ( C = 1 / R ): C xy u x = · u y ; u a = 1 , u b = 0 . � z C xz � y

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Reversible chains and resistors Irreducible Markov chain: on Ω , a � = b , x ∈ Ω , h x : = P x { τ a < τ b } ( τ is the hitting time ) is harmonic: h x = P xy h y , h a = 1 , h b = 0 . � y y a x b y a x b R xy R yb 1 V R ax 0 V i Electric resistor network: the voltage u is harmonic ( C = 1 / R ): C xy u x = · u y ; u a = 1 , u b = 0 . � z C xz � y

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Reversible chains and resistors Thus, C xy P xy = z C xz �

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Reversible chains and resistors Thus, C xy = : C xy P xy = . z C xz C x � Stationary distribtuion: � C x = µ x .

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Reversible chains and resistors Thus, C xy = : C xy P xy = . z C xz C x � Stationary distribtuion: � C x = µ x . Notice µ x P xy = C xy = C yx = µ y P yx , so the chain is reversible.

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Thomson, Dirichlet principles Thomson principle: R 1 1 Q The physical unit current is the unit flow that minimizes the sum of the ohmic power losses � i 2 R .

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Thomson, Dirichlet principles Thomson principle: R 1 1 Q The physical unit current is the unit flow that minimizes the sum of the ohmic power losses � i 2 R . Dirichlet principle: u 1 V 0 V R Q The physical voltage is the function that minimizes the ohmic power losses � ( ∇ u ) 2 / R .

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Monotonicity, transience, recurrence The monotonicity property: Between any disjoint sets of vertices, the effective resistance is a non-decreasing function of the individual resistances.

Reversible Irreversible Properties Thomson, Dirichlet Monotonicity Monotonicity, transience, recurrence The monotonicity property: Between any disjoint sets of vertices, the effective resistance is a non-decreasing function of the individual resistances. � can be used to prove transience-recurrence by reducing the graph to something manageable in terms of resistor networks.

Reversible Irreversible Properties The part Chain Network The part i ∗ λ yx y x R / 2 R / 2 Voltage amplifier: keeps the current, multiplies the potential. ( u x − i · R 2 ) · λ yx − i · R 2 = u y

Reversible Irreversible Properties The part Chain Network The part i ∗ λ yx y x R / 2 R / 2 Voltage amplifier: keeps the current, multiplies the potential. ( u x − i · R 2 ) · λ yx − i · R 2 = u y

Reversible Irreversible Properties The part Chain Network The part i ∗ λ yx y x R / 2 R / 2 Voltage amplifier: keeps the current, multiplies the potential. ( u x − i · R 2 ) · λ yx − i · R 2 = u y

Reversible Irreversible Properties The part Chain Network The part i ∗ λ yx y x R / 2 R / 2 Voltage amplifier: keeps the current, multiplies the potential. ( u x − i · R 2 ) · λ yx − i · R 2 = u y

Reversible Irreversible Properties The part Chain Network The part i ∗ λ yx y x R / 2 R / 2 Voltage amplifier: keeps the current, multiplies the potential. ( u x − i · R 2 ) · λ yx − i · R 2 = u y

Reversible Irreversible Properties The part Chain Network The part i ∗ λ yx y x R / 2 R / 2 Voltage amplifier: keeps the current, multiplies the potential. ( u x − i · R 2 ) · λ yx − i · R 2 = u y 1 It is clear that λ yx = . λ xy

Reversible Irreversible Properties The part Chain Network Harmonicity ∗ λ xb ∗ λ xa a x b R xa / 2 R xa / 2 R xb / 2 R xb / 2 D xy γ xy u x = � · u y D x y with λ xy = 1 � γ xy = , γ yx D xy = 2 γ xy C xy ( λ xy + 1 ) = D yx , D x = D xz γ zx . � z

Reversible Irreversible Properties The part Chain Network Harmonicity ∗ λ xb ∗ λ xa a x b R xa / 2 R xa / 2 R xb / 2 R xb / 2 D xy γ xy u x = � · u y D x y with λ xy = 1 � γ xy = , γ yx D xy = 2 γ xy C xy ( λ xy + 1 ) = D yx , D x = D xz γ zx . � z D xy γ xy u x = � · u y D xy = D yx 1 γ xy = y D x γ yx

Reversible Irreversible Properties The part Chain Network From network to chain Irreducible Markov chain: on Ω , a � = b , x ∈ Ω , h x : = P x { τ a < τ b } ( τ is the hitting time ) is harmonic: h x = P xy h y , h a = 1 , h b = 0 . � y D xy γ xy u x = · u y , u a = 1 , u b = 0 . � D x y D xy γ xy u x = � · u y D xy = D yx 1 γ xy = y D x γ yx

Reversible Irreversible Properties The part Chain Network From network to chain Irreducible Markov chain: on Ω , a � = b , x ∈ Ω , h x : = P x { τ a < τ b } ( τ is the hitting time ) is harmonic: h x = P xy h y , h a = 1 , h b = 0 . � y D xy γ xy u x = · u y , u a = 1 , u b = 0 . � D x y P xy = D xy γ xy . D x D xy γ xy u x = � · u y D xy = D yx 1 γ xy = y D x γ yx

Reversible Irreversible Properties The part Chain Network From network to chain Irreducible Markov chain: on Ω , a � = b , x ∈ Ω , h x : = P x { τ a < τ b } ( τ is the hitting time ) is harmonic: h x = P xy h y , h a = 1 , h b = 0 . � y D xy γ xy u x = · u y , u a = 1 , u b = 0 . � D x y P xy = D xy γ xy . D x P xy = D xy γ xy D xy = D yx 1 γ xy = D x γ yx

Reversible Irreversible Properties The part Chain Network From chain to network Stationary distribution: � D x = µ x . P xy = D xy γ xy = D xy γ xy D x µ x µ x P xy · µ y P yx = D 2 xy ; µ x P xy = γ 2 xy = λ xy . µ y P yx P xy = D xy γ xy D xy = D yx 1 γ xy = D x γ yx

Reversible Irreversible Properties The part Chain Network From chain to network Stationary distribution: � D x = µ x . P xy = D xy γ xy = D xy γ xy D x µ x µ x P xy · µ y P yx = D 2 xy ; µ x P xy = γ 2 xy = λ xy . µ y P yx Reversed chain: Replace P xy by ˆ P xy = P yx · µ y µ x . � D xy stays, λ xy reverses to λ yx . P xy = D xy γ xy D xy = D yx 1 γ xy = D x γ yx

Reversible Irreversible Properties The part Chain Network Markovian network u x = P xz u z ; P xz = 1 � � z z u x ≡ const. is a solution of the network with no external sources. This is now nontrivial.

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet Effective resistance Suppose u a , u b given, the solution is { u x } x ∈ Ω and { i xy } x ∼ y ∈ Ω . Current i a = i ax � x ∼ a flows in the network at a .

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet Effective resistance Suppose u a , u b given, the solution is { u x } x ∈ Ω and { i xy } x ∼ y ∈ Ω . Current i a = i ax � x ∼ a flows in the network at a . � Then i a 1 = const. = : C eff ab = . u a − u b R eff ab

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet What works ... the analogy with P { τ a < τ b } .

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet What works ... the analogy with P { τ a < τ b } . 1 cap ( A , B ) = C eff for all sets A , B AB = R eff AB

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet What works ... the analogy with P { τ a < τ b } . 1 cap ( A , B ) = C eff for all sets A , B AB = R eff AB Theorem Commute time = R eff · all conductances.

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet Nonmonotonicity ∗ 1 / 5 ∗ 5 / 13 3 / 2 3 / 2 2 / 2 2 / 2 R ∗ 13 / 5 ∗ 5 3 / 2 3 / 2 2 / 2 2 / 2 R eff = 27 1296 14 + 1225 R + 2268 .

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet Nonmonotonicity ∗ 1 / 5 ∗ 5 / 13 3 / 2 3 / 2 2 / 2 2 / 2 R ∗ 13 / 5 ∗ 5 3 / 2 3 / 2 2 / 2 2 / 2 R eff = 27 1296 14 + 1225 R + 2268 . �

Reversible Irreversible Properties Effective Works... Nonmonotonicity Dirichlet Dirichlet principle Classical case: