Random walks on finite networks Andr e Schumacher - PowerPoint PPT Presentation

Random walks on finite networks Andr´ e Schumacher <schumach@tcs.hut.fi> February 06, 2006

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [1] Overview • Short review of recent electric network models • Model of electric networks with arbitrary resistors • Markov chains for such networks • Interpretation of voltage • Interpretation of current Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [2] Review • Random Walks and harmonic functions in one and two dimensions • Uniqueness and Maximum Principle in one and two dimensions • Four ways of finding the harmonic function ( ≡ solution to the Dirichlet problem): 1. Monte Carlo method 2. Method of relaxations 3. Linear equations 4. Markov chains → So far, the model for electric networks only considered unit resistor values! Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [3] Network Model 1 1 X = R 1 = C 1 1 = R 1 = C 1 a b a b ☎ ✆ ✆ ☎ ☎ ✆ ☎ ✆ ☎ ✆ ✞ ✝ ✞ ✝ ✝ ✞ ✝ ✞ ✞ ✝ ✝ ✞ ☎ ✆ ☎ ✆ ☎ ✆ ✆ ☎ ✆ ☎ ✝ ✞ ✞ ✝ ✞ ✝ ✞ ✝ ✞ ✝ ✝ ✞ ✆ ☎ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✂ ✄ ✄ ✂ ✂ ✄ ✄ ✂ ✄ ✂ ✂ ✄ ✁ � ✁ � � ✁ � ✁ � ✁ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ � ✁ ✁ � � ✁ ✁ � ✁ � c c 1 1 Y = R 2 = C 2 1 = R 2 = C 2 → Rather than considering the resistor values R xy , their reciprocal, the conductance C xy is used. → We consider an electric network to be a connected, weighted, undirected graph. Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [4] Random Walk ( := Markov chain Model) Definition: We define a random walk on a graph G modeling a resistor network to be a Markov chain with transition probabilities P xy : P xy := C xy � C x := C xy C x y C 1 p ab = C 1+ C 2 C 1 b a a b p ba = C 1 ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✄ ✂ C 1 = 1 ✄ ✂ ✄ ✂ ✂ ✄ ✄ ✂ ✄ ✂ � ✁ ✁ � ✁ � ✁ � � ✁ ✁ � ✁ � ✁ � ✁ � ✁ � C 2 C 2 c p ac = c C 1+ C 2 p ca = C 2 C 2 = 1 Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [5] Terminology Definition: A Markov chain in which it is possible to reach every state from any other state is called ergodic . Lemma: For an ergodic Markov chain, there is a unique probability vector w that is a fixed vector for P (left eigenvector with eigenvalue 1), i.e. it holds that wP = w . For our random walk on the resistor network: w x = C x � C = C x C x Definition: An ergodic Markov chain for which the following holds is called reversible : w x ∗ P xy = w y ∗ P yx Lemma: If P is any reversible ergodic Markov chain, then P is the transition matrix for a random walk on an electric network with C xy := w x ∗ P xy . Special case: ∀ x, y : C xy := c ( simple random walk ) Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [6] Probabilistic Interpretation of Voltage (1/3) • Let G be a network of resistors. Like before, we associate a voltage v x to each node x and a current i xy to each edge ( x, y ) . Let v a = 1 and v b = 0 . • The following two laws are valid for “real” voltage and current and therefore have to be considered here, too: Ohm’s Law: i xy = v x − v y = ( v x − v y ) C xy ⇒ i xy = − i yx R xy Kirchhoff’s Law: � i xy = 0 y C xy � = ⇒ v x = v y = ⇒ Voltage v x is harmonic over all points x � = a, b C x y Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [7] Probabilistic Interpretation of Voltage (2/3) Proof: Ohm & Kirchhoff ⇒ � ( v x − v y ) C xy = 0 y C xy � � ⇒ v x = = x � = a, b v y P xy v y C x y y ⇒ v x harmonic for P ( Pv x = v x ) for all x � = a, b Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [8] Probabilistic Interpretation of Voltage (3/3) • Let h x be the probability that starting at state x, the Markov chain/the random walker given by P (recall: P xy := C xy C x ) reaches first state a before reaching b . • Then h x harmonic at all points x � = a, b , v a = h a = 1 and v b = h b = 0 . • Modifying P to P by defining a and b to be absorbing states it follows by the uniqueness principle that h x = v x and both are solutions to the Dirichlet problem. Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [9] Probabilistic Interpretation of Current (1/2) • Naive idea: Assume that (electrically charged) particles enter the network at point/node a and traverse edges until they eventually reach point b and leave the network. • Following the course of a single particle, we regard the current i xy to be the expected number of edge traversals x → y (reverse traversals are negatives). • The particle/random walker starts at a and keeps going in the event it returns to this point. Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [10] Probabilistic Interpretation of Current (2/2) • Let u x be the expected number of visits to state x before stating state b . Then one can show (using the reversibility of P and u x = � y u y P yx ): u x u y � = = v x P xy C x C y y The last equation holds because the left side function is harmonic for x � = a, b and has the same boundary values as v x . • Ohm’s law implies: i xy = u x P xy − u y P yx • However, the current i xy is only proportional to the current flowing when a unit voltage is applied → the currents i xy have to be normalized such that � y i ay = � y i yb = 1 . Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [11] Effective Resistance / Escape Probability (1/2) v a i a a := R eff i a � ✁ ✁ � � ✁ ✁ � ✁ � ✁ � � ✁ ✁ � ✁ � � ✁ ✆ ☎ � ✁ ✂ ✄ ✄ ✂ ☎ ✆ ✂ ✄ ✄ ✂ ☎ ✆ ☎ ✆ ☎ ✆ R 2 R 3 ✞ ✝ ✝ ✞ ✝ ✞ ✝ ✞ R 1 ✝ ✞ ✝ ✞ = R 1 + + R 4 ✞ ✝ ✝ ✞ ✞ ✝ ✝ ✞ ✝ ✞ ✝ ✞ R 2 + R 3 ✞ ✝ ✝ ✞ ✝ ✞ ✝ ✞ ✞ ✝ ✝ ✞ 1 = ☞ ✌ ☞ ✌ ✟ ✠ ✟ ✠ ✠ ✟ C eff ☞ ✌ ☞ ✌ ✟ ✠ ✟ ✠ ✠ ✟ ☞ ✌ ✌ ☞ ✟ ✠ ✟ ✠ ✠ ✟ v a R 2 ✌ ☞ ☞ ✌ ✠ ✟ ✟ ✠ ✠ ✟ R 3 ✌ ☞ ✌ ☞ ✟ ✠ ✟ ✠ ✠ ✟ ✌ ☞ ✌ ☞ ✟ ✠ ✠ ✟ ✠ ✟ ✌ ☞ ☞ ✌ ✠ ✟ ✟ ✠ ✟ ✠ ✌ ☞ ☞ ✌ ✠ ✟ ✟ ✠ ✠ ✟ ✌ ☞ ✌ ☞ ✟ ✠ ✟ ✠ ✠ ✟ Let v a = 1 and let p esc be the probability that the random walker starting at a reaches b before ☛ ✡ ✡ ☛ ☛ ✡ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ R 4 ✡ ☛ ✡ ☛ returning to a . Then: ☛ ✡ ✡ ☛ ☛ ✡ ✡ ☛ ☛ ✡ ✡ ☛ ☛ ✡ ✡ ☛ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ p esc = C eff b C a Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [12] Escape Probability (2/2) Proof: 1 v a = i a C eff ⇒ C eff = for v a = 1 i a C ay � � = (1 − v y ) C ay = i a C ay − v y C a C a y y � = C a (1 − P ay v y ) y ⇒ i a = C a p esc C eff ⇒ p esc = C a Andr´ e Schumacher

Doyle & Snell 1.3.1-1.3.4 Random walks on finite networks [13] End Thank you for your attention. . . • < Questions? / Discussion > • < Break > • < Exercises > Andr´ e Schumacher