COMS 6998-06 Network Theory Week 5: February 21, 2008

1. COMS 6998-06 Network TheoryWeek 5: February 21, 2008 Dragomir R. Radev Thursdays, 6-8 PM 233 Mudd Spring 2008

2. (8) Random walks and electrical networks

3. Random walks • Stochastic process on a graph • Transition matrix E • Simplest case: a regular 1-D graph 0 1 2 3 4 5

4. Gambler’s ruin • A has N pennies and B has M pennies. • At each turn, one of them wins a penny with a probability of 0.5 • Stop when one of them loses all his money.

5. Harmonic functions • Harmonic functions: • P(0) = 0 • P(N) = 1 • P(x) = ½*p(x-1)+ ½*p(x+1), for 0<x<N • (in general, replace ½ with the bias in the walk)

6. Simple electrical circuit 0 1 2 3 4 5 V(0)=0 V(N)=1

7. Arbitrary resistances

8. The Maximum principle • Let f(x) be a harmonic function on a sequence S. • Theorem: • A harmonic function f(x) defined on S takes on its maximum value M and its minimum value m on the boundary. • Proof: • Let M be the largest value of f. Let x be an element of S for which f(x)=M. Then f(x+1)=f(x-1)=M. If x-1 is still an interior point, continue with x-2, etc. In the worst case, reach x=0, for which f(x)=M.

9. The Uniqueness principle • Let f(x) be a harmonic function on a sequence S. • Theorem: • If f(x) and g(x) are harmonic functions on S such that f(x)=g(x) on the boundary points B, then f(x)=g(x) for all x. • Proof: • Let h(x)=f(x)-g(x). Then, if x is an interior point, and h is harmonic. But h(x)=0 for x in B, and therefore, by the Maximum principle, its minimal and maximal values are both 0. Thus h(x)=0 for all x which proves that f(x)=g(x) for all x.

10. How to find the unique solution? • Try a linear function: f(x)=x/N. • This function has the following properties: • f(0)=0 • f(N)=1 • (f(x-1)+f(x+1))*1/2=x/N=f(x)

11. Reaching the boundary • Theorem: • The random walker will reach either 0 or N. • Proof: • Let h(x) be the probability that the walker never reaches the boundary. Thenh(x)=1/2*h(x+1)+1/2*h(x-1),so h(x) is harmonic. Also h(0)=h(N)=0. According to the maximum principle, h(x)=0 for all x.

12. Number of steps to reach the boundary • m(0)=0 • m(N)=0 • m(x)=1/2m(x+1)+1/2m(x-1) • The expected number of steps until a one dimensional random walk goes up to b or down to -a is ab. • Examples: (a=1,b=1); (a=2,b=2) • (also: the displacement varies as sqrt(t) where t is time).

13. Fair games • In the penny game, after one iteration, the expected fortune is ½(k-1)+1/2(k+1)=k • Fair game = martingale • Now if A has x pennies out of a total of N, his final fortune is:(1-p(x)).0+p(x).N=p(x).N • Is the game fair if A can stop when he wants? No – e.g., stop playing when your fortune reaches $x.

14. (9) Method of relaxations and other methods for computing harmonic functions

15. 2-D harmonic functions 0 0 x y 1 z 1

16. The original Dirichlet problem U=1 U=0 • Distribution of temperature in a sheet of metal. • One end of the sheet has temperature t=0, the other end: t=1. • Laplace’s differential equation: • This is a special (steady-state) case of the (transient) heat equation : • In general, the solutions to this equation are called harmonic functions.

17. Learning harmonic functions • The method of relaxations • Discrete approximation. • Assign fixed values to the boundary points. • Assign arbitrary values to all other points. • Adjust their values to be the average of their neighbors. • Repeat until convergence. • Monte Carlo method • Perform a random walk on the discrete representation. • Compute f as the probability of a random walk ending in a particular fixed point. • Linear equation method • Eigenvector methods • Look at the stationary distribution of a random walk

18. Monte Carlo solution • Least accurate of all. Example: 10,000 runs for an accuracy of 0.01

19. Example • x=1/4*(y+z+0+0) • y=1/2*(x+1) • z=1/3*(x+1+1) • Ax=u • X=A-1u

20. Effective resistance • Series: R=R1+R2 • Parallel: C=C1+C2 1/R=1/R1+1/R R=R1R2/(R1+R2)

21. Example • Doyle/Snell page 45

22. 1 Ω Electrical networks and random walks c • Ergodic (connected) Markov chain with transition matrix P 1 Ω 1 Ω w=Pw b a 0.5 Ω 0.5 Ω d From Doyle and Snell 2000

23. 1 Ω Electrical networks and random walks c 1 Ω 1 Ω b a 0.5 Ω 0.5 Ω • vxis the probability that a random walk starting at x will reach a before reaching b. d • The random walk interpretation allows us to use Monte Carlo methods to solve electrical circuits. 1 V

24. Energy-based interpretation • The energy dissipation through a resistor is • Over the entire circuit, • The flow from x to y is defined as follows: • Conservation of energy

25. Thomson’s principle • One can show that: • The energy dissipated by the unit current flow (for vb=0 and for ia=1) is Reff. This value is the smallest among all possible unit flows from a to b (Thomson’s Principle)

26. Eigenvectors and eigenvalues • An eigenvector is an implicit “direction” for a matrix where v (eigenvector)is non-zero, though λ (eigenvalue) can be any complex number in principle • Computing eigenvalues:

27. Eigenvectors and eigenvalues • Example: • Det (A-lI) = (-1-l)*(-l)-3*2=0 • Then: l+l2-6=0; l1=2; l2=-3 • For l1=2: • Solutions: x1=x2

28. Stochastic matrices • Stochastic matrices: each row (or column) adds up to 1 and no value is less than 0. Example: • The largest eigenvalue of a stochastic matrix E is real: λ1 = 1. • For λ1, the left (principal) eigenvector is p, the right eigenvector = 1 • In other words, GTp = p.

29. Markov chains • A homogeneous Markov chain is defined by an initial distribution x and a Markov kernel E. • Path = sequence (x0, x1, …, xn).Xi = xi-1*E • The probability of a path can be computed as a product of probabilities for each step i. • Random walk = find Xjgiven x0, E, and j.

30. Stationary solutions • The fundamental Ergodic Theorem for Markov chains [Grimmett and Stirzaker 1989] says that the Markov chain with kernel E has a stationary distribution p under three conditions: • E is stochastic • E is irreducible • E is aperiodic • To make these conditions true: • All rows of E add up to 1 (and no value is negative) • Make sure that E is strongly connected • Make sure that E is not bipartite • Example: PageRank [Brin and Page 1998]: use “teleportation”

31. t=0 1 6 8 2 7 t=1 5 3 4 Example This graph E has a second graph E’(not drawn) superimposed on it:E’ is the uniform transition graph.

32. Eigenvectors • An eigenvector is an implicit “direction” for a matrix. Ev = λv, where v is non-zero, though λ can be any complex number in principle. • The largest eigenvalue of a stochastic matrix E is real: λ1 = 1. • For λ1, the left (principal) eigenvector is p, the right eigenvector = 1 • In other words, ETp = p.

33. Computing the stationary distribution functionPowerStatDist (E): begin p(0) = u; (or p(0) = [1,0,…0]) i=1; repeat p(i) = ETp(i-1) L = ||p(i)-p(i-1)||1; i = i + 1; untilL <  returnp(i) end Solution for thestationary distribution Convergence rate is O(m)

34. t=0 1 6 8 2 7 t=1 5 3 4 t=10 Example

35. More dimensions • Polya’s theorem says that a 1-D random walk is recurrent and that a 2-D walk is also recurrent. However, a 3-D walk has a non-zero escape probability (p=0.66). • http://mathworld.wolfram.com/PolyasRandomWalkConstants.html