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Random Walks. Ben Hescott CS591a1 November 18, 2002. Random Walk Definition.

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Random Walk Definition

- Given an undirected, connected graph G(V,E) with |V| = n, |E| = m a random “step” in G is a move from some node u to a randomly selected neighbor v. A random walk is a sequence of these random steps starting from some initial node.

Points to note

- Processes is discrete
- G is not necessarily planar
- G is not necessarily fully connected
- A walk can back in on itself
- Can consider staying in same place as a move

Questions

- How many steps to get from u to v
- How many steps to get back to initial node
- How many steps to visit every node
- Easy questions to answer if we consider a simple example

Regular Graphs

- The expected number of steps to get from vertex u to v in a regular graph is n-1,
- The expected number of steps to get back to starting point is n for a regular graph.
- Expected number of steps to visit every node in a regular graph is

Triangle Example

- Consider probabilities of being at a particular vertex at each step in walk.
- Each of these can be consider a vector,

Transition Matrix

- We can use a matrix to represent transition probabilities, consider adjacency matrix A and diagonal matrix, D, with entries 1/d(i) where d(i) is degree of node i. Then we can define matrix M = DA
- For triangle d(i) = 2 so M =
- Note for triangle Pr[a to b] = Pr[b to a]

Markov Chains - Generalized Random Walks

- A Markov Chain is a stochastic process defined on a set of states with matrix of transition probabilities.
- The process is discrete, namely it is only in one state at given time step (0, 1, 2, …)
- Next move does not depend on previous moves, formally

Markov Chain Definitions

- Define vector
- where the i-th entry is the probability that the chain is in state t
- Note:
- Notice that we can then calculate everything given q0 and P.

More Definitions

- Consider question where am I after t steps, define t-step probability
- Question is this my first time at node j?
- Consider probability visit state j at some time t>0 when started at state i.
- Consider how many steps to get from state i to j. given ,otherwise

Even More Definitions

- Consider fii
- State i is called transient fii < 1
- State i is called persistent if fii = 1
- If state i is persistent and hii is infinite then i is null-persistent
- If state i is persistent and hii is not infinite then i is non-null-persistent
- Turns out every Markov Chain is either transient or non-null-persistent

Almost there

- A strong component of a directed graph G is a subgraph C of G where for each edge eij there is a directed path from i to j and from j to i.
- A Markov Chain is irreducible if underlying graph G consists of a single strong component.
- A stationary distribution for a Markov Chain with transition matrix P is distribution s.t. P =
- The periodicity of state i is max int T for which there is a q0 and a>0 s.t. for all t, if then t is in arithmetic progression A state is periodic if T>1 and aperiodic otherwise
- An ergodic Markov Chain is one where all states are aperiodic and non-null persistent

Fundamental Theorem of Markov Chains

- Given any irreducible, finite, aperoidic Markov Chain then all of the following hold
- The chain is ergodic
- There is a unique stationary distribution where for
- for
- Given N(i,t) number of time chain visits state i in t steps then

Random Walk is a Markov Chain

- Consider G a connected, non-bipartite, undirected graph with |V| = n, |E| = m. There is a corresponding Markov Chain
- Consider states of chain to be set of vertices
- Define transitional matrix to be

Interesting facts

- MG is irreducible since G is connected and undirected
- Notice the periodicity is the gcd of all cycles in G - closed walk
- Smallest walk is 2 go one step and come back
- Since G is non-bipartite then there is odd length cycle
- GCD is then 1 so MG is aperiodic

Fundamental Theorem Holds

- So we have a stationary distribution, P =
- But
- Good news,
- Also get

Hitting Time

- Generally hij, the expected number of steps needed before reaching node j when starting from node i is the hitting time.
- The commute time is the expected number of steps to reach node j when starting from i and then returning back to i.
- The commute is bounded by 2m
- We can express hitting time in terms of commute time as

Lollipop

- Hitting time from i to j not necessarily same a time from j to i. Consider the kite or lollipop graph.
- Here

Cover Time

- How long until we visit every node?
- The expected number of steps to reach every node in a graph G starting from node i is called the cover time, Ci(G)
- Consider the maximum cover time over all nodes
- [Matthews]The maximum cover time of any graph with n nodes is (1+1/2+…+1/n) times the maximum hitting time between any two nodes - 2m(n-1)

Coupon Collector

- Want to collect n different coupons and every day Stop and Shop sends a different coupon at random - how long do have to wait before you can by food?
- Consider cover time on on a complete graph, here cover time is O(nlgn)

Mixing Rate

- Going back to probability, could ask how quickly do we converge to the stationary (limiting) distribution? We call this rate the mixing rate of the random walk.
- We saw
- How fast

Mixing with the Eigenvalues

- How do we calculate - yes eigenvalues!
- Since we are considering probability transitions as a matrix why not use spectral techniques
- Need graph G to be non-bipartite
- First need to make sure P is symmetric, not true unless G is regular

More decomposition

- Need to make P symmetric.
- Recall that P = DA, where D is diagonal matrix with entries of d(u), degree of node u
- Consider
- Claim this let us work with spectral since
- Now mixing rate is

Graph Connectivity

- Recall the graph connectivity problem, Given undirected graph G(V,E) want to find out if node s is connected to node t.
- Can do this in deterministic polynomial time.
- But what about space, would like to do this in a small amount of space.
- Recall hitting time of a graph is at most n3
- Try a walk of 2n3 steps to get there, needs only lg(n) space to count number of steps.

Sampling

- So what’s the fuss?
- We can use random walks to sample, since we have the powerful notion of stationary distribution and on a regular graph this is uniform distribution, we can get at random elements.
- More importantly we can get a random sample from an exponentially large set.

Covering Problems

- Jerrun, Valiant, and Vazirani - Babai product estimator, or enumeration - self reducibilty
- Given V a set and V1, V2, V3, …,Vm subsets
- For all i, |Vi| is polynomial time computable
- Can sample from V uniformly at random
- For all v in V, can determine efficiently that v is in Vi
- Can we get size of union of subsets, or can we enumerate V in polynomial time

Permanent

- Want to count the number of perfect matchings in a bipartite graph.
- This is the permanent of a matrix
- Given a n x n matrix A the permanent is
- This is #P-complete

Commercial Break

- #P is the coolest class, defined as counting class.
- Consider the number of solutions to the problem
- Hard - [Toda] PH is contained in #P
- Want to show hyp-PH is also contained in #P

How to use random walk for permanent approximation

- Given graph G with d(u) > n/2 want to generate a random perfect matching
- First notice that input graph is bipartite.
- Instead consider graph of perfect matchings
- Let each node be a perfect matching in G - problem is how to connect
- Need to consider near-perfect matchings a matching with n/2-1 edges

Permanent Cont.

- Connect near perfect matchings with an edge if they have n/2-2 edges in common and connect a perfect matching to all of the near perfect matchings contained in it to create a graph H.
- Notice degree of H is bounded by 3n
- Walk (randomly) a polynomial number of steps in H - if node is a perfect matching -good - otherwise try again.

Volume

- Want to be able to calculate the volume of a convex body in n dimensions
- Computing convex polytope is #P-Hard
- No fear randomization is here
- Want to be able to fit this problem into enumeration problem

Volume Cont.

- Given C convex body in n dimensions, assume that C contains the origin
- Further assume that C contains the unit ball and itself is contained in a ball of radius r<n3/2.
- Define Ci = intersection of C and ball around origin of radius

Volume Cont.

- Then we get
- And we know Vol(Ci)
- Now only need to be able to get a element uniformly at random - use a walk
- Difficult to do walk, create a grid and walk from intersections of Ci’s
- Stationary distribution is not uniform but a distribution with density function proportional to local conductance

Partial Reference List

- L. Lovasz. Random Walks on Graphs: A survey. Combinatorics Paul Erdos is Eighty Volume 2, p1-46
- R. Motwani, P. Raghavan Randomized Algorithms. Cambridge University 1995

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