Random Walks

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# Random Walks - PowerPoint PPT Presentation

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

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### Random Walks

Ben Hescott

CS591a1

November 18, 2002

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