Parallel random walks

Brian Moffat

Outline

• What are random walks• What are Markov chains• What are cover/hitting/mixing times• Speed ups for different graphs• Implementation

What is a random walk

• An agent which traverses a graph randomly• Each step randomly goes from node A to a

random neighbour A’

Advantages of Random walk

• Easy to implement• No knowledge of underlying graph required• Little memory footprint

Disadvantages of random walk

• Unpredictable cover and hitting time• Worst case of infinite

Uses of random walks

• Sampling massive graphs– Social networks

• Estimating the size of an unknown graph• Simplified model of brownian motion

What is a markov chain

• A probability model involving states• Each state has probabilities that determine

the transitions to neighbouring state• Can be represented as a matrix• Similar to a random walk

The Times

• Cover time– Expected time for a random walk to visit every

node in a graph

• Hitting time– Expected time for a random walk to visit a specific

vertex starting from a specific vertex

• Mixing time– time before a markov chain hits the steady state

Mixing time

• Time for a markov chain to converge to a steady state

• Steady state is when time i and time i+1 are less then ε apart

Mixing times of graphs

• Fast mixing times– Complete graphs

• Slow mixing times– Barbell graph

Markov chains and graphs

• a random walk on a graph can be easily represented as a Markov chain

• Expected Cover time can be calculated using the Markov time

Calculating cover Time

Speed ups

• Three kinds of speed ups• Logarithmic– cycles

• Linear– cliques

• Exponential– Barbell graph

Logarithmic

Linear speed up

• Cliques have a linear speed up of k for k ≤ n random walks

Exponential speed up

• Barbell graphs have an exponential speed up when the random walks are started on the center node.

Barbell

• The barbell graph is 2 cliques connected by 1 path with a node in between known as vc

• Once a random walk is in a given node there is 1 exit with 1/(n/2) probability of leaving the clique and ((n/2)-1/(n/2)) probability of staying in the clique

• This bottleneck causes a massive slow down in the cover time

Barbell graph parallel

• For k=20 ln n the expected cover time starting from vc is O(n)

• Proof: with high probability the following will not happen– In one of the cliques there are less than 4 ln n

walks after the first step– During the first 10n steps at least 2ln n vertices

return to the center– One of the cliques is not covered within the first

10n steps

Barbell graph parallel

• Starting node effects speed up of parallel walks

• Best nodes are spread out• Best single node is the center node• Worst is in a clique

Implementation

• Intel cilk plus• Graphs represented as arrays of booleans• Random walk stores array index for which

vertex its at• Randomly steps to another index based on

graph type

Questions

• What property of a barbell graph that allows exponential speed up?

• What other kinds of graphs could have an exponential speed up?