1. Parallel random walks Brian Moffat

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

3. What is a random walk An agent which traverses a graph randomly Each step randomly goes from node A to a random neighbour A’

4. Advantages of Random walk Easy to implement No knowledge of underlying graph required Little memory footprint

5. Disadvantages of random walk Unpredictable cover and hitting time Worst case of infinite

6. Uses of random walks Sampling massive graphs – Social networks Estimating the size of an unknown graph Simplified model of brownian motion

7. 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

8. 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

9. 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

10. Mixing times of graphs Fast mixing times – Complete graphs Slow mixing times – Barbell graph

11. 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

12. Calculating cover Time

13. Speed ups Three kinds of speed ups Logarithmic – cycles Linear – cliques Exponential – Barbell graph

14. Logarithmic

15. Linear speed up Cliques have a linear speed up of k for k ≤ n random walks

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

17. Barbell The barbell graph is 2 cliques connected by 1 path with a node in between known as v c 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

18. Barbell graph parallel For k=20 ln n the expected cover time starting from v c 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

19. 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

20. 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

21. Questions What property of a barbell graph that allows exponential speed up? What other kinds of graphs could have an exponential speed up?