Presented "Random Walk on Graphs" in the reading group for Knoesis. Specifically for Recommendation Context. Referred: Purnamrita Sarkar, Random Walks on Graphs: An Overview

1. Random Walk on Graphs
Pavan Kapanipathi
Reading Group (Kno.e.sis)
Referred: Purnamrita Sarkar, Random
Walks on Graphs: An Overview

2. Agenda
• Introduction
– Motivation
• Background
– Graphs
– Matrices
• Random Walk
– PageRank
– Personalized PageRank
– Topic Sensitive PageRank
• Applications
– Specifically in Recommender Systems

3. Random Walk
A drunk man will find his way home, but a drunk
bird may get lost forever.

4. Motivation: Link prediction in social
networks
4

5. Motivation: Basis for recommendation
5

6. Since I had very less slides and More
time – Graphs
• Undirected Graphs

7. Since I had very less slides and more
time in hand-- Graphs
• Directed Graphs

8. Since I had very less slides and more
time in hand – Matrix
Rows
Columns
i
j
i
j
k
i,j
k

9. Adjacency and Transition Matrix
Transition matrix P
Adjacency matrix A
1
1
1
1
1
1
1/2
1/2
9

10. Markov Property (Basic)
• Given the present state, the future and past
states are independent

11. Stochastic
• Wikipedia: In probability theory, a purely
stochastic system is one whose state is nondeterministic so that the subsequent state of
the system is determined probabilistically.
1
1
• Matrix
– Stochastic Row/Column?
1/2
1/2

12. Random Walk on Graphs

13. Random Walk on Graphs

14. Random Walk on Graphs

15. Random Walk on Graphs
The random sequence of points selected this
way is a random walk on the graph

16. Again: Transition Matrix
j
k
i
i
j
k
Transition matrix P
Probability?
1
1
1/2
1/2
16

17. Probability Distributions
• xt(i) = probability that the surfer is at node i at
time t
• xt+1(i) = ∑j(Probability of being at node j)*Pr(j->i)
=∑jxt(j)*P(j,i)
• xt+1 = xtP = xt-1*P*P= xt-2*P*P*P = …=x0 Pt
Matrix Multiplication?
• What happens when the surfer keeps walking for
a long time?
17

18. Property of Adjacency Matrix
Adjacency matrix A
1
1
1
1

19. What is a stationary distribution?
Intuitively and Mathematically
• The stationary distribution at a node is related to the amount
of time a random walker spends visiting that node.
• Remember that we can write the probability distribution at a
node as
– xt+1 = xtP
• For the stationary distribution v0 we have
– v0 = v0 P
• Whoa! that’s just the left eigenvector of the transition matrix !
19

20. Eigen Value and Eigen Vector?

21. Interesting questions
• Does a stationary distribution always exist? Is it unique?
– Yes, if the graph is “well-behaved”.
• What is “well-behaved”?
– We shall talk about this soon.
• How fast will the random surfer approach this stationary
distribution?
– Mixing Time!
21

22. Well behaved graphs
• Irreducible: There is a path from every node to every other
node.
What about connected undirected Graph?
Irreducible
Not irreducible
22

23. Well behaved graphs
• Aperiodic: The GCD of all cycle lengths is 1. The GCD is also
called period.
Periodicity is 3
Aperiodic
23

24. Implications of the Perron Frobenius Theorem
• If a markov chain is irreducible and aperiodic then the largest
eigenvalue of the transition matrix will be equal to 1 and all
the other eigenvalues will be strictly less than 1.
– Let the eigenvalues of P be {σi| i=0:n-1} in non-increasing order of
σi .
– σ0 = 1 > σ1 > σ2 >= ……>= σn
• These results imply that for a well behaved graph there exists
an unique stationary distribution.
• More details when we discuss pagerank.
24

25. Some fun stuff about undirected graphs
• A connected undirected graph is irreducible
• A connected non-bipartite undirected graph has a stationary
distribution proportional to the degree distribution!
• Makes sense, since larger the degree of the node more likely a
random walk is to come back to it.
25

26. Proximity measures from random walks
b
a
• How long does it take to hit node b in a random walk starting
at node a? --- Hitting time.
• How long does it take to hit node b and come back to node a?
--- Commute time.
26

27. Hitting and Commute times
b
a
• Hitting time from node i to node j
– Expected number of hops to hit node j starting at node i.
– Is not symmetric. h(a,b) > h(a,b)
– h(i,j) = 1 + ΣkЄnbs(A) p(i,k)h(k,j)
27

28. Hitting and Commute times
b
a
• Commute time between node i and j
– Is expected time to hit node j and come back to i
– c(i,j) = h(i,j) + h(j,i)
– Is symmetric. c(a,b) = c(b,a)
28

29. Random Walk (versions)
• PageRank
– Personalized PageRank
– Topic Sensitive PageRank
• Recommender Systems (My interests)

30. Recommender Networks
• For a customer node i define similarity as
– H(i,j)
– C(i,j)
– Or the cosine similarity

Lij

Lii Ljj
• Now the question is how to compute these quantities quickly
for very large graphs.
– Fast iterative techniques (Brand 2005)
– Fast Random Walk with Restart (Tong, Faloutsos 2006)
– Finding nearest neighbors in graphs (Sarkar, Moore 2007)
30

31. PageRank (Initial)
• Intuition
– PageRank of “A” is higher if the pages that links to
“A” has higher PageRank
• User behavior where a surfer clicks on links at random
with no regard towards content
– One page's PageRank is not completely passed on
to a page it links to, but is divided by the number
of links on the page.

32. PageRank
• Intuitively
v (i ) 
v (j )
i degout ( j )
j

• v works out to be the stationary distribution of
the markov chain corresponding to the web.

33. Pagerank & Perron-frobenius
• Perron Frobenius only holds if the graph is irreducible and
aperiodic.
• But how can we guarantee that for the web graph?
– Do it with a small restart probability c.
• At any time-step the random surfer
– jumps (teleport) to any other node with probability c
– jumps to its direct neighbors with total probability 1-c.
~
P  (1  c )P  cU
Uij 
1
n
i , j
~
P  cP  (1  c)U
U ij 
1
i, j
n
33

34. Pagerank
• We are looking for the vector v s.t.
v  (1  c ) vP  cr
• r is a distribution over web-pages.
• If r is the uniform distribution we get pagerank.
• What happens if r is non-uniform?
Personalization
34

35. Personalized Pagerank1,2,3
• The only difference is that we use a non-uniform teleportation
distribution, i.e. at any time step teleport to a set of
webpages.
• In other words we are looking for the vector v s.t.
v  (1  c ) vP  cr
• r is a non-uniform preference vector specific to an user.
• v gives “personalized views” of the web.
1. Scaling Personalized Web Search, Jeh, Widom. 2003
2. Topic-sensitive PageRank, Haveliwala, 2001
3. Towards scaling fully personalized pagerank, D. Fogaras and B. Racz, 2004
35

36. Topic-sensitive pagerank (Haveliwala’01)
• Divide the webpages into 16 broad categories
• For each category compute the biased personalized pagerank
vector by uniformly teleporting to websites under that
category.
• At query time the probability of the query being from any of
the above classes is computed, and the final page-rank vector
is computed by a linear combination of the biased pagerank
vectors computed offline.
36

37. Random Walk for Recommendations
• Collaborative Filtering by Shang et.al
• Graph
– Vertices: Users (U), Items (I), Item Information (T) and
User Profiles (P)
– Edges with weights
•
•
•
•
u has a rating for i
i has a tag for t
u belongs to profile category p
u to u if they are connected in the social network
• Edge weights assignments

38. Random Walk for Recommendations
Connectivity/Transition
Generally 0.85
Preference Vector

39. Most of it from
• Purnamrita Sarkar, Random Walks on Graphs:
An Overview
• Random Walks on Graphs: A Survey, Laszlo
Lov'asz
• OBVIOUSLY: Wikipedia :D
• Random Walk on Graphs: Ankit Agarwal

40. Thanks