random walk on graphs
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Random Walk on Graphs
Pavan Kapanipathi Reading Group (Kno.e.sis)
Referred: Purnamrita Sarkar, Random Walks on Graphs: An Overview
Agenda
• Introduction – Motivation
• Background– Graphs– Matrices
• Random Walk– PageRank– Personalized PageRank– Topic Sensitive PageRank
• Applications– Specifically in Recommender Systems
Random Walk
A drunk man will find his way home, but a drunk bird may get lost forever.
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Motivation: Link prediction in social networks
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Motivation: Basis for recommendation
Since I had very less slides and More time – Graphs
• Undirected Graphs
Since I had very less slides and more time in hand-- Graphs
• Directed Graphs
Since I had very less slides and more time in hand – Matrix
Rows
Colu
mns
i
i
j
j
k
k
i,j
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Adjacency matrix A Transition matrix P
1
1
11
1
1/2
1/21
Adjacency and Transition Matrix
Markov Property (Basic)
• Given the present state, the future and past states are independent
Stochastic
• Wikipedia: In probability theory, a purely stochastic system is one whose state is non-deterministic so that the subsequent state of the system is determined probabilistically.
• Matrix – Stochastic Row/Column?
1
1/2
1/21
Random Walk on Graphs
Random Walk on Graphs
Random Walk on Graphs
Random Walk on Graphs
The random sequence of points selected this way is a random walk on the graph
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Transition matrix P
1
1/2
1/21
Again: Transition Matrix
Probability?
i
i j
j
k
k
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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?
Property of Adjacency Matrix
What does AxA () represent in Graph?
Similarly Transitional Matrix?
Adjacency matrix A
1
1
11
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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 !
Eigen Value and Eigen Vector?
The basic equation is Ax = x. The number is an eigenvalue of A.
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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!
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Well behaved graphs
• Irreducible: There is a path from every node to every other node.
Irreducible Not irreducible
What about connected undirected Graph?
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Well behaved graphs
• Aperiodic: The GCD of all cycle lengths is 1. The GCD is also called period.
AperiodicPeriodicity is 3
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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.
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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.
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Proximity measures from random walks
• 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.
ab
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Hitting and Commute times
• 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)
ab
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Hitting and Commute times
• 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)
ab
Random Walk (versions)
• PageRank– Personalized PageRank– Topic Sensitive PageRank
• Recommender Systems (My interests)
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Recommender Networks
• For a customer node i define similarity as– H(i,j)– C(i,j)– Or the cosine similarity
• 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)
jjii
ij
LL
L
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.
PageRank
• Intuitively
• v works out to be the stationary distribution of the markov chain corresponding to the web.
ij
out j
jviv
)(deg
)()(
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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.
jin
cc
ij ,
)(~
1
1
U
UPP
jin
cc
ij ,1
)1(~
U
UPP
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Pagerank• We are looking for the vector v s.t.
• r is a distribution over web-pages.
• If r is the uniform distribution we get pagerank.
• What happens if r is non-uniform?
crc vP)1(v
Personalization
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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.
• r is a non-uniform preference vector specific to an user.
• v gives “personalized views” of the web.
rvP)1(v cc
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
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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.
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
Random Walk for Recommendations
• Item Rank (IJCAI)Connectivity/Transition
Preference VectorGenerally 0.85
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
Thanks
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