1 random walks on graphs: an overview purnamrita sarkar, cmu shortened and modified by longin jan...
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Random Walks on Graphs:An Overview
Purnamrita Sarkar, CMU
Shortened and modified by Longin Jan Latecki
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Motivation: Link prediction in social networks
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Motivation: Basis for recommendation
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Motivation: Personalized search
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Why graphs?
The underlying data is naturally a graph
Papers linked by citation Authors linked by co-authorship Bipartite graph of customers and products Web-graph Friendship networks: who knows whom
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What are we looking for
Rank nodes for a particular query Top k matches for “Random Walks” from
Citeseer Who are the most likely co-authors of “Manuel
Blum”. Top k book recommendations for Purna from
Amazon Top k websites matching “Sound of Music” Top k friend recommendations for Purna when
she joins “Facebook”
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Talk Outline Basic definitions
Random walks Stationary distributions
Properties Perron frobenius theorem Electrical networks, hitting and commute times
Euclidean Embedding
Applications Pagerank
Power iteration Convergencce
Personalized pagerank Rank stability
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Definitions
nxn Adjacency matrix A. A(i,j) = weight on edge from i to j If the graph is undirected A(i,j)=A(j,i), i.e. A is symmetric
nxn Transition matrix P. P is row stochastic P(i,j) = probability of stepping on node j from node i = A(i,j)/∑iA(i,j)
nxn Laplacian Matrix L. L(i,j)=∑iA(i,j)-A(i,j) Symmetric positive semi-definite for undirected graphs Singular
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Definitions
Adjacency matrix A Transition matrix P
1
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What is a random walk
1
1/2
1/21
t=0
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What is a random walk
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1/2
1/21
1
1/2
1/21
t=0 t=1
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What is a random walk
1
1/2
1/21
1
1/2
1/21
t=0 t=1
1
1/2
1/21
t=2
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What is a random walk
1
1/2
1/21
1
1/2
1/21
t=0 t=1
1
1/2
1/21
t=2
1
1/2
1/21
t=3
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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
Compute x1 for x0 =(1,0,0)’.
1
1/2
1/21
1
2
3
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Stationary Distribution
What happens when the surfer keeps walking for a long time?
When the surfer keeps walking for a long time
When the distribution does not change anymore i.e. xT+1 = xT
For “well-behaved” graphs this does not depend on the start distribution!!
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What is a stationary distribution? Intuitively and Mathematically
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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.
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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
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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
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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 !
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Talk Outline Basic definitions
Random walks Stationary distributions
Properties Perron frobenius theorem Electrical networks, hitting and commute times
Euclidean Embedding
Applications Pagerank
Power iteration Convergencce
Personalized pagerank Rank stability
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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
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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
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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.
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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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PageRankPage, Lawrence and Brin, Sergey and Motwani, Rajeev and Winograd, Terry The PageRank Citation Ranking: Bringing Order to the Web. Technical Report. Stanford InfoLab, 1999.
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Pagerank (Page & Brin, 1998) Web graph: if i is connected to j and 0
otherwise
An webpage is important if other important pages point to it.
Intuitively
v works out to be the stationary distribution of the Markov chain corresponding to the web: v = v P, where for example
i kiki
out P
iv
i
ivkv
,
)(
)(deg
)()(
02
1
2
1100
010
0)3(deg
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)3(deg
1)2(deg
100
0)1(deg
10
outout
out
out
P
1
1/2
1/21
)(deg
1, iP
outji
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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
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Power iteration
Power Iteration is an algorithm for computing the stationary distribution.
Start with any distribution x0
Keep computing xt+1 = xtP
Stop when xt+1 and xt are almost the same.
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Power iteration Why should this work?
Write x0 as a linear combination of the left eigenvectors {v0, v1, … , vn-1} of P
Remember that v0 is the stationary distribution.
x0 = c0v0 + c1v1 + c2v2 + … + cn-1vn-1
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Power iteration Why should this work?
Write x0 as a linear combination of the left eigenvectors {v0, v1, … , vn-1} of P
Remember that v0 is the stationary distribution.
x0 = c0v0 + c1v1 + c2v2 + … + cn-1vn-1
c0 = 1 . WHY? (see next slide)
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Convergence Issues1
Lets look at the vectors x for t=1,2,…
Write x0 as a linear combination of the eigenvectors of P
x0 = c0v0 + c1v1 + c2v2 + … + cn-1vn-1
c0 = 1 . WHY?
Remember that 1is the right eigenvector of P with eigenvalue 1, since P is stochastic. i.e. P*1T = 1T. Hence vi1T = 0 if i≠0.
1 = x*1T = c0v0*1T = c0 . Since v0 and x0 are both distributions
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Power iteration
v0 v1 v2 ……. vn-1
1 c1 c2 cn-1
0x
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Power iteration
v0 v1 v2 ……. vn-1
σ0 σ1c1 σ2c2 σn-1cn-1
~
01 Pxx
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Power iteration
v0 v1 v2 ……. vn-1
σ02 σ1
2c1 σ22c2 σn-1
2cn-1
2~
0
~
12 PxPxx
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Power iteration
v0 v1 v2 ……. vn-1
σ0t σ1
t c1 σ2
t c2 σn-1
t cn-1
t
t
~
0 Pxx
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Power iteration
v0 v1 v2 ……. vn-1
1 σ1t c1 σ2
t c2 σn-1
t cn-1
σ0 = 1 > σ1 ≥…≥ σn
t
t
~
0 Pxx
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Power iteration
v0 v1 v2 ……. vn-1
1 0 0 0
σ0 = 1 > σ1 ≥…≥ σnx
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Convergence Issues
Formally ||x0Pt – v0|| ≤ |λ|t
λ is the eigenvalue with second largest magnitude
The smaller the second largest eigenvalue (in magnitude), the faster the mixing.
For λ<1 there exists an unique stationary distribution, namely the first left eigenvector of the transition matrix.
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Pagerank and convergence The transition matrix pagerank really uses is
The second largest eigenvalue of can be proven1 to be ≤ (1-c)
Nice! This means pagerank computation will converge fast.
1. The Second Eigenvalue of the Google Matrix, Taher H. Haveliwala and Sepandar D. Kamvar, Stanford University Technical Report, 2003.
~
P
UP)1(P~
cc
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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
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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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Rank stability
How does the ranking change when the link structure changes?
The web-graph is changing continuously.
How does that affect page-rank?
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Rank stability1 (On the Machine Learning papers from the CORA2 database)
1. Link analysis, eigenvectors, and stability, Andrew Y. Ng, Alice X. Zheng and Michael Jordan, IJCAI-01
2. Automating the contruction of Internet portals with machine learning, A. Mc Callum, K. Nigam, J. Rennie, K. Seymore, In Information Retrieval Journel, 2000
Rank on 5 perturbed datasets by deleting 30% of the papers
Rank on the entire database.
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Rank stability
Ng et al 2001: Theorem: if v is the left eigenvector of . Let
the pages i1, i2,…, ik be changed in any way, and let v’ be the new pagerank. Then
So if c is not too close to 0, the system would be rank stable and also converge fast!
UPP cc )(~
1
~
P
c
ik
j j )(||'||
11
vvv
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Conclusion Basic definitions
Random walks Stationary distributions
Properties Perron frobenius theorem
Applications Pagerank
Power iteration Convergencce
Personalized pagerank Rank stability
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Acknowledgements Andrew Moore
Gary Miller Check out Gary’s Fall 2007 class on “Spectral Graph Theory,
Scientific Computing, and Biomedical Applications” http://www.cs.cmu.edu/afs/cs/user/glmiller/public/Scientific-Co
mputing/F-07/index.html
Fan Chung Graham’s course on Random Walks on Directed and Undirected Graphs http://www.math.ucsd.edu/~phorn/math261/
Random Walks on Graphs: A Survey, Laszlo Lov'asz Reversible Markov Chains and Random Walks on Graphs, D
Aldous, J Fill Random Walks and Electric Networks, Doyle & Snell