Post on 31-May-2020
Link Analysis
Leonid E. Zhukov
School of Data Analysis and Artificial IntelligenceDepartment of Computer Science
National Research University Higher School of Economics
Structural Analysis and Visualization of Networks
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 1 / 21
Lecture outline
1 Graph-theoretic definitions
2 Web search engines
3 Web page ranking algorithms– Pagerank– HITS
4 The Web as a graph
5 PageRank beyond the web
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 2 / 21
Graph theory
Graph G (E ,V ), |V | = n, |E | = mAdjacency matrix An×n, Aij , edge i → j
Graph is directed, matrix is non-symmetric: AT 6= A, Aij 6= Aji
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 3 / 21
Graph theory
sinks: zero out degree nodes, kout(i) = 0, absorbing nodes
sources: zero in degree nodes, kin(i) = 0
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 4 / 21
Graph theory
Graph is strongly connected if every vertex is reachable form everyother vertex.
Strongly connected components are partitions of the graph intosubgraphs that are strongly connected
In strongly connected graphs there is a path is each direction betweenany two pairs of vertices
image from Wikipedia
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 5 / 21
Graph theory
A directed graph is aperiodic if the greatest common divisor of thelengths of its cycles is one (there is no integer k >1 that divides thelength of every cycle of the graph)
image from Wikipedia
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 6 / 21
Web search engine
”The Anatomy of a Large-Scale Hypertextual Web Search Engine”
Sergey Brin and Lawrence Page, 1998
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 7 / 21
Web as a graph
Hyperlinks - implicit endorsements
Web graph - graph of endorsements (sometimes reciprocal)
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 8 / 21
Ranking on directed graph
iteratively update
ri ←∑
j∈N(i)
rj =∑j
Aji rj
r t+1i =
∑j
Aji rtj , with r t=0
j = r0j
rt+1 = AT rt , rt=0 = r0
norm ||rt+1|| ≥ ||rt ||
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 9 / 21
Ranking on directed graph
Absorbing nodes
Source nodes
Cycles
rt+1 = AT rt , rt=0 = r0
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 10 / 21
PageRank
”PageRank can be thought of as a model of user behavior. We assume there is a”random surfer” who is given a web page at random and keeps clicking on links,never hitting ”back” but eventually gets bored and starts on another randompage. The probability that the random surfer visits a page is its PageRank.”
PR(A) = (1− d) + d(PR(T1)/C (T1) + ...+ PR(Tn)/C (Tn))
Sergey Brin and Larry Page, 1998
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 11 / 21
Random walk
Random walk on a directed graph
pt+1i =
∑j∈N(i)
ptjdoutj
=∑j
Aji
doutj
pj
Dii = diag{douti }
pt+1 = (D−1A)Tpt
pt+1 = PTpt
Markov chain with transition probability matrix P = D−1A
limt→∞
pt = π
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 12 / 21
Perron-Frobenius Theorem
Perron-Frobenius theorem (Fundamental Theorem of Markov Chains)If matrix is
stochastic (non-negative and rows sum up to one, describes Markovchain)
irreducible (strongly connected graph)
aperiodic
then∃ limt→∞
p̄t = π̄
and can be found as a left eigenvector
π̄P = π̄, where ||π̄||1 = 1
π̄ - stationary distribution of Markov chain, raw vectorOscar Perron, 1907, Georg Frobenius,1912.
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 13 / 21
PageRank
Transition matrix:
P = D−1A
Stochastic matrix:
P′ = P +seT
nPageRank matrix:
P′′ = αP′ + (1− α)eeT
n
Eigenvalue problem (choose solution with λ = 1):
P′′Tp = λp
Notations:e - unit column vector, s - absorbing nodes indicator vector (column)
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 14 / 21
PageRank computations
Eigenvalue problem (λ = 1, ||p||1 = pTe = 1):(αP′ + (1− α)
eeT
n
)T
p = λp
p = αP′Tp + (1− α)e
n
Power iterations:p← αP′Tp + (1− α)
e
n
Sparse linear system:
(I− αP′T )p = (1− α)e
n
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 15 / 21
Graph structure of the web
Bow tie structure of the web
Andrei Broder et al, 1999
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 16 / 21
Hubs and Authorities (HITS)
Citation networks. Reviews vs original research (authoritative) papers
authorities, contain useful information, aihubs, contains links to authorities, hi
Mutual recursion
Good authorities reffered bygood hubs
ai ←∑j
Ajihj
Good hubs point to goodauthorities
hi ←∑j
Aijaj
Jon Kleinberg, 1999
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 17 / 21
HITS
System of linear equations
a = αATh
h = βAa
Symmetric eigenvalue problem
(ATA)a = λa
(AAT )h = λh
where eigenvalue λ = (αβ)−1
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 18 / 21
HITS
Focused subgraph of WWW
Jon Kleinberg, 1999
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 19 / 21
PageRank beyond the Web
from David Gleich
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 20 / 21
References
The Anatomy of a Large-Scale Hypertextual Web Search Engine,Sergey Brin and Lawrence Page, 1998
Authoritative Sources in a Hyperlinked Environment. Jon M.Kleinberg, Proc. 9th ACM-SIAM Symposium on Discrete Algorithms,1999
Graph structure in the Web, Andrei Broder et all. Procs of the 9thinternational World Wide Web conference, 2000
A Survey of Eigenvector Methods of Web Information Retrieval. AmyN. Langville and Carl D. Meyer, 2004
PageRank beyond the Web. David F. Gleich, arXiv:1407.5107, 2014
Leonid E. Zhukov (HSE) Lecture 6 17.02.2015 21 / 21