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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank, Centrality Measures and Consensus: A Systems and Control Viewpoint
Roberto Tempo
CNR-IEIIT
Consiglio Nazionale delle Ricerche
Politecnico di Torino
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank Problem
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Website of Umberto Eco
Author of
“The Name of the
Rose” translated
in more than
40 languages
Looking for
“Umberto Eco”
in Google we
find the websiteICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank is a numerical value in the interval [0,1]
Using a PageRank checker we compute
“PageRank is Google’s view of the importance of this page”
“PageRank reflects our view of the importance of Web pages byconsidering more than 500 million variables and 2 billion terms.Pages that are considered important receive a higher PageRankand are more likely to appear at the top of the search results”
PageRank for Umberto Eco
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank is used in search engines to indicate theimportance of the page currently visited
PageRank has broader utility than search engines
Used in various areas for ranking other “objects”
o scientific journals (Eigenfactor)o economic complexityo cancer biology and protein detectiono ranking top sport playerso …
PageRank
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
Network consisting of servers (nodes) connected bydirected communication links
Web surfer moves along randomly following thehyperlink structure
When arriving at a page with several outgoing links, oneis chosen at random, then the random surfer moves to anew page, and so on…
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
Web representation with incoming and outgoing links
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
Pick an outgoing link at random
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
Arriving at a new web page
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
Pick another outgoing link at random
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model
If a page is important then it is visited more often...
The time the random surfer spends on a page is ameasure of its importance
If important pages point to your page, then your pagebecomes important because it is visited often
What is the probability that your page is visited?
Need to rank the pages in order of importance forfacilitating the web search
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Graph Representation
1 2
34
5
Directed graph with nodes (pages) and links representing the web
Graph is not necessarily strongly connected (from 5 you cannot reach other pages)
Graph is constructed using crawlers and spiders moving continuously along the web
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
For each node we count the number of outgoing links and normalize their sum to 1
Hyperlink matrix is a nonnegative (column) substochastic matrix
Hyperlink Matrix
1 2
3 4
5
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Hyperlink Matrix
1 2
3 4
5
0 1 0 0 0
1 / 3 0 0 1 / 2 0
1 / 3 0 0 1 / 2 0
1 / 3 0 0 0 0
0 0 1 0 0
A
4
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Page 5 is a Dangling Node
1 2
34
5
0 1 0 0 0
1 / 3 0 0 1 / 2 0
1 / 3 0 0 1 / 2 0
1 / 3 0 0 0 0
0 0 1 0 0
A
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Dangling nodes are pageshaving no outgoing link
Problem not well-posed
Probability of page 5increases and increases…probability of the otherpages decreases anddecreases…
Eventually we obtain p5 = 1and p1 = p2 = p3 = p4 = 0
Dangling Nodes
1 2
34
5
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Dangling Nodes
Example: pdf file with no
hyperlink
Benchmark: Web Lincoln
University, New Zealand
3756 nodes
31718 total #outgoing links
H. Ishii, R. Tempo (2014)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Dangling Nodes
3255 dangling nodes (85%)
Red dots outgoing
links toward dangling
nodes
Blue dots are normal
links
White area corresponds
to no-links
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Dangling Nodes
Random surfer gets stuck when visiting a pdf file
In this case the “back button” of the browser is used
Easy fix: Add new links to make the matrix stochastic
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Easy Fix: Add New Link
1 2
3 4
5
We add a new outgoinglink from page 5 to page 3
0 1 0 0 0
1/ 3 0 0 1/ 2 0
1/ 3 0 0 1/ 2
1/ 3 0 0 0
0
1
0
0 0 1 0
A
In the benchmark this fix increases the #links from to 31718 to 40646
5
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Assumption: No Dangling Nodes
This is a web modeling problem
Assume that there are no dangling nodes or add artificiallinks
Hyperlink matrix A is a nonnegative stochastic matrix(instead of substochastic)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Random Surfer Model and Markov Chains
Random surfer model is represented as a Markov chain
where x(k) is a probability vector
x(k) [0,1]n and ∑i xi(k) = 1
xi(k) represents the importance of the page i at time k
( 1) ( )x k Ax k
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convergence of the Markov Chain
Question: Does the Markov chain converge to astationary value
x(k) x* for k
representing the probability that the pages are visited?
Answer: No
Example:0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
A
0
1(0)
0
0
x
x1(k)
k
1 -
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Teleportation Model
Recall that the matrix A is a nonnegative stochasticmatrix
We introduce a different model
Teleportation: After a while the random surfer gets boredand decides to “jump” to another page not directlyconnected to that currently visited
New page may be geographically or content-basedlocated far away
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Recall the Random Surfer Model
Web representation with incoming and outgoing links
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Recall the Random Surfer Model
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CNR-IEIIT Teleportation Model
We are “teleported” to a web page located far away
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model Again
Pick another outgoing link at random
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Random Surfer Model Again
Pick another outgoing link at random
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Teleportation Model Again
We are teleported to another web page located far away
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convex Combination of Matrices
Teleportation model is represented as a convexcombination of matrices A and S/n
S = 1 1T is a rank-one matrix
1 vector with all entries equal to one
Consider a matrix M defined as
M = (1 - m) A + m/n S m (0,1)
where n is the number of pages
The value m = 0.15 is used at Google
1 1
1 1
S
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Matrix M
M is a convex combination of two nonnegativestochastic matrices and m (0,1)
M is a strictly positive stochastic matrix
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convergence of the Markov Chain
Consider the Markov chain
x(k+1) = M x(k)
where M is a strictly positive stochastic matrix
If ∑i xi(0) = 1 convergence is guaranteed by PerronTheorem
x(k) x* for k
x* = M x* = [(1 - m) A + m/n S] x* m (0,1)
Corresponding graph is strongly connected
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank: Bringing Order to the Web
Rank n web pages in order of importance
Ranking is provided by x*
PageRank x* of the hyperlink matrix M is defined as
x*=M x* where x* [0,1]n and ∑i xi* = 1
S. Brin, L. Page (1998)S. Brin, L. Page, R. Motwani, T. Winograd (1999)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank: Bringing Order to the Web
Rank n web pages in order of importance
Ranking is provided by x*
PageRank x* of the hyperlink matrix M is defined as
x*=M x* where x* [0,1]n and ∑i xi* = 1
x* is the stationary distribution of the Markov Chain(steady-state probability that pages are visited is x* )
x* is a nonnegative unit eigenvector corresponding to theeigenvalue 1 of M
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Positive Stochastic Matrices and Perron Theorem
The eigenvalue 1 of M is a simple eigenvalue and anyother eigenvalue (possibly complex) is strictly smallerthan 1 in absolute value
There exists an eigenvector x* of M with eigenvalue 1such that all components of x* are positive
M is irreducible and the corresponding graph is stronglyconnected (for any two vertices i,j there is a sequence ofedges which connects i to j)
Stationary probability vector x* is the eigenvector of M
x* exists and it is unique
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Two vertices i, j V of an undirected graph G V ,E)are connected if there is a path between i and j
A graph is connected if every pair of vertices in thegraph is connected
A directed graph is a graph where the edges have adirection associated with them
Oriented graphs are directed graphs with no self-loops,no multiple adjacencies and no 2-cycles
Undirected, Directed, Oriented Graphs
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
A directed graph is called weakly connected if replacingall its directed edges with undirected edges produces aconnected (undirected) graph
A graph is said to be strongly connected if every vertexis reachable from every other vertex
Weakly and Strongly Connected Graphs
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank Computation
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank Computation withPower Method
PageRank is computed with the power method
x(k+1) = M x(k)
If ∑i xi(0) = 1 convergence is guaranteed because M is astrictly positive stochastic matrix (Perron Theorem)
x(k) x* for k
PageRank computation requires 50-100 iterations (40 inthe benchmark)
This computation takes about a week and it is performedcentrally at Google once a month
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Why m = 0.15?
Asymptotic rate of convergence of power method isexponential and depends on the ratio
We have
1(M) = 1 | 2(M) | ≤ 1 - m = 0.85
For Nit= 50 we have 0.8550≈ 2.95 10-4
For Nit= 100 we have 0.85100≈ 8.74 10-8
Larger m implies faster convergence, but numericallyunstable
2
1
| λ |
| λ | 1
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank Computation with Power Method
1 4
23
0.038 0.037 0.037 0.321
0.887 0.037 0.462 0.321
0.037 0.462 0.037 0.321
0.037 0.462 0.462 0.037
M
0 0 0 1/ 3
1 0 1/ 2 1/ 3
0 1/ 2 0 1/ 3
0 1/ 2 1/ 2 0
A
T* 0.12 0.33 0.26 0.29x
0.15m
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Size of the Web
The size of M is more than 8 billion (and it isincreasing)!
Sparsity in the web: 1020 entries
1012 non-zero entries
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Why m = 0.15?Answer 2: Sensitivity
Study sensitivity wrt m taking
We obtain for i = 1, 2,…, n
A deeper analysis shows that if 2(M) is close to 1(M)then x*(m) is very sensitive for small m
Conclusion: m=0.15 is a good compromise
*d( )
d ix mm
*d 1( ) 6.66
d ix mm m
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Columbia River, The Dalles, Oregon
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Distributed Viewpoint
More and more computing power is needed…
… the distributed viewpoint because global informationabout the network is difficult to obtain
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Randomized Decentralized Algorithms
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Main idea: Develop Las Vegas randomized decentralized
algorithms for computing PageRank
Randomized Decentralized Algorithms for PageRank Computation
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Monte Carlo and Las Vegas Randomized Algorithms
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Monte Carlo and Las Vegas
Monte Carlo was invented by Metropolis, Ulam, vonNeumann, Fermi, … in the fourties (Manhattan project)
Metropolis Fermi Ulam, Feymann, von Neumann
Las Vegas first appeared in computer science in the lateseventies
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Randomized Algorithm: Definition
Randomized Algorithm (RA): An algorithm that makesrandom choices during its execution to produce a result
Example: Matlab code
set_r =1:0.01:3;for k =1:length(set_r)
if (rand > 0.5) then a_opt(k) = hel(k);else a_opt(k) = 3.7;end if
a_lin(k) =(e/(e-1))*r;a_sub(k) =(a/(a-1))*(r+log(a)-1);
end
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Las Vegas Randomized Algorithm
Las Vegas Randomized Algorithm (LVRA): Arandomized algorithm that always produces correctresults, the only variation from one run to another is therunning time
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Example of Las Vegas: Discrete Random Variables
q1 q2 q3 q4 q5 q6 q7 q8 q9 q10
Consider discrete random variables
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Example of Las Vegas: Discrete Random Variables
q1 q2 q3 q4 q5 q6 q7 q8 q9 q10
Consider discrete random variables
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Example of Las Vegas: Discrete Random Variables
q1 q2 q3 q4 q5 q6 q7 q8 q9 q10
Consider discrete random variables
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Las Vegas Randomized Algorithms
Las Vegas Randomized Algorithm (LVRA): Alwaysgive the correct solution
They are also called zero-sided randomized algorithms
The solution obtained with a LVRA is probabilistic, so“always” means with probability one
Running time may be different from one run to another
We study the average running time
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Las Vegas Viewpoint
Consider discrete random variables
The sample space is discrete and MN possible choicescan be made
In the binary case we have 2N
Finding maximum requires ordering the 2N choices
Las Vegas can be used for ordering real numbers
Example: RQS
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Decentralized Communication Protocol - Randomization
1 4
23
Gossip communication protocol:
1. at time k randomly selectpage i (i=4) for PageRankupdate
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Decentralized Communication Protocol – Outgoing Links
1 4
23
Gossip communication protocol:
1. at time k randomly selectpage i (i=4) for PageRankupdate
2. send PageRank value ofpage i to the outgoing pagesthat are linked to page i
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Decentralized Communication Protocol – Incoming Links
1 4
23
Gossip communication protocol:at time k randomly select page i(i=4) for PageRank update
3. request PageRank values fromincoming pages that arelinked to page i
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Global Clock
The pages taking action are determined via a stochasticprocess θ(k) {1, …, n}
θ(k) is assumed to be i.i.d. with uniform probability
Prob{θ(k)=i} = 1/n
If at time k θ(k) = i then page i initiates PageRank update
θ(k) is a global clock known to all the pages
Clock synchronization problem
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Randomized Update Scheme
Consider the randomized update scheme
x(k+1) = Aθ(k) x(k)
where Aθ(k) are the decentralized link matrices related tothe matrix A
This is a Markovian jump system
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CNR-IEIIT Decentralized Link Matrices - 1
0 0 0 1/ 3
1 0 1/ 2 1/ 3
0 1/ 2 0 1/ 3
0 1/ 2 1/ 2 0
A
4
1/ 3
1/ 3
1/ 3
0
A
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Decentralized Link Matrices - 2
0 0 0 1/ 3
1 0 1/ 2 1/ 3
0 1/ 2 0 1/ 3
0 1/ 2 1/ 2 0
A
4
0 0 1/ 3
0 0 1/ 3
0 0 1/ 3
0 0 0 0
A
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Decentralized Link Matrices - 3
0 0 0 1/ 3
1 0 1/ 2 1/ 3
0 1/ 2 0 1/ 3
0 1/ 2 1/ 2 0
A
4
1 0 0 1/ 3
0 1 0 1/ 3
0 0 1 1/ 3
0 0 0 0
A
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Decentralized Link Matrices - 4
0 0 0 1/ 3
1 0 1/ 2 1/ 3
0 1/ 2 0 1/ 3
0 1/ 2 1/ 2 0
A
4
1 0 0 1/ 3
0 1 0 1/ 3
0 0 1 1/ 3
0 0 0 0
A
3
1 0 0 0
0 1 1/ 2 0
0 0 0 0
0 0 1/ 2 1
A
Matrices Ai are very sparse
n -1 diagonal identity entries
n non-diagonal entries
Linear # entries
Algorithm easily implementable
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Properties of the Algorithm: Average Matrix Aave
Average matrix Aave = E[ Aθ(k) ]
Taking uniform distribution in the stochastic process θ(k)we have
Aave = 1/n ∑i Ai
Lemma (properties of the average matrix Aave )
(i) Aave = 2/n A + (1-2/n) I
(ii) Matrices A and Aave have the same eigenvectorcorresponding to the eigenvalue 1
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Modified Randomized Update Scheme
Remark: Need to work with the teleportation model(positive stochastic matrix M)
Consider the modified randomized update scheme
x(k+1) = Mθ(k) x(k)
where Mθ(k) are the modified decentralized link matricescomputed as
Mi = (1- r) Ai + r/n S i = 1, 2, …, n
and r (0,1) is a design parameter
r = 2m/(n – mn + 2m)
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Properties of the Algorithm:Average Matrix Mave
Average matrix Mave = E[ Mθ(k) ]
Define r = 2m/(n – mn + 2m)
Lemma (properties of the average matrix Mave)
(i) r (0,1) and r < m =0.15
(ii) Mave = r/m M + (1-r/m) I
(iii) Eigenvalue 1 for Mave is the unique eigenvalue ofmaximum modulus and PageRank is the correspondingeigenvector
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Undesired Oscillations
The state x(k) oscillates and there is no convergence to afixed point…
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Time-Averaging
Time averaging was introduced in the seventies toaccelerate convergence of stochastic approximationalgorithms
With time-averaging we remove oscillations
0
1( ) ( )
1
k
i
y k x ik
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convergence Properties
Theorem (convergence properties)
The time average y(k) of the modified randomizedupdate scheme converges to PageRank x* in MSE
E[ ||y(k) - x*||2 ] 0 for k
provided that ∑i xi(0) = 1
Proof: Based on the theory of ergodic matrices
H. Ishii, R. Tempo (2010)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Comments
Time average y(k) can be computed recursively as afunction of y(k-1)
Because of averaging convergence rate is 1/k
Sparsity of the matrix A is preserved because
x(k+1) = M x(k) = (1- r) A x(k) + r/n 1
where 1 is a vector with all entries equal to one0 0 0 1 / 3
1 0 1 / 2 1 / 3
0 1 / 2 0 1 / 3
0 1 / 2 1 / 2 0
A
0.038 0.037 0.037 0.321
0.887 0.037 0.462 0.321
0.037 0.462 0.037 0.321
0.037 0.462 0.462 0.037
M =
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Extensions - 1
Extensions to simultaneous
update of multiple web pages
with Bernoulli process
PageRank computation
with lossy communication
Other convergence properties of the algorithm hold
(almost-sure)
H. Ishii, R. Tempo and E. W. Bai (2012)W.-X. Zhao, H.-F. Chen and H. Fang (2013)
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CNR-IEIIT Simulation Results for the Benchmark
There are two clusters of pages with dense link structurescorresponding to higher PageRank
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Ranking Control Journals
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CNR-IEIIT ISI Web of Knowledge
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CNR-IEIIT Ranking Journals: Impact Factor
Impact Factor IF
Census period (2013) of one year and a window period (2011-2012) of two years
Remark: Impact Factor is a “flat criterion” (it does not take into account where the citations come from)
2013 IF2011- 2012
2011- 2012
number citations in 2013 of articles published in
number of articles published in
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT ISI Web of Knowledge
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Ranking Journals: Eigenfactor
Eigenfactor EF
Ranking journals using ideas from PageRankcomputation in Google
In Eigenfactor journals are considered influential if theyare cited often by other influential journals
What is the probability that a journal is cited?
C. T. Bergstrom (2007)
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CNR-IEIIT Random Reader Model
“Imagine that a researcher is to spend all eternity in the libraryrandomly following citations within scientific periodicals. Theresearcher begins by picking a random journal in the library.From this volume a random citation is selected. The researcherthen walks over to the journal referenced by this citation. Theprobability of doing so corresponds to the fraction of citationslinking the referenced journal to the referring one, conditioned onthe fact that the researcher starts from the referring journal. Fromthis new volume the researcher now selects another randomcitation and proceeds to that journal. This process is repeated adinfinitum.”
A. D. West, T. C. Bergstrom, C. T. Bergstrom (2010)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Random Reader Model: Graph Representation
Directed graph with journals (nodes) and citations (links)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Random Reader Model:Graph Representation
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CNR-IEIIT
Random Reader Model:Graph Representation
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Random Reader Model:Graph Representation
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Random Reader Model:Graph Representation
Directed graph with journals (nodes) and citations (links)
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CNR-IEIIT
Self-citations are omitted
Normalization to obtain a column substochastic matrix
(note: if 0/0 we set aij= 0)
0
0
0
A
Cross-citation Matrix - 1
i j
i jk jk
aa
a
2013
2008 - 2012
a number citations in from journal j to ijarticles published in journal i in
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
0
0
0
0
A
Black Holes
1 2
34
5
Black holes: journals that do not cite any other journalColumns with entries equal to zeroA substochastic matrix
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Article vector v
vi is the fraction of all published articles coming fromjournal i during the window period 2008-2012
Article vector v is a stochastic vector
Total number of journals n = 8400
Article Vector
2008 - 2012
2008 - 2012i
number articles published by journal i in v
number articles published by all journals in
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
* * * * 0
* * * * 0
* * * * 0
* * * * 0
* * * * 0
A
Cross-citation Matrix – 2
1 2
34
Replace A with stochastic matrix introducing the article vector v
A
5
1v2v
3v 4v
5v
1
2
3
4
5
* * * *
* * * *
* * * *
* * * *
* * * *
v
v
A v
v
v
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Replace substochastic matrix A with stochastic matrix
introducing the article vector v
1
2
3
4
5
* * * *
* * * *
* * * *
* * * *
* * * *
v
v
A v
v
v
* * * * 0
* * * * 0
* * * * 0
* * * * 0
* * * * 0
A
Cross-citation Matrix - 2
A
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Random Reader Model and Markov Chains
Rank n = 8336 journals in order of importance
Random reader model represented as a Markov chain
where x(k) [0,1]n and ∑i xi(k) = 1 is the journalinfluence vector at step k
xi(k) represents the importance of the journal i
Question: Does the Markov chain converge to astationary value representing the probability of citation?
( 1) ( )x k Ax k
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ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convergence of the Markov Chain - 1
Answer: No (recall previous example)
Need to replace with the matrix M obtaining theEigenfactor equation
where is a weighting matrix
T(1 - ) 0.15M m A m v m 1
A
1 1T
n n
v v
v
v v
1
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Teleportation Model
This avoids being trapped in a small set of journals
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convergence of the Markov Chain - 2
Consider the Markov chain
x(k+1) = M x(k)
If ∑i xi(0) = 1 convergence is guaranteed by PerronTheorem because M is a positive stochastic matrix
x(k) x* for k
The corresponding graph is strongly connected
We have x* = M x*
Journal influence vector x* exists and it is unique
Eigenvector corresponding to the simple eigenvalue 1ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
0
0
0
0
A
0
1
0
0
b
To preserve sparsity of A we write the journal influencevector iteration using power method
where b is the black holes vector having 1 for entriescorresponding to black holes and 0 elsewhere
PageRank Iteration
T( 1) (1- ) ( ) [(1- ) ( ) ]x k m Ax k m b x k +m v
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Eigenfactor EF
Journal influence vector x* is the steady-state fraction oftime spent reading each journal
x* provides weights to the citation values
Eigenfactor EFj of journal j is the
percentage of the total weighted
citations that journal j receives
from all 8336 journals
Article influence score AIj
100 ( *)
( *) j
ji
i
Ax
Ax
EF
0.01
j
jjv
EF
AI
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
2013 Impact Factor 2013 EigenfactorTM
3.4 IEEE CSM 1 Automatica 0.052
3.2 IEEE TAC 2 IEEE TAC 0.051
3.1 Automatica 3 SIAM J Contr & Opt 0.016
2.6 Int J Rob Nonlin Contr 4 Syst & Contr Lett 0.013
2.5 IEEE TCST 5 IEEE TCST 0.013
2.2 J Proc Contr 6 Int J Contr 0.009
1.9 Contr Eng Pract 7 Int J Rob Nonlin Contr 0.009
1.9 Syst & Contr Lett 8 J Proc Contr 0.008
1.4 SIAM J Contr & Opt 9 Contr Eng Pract 0.008
1.2 Math Contr Sign Sys 10 IEEE CSM 0.004
1.1 Int J Contr 11 Europ J Contr 0.002
0.8 Europ J Contr 12 Math Contr Sig Sys 0.001
18
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Distributed Average Consensus
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Distributed Average Consensus
General problem: Consider a set of N agents each
having a numerical value
This numerical value is communicated to the
neighboring agents iteratively
Consensus: All agents eventually reach a common
value (average of the initial value)
Applications: Sensor networks, load balancing, multi-
vehicle coordination, UAVs, …
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Graph Formulation
Consider a network of N nodes and the graph (V, E)
E is the set of edges V is the set of nodes
graph is undirected
and connected
(no directions, no
self-loops, no
multiple edges)
at time k each node i has a scalar value xi(k); initial value is xi(0)
x5(k)
x1(k)
x4(k)
x3(k)
x2(k)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Distributed Average Consensus
Objective: Derive a randomized algorithm (MC or LV)
such that
1. Nodes update the values xi(k) using neighbor
information
2. The values of the nodes converge to the average of
initial values xi(0)
Three different cases depending on the range of node
values: Real, integer (quantized) and binary values
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Real/Quantized/Binary Agent Values
Real and quantized case are very popular[1]
Monte Carlo (MCMC Markov Chain Monte Carlo)
algorithms are developed in these cases
The binary value case is not really studied in the
systems and control community…
Paradigm: Byzantine Agreement Problem
[1] P.J. Antsaklis, J. Baillieul (2007)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Byzantine Agreement Problem (ByzAgr)
19
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Byzantium, 1453 AD
Some history: In the year 1453 AD, the city of
Byzantium is under siege… powerful Ottoman
battalions are camped around the city on both sides of
the Bosporus, poised to launch the final attack…
The generals, located in several camps, are trying to
reach an agreement… they can communicate thanks to
the messenger service of the Ottoman Army…
Messages can be delivered certifying the identity of
the sender and preserving its content…
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Byzantium and the Bosporus
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Loyal and Traitor Generals
Loyal generals are trying to reach an agreement on
when they should launch the final attack, but…
Traitor generals are secretly conspiring… their aim is
to confuse the loyal generals so that an insufficient
number of generals is deceived into attacking…
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Byzantine Agreement Problem (ByzAgr)[1]
Find a distributed consensus algorithm satisfying thefollowing conditions:
1. Non-triviality: If all generals have the same input, allloyal generals take a decision equal to the input
2. Agreement: All loyal generals should agree on thedecision
3. Limited dithering: Eventually all loyal generals shouldcome to a decision
[1] L. Lamport, R. Shostak, M. Pease (1982)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Agents and Communication Mechanism
Synchronous: The agents have synchronized clocks
Reliable: A message is guaranteed to be delivered
Authenticated: Identity of the sender is known
Faulty agents: Hard to identify because they behave
maliciously instead of simply crashing; they may
collude to maximize damage
Point-to-point communication: Underlying topology is
that of a undirected and connected graph
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Binary Consensus
Consider N binary agents xi(k) {0, 1} for all i, k
There are Nf < N faulty agents
Binary consensus is achieved if each agent determines
a binary decision value yi {0, 1} such that:
1. All non-faulty agents arrive at the same decision value
2. If all agents have the same initial value xi(0), then the
non-faulty agents reach a common decision yi = xi(0)
20
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Distributed Agreement Protocol
Distributed agreement protocol proceeds in a sequence
of rounds
At each round each agent sends a binary message
(called vote, e.g. attack or retrieve) to the other agents
Non-faulty agents send the same vote to all the other
agents; faulty agents may send different votes to
different agents
Round is concluded when all agents receive a vote
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Randomized Algorithm for ByzAgr
First randomized algorithm is due to Rabin[1]
There is a global coin toss performed by a trusted
party consisting of
heads or tails
The result of the global coin toss is correctly
transmitted to all agents
All agents equally contribute to the decision
Majority vote: Count the number of votes received[1] M.O. Rabin (1983)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Rabin’s Randomized Algorithm
Nf < N/8; N is a multiple of 8; L = (5N/8) + 1; H = (3N/4) + 1; G = 7N/8
input: vote, output: decision yi, for each round do
1. broadcast vote, receive votes from all other agents
2. set mi(k) ← majority value (0, 1) received
3. set ti(k) ← number of occurrences of mi(k)
4. if coin = heads, then set t(k) ← L, else t(k) ← H
5. if ti(k) t(k), then set vote ← mi(k), else vote ← 0
6. if ti(k) G, then set yi ← mi(k)
¯
¯ ¯
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Main Theorem for ByzAgr
Main Theorem[1]
- Binary consensus is achieved with probability one for
each initial condition
- Expected number of steps is a constant
[1] M.O. Rabin (1983)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT ByzAgr and LVRA
Rabin’s algorithm always (probabilistically) gives a
correct output
Number of rounds is a random variable because the
algorithm is based on randomization
The expected number of steps to reach consensus is a
constant
This algorithm is a LVRA
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Randomized vs Deterministic Algorithms for ByzAgr
Deterministic algorithms requires Nf steps
Impossibility Theorem[1]
If there is no synchronized clock for all agents and if
the processing speed of the agents is different, then no
deterministic algorithm can achieve consensus (even in
the presence of a single faulty agent)
[1] M.J. Fisher, N.A. Lynch, M.S. Paterson (1985)
21
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Consensus and PageRank
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Graph of Agents
Consider a graph of agents which communicate using adecentralized communication pattern (similar to that forPageRank)
The value of agent i at time k is xi
Values of agents may be updated using a LVRA
x(k+1) = Aθ(k) x(k)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Decentralized Communication Pattern
Stochastic process θ(k) {1, …, d} where d is thenumber of communication patterns
θ(k) is assumed to be i.i.d. with uniform probability
Prob{θ(k)=i} = 1/d
If at time k θ(k) = i then agent i initiates update
Technical differences with communication protocol forPageRank but the ideas are similar
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Communication Pattern for Agent 1
1 4
2
1
* 0 0 *
* * 0 0
0 0 * 0
0 0 0 *
A
3
A1 is row stochastic1
1/ 2 0 0 1/ 2
1/ 2 1/ 2 0 0
0 0 1 0
0 0 0 1
A
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Consensus
We say that consensus w.p.1 is achieved if for any initialcondition x(0) we have
Prob
for all agents i, j lim ( ) ( ) 0 1i j
kx k x k
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Convergence Properties
Lemma (convergence properties)
Assuming that the graph is strongly connected, thescheme
x(k+1) = Aθ(k) x(k)
achieves consensus w.p.1
Prob
for all agents i, j and for any initial condition x(0)
Y. Hatano, M. Mesbahi (2005); C. W. Wu (2006)A. Tahbaz-Salehi, A. Jadbabaie (2008)
lim ( ) ( ) 0 1i jk
x k x k
22
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT PageRank and Consensus
Consensus PageRank
all agent values become equal page values converge to constant
strongly connected agent graph web is not strongly connected
Ai are row stochastic matrices Ai and Mi are column stochastic
self-loops in the agent graph no self-loops in the web
convergence for any x(0) x(0) stochastic vector
time averaging y(k) not necessary averaging y(k) required
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Economic Complexity
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Economic Complexity
Introduced in the paper
Hidalgo C. and Hausmann R. “The Building Blocks ofEconomic Complexity,” Proceedings of the NationalAcademy of Science, 106, 2009
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Bipartite Graphs - 1
Bipartite graph representing countries and products
countries v productsxv={v1,v2,v3,v4,v5} xu={u1,u2,u3,u4}
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Bipartite Graphs - 2
Bipartite graph representing countries and products
countries v products
xv={v1,v2,v3,v4,v5} xu={u1,u2,u3,u4}
Construct a binary cross-product matrix M
Compute PageRank xv* and xu* representing the fitnessof countries and the quality of products
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Economic Complexity
Tripartite graph representing countries, capabilities andproducts
theory of
hidden
capabilities
Study convergence of algorithms for PageRank
xv(k+1) = xv(k) and xu(k+1) = xu(k)
23
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Distributed Algorithms for Web Aggregation
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Aggregation, PageRank and Complexity
Aggregation is a tool to break complexity
Applications:
o webo power grido biologyo social networkso journalso e-commerce
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT PageRank and Web Aggregation
View the web as a network of agents connected via links
Web aggregation
Approximate computation
of PageRank to reduce
computation cost and
communication at servers
Guaranteed bounds on the
approximation error
H. Ishii, R. Tempo, E.-W. Bai (2012)ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Graph Representation
1 4
2
6
3
5
0 1/2 0 0 0 0
1/ 2 0 1/3 0 0 0
0 1/2 0 1/3 0 0
1/ 2 0 1/3 0 0 1/2
0 0 0 1/3 0 1/2
0 0 1/3 1/3 1 0
A
T* 0.0614 0.0857 0.122 0.214 0.214 0.302x
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Web Aggregation
1 4
2
6
3
5
0 1/2 0 0 0 0
1/ 2 0 1/3 0 0 0
0 1/2 0 1/3 0 0
1/ 2 0 1/3 0 0 1/2
0 0 0 1/3 0 1/2
0 0 1/3 1/3 1 0
A
T* 0.0614 0.0857 0.122 0.214 0.214 0.302x U1
U2
U3
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Aggregation of Agents
1 4
2
6
3
5
U1
U2
U3
U3
U2
U1
24
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Web Aggregation
Most links are intra‐host ones
Two-step approach:
1. Global step: Compute total PageRank for each group
2. Local step: Distribute value among group members
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Web Aggregation
Singular perturbation
methods for PageRank
computation
Web pages having many external outlinks are consideredas single groups
number external outlinksδ
number outlinks
J.H. Chow, P.V. Kokotovic (1985)H. Ishii, R. Tempo, E.W. Bai (2012)
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Coordinate Change for PageRank
where is the aggregated PageRank (based on groupvalue), V1 and V2 are block diagonal matrices (afterreordering)
Objective: design a distributed randomized algorithm forcomputing approximately PageRank x* based on thecomputation of
* *x Vx*
1* *1*
22
Vxx x
Vx
*1x
*1x
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
PageRank equation
x* = (1- m) A x* + m/n 1
Internal: block diag stochastic
External: block diag norm bounded by 2
internalcommunications in the same group
external communications
inter-groups
PageRank Equation: Internal/External Communications
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
11
21 2
* *1 12
2
1* *2 2
(1 )0
A
A A
ux xA mm
nx x
New PageRank equation
where u = V1 1
is approximate PageRank
Approximate PageRank Equation
A A1 111A AI2 221 22
0(1 )
0
ux xA mm
nx xA A
Ax
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
A11 1
A1 (( 1) (1 ))
0A x
umx k m
nk
1. Group values: reduced order computation
2. Iteration
communications inter-groups
communications inter-groups
Aggregated PageRank Algorithm - 1
A A 1A A22 12 22 1( 1) ( ) ((1 )[ 1 )( ) ]x k x k m I- A A x km
A22A
A11 1 ( )A x k
A21 1 ( )A x k
communications
in the same group
25
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
3. Inverse state transformation
Theorem (convergence properties):
xA(k) xA for k in MSE sense provided that
xA (0) is a stochastic vector
1A A( ) ( )Vx k x k
Aggregated PageRank Algorithm - 2
1V
block diagonal matrix
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Approximation Error
Approximate PageRank
Theorem (approximation error):
Given ε > 0, take
then
A A1 111A AI2 221 22
0(1 )
0
ux xA mm
nx xA A
A 1 Ax V x
number external outlinks 4δ
number outlinks 4(1 )(1 )m
* A1|| - ||x x
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
Centrality Measures
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
How central are specific nodes or edges in a network?
PageRank and Network Centrality Measures
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Other Centrality Measures
Degree
Closeness
Betweenness
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Degree Centrality
Degree centrality: for each node count the number ofincoming links
26
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Closeness Centrality
Closeness centrality: a node is more central if it is closerto most of the other nodes
Defined as the “distance” from all the other nodes
2 => 1 dist = 1
3 => 1 dist = 2
4 => 1 dist = 3
5 => 1 dist = 5
6 => 1 dist = 4
total = 15ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Closeness Centrality
Closeness centrality: a node is more central if it is closerto most of the other nodes
Defined as the “distance” from all the other nodes
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Betweenness Centrality
Betweenness centrality ranks a node higher if it belongsto many paths between other nodes
1
# 1, 1B
# , 1
shortest paths i j passing through i j
shortest paths i j i j
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Betweenness Centrality
2 => 3 0 4 => 5 0 4 => 6 0 5 => 6 0
2 => 4 1/2 total = 1/2 + 1/3 = 5/6
2 => 5 1/3
2 => 6 0
3 => 4 0
3 => 5 0
3 => 6 0
1
# 1, 1B
# , 1
shortest paths i j passing through i j
shortest paths i j i j
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Betweenness Centrality
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Comparison of Centrality Measures
27
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
On-Going Work: Distributed Randomized Algorithms
Design distributed randomized algorithms for
centrality measures
Degree and closeness are “easy”
Betweenness computation is very hard for general
graphs
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
On-Going Work: Temporal Networks
Fixed number of nodes
Time-varying links
Need a new definition of PageRank
Design distributed randomized algorithms
Establish convergence properties
Goal: Handle web spam effectively
Avoid deliberate manipulation of search engines
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT
References
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Key References
H. Ishii and R. Tempo, “The PageRank Problem, Multi-Agent Consensus and Web Aggregation,,” IEEE CSM,2014
R. Tempo, G. Calafiore, F. Dabbene, “RandomizedAlgorithms for Analysis and Control of UncertainSystems, with Applications,” Second Edition,Springer-Verlag, London, 2013
… more references on my website…
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT PageRank (Other References)
O. Fercoq, M. Akian, M. Bouhtou, S. Gaubert, “PageRank,Markov Chains and Ergodic Control,” IEEE TAC, 2013
W.-X. Zhao, H.-F. Chen, H.T. Fang, “Almost-Sure Convergenceof the Randomized Scheme,” IEEE TAC, 2013
A.V. Nazin and B.T. Polyak. “Randomized Algorithm toDetermine the Eigenvector of a Stochastic Matrix,” Automationand Remote Control, 2011
ICT International Doctoral School, Trento @RT 2015
CNR-IEIIT Eigenfactor References
J.D. West, T.C. Bergstrom, C.T. Bergstrom, “The EigenfactorMetrics: A Network Approach to Assessing Scholarly Journals,”College of Research Libraries, 2010
C.T. Bergstrom, “Eigenfactor: Measuring the Value and Prestigeof Scholarly Journals,” C&RL News, 2007
L. Leydesdorff, “How are New Citation-Based Journal IndicatorsAdding to the Bibliometric Toolbox?,” Journal of the AmericanSociety for Information Science and Technology, 2008