Spectral properties of Google Matrix
(towards two-dimensional search engines)
Leonardo Ermann
CNEA-CONICET, Buenos Aires, Argentina
Dima Shepelyansky (Toulouse)
Klaus Frahm (Toulouse)
Alexei Chepelianskii (Cambridge)
November 5th 2013, “Workshop on Dynamic Networks 2013” Buenos Aires, Argentina
Motivations in complex networks
Outline
• Google matrix introduction
• PageRank and CheiRank
• Examples of 2-d rank
(Wikipedia, Univ., etc.)
• World Trade Network
• Other applications
(Ulam networks, FWL, communities)
• Conclusions
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
PageRank - Google Matrix
*Brin and Page (1998)
Spectral Indices
directed network
“Spectral properties of Google matrix”, L. Ermann
•
•
•
•
directed networks
easy to compute
incoming links
non-local properties
adjacency matrix
November 5th 2013, WDN13 Buenos Aires
PageRank - Google Matrix
*Brin and Page (1998)
weighted adjacency matrix and dangling nodes
directed network
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
PageRank - Google Matrix
PageRank
Google Matrix
directed network
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Google Matrix examples
L.E, Chepelianskii and Shepelyansky, Jour. Phys. A 45, 275101 (2012).
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
G*, CheiRank and spectrum
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
PageRank - CheiRank analysis
KP-P* correlator
K-K* plane
persona
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Global 2-d examples
Cambridge
(2006)
N∼2.105
Oxford
(2006)
N∼2.105
Linux Kernel
V2.4
N∼3.105
wikipedia
N∼2.106
“Spectral properties of Google matrix”, L. Ermann
10
November 5th 2013, WDN13 Buenos Aires
100
UCLA
Stanford
St Andrews
NYU
80
countries
TAMU
PSU
Oxford
universities
Virginia
MSU
Harvard
Manchester SU
Maryland
60
Queen
Cambridge
Minnesota
K*
WUSTL
Georgia
UM
Heidelberg
UNC
40
GT
Tufts
Caltech
Yale
Toronto
Rutgers
Cornell
20
Michigan
MIT
USC
Berkeley
Columbia
Georgetown
CMU
Brown
Northwestern
FSU
physicists
Sakharov
80
Gauss
Stokes
Bethe
Hamilton
Dirac
Boltzmann
Marconi
Planck
60
Curie
Euler
Lagrange
K*
40
Fermi
Archimedes
Maxwell
Rutherford
Neumann
Kelvin
Faraday Hooke
Bell
Galilei
Kepler
Alhazen
20
Edison
Newton
Avicenna
Franklin
Oppenheimer
40
Penrose
Noether
Wheeler
von BraunMay
Dyson
Gibbs
Eddington
Kuo
40 K L.60
“Spectral properties of 20
Google matrix”,
Ermann 80
100
Davis
Gross
Gabor Schawlow
KapitsaHertz
Franck
Compton
Kilby
BardeenCockcroft
Bragg
Chadwick
Prigogine
Lawrence
Landau
Von Laue
Shockley
Michelson
Rayleigh
Millikan
40
Appleton
Lee
Wilczek
Leggett
Yang
Gell-Mann
Chandrasekhar
Onnes
Thomson
Bohr
Sakharov
Chu
Bethe
Marconi
Dirac
Planck
Curie Wigner
Born
Pauli Weinberg
Fermi
Debye Raman
Salam
Heisenberg
Feynman
Einstein
20
Hodgkin
Ryle
Nobel
laureates
in physics
Lederman
Townes
Schwinger
20
Heisenberg
Biruni
Feynman
Einstein
Tesla
hawking
60
K*
Abdus Salam
Aristotle
Leibniz
Rabi
80
Weinberg
100
van del Walls
Purcell
de Broglie
Bragg
Chu
Wigner
Pauli Teller
80
Wieman
Lorentz
Born
60
Wien
Hahn
Huygens
K
Cornell
Tomonaga
Curie
Boyle
40
100
Meitner
Watt
Pitt
Florida
20
100
Durham
Smoot
Nambu
Blackett
Bothe
AlvarezBridgman
802013, WDN13
100
K 60
November
5th
Buenos Aires
Google Matrix of the WTN
United Nation Commodities Trade Network all countries of UN, from 1962 to 2011, all commodities or some specific products
Money Matrix
Mi,j = U $S(j ! i)
G
GP = P
M
G*
ExportRank
PageRank
ImportRank
CheiRank
G⇤ P ⇤ = P ⇤
(K̃, K̃ ⇤ )
(K, K⇤)
ImportRank, ExportRank
PageRank, CheiRank
L. Ermann and D.L. Shepelyansky, APPA,Vol. 120, A-158 (2011), arXiv:1103.5027,
“Spectral properties of Google matrix”, L. Ermann
http://www.quantware.ups-tlse.fr/QWLIB/tradecheirank
November 5th 2013, WDN13 Buenos Aires
G and G* matrices of the WTN (2008)
G
G*
crude petroleum
all commodities
M
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
PageRank, CheiRank and Spectrum
PageRank, CheiRank, ImportRank, ExportRank
↵ = 0.85
↵ = 0.5
Spectra
↵=1
Zipf law P~1/K
all commodities
crude petroleum
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
2d ranking “all commodities”
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
PageRank, CheiRank vs. ImportRank, ExportRank
countries are treated on equal democratic ground G-20 ~ 74%
K̃ ⇤ = 11 ! K ⇤ = 16
K̃ ⇤ = 13 ! K ⇤ > 20
K̃ ⇤ = 15 ! K ⇤ = 11
K̃ ⇤ = 19 ! K ⇤ = 12
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
2d ranking: crude petroleum
Russia K̃ ⇤ = 2 ! K ⇤ = 1
Iran K̃ ⇤ = 5 ! K ⇤ = 14
Kazakhstan K̃ ⇤ = 12 ! K ⇤ = 2
its trade network in this product is better and broader than the one of Saudi Arabia (1st)
its trade network is restricted to a small number of nearby countries.
is practically the only country which sells crude petroleum to the CheiRank leader in this product Russia.
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
2d ranking: barley and cars
Ukraine (K* 1st to 6th)
USA (K* 8th to 3rd)
“Spectral properties of Google matrix”, L. Ermann
France (K* 7th to 3rd)
Thailand (K* 19th to 10th)
November 5th 2013, WDN13 Buenos Aires
2d ranking evolution
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Other applications
of Google Matrix
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Ulam Networks
Chirikov standard map
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Fractal Weyl law
Standard map
L.E and Shepelyansky, Eur. Phys. J. B 75, 299 (2010)
Linux Kernel architecture network
L.E, Chepelianskii and Shepelyansky, Eur. Phys. J. B 79, 115 (2011)
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Wikipedia communities in G eigenstates
L.E, Frahm and Shepelyansky, Eur. Phys. J. B 86, 193 (2013)
“Spectral properties of Google matrix”, L. Ermann
November 5th 2013, WDN13 Buenos Aires
Conclusions
• Google Matrix analysis of different networks
(www: wikipedia, universities, softwares, etc.)
• CheiRank and 2-d rankings
• WTN example •Other applications:
Ulam Networks
Fractal Weyl law in maps and Linux Kernel
Communities in Wikipedia
•
•
•
Thank You