Detecting topological patterns in protein networks

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Lecture 4
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• Modules/communities
• in networks

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What is a module?

• Nodes in a given module (or community group or a
  functional unit) tend to connect with other nodes
  in the same module
• Biology proteins of the same function (e.g. DNA
  repair) or sub-cellular localization (e.g.
  nucleus)
• WWW websites on a common topic (e.g. physics)
  or organization (e.g. EPFL)
• Internet Autonomous systems/routers by
  geography (e.g. Switzerland) or domain (e.g.
  educational or military)

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Sometimes easy to discover
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Sometimes hard
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Hierarchical clustering

• calculating the similarity weight Wij for all
  pairs of vertices (e.g. of independent paths i
  ? j)
• start with all n vertices disconnected
• add edges between pairs one by one in order of
  decreasing weight
• result nested components, where one can take a
  slice at any level of the tree

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Girvan Newman (2002) betweenness clustering

• Betweenness of and edge i -- j is the of
  shortest paths going through this edge
• Algorithm
• compute the betweenness of all edges
• remove edge with the lowest betweenness
• recalculate betweenness
• Caveats
• Betweenness needs to be recalculated at each step
• very expensive all pairs shortest path O(N3)
• may need to repeat up to N times
• does not scale to more than a few hundred nodes,
  even with the fastest algorithms

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• Using random walks/diffusion to discover modules
  in networks

K. Eriksen, I. Simonsen, S. Maslov, K. Sneppen,
PRL 90, 148701(2003)
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Why diffusion?

• Any dynamical process would equilibrate faster on
  modules and slower between modules
• Thus its slow modes reveal modules
• Diffusion is the simplest dynamical process
  (people also use others like Ising/Potts model,
  etc.)

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Random walkers on a network

• Study the behavior of many VIRTUAL random walkers
  on a network
• At each time step each random walker steps on a
  randomly selected neighbor
• They equilibrate to a steady state ni ki
  (solid state physics ni const)
• Slow modes of equilibration to the steady state
  allow to detect modules in a network

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Matrix formalism
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Eigenvectors of the transfer matrix Tij
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Similarity transformation

• Matrix Tij is asymmetric ?
• Could in principle result to complex
  eigenvalues/eigenvectors
• Luckily, Sij1/(?Ki ?Kj) has the same eigenvalues
  and eigenvectors vi /?Ki
• Known as similarity transformation

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Density of states ?(?)

• filled circles real AS-network
• empty squares degree-preserving randomized
  version

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Participation ratio PR(?) ?i1/(v(?)i)4
250
200
150
Participation Ratio
100
50
0
-1
-0.5
0
0.5
1
l
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US Military
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2 0.9626 RU RU RU RU CA RU RU
?? ?? US US US US ?? (US
Department of Defence) 3 0.9561 ?? FR FR FR
?? FR ?? RU RU RU ?? ?? RU ??
4 0.9523 US ?? US ?? ?? ?? ?? (US Navy)
NZ NZ NZ NZ NZ NZ NZ 5. 0.9474
KR KR KR KR KR ?? KR UA UA UA
UA UA UA UA
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Hacked Ford AS
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• Using random walks/diffusion to rank information
  networks

e.g. Googles PageRank made it 160 billion
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Information networks

• 3x105 Phys Rev articles connected by 3x106
  citation links
• 1010 webpages in the world
• To find relevant information one needs to
  efficiently search and rank!!

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Ranking webpages

• Assign an importance factor Gi to every webpage
  
• Given a keyword (say jaguar) find all the pages
  that have it in their text and display them in
  the order of descending Gi.
• One solution still used in scientific publishing
  is GiKin(i) (the number of incoming links), but
  
• Too democratic It doesnt take into account the
  importance of nodes sending links
• Easy to trick and artificially boost the ranking
  (for the WWW)

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How Google works

• Googles recipe (circa 1998) is to simulate the
  behavior of many virtual random surfers
• PageRank Gi the number of virtual hits the
  page gets. It is also the steady state number
  of random surfers at a given page
• Popular pages send more surfers your way ?
  PageRank Kin is weighted by the popularity of a
  webpage sending each hyperlink
• Surfers get bored following links ? with
  probability ?0.15 a surfer jumps to a randomly
  selected page (not following any hyperlinks)

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• How communities in the WWW influence Google
  ranking

H. Xie, K.-K. Yan, SM, cond-mat/0409087 physics/05
10107 Physica A 373 (2007) 831836
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How do WWW communities influence their average Gi?

• Pages in a web-community preferentially link to
  each other. Examples
• Pages from the same organization (e.g. EPFL)
• Pages devoted to a common topic (e.g. Physics)
• Pages in the same geographical location (e.g
  Switzerland)
• Naïve argument communities trap random surfers
  to spend more time inside ? they should increase
  the average Google ranking of the community

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Test of a naïve argument
Community 1
log10(ltGgtc)
Community 2
of intra-community links

• Naïve argument is wrong it could go either way

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Eww
Ecc
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• Gc average Google rank of pages in the
  community Gw ? 1 in the outside world
• Ecw Gc/ltKoutgtc current from C to W
• It must be equal to Ewc Gw/ltKoutgtw
  current from W to C
• Thus Gc depends on the ratio between Ecw and
  Ewc the number of edges (hyperlinks) between
  the community and the world

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Balancing currents for nonzero ?

• Jcw(1- ?) Ecw Gc/ltKoutgtc ? Gc Nc current
  from C to W
• It must be equal to Jcw(1- ?) Ewc Gw/ltKoutgt
  ? Gw Nw(Nc/Nw) current from W to C

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What are the consequences?

• For very isolated communities (Ecw/E(r)cwlt? and
  Ewc/E(r)wclt?) one has Gc1. Their Google rank is
  decoupled from the outside world!
• Overall range ? ltGclt1/?

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WWW - the empirical data

• We have data for 10 US universities ( all UK
  and Australian Universities)
• Looked closely at UCLA and Long Island University
  (LIU)
• UCLA has different departments
• LIU has 4 campuses

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?0.15
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?0.001
Abnormally high PageRank
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Top PageRank LIU websites for ?0.001 dont make
sense

• 1 www.cwpost.liu.edu/cwis/cwp/edu/edleader/highe
  r_ed/ hear.html'
• 5 /higher_ed/ index.html
• 9 /higher_ed/courses.html

Strongly connected component
World
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What about citation networks?

• Better use ?0.5 instead of ?0.15 people dont
  click through papers as easily as through
  webpages
• Time arrow papers only cite older papers Small
  values of ? give older papers unfair advantage
• New algorithm CiteRank (as in PageRank). Random
  walkers start from recent papers exp(-t/?d)

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Summary

• Diffusion and modules (communities) in a network
  affect each other
• In the hardware part of the Internet
  (Autonomous systems or routers ) diffusion allows
  one to detect modules
• In the software part
• Diffusion-like process is used for ranking
  (Googles PageRank)
• WWW communities affect this ranking in a
  non-trivial way

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THE END
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Part 2 Opinion networks
"Extracting Hidden Information from Knowledge
Networks", S. Maslov, and Y-C. Zhang, Phys. Rev.
Lett. (2001). "Exploring an opinion network for
taste prediction an empirical study", M.
Blattner, Y.-C. Zhang, and S. Maslov, in
preparation.
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Predicting customers tastes from their opinions
on products

• Each of us has personal tastes
• Information about them is contained in our
  opinions on products
• Matchmaking opinions of customers with tastes
  similar to mine could be used to forecast my
  opinions on untested products
• Internet allows to do it on large scale (see
  amazon.com and many others)

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Opinion networks
Opinions of movie-goers on movies
WWW
Other webpages
1
opinion
1
Movies
1
Webapges
Customers
2
1
2
2
3
2
3
3
4
3
4
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Storing opinions
Matrix of opinions ?IJ
Network of opinions
Movies
1
2
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Customers
1
2
8
2
3
8
1
3
4
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Using correlations to reconstruct customers
tastes

• Similar opinions ? similar tastes
• Simplest model
• Movie-goers ? M-dimensional vector of tastes TI
• Movies ? M-dimensional vector of features FJ
• Opinions ? scalar product
• ?IJ TI?FJ

Movies
2
1
1
Customers
9
1
2
8
2
3
8
3
4
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Loop correlation

• Predictive power 1/M(L-1)/2
• One needs many loops to best reconstruct
  unknown opinions

L5 known opinions Predictive power of an
unknown opinion is 1/M2
An unknown opinion
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Main parameter density of edges

• The larger is the density of edges p the easier
  is the prediction
• At p1 ? 1/N (NNcostomersNmovies) macroscopic
  prediction becomes possible. Nodes are connected
  but vectors TI and FJ are not fixed ordinary
  percolation threshold
• At p2 ? 2M/N gt p1 all tastes and features (TI
  and FJ) can be uniquely reconstructed rigidity
  percolation threshold

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Real empirical data (EachMovie dataset) on
opinions of customers on movies 5-star ratings
of 1600 movies by 73000 users 1.6 million
opinions!
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Spectral properties of ?

• For MltN the matrix ?IJ has N-M zero eigenvalues
  and M positive ones ? R ? R.
• Using SVD one can diagonalize R U ? D ? V
  such that matrices V and U are orthogonal V ? V
  1, U ? U 1, and D is diagonal. Then ? U ?
  D2? U
• The amount of information contained in ?
  NM-M(M-1)/2 ltlt N(N-1)/2 - the of off-diagonal
  elements

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Recursive algorithm for the prediction of unknown
opinions

• Start with ?0 where all unknown elements are
  filled with lt?gt (zero in our case)
• Diagonalize and keep only M largest eigenvalues
  and eigenvectors
• In the resulting truncated matrix ?0 replace
  all known elements with their exact values and go
  to step 1

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Convergence of the algorithm

• Above p2 the algorithm exponentially converges
  to theexact values of unknown elements
• The rate of convergence scales as (p-p2)2

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Reality check sources of errors

• Customers are not rational! ?IJ rI?bJ
  ?IJ(idiosyncrasy)
• Opinions are delivered to the matchmaker through
  a narrow channel
• Binary channel SIJ sign(?IJ) 1 or 0 (liked or
  not)
• Experience rated on a scale 1 to 5 or 1 to 10 at
  best
• If number of edges K, and size N are large,
  while M is small these errors could be reduced

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How to determine M?

• In real systems M is not fixed there are always
  finer and finer details of tastes
• Given the number of known opinions K one should
  choose Meff ? K/(NreadersNbooks) so that systems
  are below the second transition p2 ? tastes
  should be determined hierarchically

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Avoid overfitting

• Divide known votes into training and test sets
• Select Meff so that to avoid overfitting !!!

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Knowledge networks in biology

• Interacting biomolecules key and lock principle
• Matrix of interactions (binding energies) ?IJ
  kI?lJ lI?kJ
• Matchmaker (bioinformatics researcher) tries to
  guess yet unknown interactions based on the
  pattern of known ones
• Many experiments measure SIJ ?(?IJ-?th)

k(1)
k(2)
l(2)
l(1)
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THE END