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Lecture 4

Lecture 4. Modules/communities in networks. 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)

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Lecture 4

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  1. Lecture 4

  2. Modules/communities in networks

  3. 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)

  4. Sometimes easy to discover

  5. Sometimes hard

  6. 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

  7. 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

  8. Using random walks/diffusion to discover modules in networks K. Eriksen, I. Simonsen, S. Maslov, K. Sneppen, PRL 90, 148701(2003)

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

  10. 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

  11. Matrix formalism

  12. Eigenvectors of the transfer matrix Tij

  13. Similarity transformation • Matrix Tij is asymmetric • Could in principle result to complex eigenvalues/eigenvectors • Luckily, Sij=1/(Ki Kj) has the same eigenvalues and eigenvectors vi /Ki • Known as similarity transformation

  14. Density of states () • filled circles –real AS-network • empty squares – degree-preserving randomized version

  15. Participation ratio: PR()=i1/(v()i)4 250 200 150 Participation Ratio 100 50 0 -1 -0.5 0 0.5 1 l

  16. Russia US Military

  17. 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

  18. Hacked Ford AS

  19. Using random walks/diffusion to rank information networks e.g. Google’s PageRank made it 160 billion $

  20. 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!!

  21. 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 Gi=Kin(i)(the number of incoming links), but: • Too democratic: It doesn’t take into account the importance of nodes sending links • Easy to trick and artificially boost the ranking (for the WWW)

  22. How Google works • Google’s 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 ~ Kinis weighted bythe 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)

  23. How communities in the WWW influence Google ranking H. Xie, K.-K. Yan, SM, cond-mat/0409087 physics/0510107 Physica A 373 (2007) 831–836

  24. 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

  25. Test of a naïve argument Naïve argument is wrong: it could go either way Community #1 Community #2 log10(<G>c) # of intra-community links

  26. Eww Ecc

  27. Gc – average Google rank of pages in the community; Gw  1 – in the outside world • Ecw Gc/<Kout>c– current from C to W • It must be equal to: Ewc Gw/<Kout>w– current from W to C • Thus Gcdepends on the ratio between Ecw and Ewc– the number of edges (hyperlinks) between the community and the world

  28. Balancing currents for nonzero  • Jcw=(1- ) Ecw Gc/<Kout>c + Gc Nc – current from C to W • It must be equal to: Jcw=(1- )Ewc Gw/<Kout> + Gw Nw(Nc/Nw)– current from W to C

  29. What are the consequences? • For very isolated communities(Ecw/E(r)cw< and Ewc/E(r)wc<) one has Gc=1. Their Google rank is decoupled from the outside world! • Overall range: <Gc<1/

  30. 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

  31. =0.15

  32. =0.001 Abnormally high PageRank

  33. Top PageRank LIU websites for =0.001 don’t make sense • #1 www.cwpost.liu.edu/cwis/cwp/edu/edleader/higher_ed/ hear.html' • #5 …/higher_ed/ index.html • #9 …/higher_ed/courses.html Strongly connected component World

  34. What about citation networks? • Better use =0.5 instead of =0.15: people don’t 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)

  35. 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 (Google’s PageRank) • WWW communities affect this ranking in a non-trivial way

  36. THE END

  37. 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.

  38. 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)

  39. 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

  40. Storing opinions Matrix of opinions IJ Network of opinions Movies 1 2 9 Customers 1 2 8 2 3 8 3 1 4

  41. Using correlations to reconstruct customer’s 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 1 9 2 8 2 3 8 3 4

  42. Loop correlation • Predictive power 1/M(L-1)/2 • One needs many loops to best reconstruct unknown opinions L=5 known opinions: Predictive power of an unknown opinion is 1/M2 An unknown opinion

  43. Main parameter: density of edges • The larger is the density of edgesp the easier is the prediction • At p1 1/N (N=Ncostomers+Nmovies) macroscopic prediction becomes possible. Nodes are connected but vectors TI andFJ are not fixed: ordinary percolation threshold • At p2 2M/N> p1 all tastes and features (TI andFJ) can be uniquely reconstructed: rigidity percolation threshold

  44. Real empirical data (EachMovie dataset) on opinions of customers on movies: 5-star ratings of 1600 movies by 73000 users 1.6 million opinions!

  45. Spectral properties of  • For M<N the matrix IJhas N-M zero eigenvalues and M positive ones:  = R  R+. • Using SVD one can “diagonalize” R = U  D  V+such that matrices VandU 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 << N(N-1)/2 - the # of off-diagonal elements

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