Network Essence: PageRank Completion and Centrality-Conforming Markov Chains



Jiří Matoušek (1963–2015) had many breakthrough contributions in mathematics and algorithm design. His milestone results are not only profound but also elegant. By going beyond the original objects—such as Euclidean spaces or linear programs—Jirka found the essence of the challenging mathematical/algorithmic problems as well as beautiful solutions that were natural to him, but were surprising discoveries to the field.

In this short exploration article, I will first share with readers my initial encounter with Jirka and discuss one of his fundamental geometric results from the early 1990s. In the age of social and information networks, I will then turn the discussion from geometric structures to network structures, attempting to take a humble step towards the holy grail of network science, that is to understand the network essence that underlies the observed sparse-and-multifaceted network data. I will discuss a simple result which summarizes some basic algebraic properties of personalized PageRank matrices. Unlike the traditional transitive closure of binary relations, the personalized PageRank matrices take “accumulated Markovian closure” of network data. Some of these algebraic properties are known in various contexts. But I hope featuring them together in a broader context will help to illustrate the desirable properties of this Markovian completion of networks, and motivate systematic developments of a network theory for understanding vast and ubiquitous multifaceted network data.


  1. 1.
    K.V. Aadithya, B. Ravindran, T. Michalak, N. Jennings, Efficient computation of the Shapley value for centrality in networks, in Internet and Network Economics. Volume 6484 of Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2010), pp. 1–13Google Scholar
  2. 2.
    E. Abbe, C. Sandon, Recovering communities in the general stochastic block model without knowing the parameters. CoRR, abs/1506.03729 (2015)Google Scholar
  3. 3.
    P.K. Agarwal, S. Har-Peled, K.R. Varadarajan, Geometric approximation via coresets, in Combinatorial and Computational Geometry, MSRI (Cambridge University Press, Cambridge, 2005), pp. 1–30MATHGoogle Scholar
  4. 4.
    E.M. Airoldi, T.B. Costa, S.H. Chan, Stochastic blockmodel approximation of a graphon: theory and consistent estimation, in 27th Annual Conference on Neural Information Processing Systems 2013 (2013), pp. 692–700Google Scholar
  5. 5.
    N. Alon, V. Asodi, C. Cantor, S. Kasif, J. Rachlin, Multi-node graphs: a framework for multiplexed biological assays. J. Comput. Biol. 13(10), 1659–1672 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Altman, M. Tennenholtz, An axiomatic approach to personalized ranking systems. J. ACM 57(4), 26:1–26:35 (2010)Google Scholar
  7. 7.
    N. Amenta, M. Bern, D. Eppstein, S.-H. Teng, Regression depth and center points. Discret. Comput. Geom. 23(3), 305–323 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    R. Andersen, C. Borgs, J. Chayes, J. Hopcraft, V.S. Mirrokni, S.-H. Teng, Local computation of PageRank contributions, in Proceedings of the 5th International Conference on Algorithms and Models for the Web-Graph, WAW’07 (Springer, 2007), pp. 150–165Google Scholar
  9. 9.
    R. Andersen, F. Chung, K. Lang, Using PageRank to locally partition a graph. Internet Math. 4(1), 1–128 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    K.J. Arrow, Social Choice and Individual Values, 2nd edn. (Wiley, New York, 1963)MATHGoogle Scholar
  11. 11.
    M.F. Balcan, C. Borgs, M. Braverman, J.T. Chayes, S.-H. Teng, Finding endogenously formed communities, in SODA (2013), pp. 767–783Google Scholar
  12. 12.
    J. Batson, D.A. Spielman, N. Srivastava, S.-H. Teng, Spectral sparsification of graphs: theory and algorithms. Commun. ACM 56(8), 87–94 (2013)CrossRefGoogle Scholar
  13. 13.
    A. Bavelas, Communication patterns in task oriented groups. J. Acoust. Soc. Am. 22(6), 725–730 (1950)CrossRefGoogle Scholar
  14. 14.
    P. Bonacich, Power and centrality: a family of measures. Am. J. Soc. 92(5), 1170–1182 (1987)CrossRefGoogle Scholar
  15. 15.
    P. Bonacich, Simultaneous group and individual centralities. Soc. Netw. 13(2), 155–168 (1991)CrossRefGoogle Scholar
  16. 16.
    S.P. Borgatti, Centrality and network flow. Soc. Netw. 27(1), 55–71 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S.P. Borgatti, M.G. Everett, A graph-theoretic perspective on centrality. Soc. Netw. 28(4), 466–484 (2006)CrossRefGoogle Scholar
  18. 18.
    C. Borgs, J. Chayes, L. Lovász, V.T. Sós, B. Szegedy, K. Vesztergombi, Graph limits and parameter testing, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC’06 (2006), pp. 261–270Google Scholar
  19. 19.
    C. Borgs, J.T. Chayes, A. Marple, S. Teng, An axiomatic approach to community detection, in Proceedings of the ACM Conference on Innovations in Theoretical Computer Science, ITCS’16 (2016), pp. 135–146Google Scholar
  20. 20.
    C. Borgs, J.T. Chayes, A.D. Smith, Private graphon estimation for sparse graphs, in Annual Conference on Neural Information Processing Systems (2015), pp. 1369–1377Google Scholar
  21. 21.
    S.J. Brams, M.A. Jones, D.M. Kilgour, Dynamic models of coalition formation: Fallback vs. build-up, in Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge, TARK’03 (2003), pp. 187–200Google Scholar
  22. 22.
    S. Brin, L. Page, The anatomy of a large-scale hypertextual Web search engine. Comput. Netw. 30(1–7), 107–117 (1998)Google Scholar
  23. 23.
    M. Caesar, J. Rexford, BGP routing policies in ISP networks. Netw. Mag. Global Internetwkg. 19(6), 5–11 (2005)Google Scholar
  24. 24.
    J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, ed by R.C. Gunning (Princeton University Press, Princeton, 1970), pp. 195–199Google Scholar
  25. 25.
    W. Chen, S.-H. Teng, Interplay between social influence and network centrality: a comparative study on Shapley centrality and single-node-influence centrality, in Proceedings of the 26th International Conference on World Wide Web, WWW, Perth (ACM, 2017), pp. 967–976Google Scholar
  26. 26.
    X. Chen, X. Deng, S.-H. Teng, Settling the complexity of computing two-player nash equilibria. J. ACM 56(3), 14:1–14:57 (2009)Google Scholar
  27. 27.
    D. Cheng, Y. Cheng, Y. Liu, R. Peng, S.-H. Teng, Efficient sampling for Gaussian graphical models via spectral sparsification, in Proceedings of the 28th Conference on Learning Theory, COLT’05 (2015)Google Scholar
  28. 28.
    F.R.K. Chung, Spectral Graph Theory (CBMS Regional Conference Series in Mathematics, No. 92) (American Mathematical Society, 1997)Google Scholar
  29. 29.
    K. Clarkson, D. Eppstein, G.L. Miller, C. Sturtivant, S.-H. Teng, Approximating center points with and without linear programming, in Proceedings of 9th ACM Symposium on Computational Geometry (1993), pp. 91–98Google Scholar
  30. 30.
    L. Danzer, J. Fonlupt, V. Klee, Helly’s theorem and its relatives. Proc. Symp. Pure Math. Am. Math. Soc. 7, 101–180 (1963)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    C. Daskalakis, P.W. Goldberg, C.H. Papadimitriou, The complexity of computing a nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    P. Domingos, M. Richardson, Mining the network value of customers, in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’01 (2001), pp. 57–66, CoRR, abs/cond-mat/0502230Google Scholar
  33. 33.
    L. Donetti, P.I. Hurtado, M.A. Munoz, Entangled networks, synchronization, and optimal network topology (2005)MATHGoogle Scholar
  34. 34.
    H. Edelsbrunner, Algorithms in Combinatorial Geometry (Springer, New York, 1987)CrossRefMATHGoogle Scholar
  35. 35.
    H. Eulau, The columbia studies of personal influence: social network analysis. Soc. Sci. Hist. 4(02), 207–228 (1980)Google Scholar
  36. 36.
    M.G. Everett, S.P. Borgatti, The centrality of groups and classes. J. Math. Soc. 23(3), 181–201 (1999)CrossRefMATHGoogle Scholar
  37. 37.
    K. Faust, Centrality in affiliation networks. Soc. Netw. 19(2), 157–191 (1997)CrossRefGoogle Scholar
  38. 38.
    D. Feldman, M. Langberg, A unified framework for approximating and clustering data, in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC’11 (2011), pp. 569–578Google Scholar
  39. 39.
    M. Fiedler, Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)MathSciNetMATHGoogle Scholar
  40. 40.
    M. Fiedler, A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory. Czechoslov. Math. J. 25(100), 619–633 (1975)MathSciNetMATHGoogle Scholar
  41. 41.
    L.C. Freeman, A set of measures of centrality based upon betweenness. Sociometry 40, 35–41 (1977)CrossRefGoogle Scholar
  42. 42.
    L.C. Freeman, Centrality in social networks: conceptual clarification. Soc. Netw. 1(3), 215–239 (1979)MathSciNetCrossRefGoogle Scholar
  43. 43.
    D. Gale, L.S. Shapley, College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    R. Ghosh, S.-H. Teng, K. Lerman, X. Yan, The interplay between dynamics and networks: centrality, communities, and Cheeger inequality, in Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’14 (2014), pp. 1406–1415Google Scholar
  45. 45.
    J.R. Gilbert, G.L. Miller, S.-H. Teng, Geometric mesh partitioning: implementation and experiments. SIAM J. Sci. Comput. 19(6), 2091–2110 (1998)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    D. Gusfield, R.W. Irving, The Stable Marriage Problem: Structure and Algorithms (MIT Press, Cambridge, 1989)MATHGoogle Scholar
  47. 47.
    Q. Han, K.S. Xu, E.M. Airoldi, Consistent estimation of dynamic and multi-layer block models. CoRR, abs/1410.8597 (2015)Google Scholar
  48. 48.
    S. Hanneke, E.P. Xing, Network completion and survey sampling, in AISTATS, ed. by D.A.V. Dyk, M. Welling. Volume 5 of JMLR Proceedings (2009), pp. 209–215Google Scholar
  49. 49.
    T. Haveliwala, Topic-sensitive Pagerank: a context-sensitive ranking algorithm for web search. Trans. Knowl. Data Eng. 15(4), 784–796 (2003)CrossRefGoogle Scholar
  50. 50.
    M. Hubert, P.J. Rousseeuw, The catline for deep regression. J. Multivar. Anal. 66, 270–296 (1998)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    L. Katz, A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953)CrossRefMATHGoogle Scholar
  52. 52.
    M.J. Kearns, M.L. Littman, S.P. Singh, Graphical models for game theory, in Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence, UAI’01 (2001), pp. 253–260Google Scholar
  53. 53.
    J.A. Kelner, Y.T. Lee, L. Orecchia, A. Sidford, An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations, in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’14 (2014), pp. 217–226Google Scholar
  54. 54.
    J.A. Kelner, L. Orecchia, A. Sidford, Z.A. Zhu, A simple, combinatorial algorithm for solving SDD systems in nearly-linear time, in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC’13 (2013), pp. 911–920Google Scholar
  55. 55.
    D. Kempe, J. Kleinberg, E. Tardos, Maximizing the spread of influence through a social network, in KDD’03 (ACM, 2003), pp. 137–146Google Scholar
  56. 56.
    M. Kim, J. Leskovec, The network completion problem: inferring missing nodes and edges in networks, in SDM (SIAM/Omnipress, 2011), pp. 47–58Google Scholar
  57. 57.
    M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter, Multilayer networks. CoRR, abs/1309.7233 (2014)Google Scholar
  58. 58.
    D. Koller, N. Friedman, Probabilistic Graphical Models: Principles and Techniques – Adaptive Computation and Machine Learning (The MIT Press, Cambridge, 2009)Google Scholar
  59. 59.
    I. Koutis, G. Miller, R. Peng, A nearly-mlogn time solver for SDD linear systems, in 2011 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2011), pp. 590–598Google Scholar
  60. 60.
    K. Lerman, S. Teng, X. Yan, Network composition from multi-layer data. CoRR, abs/1609.01641 (2016)Google Scholar
  61. 61.
    N. Linial, E. London, Y. Rabinovich, The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    L. Lovász, M. Simonovits, Random walks in a convex body and an improved volume algorithm. RSA: Random Struct. Algorithms 4, 359–412 (1993)MathSciNetMATHGoogle Scholar
  63. 63.
    F. Masrour, I. Barjesteh, R. Forsati, A.-H. Esfahanian, H. Radha, Network completion with node similarity: a matrix completion approach with provable guarantees, in Proceedings of the IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM’15 (ACM, 2015), pp. 302–307Google Scholar
  64. 64.
    J. Matoušek, Approximations and optimal geometric divide-and-conquer, in Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, STOC’91 (1991), pp. 505–511Google Scholar
  65. 65.
    J. Matoušek, M. Sharir, E. Welzl, A subexponential bound for linear programming, in Proceedings of the Eighth Annual Symposium on Computational Geometry, SCG’92 (1992), pp. 1–8Google Scholar
  66. 66.
    T.P. Michalak, K.V. Aadithya, P.L. Szczepanski, B. Ravindran, N.R. Jennings, Efficient computation of the Shapley value for game-theoretic network centrality. J. Artif. Int. Res. 46(1), 607–650 (2013)MathSciNetMATHGoogle Scholar
  67. 67.
    G.L. Miller, S.-H. Teng, W. Thurston, S.A. Vavasis, Separators for sphere-packings and nearest neighbor graphs. J. ACM 44(1), 1–29 (1997)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    G.L. Miller, S.-H. Teng, W. Thurston, S.A. Vavasis, Geometric separators for finite-element meshes. SIAM J. Sci. Comput. 19(2), 364–386 (1998)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    J. Nash, Equilibrium points in n-person games. Proc. Natl. Acad. USA 36(1), 48–49 (1950)MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    J. Nash, Noncooperative games. Ann. Math. 54, 289–295 (1951)CrossRefGoogle Scholar
  71. 71.
    M. Newman, Networks: An Introduction (Oxford University Press, Inc., New York, 2010)CrossRefMATHGoogle Scholar
  72. 72.
    L. Page, S. Brin, R. Motwani, T. Winograd., The Pagerank citation ranking: bringing order to the Web, in Proceedings of the 7th International World Wide Web Conference (1998), pp. 161–172Google Scholar
  73. 73.
    S. Paul, Y. Chen, Community detection in multi-relational data with restricted multi-layer stochastic blockmodel. CoRR, abs/1506.02699v2 (2016)Google Scholar
  74. 74.
    L. Peel, D.B. Larremore, A. Clauset, The ground truth about metadata and community detection in networks. CoRR, abs/1608.05878, 2016Google Scholar
  75. 75.
    R. Peng, Approximate undirected maximum flows in O(mpolylog(n)) time, in Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’16 (2016), pp. 1862–1867Google Scholar
  76. 76.
    M. Piraveenan, M. Prokopenko, L. Hossain, Percolation centrality: quantifying graph-theoretic impact of nodes during percolation in networks. PLoS ONE 8(1) (2013)Google Scholar
  77. 77.
    Y. Rekhter, T. Li, A Border Gateway Protocol 4. IETF RFC 1771 (1995)Google Scholar
  78. 78.
    M. Richardson, P. Domingos, Mining knowledge-sharing sites for viral marketing, in Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’02 (2002), pp. 61–70Google Scholar
  79. 79.
    A.E. Roth, The evolution of the labor market for medical interns and residents: A case study in game theory. J. Politi. Econ. 92, 991–1016 (1984)Google Scholar
  80. 80.
    G. Sabidussi, The centrality index of a graph. Psychometirka 31, 581–606 (1996)MathSciNetCrossRefMATHGoogle Scholar
  81. 81.
    L.S. Shapley, A value for n-person games, in Contributions to the Theory of Games II, ed. by H. Kuhn, A.W. Tucker (Princeton University Press, Princeton, 1953), pp. 307–317Google Scholar
  82. 82.
    L.S. Shapley, Cores of convex games. Int. J. Game Theory 1(1), 11–26 (1971)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    J. Sherman, Nearly maximum flows in nearly linear time, in Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS’13 (2013), pp. 263–269Google Scholar
  84. 84.
    D.A. Spielman, S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes. Linear Algebra Appl. 421(2–3), 284–305 (2007)MathSciNetCrossRefMATHGoogle Scholar
  85. 85.
    D.A. Spielman, S.-H. Teng, Spectral sparsification of graphs. SIAM J. Comput. 40(4), 981–1025 (2011)MathSciNetCrossRefMATHGoogle Scholar
  86. 86.
    D.A. Spielman, S.-H. Teng, A local clustering algorithm for massive graphs and its application to nearly linear time graph partitioning. SIAM J. Comput. 42(1), 1–26 (2013)MathSciNetCrossRefMATHGoogle Scholar
  87. 87.
    D.A. Spielman, S.-H. Teng, Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM J. Matrix Anal. Appl. 35(3), 835–885 (2014)MathSciNetCrossRefMATHGoogle Scholar
  88. 88.
    S.-H. Teng, Points, Spheres, and Separators: A Unified Geometric Approach to Graph Partitioning. PhD thesis, Advisor: Gary Miller, Carnegie Mellon University, Pittsburgh, 1991Google Scholar
  89. 89.
    S.-H. Teng, Scalable algorithms for data and network analysis. Found. Trends Theor. Comput. Sci. 12(1–2), 1–261 (2016)MathSciNetCrossRefMATHGoogle Scholar
  90. 90.
    H. Tverberg, A generalization of Radon’s theorem. J. Lond. Math Soc. 41, 123–128 (1966)MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    V.N. Vapnik, The Nature of Statistical Learning Theory (Springer, New York, 1995)CrossRefMATHGoogle Scholar
  92. 92.
    V.N. Vapnik, A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971)CrossRefMATHGoogle Scholar
  93. 93.
    H.P. Young, An axiomatization of Borda’s rule. J. Econ. Theory 9(1), 43–52 (1974)MathSciNetCrossRefGoogle Scholar
  94. 94.
    A.Y. Zhang, H.H. Zhou, Minimax rates of community detection in stochastic block models. CoRR, abs/1507.05313 (2015)Google Scholar

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© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Computer Science and Mathematics, USCLos AngelesUSA

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