First-Order Model-Checking in Random Graphs and Complex Networks

Authors Jan Dreier , Philipp Kuinke , Peter Rossmanith



PDF
Thumbnail PDF

File

LIPIcs.ESA.2020.40.pdf
  • Filesize: 0.55 MB
  • 23 pages

Document Identifiers

Author Details

Jan Dreier
  • Department of Computer Science, RWTH Aachen University, Germany
Philipp Kuinke
  • Department of Computer Science, RWTH Aachen University, Germany
Peter Rossmanith
  • Department of Computer Science, RWTH Aachen University, Germany

Cite AsGet BibTex

Jan Dreier, Philipp Kuinke, and Peter Rossmanith. First-Order Model-Checking in Random Graphs and Complex Networks. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 40:1-40:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.40

Abstract

Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central role in today’s society. Algorithmic meta-theorems show that many problems can be solved efficiently. Since logic is a powerful tool to model problems, it has been used to obtain very general meta-theorems. In this work, we consider all problems definable in first-order logic and analyze which properties of complex networks allow them to be solved efficiently. The mathematical tool to describe complex networks are random graph models. We define a property of random graph models called α-power-law-boundedness. Roughly speaking, a random graph is α-power-law-bounded if it does not admit strong clustering and its degree sequence is bounded by a power-law distribution with exponent at least α (i.e. the fraction of vertices with degree k is roughly O(k^{-α})). We solve the first-order model-checking problem (parameterized by the length of the formula) in almost linear FPT time on random graph models satisfying this property with α ≥ 3. This means in particular that one can solve every problem expressible in first-order logic in almost linear expected time on these random graph models. This includes for example preferential attachment graphs, Chung-Lu graphs, configuration graphs, and sparse Erdős-Rényi graphs. Our results match known hardness results and generalize previous tractability results on this topic.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • random graphs
  • average case analysis
  • first-order model-checking

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Réka Albert, Hawoong Jeong, and Albert-László Barabási. Internet: Diameter of the world-wide web. Nature, 401(6749):130, 1999. Google Scholar
  2. Sanjeev Arora and Boaz Barak. Computational complexity: A modern approach. Cambridge University Press, 2009. Google Scholar
  3. Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999. Google Scholar
  4. Thomas Bläsius, Tobias Friedrich, and Anton Krohmer. Hyperbolic random graphs: Separators and treewidth. In 24th Annual European Symposium on Algorithms (ESA 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  5. Hans L Bodlaender, Fedor V Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M Thilikos. (Meta) kernelization. Journal of the ACM (JACM), 63(5):44, 2016. Google Scholar
  6. Andrej Bogdanov and Luca Trevisan. Average-Case Complexity. Foundations and Trends in Theoretical Computer Science, 2(1):1-106, 2006. Google Scholar
  7. Béla Bollobás, Oliver Riordan, Joel Spencer, and Gábor Tusnády. The degree sequence of a scale-free random graph process. Random Structures & Algorithms, 18(3):279-290, May 2001. Google Scholar
  8. Béla Bollobás and Oliver M Riordan. Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet, pages 1-34, 2003. Google Scholar
  9. Béla Bollobás. Random Graphs. Cambridge University Press, 2nd edition, 2001. Google Scholar
  10. Anna D. Broido and Aaron Clauset. Scale-free networks are rare. Nature communications, 10(1):1017, 2019. Google Scholar
  11. Elisabetta Candellero and Nikolaos Fountoulakis. Clustering and the hyperbolic geometry of complex networks. Internet Mathematics, 12(1-2):2-53, 2016. Google Scholar
  12. Fan Chung and Linyuan Lu. The average distances in random graphs with given expected degrees. Proc. of the National Academy of Sciences, 99(25):15879-15882, 2002. Google Scholar
  13. Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree sequences. Annals of Combinatorics, 6(2):125-145, 2002. Google Scholar
  14. Fan Chung and Linyuan Lu. Complex graphs and networks, volume 107. American Math. Soc., 2006. Google Scholar
  15. Aaron Clauset, Cosma Rohilla Shalizi, and Mark E. J. Newman. Power-Law Distributions in Empirical Data. SIAM Review, 51(4):661-703, 2009. Google Scholar
  16. Bruno Courcelle. The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. Google Scholar
  17. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  18. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  19. Anuj Dawar, Martin Grohe, and Stephan Kreutzer. Locally Excluding a Minor. In Proceedings of the 22nd Symposium on Logic in Computer Science, pages 270-279, 2007. Google Scholar
  20. Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, November 2005. URL: https://doi.org/10.1145/1101821.1101823.
  21. Erik D. Demaine and M. Hajiaghayi. The bidimensionality theory and its algorithmic applications. Comput. J., 51(3):292-302, 2008. Google Scholar
  22. Erik D. Demaine, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar, and Blair D. Sullivan. Structural sparsity of complex networks: Bounded expansion in random models and real-world graphs. J. Comput. Syst. Sci., 105:199-241, 2019. URL: https://doi.org/10.1016/j.jcss.2019.05.004.
  23. R. Diestel. Graph Theory. Springer, Heidelberg, 2010. Google Scholar
  24. Sander Dommers, Remco van der Hofstad, and Gerard Hooghiemstra. Diameters in preferential attachment models. Journal of Statistical Physics, 139(1):72-107, 2010. Google Scholar
  25. Rod G. Downey, Michael R. Fellows, and Udayan Taylor. The Parameterized Complexity of Relational Database Queries and an Improved Characterization of W[1]. DMTCS, 96:194-213, 1996. Google Scholar
  26. Jan Dreier, Philipp Kuinke, and Peter Rossmanith. First-order model-checking in random graphs and complex networks, 2020. URL: http://arxiv.org/abs/2006.14488.
  27. Jan Dreier, Philipp Kuinke, and Peter Rossmanith. Maximum shallow clique minors in preferential attachment graphs have polylogarithmic size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), volume 176 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  28. Jan Dreier and Peter Rossmanith. Hardness of FO model-checking on random graphs. In 14th International Symposium on Parameterized and Exact Computation, IPEC 2019, September 11-13, 2019, Munich, Germany, volume 148 of LIPIcs, pages 11:1-11:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.IPEC.2019.11.
  29. Jan Dreier and Peter Rossmanith. Motif counting in preferential attachment graphs. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2019, December 11-13, 2019, Bombay, India, volume 150 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2019.13.
  30. Zdenek Dvořak, Daniel Král, and Robin Thomas. Deciding First-Order Properties for Sparse Graphs. In Proceedings of the 51st Conference on Foundations of Computer Science, pages 133-142, 2010. Google Scholar
  31. P. Erdős and A. Rényi. On random graphs. Publicationes Mathematicae, 6:290-297, 1959. Google Scholar
  32. Ronald Fagin. Probabilities on finite models 1. The Journal of Symbolic Logic, 41(1):50-58, 1976. Google Scholar
  33. Matthew Farrell, Timothy D Goodrich, Nathan Lemons, Felix Reidl, Fernando Sánchez Villaamil, and Blair D Sullivan. Hyperbolicity, degeneracy, and expansion of random intersection graphs. In International Workshop on Algorithms and Models for the Web-Graph, pages 29-41. Springer, 2015. Google Scholar
  34. Jörg Flum, Markus Frick, and Martin Grohe. Query Evaluation via Tree-Decompositions. Journal of the ACM (JACM), 49(6):716-752, 2002. Google Scholar
  35. Jörg Flum and Martin Grohe. Fixed-Parameter Tractability, Definability, and Model-Checking. SIAM Journal on Computing, 31(1):113-145, 2001. Google Scholar
  36. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M Thilikos. Bidimensionality and kernels. In Proc. of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 503-510, 2010. Google Scholar
  37. Markus Frick and Martin Grohe. Deciding first-order properties of locally tree-decomposable structures. Journal of the ACM (JACM), 48(6):1184-1206, 2001. Google Scholar
  38. Markus Frick and Martin Grohe. The complexity of first-order and monadic second-order logic revisited. Annals of pure and applied logic, 130(1-3):3-31, 2004. Google Scholar
  39. Haim Gaifman. On local and non-local properties. In Studies in Logic and the Foundations of Mathematics, volume 107, pages 105-135. Elsevier, 1982. Google Scholar
  40. Jakub Gajarský, Petr Hliněnỳ, Jan Obdrzálek, Daniel Lokshtanov, and M. S. Ramanujan. A new perspective on FO model checking of dense graph classes. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016, pages 176-184, 2016. URL: https://doi.org/10.1145/2933575.2935314.
  41. Yong Gao. Treewidth of Erdős-Rényi random graphs, random intersection graphs, and scale-free random graphs. Discrete Applied Mathematics, 160(4-5):566-578, 2012. Google Scholar
  42. Yu V Glebskii, DI Kogan, MI Liogon'kii, and VA Talanov. Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics and Systems Analysis, 5(2):142-154, 1969. Google Scholar
  43. Anna Goldenberg, Alice X. Zheng, Stephen E. Fienberg, Edoardo M. Airoldi, et al. A survey of statistical network models. Foundations and Trends in Machine Learning, 2(2):129-233, 2010. Google Scholar
  44. Martin Grohe. Generalized model-checking problems for first-order logic. In Annual Symposium on Theoretical Aspects of Computer Science, pages 12-26. Springer, 2001. Google Scholar
  45. Martin Grohe. Logic, graphs, and algorithms. Logic and Automata, 2:357-422, 2008. Google Scholar
  46. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. J. ACM, 64(3), 2017. Google Scholar
  47. Michał Karoński, Edward R. Scheinerman, and Karen B. Singer-Cohen. On random intersection graphs: The subgraph problem. Combinatorics, Probability and Computing, 8(1-2):131-159, 1999. Google Scholar
  48. Carol Karp. The first order properties of products of algebraic systems. fundamenta mathematicae. Journal of Symbolic Logic, 32(2):276–276, 1967. URL: https://doi.org/10.2307/2271704.
  49. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. ACM Transactions on Algorithms (TALG), 12(2):21, 2016. Google Scholar
  50. Jon Kleinberg. The Small-World Phenomenon: An Algorithmic Perspective. In Proceedings of the 32nd Symposium on Theory of Computing, pages 163-170, 2000. Google Scholar
  51. Jon M. Kleinberg. Navigation in a small world. Nature, 406(6798):845-845, 2000. Google Scholar
  52. Stephan Kreutzer. Algorithmic meta-theorems. In International Workshop on Parameterized and Exact Computation, pages 10-12. Springer, 2008. Google Scholar
  53. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010. Google Scholar
  54. Leonid A. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285-286, 1986. Google Scholar
  55. Johann A. Makowsky. Algorithmic uses of the feferman-vaught theorem. Annals of Pure and Applied Logic, 126(1-3):159-213, 2004. Google Scholar
  56. Stanley Milgram. The small world problem. Psychology Today, 2(1):60-67, 1967. Google Scholar
  57. Ron Milo, Shai Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri Chklovskii, and Uri Alon. Network motifs: simple building blocks of complex networks. Science, 298(5594):824-827, 2002. Google Scholar
  58. Alan Mislove, Massimiliano Marcon, Krishna P Gummadi, Peter Druschel, and Bobby Bhattacharjee. Measurement and analysis of online social networks. In Proc. of the 7th ACM SIGCOMM Conference on Internet Measurement, pages 29-42. ACM, 2007. Google Scholar
  59. M. Molloy and B. A. Reed. The size of the giant component of a random graph with a given degree sequence. Combin., Probab. Comput., 7(3):295-305, 1998. Google Scholar
  60. Michael Molloy and Bruce Reed. A critical point for random graphs with a given degree sequence. Random Structures & Algorithms, 6(2-3):161-180, 1995. Google Scholar
  61. Paul D. Seymour N. Robertson. Graph minors XVI. Excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89:43-76, 2003. Google Scholar
  62. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity. Springer, 2012. Google Scholar
  63. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion I. Decompositions. European Journal of Combinatorics, 29(3):760-776, 2008. Google Scholar
  64. Derek de Solla Price. A general theory of bibliometric and other cumulative advantage processes. Journal of the American society for Information science, 27(5):292-306, 1976. Google Scholar
  65. Nataša Pržulj. Biological network comparison using graphlet degree distribution. Bioinformatics, 23(2):e177-e183, 2007. Google Scholar
  66. Katarzyna Rybarczyk. Diameter, connectivity, and phase transition of the uniform random intersection graph. Discrete Mathematics, 311(17):1998-2019, 2011. Google Scholar
  67. Satu Elisa Schaeffer. Graph clustering. Computer Science Review, 1(1):27-64, 2007. Google Scholar
  68. Nicole Schweikardt, Luc Segoufin, and Alexandre Vigny. Enumeration for FO queries over nowhere dense graphs. In Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, Houston, TX, USA, June 10-15, 2018, pages 151-163. ACM, 2018. URL: https://doi.org/10.1145/3196959.3196971.
  69. Detlef Seese. Linear time computable problems and first-order descriptions. Math. Struct. in Comp. Science, 6:505-526, 1996. Google Scholar
  70. Joel Spencer. The strange logic of random graphs, volume 22. Springer Science & Business Media, 2013. Google Scholar
  71. Larry J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1-22, 1976. Google Scholar
  72. Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’networks. nature, 393(6684):440, 1998. Google Scholar
  73. Konstantin Zuev, Marián Boguná, Ginestra Bianconi, and Dmitri Krioukov. Emergence of soft communities from geometric preferential attachment. Scientific reports, 5:9421, 2015. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail