Document

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

## File

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

## Cite As

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

## References

1. Réka Albert, Hawoong Jeong, and Albert-László Barabási. Internet: Diameter of the world-wide web. Nature, 401(6749):130, 1999.
2. Sanjeev Arora and Boaz Barak. Computational complexity: A modern approach. Cambridge University Press, 2009.
3. Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999.
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.
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.
6. Andrej Bogdanov and Luca Trevisan. Average-Case Complexity. Foundations and Trends in Theoretical Computer Science, 2(1):1-106, 2006.
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.
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.
9. Béla Bollobás. Random Graphs. Cambridge University Press, 2nd edition, 2001.
10. Anna D. Broido and Aaron Clauset. Scale-free networks are rare. Nature communications, 10(1):1017, 2019.
11. Elisabetta Candellero and Nikolaos Fountoulakis. Clustering and the hyperbolic geometry of complex networks. Internet Mathematics, 12(1-2):2-53, 2016.
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.
13. Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree sequences. Annals of Combinatorics, 6(2):125-145, 2002.
14. Fan Chung and Linyuan Lu. Complex graphs and networks, volume 107. American Math. Soc., 2006.
15. Aaron Clauset, Cosma Rohilla Shalizi, and Mark E. J. Newman. Power-Law Distributions in Empirical Data. SIAM Review, 51(4):661-703, 2009.
16. Bruno Courcelle. The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990.
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.
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.
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.
24. Sander Dommers, Remco van der Hofstad, and Gerard Hooghiemstra. Diameters in preferential attachment models. Journal of Statistical Physics, 139(1):72-107, 2010.
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.
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.
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.
31. P. Erdős and A. Rényi. On random graphs. Publicationes Mathematicae, 6:290-297, 1959.
32. Ronald Fagin. Probabilities on finite models 1. The Journal of Symbolic Logic, 41(1):50-58, 1976.
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.
34. Jörg Flum, Markus Frick, and Martin Grohe. Query Evaluation via Tree-Decompositions. Journal of the ACM (JACM), 49(6):716-752, 2002.
35. Jörg Flum and Martin Grohe. Fixed-Parameter Tractability, Definability, and Model-Checking. SIAM Journal on Computing, 31(1):113-145, 2001.
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.
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.
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.
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.
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.
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.
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.
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.
45. Martin Grohe. Logic, graphs, and algorithms. Logic and Automata, 2:357-422, 2008.
46. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. J. ACM, 64(3), 2017.
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.
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.
50. Jon Kleinberg. The Small-World Phenomenon: An Algorithmic Perspective. In Proceedings of the 32nd Symposium on Theory of Computing, pages 163-170, 2000.
51. Jon M. Kleinberg. Navigation in a small world. Nature, 406(6798):845-845, 2000.
52. Stephan Kreutzer. Algorithmic meta-theorems. In International Workshop on Parameterized and Exact Computation, pages 10-12. Springer, 2008.
53. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010.
54. Leonid A. Levin. Average case complete problems. SIAM Journal on Computing, 15(1):285-286, 1986.
55. Johann A. Makowsky. Algorithmic uses of the feferman-vaught theorem. Annals of Pure and Applied Logic, 126(1-3):159-213, 2004.
56. Stanley Milgram. The small world problem. Psychology Today, 2(1):60-67, 1967.
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.
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.
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.
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.
61. Paul D. Seymour N. Robertson. Graph minors XVI. Excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89:43-76, 2003.
62. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity. Springer, 2012.
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.
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.
65. Nataša Pržulj. Biological network comparison using graphlet degree distribution. Bioinformatics, 23(2):e177-e183, 2007.
66. Katarzyna Rybarczyk. Diameter, connectivity, and phase transition of the uniform random intersection graph. Discrete Mathematics, 311(17):1998-2019, 2011.
67. Satu Elisa Schaeffer. Graph clustering. Computer Science Review, 1(1):27-64, 2007.
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.
70. Joel Spencer. The strange logic of random graphs, volume 22. Springer Science & Business Media, 2013.
71. Larry J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1-22, 1976.
72. Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’networks. nature, 393(6684):440, 1998.
73. Konstantin Zuev, Marián Boguná, Ginestra Bianconi, and Dmitri Krioukov. Emergence of soft communities from geometric preferential attachment. Scientific reports, 5:9421, 2015.