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Random Graph Modeling: A survey of the concepts

Published 21 Mar 2024 in cs.SI | (2403.14415v1)

Abstract: Random graph (RG) models play a central role in the complex networks analysis. They help to understand, control, and predict phenomena occurring, for instance, in social networks, biological networks, the Internet, etc. Despite a large number of RG models presented in the literature, there are few concepts underlying them. Instead of trying to classify a wide variety of very dispersed models, we capture and describe concepts they exploit considering preferential attachment, copying principle, hyperbolic geometry, recursively defined structure, edge switching, Monte Carlo sampling, etc. We analyze RG models, extract their basic principles, and build a taxonomy of concepts they are based on. We also discuss how these concepts are combined in RG models and how they work in typical applications like benchmarks, null models, and data anonymization.

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References (157)
  1. Abbe, E. Community detection and stochastic block models: recent developments. The Journal of Machine Learning Research 18, 1 (2017), 6446–6531.
  2. A random graph model for massive graphs. In Proceedings of the thirty-second annual ACM symposium on Theory of computing (2000), Acm, pp. 171–180.
  3. Rtg: a recursive realistic graph generator using random typing. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (2009), Springer, pp. 13–28.
  4. Statistical mechanics of complex networks. Reviews of modern physics 74, 1 (2002), 47.
  5. Large scale structure and dynamics of complex networks: from information technology to finance and natural science, vol. 2. World Scientific, 2007.
  6. Which models are used in social simulation to generate social networks? a review of 17 years of publications in jasss. In Winter Simulation Conference (WSC), 2015 (2015), IEEE, pp. 4021–4032.
  7. An, W. Fitting ergms on big networks. Social science research 59 (2016), 107–119.
  8. Synchronization reveals topological scales in complex networks. Physical review letters 96, 11 (2006), 114102.
  9. Radio cover time in hyper-graphs. In Proceedings of the 6th international workshop on foundations of mobile computing (2010), ACM, pp. 3–12.
  10. Wherefore art thou r3579x?: anonymized social networks, hidden patterns, and structural steganography. In Proceedings of the 16th international conference on World Wide Web (2007), ACM, pp. 181–190.
  11. Exploring biological network structure with clustered random networks. BMC bioinformatics 10, 1 (2009), 405.
  12. Emergence of scaling in random networks. science 286, 5439 (1999), 509–512.
  13. Deterministic scale-free networks. Physica A: Statistical Mechanics and its Applications 299, 3-4 (2001), 559–564.
  14. Dynamical Processes on Complex Networks. Cambridge University Press, 2008.
  15. Characterization and modeling of weighted networks. Physica a: Statistical mechanics and its applications 346, 1-2 (2005), 34–43.
  16. The asymptotic number of labeled graphs with given degree sequences. Journal of Combinatorial Theory, Series A 24, 3 (1978), 296–307.
  17. Learning multifractal structure in large networks. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining (2014), ACM, pp. 1326–1335.
  18. Competition-induced preferential attachment. In International Colloquium on Automata, Languages, and Programming (2004), Springer, pp. 208–221.
  19. Random graphs, models and generators of scale-free graphs. Proceedings of the Institute for System Programming of the RAS 22 (2012).
  20. The structure and dynamics of multilayer networks. Physics Reports 544, 1 (2014), 1–122.
  21. Bollobás, B. Random Graphs. No. 73. Cambridge University Press, 2001.
  22. Directed scale-free graphs. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms (2003), Society for Industrial and Applied Mathematics, pp. 132–139.
  23. Mathematical results on scale-free random graphs. Handbook of graphs and networks: from the genome to the internet (2003), 1–34.
  24. Bonato, A. A course on the web graph, vol. 89. American Mathematical Soc., 2008.
  25. Bonato, A. A survey of properties and models of on-line social networks. In Proc. of the 5th International Conference on Mathematical and Computational Models, ICMCM (2009), Citeseer.
  26. Mining large networks with subgraph counting. In 2008 Eighth IEEE International Conference on Data Mining (2008), IEEE, pp. 737–742.
  27. Geometric inhomogeneous random graphs. arXiv preprint arXiv:1511.00576 (2016).
  28. Spectra of graphs. Springer Science & Business Media, 2011.
  29. Caldarelli, G. Scale-Free Networks: Complex Webs in Nature and Technology. Oxford University Press, 01 2007.
  30. Network robustness and fragility: Percolation on random graphs. Physical review letters 85, 25 (2000), 5468.
  31. Statistical physics of social dynamics. Reviews of modern physics 81, 2 (2009), 591.
  32. Graph mining: Laws, generators, and algorithms. ACM computing surveys (CSUR) 38, 1 (2006), 2.
  33. Graph Mining: Laws, Tools, and Case Studies. Synthesis Lectures on Data Mining and Knowledge Discovery. Morgan & Claypool Publishers, 2012.
  34. R-mat: A recursive model for graph mining. In Proceedings of the 2004 SIAM International Conference on Data Mining (2004), SIAM, pp. 442–446.
  35. The average distances in random graphs with given expected degrees. Proceedings of the National Academy of Sciences 99, 25 (2002), 15879–15882.
  36. Connected components in random graphs with given expected degree sequences. Annals of combinatorics 6, 2 (2002), 125–145.
  37. Distributed generation of billion-node social graphs with overlapping community structure. In Complex Networks V. Springer, 2014, pp. 199–208.
  38. Generating random networks and graphs. Oxford University Press, 2017.
  39. Characterization of complex networks: A survey of measurements. Advances in physics 56, 1 (2007), 167–242.
  40. Emergence of a small world from local interactions: Modeling acquaintance networks. Physical Review Letters 88, 12 (2002), 128701.
  41. Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physical Review E 84, 6 (2011), 066106.
  42. Pseudofractal scale-free web. Physical review E 65, 6 (2002), 066122.
  43. Evolution of networks: From biological nets to the Internet and WWW. OUP Oxford, 2003.
  44. Learning and scaling directed networks via graph embedding. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (2017), Springer, pp. 634–650.
  45. Reproducing network structure: A comparative study of random graph generators. In Ivannikov ISPRAS Open Conference (ISPRAS), 2017 (2017), IEEE, pp. 83–89.
  46. Durrett, R. Random graph dynamics, vol. 200. Cambridge university press Cambridge, 2007.
  47. Dwork, C. Differential privacy: A survey of results. In International Conference on Theory and Applications of Models of Computation (2008), Springer, pp. 1–19.
  48. Scale-free topology of e-mail networks. Physical review E 66, 3 (2002), 035103.
  49. Darwini: Generating realistic large-scale social graphs. arXiv preprint arXiv:1610.00664 (2016).
  50. Edward M. Lazzarin, R. J. An overview of the analysis of online social networks. ? (2011).
  51. Revisiting power-law distributions in spectra of real world networks. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (2017), ACM, pp. 817–826.
  52. On random graphs i. Publ. Math. Debrecen 6 (1959), 290–297.
  53. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci 5, 1 (1960), 17–60.
  54. Heuristically optimized trade-offs: A new paradigm for power laws in the internet. In International Colloquium on Automata, Languages, and Programming (2002), Springer, pp. 110–122.
  55. On power-law relationships of the internet topology. In ACM SIGCOMM computer communication review (1999), vol. 29, ACM, pp. 251–262.
  56. Double pareto lognormal distributions in complex networks. In Handbook of Optimization in Complex Networks. Springer, 2012, pp. 55–80.
  57. Faragó, A. Structural properties of random graph models. In Proceedings of the Fifteenth Australasian Symposium on Computing: The Australasian Theory-Volume 94 (2009), Australian Computer Society, Inc., pp. 131–138.
  58. Configuring random graph models with fixed degree sequences. SIAM Review 60, 2 (2018), 315–355.
  59. On the diameter of hyperbolic random graphs. SIAM Journal on Discrete Mathematics 32, 2 (2018), 1314–1334.
  60. Introduction to random graphs. Cambridge University Press, 2015.
  61. Gilbert, E. N. Random graphs. The Annals of Mathematical Statistics 30, 4 (1959), 1141–1144.
  62. Social circles: A simple structure for agent-based social network models. Journal of Artificial Societies and Social Simulation 12, 2 (2009).
  63. Community structure in social and biological networks. Proceedings of the national academy of sciences 99, 12 (2002), 7821–7826.
  64. Moment-based estimation of stochastic kronecker graph parameters. Internet Mathematics 8, 3 (2012), 232–256.
  65. A survey of statistical network models. Foundations and Trends® in Machine Learning 2, 2 (2009), 129–233.
  66. Graph embedding techniques, applications, and performance: A survey. Knowledge-Based Systems 151 (2018), 78–94.
  67. Granovetter, M. S. The strength of weak ties. In Social networks. Elsevier, 1977, pp. 347–367.
  68. Random hyperbolic graphs: degree sequence and clustering. In International Colloquium on Automata, Languages, and Programming (2012), Springer, pp. 573–585.
  69. Multiscale network generation. In Information Fusion (Fusion), 2015 18th International Conference on (2015), IEEE, pp. 158–165.
  70. Roll: Fast in-memory generation of gigantic scale-free networks. In Proceedings of the 2016 International Conference on Management of Data (2016), ACM, pp. 1829–1842.
  71. Model-based clustering for social networks. Journal of the Royal Statistical Society: Series A (Statistics in Society) 170, 2 (2007), 301–354.
  72. Harris, J. K. An introduction to exponential random graph modeling, vol. 173. Sage Publications, 2013.
  73. Generating random graphs with tunable clustering coefficients. Physica A: Statistical Mechanics and its Applications 390, 23-24 (2011), 4577–4587.
  74. Classification of graph metrics. Delft University of Technology, Tech. Rep (2011), 1–8.
  75. Learning representations in directed networks. In International Conference on Analysis of Images, Social Networks and Texts (2015), Springer, pp. 196–207.
  76. Large cliques in a power-law random graph. Journal of Applied Probability 47, 4 (2010), 1124–1135.
  77. Secgraph: A uniform and open-source evaluation system for graph data anonymization and de-anonymization. In USENIX Security Symposium (2015), pp. 303–318.
  78. Stochastic blockmodels and community structure in networks. Physical review E 83, 1 (2011), 016107.
  79. Keusch, R. Geometric Inhomogeneous Random Graphs and Graph Coloring Games. PhD thesis, ETH Zurich, 2018.
  80. A bound for the diameter of random hyperbolic graphs. In Proceedings of the Meeting on Analytic Algorithmics and Combinatorics (2015), Society for Industrial and Applied Mathematics, pp. 26–39.
  81. The web as a graph: measurements, models, and methods. In International Computing and Combinatorics Conference (1999), Springer, pp. 1–17.
  82. Kolaczyk, E. D. Statistical Analysis of Network Data: Methods and Models, 1st ed. Springer Publishing Company, Incorporated, 2009.
  83. Kovalenko, I. Theory of random graphs. Cybernetics 7, 4 (1971), 575–579.
  84. Network growth by copying. Physical Review E 71, 3 (2005), 036118.
  85. Hyperbolic geometry of complex networks. Physical Review E 82, 3 (2010), 036106.
  86. Stochastic models for the web graph. In Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on (2000), IEEE, pp. 57–65.
  87. Kunegis, J. Konect: the koblenz network collection. In Proceedings of the 22nd International Conference on World Wide Web (2013), ACM, pp. 1343–1350.
  88. Preferential attachment in online networks: Measurement and explanations. In Proceedings of the 5th annual ACM web science conference (2013), ACM, pp. 205–214.
  89. Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. Physical Review E 80, 1 (2009), 016118.
  90. Benchmark graphs for testing community detection algorithms. Physical review E 78, 4 (2008), 046110.
  91. Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication. In European Conference on Principles of Data Mining and Knowledge Discovery (2005), Springer, pp. 133–145.
  92. Kronecker graphs: An approach to modeling networks. Journal of Machine Learning Research 11, Feb (2010), 985–1042.
  93. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD) 1, 1 (2007), 2.
  94. Statistical properties of community structure in large social and information networks. In Proceedings of the 17th international conference on World Wide Web (2008), ACM, pp. 695–704.
  95. Lovász, L. Large networks and graph limits, vol. 60. American Mathematical Soc., 2012.
  96. Systematic topology analysis and generation using degree correlations. In ACM SIGCOMM Computer Communication Review (2006), vol. 36, ACM, pp. 135–146.
  97. The rise and fall of a networked society: A formal model. Proceedings of the National Academy of Sciences 101, 6 (2004), 1439–1442.
  98. Weighted graphs and disconnected components: patterns and a generator. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining (2008), ACM, pp. 524–532.
  99. Birds of a feather: Homophily in social networks. Annual review of sociology 27, 1 (2001), 415–444.
  100. On the origin of power laws in internet topologies. ACM SIGCOMM computer communication review 30, 2 (2000), 18–28.
  101. Large-scale graph generation and big data: An overview on recent results. Bulletin of EATCS 2, 122 (2017).
  102. Superfamilies of evolved and designed networks. Science 303, 5663 (2004), 1538–1542.
  103. On the uniform generation of random graphs with prescribed degree sequences. arXiv preprint cond-mat/0312028 (2003).
  104. Network motifs: simple building blocks of complex networks. Science 298, 5594 (2002), 824–827.
  105. Molontay, R. Fractal Characterization of Complex Networks. PhD thesis, Department of Stochastics, Budapest University of Technology and Economics, 2015.
  106. Modeling the variance of network populations with mixed kronecker product graph models.
  107. Block kronecker product graph model. In Workshop on Mining and Learning from Graphs (2013).
  108. De-anonymizing social networks. In Security and Privacy, 2009 30th IEEE Symposium on (2009), IEEE, pp. 173–187.
  109. Newman, M. Networks: an introduction. Oxford university press, 2010.
  110. The structure and dynamics of networks. Princeton University Press, 2006.
  111. Newman, M. E. Assortative mixing in networks. Physical review letters 89, 20 (2002), 208701.
  112. Newman, M. E. The structure and function of complex networks. SIAM review 45, 2 (2003), 167–256.
  113. Newman, M. E. Analysis of weighted networks. Physical review E 70, 5 (2004), 056131.
  114. Newman, M. E. Modularity and community structure in networks. Proceedings of the national academy of sciences 103, 23 (2006), 8577–8582.
  115. Finding and evaluating community structure in networks. Physical review E 69, 2 (2004), 026113.
  116. Random graphs with arbitrary degree distributions and their applications. Physical review E 64, 2 (2001), 026118.
  117. Nickel, C. L. M. Random dot product graphs a model for social networks. PhD thesis, The Johns Hopkins University, 2006.
  118. Generating random graphs for the simulation of wireless ad hoc, actuator, sensor, and internet networks. Pervasive and Mobile Computing 4, 5 (2008), 597–615.
  119. Multifractal network generator. Proceedings of the National Academy of Sciences (2010).
  120. Trilliong: A trillion-scale synthetic graph generator using a recursive vector model. In Proceedings of the 2017 ACM International Conference on Management of Data (2017), ACM, pp. 913–928.
  121. Penrose, M. Random geometric graphs. No. 5. Oxford University Press, 2003.
  122. Physical theories of the evolution of online social networks: a discussion impulse. dynamical systems 22 (2012), 4.
  123. Partitioning sparse matrices with eigenvectors of graphs. SIAM journal on matrix analysis and applications 11, 3 (1990), 430–452.
  124. Raigorodsky. Models of random graphs. Litres, 2011.
  125. Hierarchical organization in complex networks. Physical Review E 67, 2 (2003), 026112.
  126. An introduction to exponential random graph (p*) models for social networks. Social networks 29, 2 (2007), 173–191.
  127. Fractal and transfractal recursive scale-free nets. New Journal of Physics 9, 6 (2007), 175.
  128. Measurement-calibrated graph models for social network experiments. In Proceedings of the 19th international conference on World wide web (2010), ACM, pp. 861–870.
  129. Detection of node group membership in networks with group overlap. The European Physical Journal B 67, 3 (2009), 277–284.
  130. Community structure and scale-free collections of erdős-rényi graphs. Physical Review E 85, 5 (2012), 056109.
  131. An in-depth study of stochastic kronecker graphs. In Data Mining (ICDM), 2011 IEEE 11th International Conference on (2011), IEEE, pp. 587–596.
  132. Mobile call graphs: beyond power-law and lognormal distributions. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining (2008), ACM, pp. 596–604.
  133. Exponential random graph modeling for complex brain networks. PloS one 6, 5 (2011), e20039.
  134. Generating scaled replicas of real-world complex networks. In International Workshop on Complex Networks and their Applications (2016), Springer, Cham, pp. 17–28.
  135. Generating constrained random graphs using multiple edge switches. Journal of Experimental Algorithmics (JEA) 16 (2011), 1–7.
  136. Taylor, R. Constrained switchings in graphs. In Combinatorial Mathematics VIII. Springer, 1981, pp. 314–336.
  137. Generate country-scale networks of interaction from scattered statistics. In The fifth conference of the European social simulation association, Brescia, Italy (2008), vol. 240.
  138. A comparative study of social network models: Network evolution models and nodal attribute models. Social Networks 31, 4 (2009), 240–254.
  139. A model for social networks. Physica A: Statistical Mechanics and its Applications 371, 2 (2006), 851–860.
  140. Van Der Hofstad, R. Random graphs and complex networks, 2014.
  141. A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models. Social Networks 31, 1 (2009), 52–62.
  142. Virus spread in networks. IEEE/ACM Transactions on Networking (TON) 17, 1 (2009), 1–14.
  143. Vázquez, A. Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations. Physical Review E 67, 5 (2003), 056104.
  144. Spontaneous emergence of complex optimal networks through evolutionary adaptation. Computers & chemical engineering 28, 9 (2004), 1789–1798.
  145. Preserving differential privacy in degree-correlation based graph generation. Transactions on data privacy 6, 2 (2013), 127.
  146. Collective dynamics of ’small-world’ networks. nature 393, 6684 (1998), 440.
  147. Waxman, B. M. Routing of multipoint connections. IEEE journal on selected areas in communications 6, 9 (1988), 1617–1622.
  148. Wegner, A. Random graphs with motifs.
  149. A spatial model for social networks. Physica A: Statistical Mechanics and its Applications 360, 1 (2006), 99–120.
  150. Fractality and scale-free effect of a class of self-similar networks. Physica A: Statistical Mechanics and its Applications 478 (2017), 31–40.
  151. Structure and overlaps of ground-truth communities in networks. ACM Transactions on Intelligent Systems and Technology (TIST) 5, 2 (2014), 26.
  152. Defining and evaluating network communities based on ground-truth. Knowledge and Information Systems 42, 1 (2015), 181–213.
  153. Randomizing social networks: a spectrum preserving approach. In proceedings of the 2008 SIAM International Conference on Data Mining (2008), SIAM, pp. 739–750.
  154. Graph generation with prescribed feature constraints. In Proceedings of the 2009 SIAM International Conference on Data Mining (2009), SIAM, pp. 966–977.
  155. Modeling the internet’s large-scale topology. Proceedings of the National Academy of Sciences 99, 21 (2002), 13382–13386.
  156. Gscaler: Synthetically scaling a given graph. In EDBT (2016), pp. 53–64.
  157. Zweig, K. A. Network analysis literacy: a practical approach to the analysis of networks. lecture notes in social networks, 2016.
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