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Random Graphs and Network Models

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Network Analysis Literacy

Part of the book series: Lecture Notes in Social Networks ((LNSN))

Abstract

One of the most important concepts in network analysis is to understand the structure of a given graph with respect to a set of suitably randomized graphs, a so-called random graph model. Structures which are found to be significantly different from those expected in the random graph model require a new random graph model which exemplifies how the structure might emerge in the complex network. In this chapter the most common random graph models are introduced: the classic Erdős-Rényi model, the small-world model by Watts and Strogatz, and the preferential attachment model by Barabási and Albert.

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Notes

  1. 1.

    In the following, if not stated otherwise, the word “graph” should always be understood as “simple graph”, i.e., not containing self-loops or multiple edges.

  2. 2.

    In computer science, an instance of a mathematical model is a concrete entity chosen according to its probability.

  3. 3.

    Chung differentiates between off-line and on-line or generative models. The first ones are those based on a fixed number of vertices, in the second type of model, at each time step edges and vertices might be added or deleted [13, p. 17].

  4. 4.

    Bollobás and Riordan write that actually this model was first published and analyzed by Gilbert in [19]. A similar model was described even earlier by Solomonoff and Rapoport: in it, every node is assigned some number k of outgoing edges each of which is connected to every other node with the same probability [39]. This can be seen as a k-out-regular random graph model.

  5. 5.

    Square brackets denote an interval of real numbers, i.e., [0, 1] means that p is any number between 0 and 1.

  6. 6.

    More detailed: If \(p = \log n/(n) + \omega (n)/(n)\) and \(\omega (n)\) is a function of n with \(\omega (n)\rightarrow \infty \) for \(n\rightarrow \infty \), then any instance of \(\mathcal G(n,p)\) is connected with high probability. Cf. [9, p. 3], [7, Theorem 7.3].

  7. 7.

    Asymptotic means that the diameter of the graph will approach this value closer and closer for increasing n.

  8. 8.

    Note that the double average is intended: it is once averaged over all pairs of nodes in each graph and then averaged over all graphs in the sample.

  9. 9.

    A lattice or grid of length l and dimension d is a set of \(l^d\) points with all possible combinations of integer coordinates from 0 to \(l-1\) in all d dimensions. The most common grid is the two-dimensional grid. The points are normally connected to their \(2\cdot d\) nearest points but principally they can be connected to an arbitrary (but constant) number of nearest points.

  10. 10.

    The diameter of a graph G is defined as the maximal distance between any two vertices in G.

  11. 11.

    Note that the size of the grid is the number of nodes in one dimension. Since we use a two-dimensional grid, the actual number of nodes in the graph is \(size^2\). Note also that each vertex is connected to its four closest neighbors, but it could be connected to any constant number of closest neighbors and the effect would still be the same.

  12. 12.

    Note that this is not a very strict definition. There were attempts to define small-worldness more strictly, e.g., Lehmann et al. [25]. However, the mental picture provoked by the definition of Watts and Strogatz is vivid and to my knowledge, there was never a debate whether a given network is a small-world or not.

  13. 13.

    For a thorough discussion of the physics behind the classic Watts-Strogatz small-world model read Barthélémy and Amaral [6] and Barrat and Weigt [5].

  14. 14.

    I will later discuss how reliable this finding is in the first place—as protein-protein-interaction networks are especially prone to mistakes, their analysis requires extra carefulness (Sect. 10.3.1).

  15. 15.

    See the glossary for an explanation of this mathematical jargon [p. 527, entry ‘limit of’].

References

  1. Amaral LAN, Scala A, Barthélémy M, Stanley HE (2000) Classes of small-world networks. Proc Natl Acad Sci 97:11149–11152

    Article  Google Scholar 

  2. Barabási A-L (2009) Scale-free networks: a decade and beyond. Science 325:412–413

    Article  MathSciNet  MATH  Google Scholar 

  3. Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    Article  MathSciNet  MATH  Google Scholar 

  4. Barabási A-L, Jeong H, Albert R (1999) The diameter of the world wide web. Nature 401:130

    Article  Google Scholar 

  5. Barrat A, Weigt M (2000) On the properties of small-world networks. Eur Phys J B 13:547–560

    Article  Google Scholar 

  6. Barthélémy M, Nunes Amaral LA (1999) Small-world networks: evidence for a crossover picture. Am Phys Soc 82(15):3180–3183

    Google Scholar 

  7. Bollobás B (2001) Random graphs. Cambridge studies in advanced mathematics, vol 73, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  8. Bollobás B (2004) Extremal graph theory, dover edn. Dover Publications Inc., Mineola (USA)

    MATH  Google Scholar 

  9. Bollobás B, Riordan OM (2003) Handbook of graphs and networks. In: Bornholdt S, Schuster H-G (eds) Mathematical results on scale-free random graphs. Springer, Heidelberg, pp 1–34

    Google Scholar 

  10. Bonacich P (2004) The invasion of the physicists. Soc Netw 26:258–288

    Google Scholar 

  11. Buchanan M (2002) Nexus: small worlds and the groundbreaking science of networks. W.W. Norton & Company, New York

    Google Scholar 

  12. Callaway DS, Hopcroft JE, Kleinberg JM, Newman MEJ, Strogatz SH (2001) Are randomly grown graphs really random? Phys Rev E 604(4):1902

    Google Scholar 

  13. Chung F, Linyuan L (2006) Complex graphs and networks. American Mathematical Society, USA

    Book  MATH  Google Scholar 

  14. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev Mod Phys 51(4):661–703

    Article  MathSciNet  MATH  Google Scholar 

  15. da Fontoura Costa L, Rodrigues FA, Travieso G, Villas Boas PR (2007) Characterization of complex networks: a survey of measurements. Adv Phys 56:167–242

    Article  Google Scholar 

  16. Dorogovtsev SN, Mendes JFF (2003) Evolution of networks. Oxford University Press

    Google Scholar 

  17. Dorogovtsev SN (2010) Lectures on complex networks. Oxford University Press, New York

    Book  MATH  Google Scholar 

  18. Faloutsos M, Faloutsos P, Faloutsos C (1999) On power-law relationships of the internet topology. Comput Commun Rev 29:251–262

    Article  MATH  Google Scholar 

  19. Gilbert EN (1959) Random graphs. Ann Math Statist 30:1141–1144

    Article  Google Scholar 

  20. Gladwell M (1999) The six degrees of Lois Weisberg. New Yorker January:52–63

    Google Scholar 

  21. Kleinberg J (2000) Navigation in a small world. Nature 406:845

    Article  Google Scholar 

  22. Kleinberg J (2000) The small-world phenomenon: an algorithmic perspective. In: Proceedings of the 32nd ACM symposium on theory of computing, pp 163–170

    Google Scholar 

  23. Krapivsky PL, Redner S, Leyvraz F (2000) Connectivity of growing random networks. Phys Rev Lett 85:4629–4632

    Article  Google Scholar 

  24. Krauth W (2006) Statistical mechanics: algorithms and computations. Oxford University Press, New York

    MATH  Google Scholar 

  25. Lehmann KA, Post HD, Kaufmann M (2006) Hybrid graphs as a framework for the small-world effect. Phys Rev E 73:056108

    Article  Google Scholar 

  26. Lusher D, Koskinen J, Robins G (eds) (2013) Exponential random graph models for social networks: theory, methods, and applications. Cambridge University Press, New York

    Google Scholar 

  27. Newman M (2000) Models of the small world: a review. J Stat Phys 101:819–841

    Article  MATH  Google Scholar 

  28. Newman ME (2010) Networks: an introduction. Oxford University Press, New York

    Book  MATH  Google Scholar 

  29. Newman MEJ, Watts DJ (1999) Renormalization group analysis of the small-world network model. Phys Lett A 263:341–346

    Article  MathSciNet  MATH  Google Scholar 

  30. Newman MEJ, Watts DJ, Strogatz SH (2002) Random graph models of social networks. Proc Natl Acad Sci USA 99:2566–2572

    Article  MATH  Google Scholar 

  31. Jupyter notebook. https://ipython.org/notebook.html

  32. Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barabási A-L (2002) Hierarchical organization of modularity in metabolic networks. Science 297:1551–1553

    Article  Google Scholar 

  33. Robins G (2011) The SAGE handbook of social network analysis. Exponential random graph models for social networks. SAGE Publications Ltd., London, pp 484–500

    Google Scholar 

  34. Scholtes I. Analysis of non-markovian temporal networks—an educational tutorial using the free python module pyTempNets. http://www.ingoscholtes.net/research/insights/Temporal_Networks.html

  35. Scholtes I, Wider N, Pfitzner R, Garas A, Tessone CJ, Schweitzer F (2014) Causality-driven slow-down and speed-up of diffusion in non-markovian temporal networks. Nat Commun 5:5024

    Article  Google Scholar 

  36. Scott J, Carrington PJ (eds) (2011) The SAGE handbook of social network analysis. SAGE Publications Ltd., London

    Google Scholar 

  37. Simon HA (1955) On a class of skew distribution functions. Biometrika 42(3–4):425–440

    Article  MathSciNet  MATH  Google Scholar 

  38. Solé RV, Pastor-Satorras R, Smith E, Kepler TB (2002) A model of large-scale proteome evolution. Adv Complex Syst 5:43–54

    Article  MATH  Google Scholar 

  39. Solomonoff R, Rapoport A (1951) Connectivity of random nets. Bull Math Biophys 13:107–117

    Article  MathSciNet  Google Scholar 

  40. van Duijn MAJ, Huisman M (2011) The SAGE handbook of social network analysis. Statistical models for ties and actors. SAGE Publications Ltd., London, pp 459–483

    Google Scholar 

  41. Vázquez A, Flammini A, Maritan A, Vespignani A (2002) Modeling of protein interaction networks. ComPlexUs 1:38–44

    Google Scholar 

  42. Watts DJ (1999) Small worlds—the dynamics of networks between order and randomness. Princeton studies in complexity, Princeton University Press

    Google Scholar 

  43. Watts DJ (2003) Six degrees–the science of a connected age. W.W. Norton & Company, New York

    Google Scholar 

  44. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442

    Article  Google Scholar 

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Authors and Affiliations

  1. TU Kaiserslautern, FB Computer Science, Graph Theory and Analysis of Complex Networks, Gottlieb-Daimler-Str. 48, 67331, Kaiserslautern, Germany

    Katharina A. Zweig

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Zweig, K.A. (2016). Random Graphs and Network Models. In: Network Analysis Literacy. Lecture Notes in Social Networks. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0741-6_6

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  • DOI: https://doi.org/10.1007/978-3-7091-0741-6_6

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