Random Graphs as Null Models

  • Katharina A. ZweigEmail author
Part of the Lecture Notes in Social Networks book series (LNSN)


In the last chapter, a qualitative comparison of various real-world structures with classic random graph models revealed that complex networks are non-random in many aspects. This chapter focuses on the question of how to quantify the statistical significance of an observed network structure with respect to a given random graph model. The chapter starts with a discussion of the statistical significance of a given percentage of reciprocal edges in a directed graph and then introduces a new random graph model in which the degree sequence(s) are maintained. Finally, the notion of network motifs is introduced.


Bipartite Graph Random Graph Common Neighbor Network Motif Degree Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adamic L, Adar E (2005) How to search a social network. Soc Netw 27(3):187–203CrossRefGoogle Scholar
  2. 2.
    Alon U (2006) An introduction to systems biology—design principles of biological circuits. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  3. 3.
    Alon U (2007) Network motifs: theory and experimental approaches. Nat Rev Genet 8:450–461CrossRefGoogle Scholar
  4. 4.
    Artzy-Randrup Y, Fleishman SJ, Ben-Tal N, Stone L (2004) Comment on “network motifs: Simple building blocks of complex networks” and “superfamilies of evolved and designed networks”. Science 305:1107cCrossRefGoogle Scholar
  5. 5.
    Berger A, Müller-Hannemann M (2010) Uniform sampling of undirected and directed graphs with a fixed degree sequence. In: Proceedings of the 36th international workshop on graph-theoretic concepts in computer scienceGoogle Scholar
  6. 6.
    Brualdi RA (1980) Matrices of zeros and ones with fixed row and column sum vectors. Linear Algebra Appl 33:159–231MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brualdi RA (2006) Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors. Discrete Math 306:3054–3062MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chung F, Linyuan L (2002) The average distances in random graphs with given expected degrees. Proc Natl Acad Sci 99(25):15879–15882MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chung F, Linyuan L (2006) Complex graphs and networks. American Mathematical Society, USACrossRefzbMATHGoogle Scholar
  10. 10.
    Cobb GW, Chen Y-P (2003) An application of Markov Chain Monte Carlo to community ecology. Am Math Monthly 110:265–288MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Conlan AJK, Eames KTD, Gage JA, von Kirchbach JC, Ross JV, Saenz RA, Gog JR (2011) Measuring social networks in British primary schools through scientific engagement. Proc R Soc Lond B 278(1711):1467–1475CrossRefGoogle Scholar
  12. 12.
    Connor EF, Simberloff D (1979) The assembly of species communities: chance or competition? Ecology 60(6):1132–1140CrossRefGoogle Scholar
  13. 13.
    de Silva E, Stumpf MPH (2005) Complex networks and simple models in biology. J R Soc Interface 2:419–430Google Scholar
  14. 14.
    Diaconis P (1985) Exploring data tables, trends and shapes. Theories of data analysis: from magical thinking through classical statistics. Wiley, pp 1–35Google Scholar
  15. 15.
    Diaconis P, Gangolli A (1994) Rectangular arrays with fixed margins. Inst Math Appl 72:15–41MathSciNetzbMATHGoogle Scholar
  16. 16.
    Erdős PL, Miklós I, Toroczkai Z (2010) A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs. Electron J Comb [electronic only], 17(1):Research Paper R66, 10 p–Research Paper R66, 10 pGoogle Scholar
  17. 17.
    Garlaschelli D, Loffredo MI (2004) Patterns of link reciprocity in directed networks. Phys Rev Lett 93:268701CrossRefGoogle Scholar
  18. 18.
    Geng L, Hamilton HJ (2006) Interestingness measures for data mining: a survey. ACM Comput Surv 38(3):9CrossRefGoogle Scholar
  19. 19.
    Gionis A, Mannila H, Mielikäinen T, Tsaparas P (2006) Assessing data mining results via swap randomization. In: Proceedings of the twelfth ACM SIGKDD international conference on knowledge discovery and data mining (KDD’06)Google Scholar
  20. 20.
    Gionis A, Mannila H, Mielikäinen T, Tsaparas P (2007) Assessing data mining results via swap randomization. ACM Trans Knowl Discovery Data 1(3):article no 14Google Scholar
  21. 21.
    Gkantsidis C, Mihail M, Zegura E (2003) The Markov chain simulation method for generating connected power law random graphs. In: Proceedings of ALENEX’03, pp 16–25Google Scholar
  22. 22.
    Gotelli NJ, Graves GR (1996) Null-models in ecology. Smithsonian Institution Press, Washington and LondonGoogle Scholar
  23. 23.
    Hakimi SL (1962) On realizability of a set of integers as degrees of the vertices of a linear graph. I. J Soc Ind Appl Math 10:496–506MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Havel V (1955) Poznámka o existenci konečných grafåu (in Czech) (A remark about the existence of finite graphs). Časopis pro pěstování matematiky 80:477–480Google Scholar
  25. 25.
    Holland PW, Leinhardt S (1970) A method for detecting structure in sociometric data. Am J Soc 76(3):492–513CrossRefGoogle Scholar
  26. 26.
    Holland PW, Leinhardt S (1976) Local structure in social networks. Sociol Methodol 7:1–45CrossRefGoogle Scholar
  27. 27.
    Horvát EÁ, Zhang JD, Uhlmann S, Sahin Ö, Zweig KA (2013) A network-based method to assess the statistical significance of mild co-regulation effects. PLOS ONE 8(9):e73413Google Scholar
  28. 28.
    Horvát E-Á, Zweig KA (2012) One-mode projections of multiplex bipartite graphs. In: Proceedings of the 2012 IEEE/ACM international conference on advances in social networks analysis and mining (ASONAM 2012)Google Scholar
  29. 29.
    Ingram PJ, Stumpf MPH, Stark J (2006) Network motifs: structure does not determine function. MBC Genomics 7:108Google Scholar
  30. 30.
    Kashtan N, Itzkovitz S, Milo R, Alon U (2004) Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs. Bioinformatics 20(11):1746–1758CrossRefGoogle Scholar
  31. 31.
    Katz L, Powell JH (1955) Measurement of the tendency toward reciprocation of choice. Sociometry 18(4):403–409CrossRefGoogle Scholar
  32. 32.
    Lehmann KA, Kaufmann M (2005) Evolutionary algorithms for the self-organized evolution of networks. In: Proceedings of the genetic and evolutionary computation conference (GECCO’05), pp 563–570Google Scholar
  33. 33.
    Leicht EA, Holme P, Newman MEJ (2006) Vertex similarity in networks. Phys Rev E 73(2):026120Google Scholar
  34. 34.
    Liben-Nowell D, Kleinberg J (2007) The link-prediction problem for social networks. J Am Soc Inf Sci Technol 58(7):1019–1031CrossRefGoogle Scholar
  35. 35.
    Marvel SA, Kleinberg J, Kleinberg RD, Strogatz SH (2011) Continuous-time model of structural balance. Proc Natl Acad Sci (published ahead of print January 3, 2011). doi: 10.1073/pnas.1013213108
  36. 36.
    Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298:824–827CrossRefGoogle Scholar
  37. 37.
    Milo R, Itzkovitz S, Kashtan N, Levitt R, Alon U (2004) Response to comment on “Network motifs: simple building blocks of complex networks” and “Superfamilies of evolved and designed networks”. Science 305:1107dCrossRefGoogle Scholar
  38. 38.
    Newman MEJ (2002) Random graphs as models of networks. Technical report, Working Paper 02-02-005 at the Santa Fe InstituteGoogle Scholar
  39. 39.
    Newman ME (2010) Networks: an introduction. Oxford University Press, New YorkGoogle Scholar
  40. 40.
    Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69(2):026113CrossRefGoogle Scholar
  41. 41.
    Newman MEJ, Forrest S, Balthrop J (2002) Email networks and the spread of computer viruses. Phys Rev E 66:035101(R)Google Scholar
  42. 42.
    North BV, Curtis D, Sham PC (2002) A note on the calculation of empirical p-values from Monte Carlo procedures. Am J Hum Genet 71(2):439–441CrossRefGoogle Scholar
  43. 43.
    Porter MA, Onnela J-P, Mucha PJ (2009) Communities in networks. Not AMS 56(9):1082–1097 and 1164–1166Google Scholar
  44. 44.
    Schlauch W, Zweig K, Horvát E-Á. Different flavors of randomness. Soc Netw Anal Mining 5:eid:36Google Scholar
  45. 45.
    Shen-Orr SS, Milo R, Mangan S, Alon U (2002) Network motifs in the transcriptional regulation network of escherichia coli. Nat Genet 31:64–68Google Scholar
  46. 46.
    Spitz A, Gimmler A, Stoeck T, Zweig KA, Horvát E-Á (2016) Assessing low intensity relationships in complex networks. PLoS ONE 11(4):e0152536Google Scholar
  47. 47.
    Taylor R (1980) Constrained switchings in graphs. Comb Math 8:314–336Google Scholar
  48. 48.
    Wasserman S, Faust K (1999) Social network analysis—methods and applications, revised, reprinted edition. Cambridge University Press, CambridgeGoogle Scholar
  49. 49.
    Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442CrossRefGoogle Scholar
  50. 50.
    Yeger-Lotem E, Sattath S, Kashtan N, Itzkovitz S, Milo R, Pinter RY, Alon U, Margalit H (2004) Network motifs in integrated cellular networks of transcription-regulation and protein-protein interaction. Proc Natl Acad Sci 101(101):5934–5939CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2016

Authors and Affiliations

  1. 1.TU Kaiserslautern, FB Computer ScienceGraph Theory and Analysis of Complex NetworksKaiserslauternGermany

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