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Different flavors of randomness: comparing random graph models with fixed degree sequences

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Abstract

When a structural characteristic of a network is measured, the observed value needs to be compared to its expected value in a random graph model to assess the statistical significance of its occurrence. The random graph model with which the observed graph is compared is chosen to be structurally similar to the real-world network in some aspects and totally random in all others. To make the analysis of the expected value amenable, the random graph model is also chosen to be as simple as possible. The most common random graph models maintain the degree sequence of the observed graph or at least approximate it. In cases where multi-edges and self-loops are not allowed, typically the fixed degree sequence model (FDSM) is used. Since it is computationally expensive, in this article, we discuss whether one of the following three approximative models can replace it: the configuration model, its simplified version (eCFG), and the mathematical approximation we term simple independence model. While the latter models are more scalable than the FDSM, we show that there are several networks for which they cannot be meaningfully applied. We investigate based on some examples whether, and if so in which cases, these approximating models can replace the computationally more involved FDSM.

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Notes

  1. http://www.weizmann.ac.il/mcb/UriAlon/.

  2. http://www-personal.umich.edu/~mejn/netdata/.

  3. http://snap.stanford.edu/data/index.html#canets.

References

  • Aghbolagh RD, Zeilemaker N, Pouwelse J, Epema D (2013) A network science perspective of a distributed reputation mechanism. In: IFIP Networking 2013

  • Alon U (2006) An introduction to systems biology—design principles of biological circuits. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    Article  MathSciNet  Google Scholar 

  • Boguá M, Pastor-Satorras R, Vespignani A (2003) Epidemic spreading in complex networks with degree correlations. In: Pastor-Satorras R, Rubi M, Diaz-Guilera A (eds) Statistical mechanics of complex networks. Lecture notes in physics, vol 625. Springer, Berlin, pp 127–147. doi:10.1007/978-3-540-44943-0_8

  • Bollobás B, Riordan O (2004) The diameter of a scale-free random graph. Combinatorica 24(1):5–34

    Article  MATH  MathSciNet  Google Scholar 

  • Brualdi RA (2006) Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors. Discret Math 306:3054–3062

    Article  MATH  MathSciNet  Google Scholar 

  • Chung F, Lu L (2002) The average distances in random graphs with given expected degrees. Proc Natl Acad Sci 99(25):15879–15882. doi:10.1073/pnas.252631999

    Article  MATH  MathSciNet  Google Scholar 

  • Cobb GW, Chen YP (2003) An application of Markov Chain Monte Carlo to community ecology. Am Math Mon 110:265–288

    Article  MATH  MathSciNet  Google Scholar 

  • de Arruda GF, Barbieri AL, Rodríguez PM, Rodrigues FA, Moreno Y, da Fontoura Costa L (2014) Role of centrality for the identification of influential spreaders in complex networks. Phys Rev E 90(032):812. doi:10.1103/PhysRevE.90.032812

    Google Scholar 

  • Erdős P, Rényi A (1959) On random graphs I. Publ Math Debrecen 6:290–297

    MathSciNet  Google Scholar 

  • Freeman LC (1977) A set of measures of centrality based upon betweenness. Sociometry 40:35–41

    Article  Google Scholar 

  • Geng L, Hamilton HJ (2006) Interestingness measures for data mining: a survey. ACM Comput Surv 38(3):9

    Article  Google Scholar 

  • Gionis A, Mannila H, Mielikäinen T, Tsaparas P (2007) Assessing data mining results via swap randomization. ACM Trans Knowl Discov Data 1(3):14

    Article  Google Scholar 

  • Goh KII, Cusick ME, Valle D, Childs B, Vidal M, Barabási ALL (2007) The human disease network. Proc Natl Acad Sci USA 104(21):8685–8690. doi:10.1073/pnas.0701361104

    Article  Google Scholar 

  • Gotelli NJ, Graves GR (1996) Null-models in ecology. Smithsonian Institution Press, Washington

    Google Scholar 

  • Katz L, Powell JH (1955) Measurement of the tendency toward reciprocation of choice. Sociometry 18(4):403–409

    Article  Google Scholar 

  • Ledermann W (ed) (1980) Handbook of applicable mathematics. Wiley, Chichester

  • Leicht EA, Holme P, Newman ME (2006) Vertex similarity in networks. Phys Rev E 73(2):026120

    Article  Google Scholar 

  • Malumbres M (2012) miRNAs versus oncogenes: the power of social networking. Mol Syst Biol 8:569

    Article  Google Scholar 

  • Mcauley J, J Leskovec (2014) Discovering social circles in ego networks. ACM Trans Knowl Discov Data 8(1):4:1–4:28. doi:10.1145/2556612

    Article  Google Scholar 

  • Milo R, Itzkovitz S, Kashtan N, Levitt R, Shen-Orr S, Ayzenshtat I, Sheffer M, Alon U (2004a) Superfamilies of evolved and designed networks. Science 303:1538–1542

    Article  Google Scholar 

  • Milo R, Kashtan N, Itzkovitz S, Newman MEJ, Alon U (2004b) Subgraphs in networks. Phys Rev E 70(058):102

    MathSciNet  Google Scholar 

  • Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298(5594):824–827. doi:10.1126/science.298.5594.824. http://www.sciencemag.org/content/298/5594/824.full.pdf

  • Molloy M, Reed B (1995) A critical point for random graphs with a given degree sequence. Random Struct Algorithms 6(2–3):161–180. doi:10.1002/rsa.3240060204

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Newman ME (2003) Mixing patterns in networks. Phys Rev E 67(2):026126. doi:10.1103/physreve.67.026126

    Article  MathSciNet  Google Scholar 

  • Newman MEJ (2010) Networks: an introduction. Oxford University Press, New York

    Book  Google Scholar 

  • Newman ME, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69(2):026113

    Article  Google Scholar 

  • Opsahl T, Panzarasa P (2009) Clustering in weighted networks. Soc Netw 31(2):155–163. doi:10.1016/j.socnet.2009.02.002

    Article  Google Scholar 

  • Shen-Orr S, Milo R, Mangan S, Alon U (2002) Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31(1):64–68. doi:10.1038/ng881

    Article  Google Scholar 

  • Traud AL, Kelsic ED, Mucha PJ, Porter MA (2011) Comparing community structure to characteristics in online collegiate social networks. SIAM Rev 53(3):526–543. doi:10.1137/080734315

    Article  MathSciNet  Google Scholar 

  • Uhlmann S, Mannsperger H, Zhang JD, Horvat EÁ, Schmidt C, Küblbeck M, Ward A, Tschulena U, Zweig K, Korf U, Wiemann S, Sahin Ö (2012) Global miRNA regulation of a local protein network: case study with the EGFR-driven cell cycle network in breast cancer. Mol Syst Biol 8:570

    Article  Google Scholar 

  • van der Hofstad R (2012) Random graphs and complex networks. Department of Mathematics and Computer Science

  • van der Hofstad R, Hooghiemstra G (2008) Universality for distances in power-law random graphs. J Math Phys 49(12):125209. doi:10.1063/1.2982927

    Article  MathSciNet  Google Scholar 

  • Wasserman S, Faust K (1994) Social network analysis., Methods and applicationsCambridge University Press, Cambridge

    Book  Google Scholar 

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

    Article  Google Scholar 

  • 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. Proceedings of the National Academy of Sciences 101(101):5934–5939

    Article  Google Scholar 

  • Zweig KA (2010) How to forget the second side of the story: a new method for the one-mode projection of bipartite graphs. In: Proceedings of the 2010 international conference on advances in social networks analysis and mining (ASONAM 2010), pp 200–207

  • Zweig KA (2014) Network representations of complex data. In: Encyclopedia of social network analysis and mining. Springer, Heidelberg

  • Zweig KA, Kaufmann M (2011) A systematic approach to the one-mode projection of bipartite graphs. Soc Netw Anal Min 1(3):187–218

    Article  Google Scholar 

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Authors and Affiliations

  1. Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Wolfgang E. Schlauch & Katharina A. Zweig

  2. Northwestern Institute on Complex Systems, Northwestern University, Evanston, USA

    Emőke Ágnes Horvát

Authors
  1. Wolfgang E. Schlauch
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  2. Emőke Ágnes Horvát
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  3. Katharina A. Zweig
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Correspondence to Wolfgang E. Schlauch.

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Schlauch, W.E., Horvát, E.Á. & Zweig, K.A. Different flavors of randomness: comparing random graph models with fixed degree sequences. Soc. Netw. Anal. Min. 5, 36 (2015). https://doi.org/10.1007/s13278-015-0267-z

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