On Scale-Free Prior Distributions and Their Applicability in Large-Scale Network Inference with Gaussian Graphical Models

  • Paul Sheridan
  • Takeshi Kamimura
  • Hidetoshi Shimodaira
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 4)


This paper concerns the specification, and performance, of scale-free prior distributions with a view toward large-scale network inference from small-sample data sets. We devise three scale-free priors and implement them in the framework of Gaussian graphical models. Gaussian graphical models are used in gene network inference where high-throughput data describing a large number of variables with comparatively few samples are frequently analyzed by practitioners. And, although there is a consensus that many such networks are scale-free, the modus operandi is to assign a random network prior. Simulations demonstrate that the scale-free priors outperform the random network prior at recovering scale-free trees with degree exponents near 2, such as are characteristic of many real-world systems. On the other hand, the random network prior compares favorably at recovering scale-free trees characterized by larger degree exponents.


Bayesian inference complex networks Gaussian graphical model Markov chain Monte Carlo prior distribution scale-free “small n large p” problem small-sample inference 


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  1. 1.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Burda, Z., Correia, J.D., Krzywicki, A.: Statistical Ensemble of Scale-Free Random Graphs. Phys. Rev. E 64, 046118+ (2001)CrossRefGoogle Scholar
  3. 3.
    Dempster, A.P.: Covariance selection. Biometrics 28, 95–108 (1972)CrossRefGoogle Scholar
  4. 4.
    Dorba, A., Hans, C., Jones, B., Nevins, J.R., West, M.: Sparse graphical models for exploring gene expression data. J. Multiv. Analysis 90, 196–212 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Erdös, P., Rényi, A.: On Random Graphs I. Pub. Math. Debrecen 6, 290–297 (1959)MATHGoogle Scholar
  6. 6.
    Jeong, H., Mason, S.P., Barabasi, A.L., Oltvai, Z.N.: Lethality and Centrality in Protein Networks. Nature 411, 41–42 (2001)CrossRefGoogle Scholar
  7. 7.
    Jones, B., Dobra, C., Carvalho, C., Hans, C., Carter, C., West, M.: Experiments in Stochastic Computation for High-Dimensional Graphical Models. Statistical Science 20(4), 388–400 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lee, D.-S., Goh, H.-I., Kahng, B., Kim, D.: Scale-Free Random Graphs and Potts Model. Pramana J. Phys. 64, 1149–1159 (2005)CrossRefGoogle Scholar
  9. 9.
    Newman, M., Barabási, A.-L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press, Princeton (2006)MATHGoogle Scholar
  10. 10.
    Schäfer, J., Strimmer, K.: An Empirical Bayes Approach to Inferring Large-Scale Gene Association Networks. Bioinformatics 21, 754–764 (2005a)CrossRefGoogle Scholar
  11. 11.
    Schäfer, J., Strimmer, K.: A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics. Stat. Appl. Genet. Mol. Biol. 4, article 32 (2005b)Google Scholar
  12. 12.
    Sheridan, P., Kamimura, T., Shimodaira, H.: Scale-Free Networks in Bayesian Inference with Applications to Bioinformatics. In: Proceedings of The International Workshop on Data-Mining and Statistical Science (DMSS 2007), Tokyo, pp. 1–16 (2007)Google Scholar
  13. 13.
    Sheridan, P., Yagahara, Y., Shimodaira, H.: A Preferential Attachment Model with Poisson Growth for Scale-Free Networks. Ann. Inst. Stat. Math. 60 (2008)Google Scholar
  14. 14.
    Solé, R.V., Pastor-Satorras, R., Smith, E., Kepler, T.B.: A Model of Large-Scale Proteome Evolution. Adv. Complex Systems 5, 43–54 (2002)CrossRefMATHGoogle Scholar
  15. 15.
    Vandewalle, N., Brisbois, F., Tordoir, X.: Non-Random Topology of Stock Markets. Quant. Finance 1, 372–374 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wagner, A.: How the Global Structure of Protein Interaction Networks Evolve. Proc. R. Soc. B 270, 457–466 (2003)CrossRefGoogle Scholar
  17. 17.
    Werhli, A.V.V., Grzegorczyk, M., Husmeier, D.: Comparative Evaluation of Reverse Engineering Gene Regulatory Networks with Relevance Networks, Graphical Gaussian Models and Bayesian Networks. Bioinformatics 22, 2523–2531 (2006)CrossRefGoogle Scholar
  18. 18.
    Wong, K., Carter, C., Kohn, R.: Efficient Estimation of Covariance Selection Models. Biometrika 90, 809–830 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering 2009

Authors and Affiliations

  • Paul Sheridan
    • 1
  • Takeshi Kamimura
    • 1
  • Hidetoshi Shimodaira
    • 1
  1. 1.Tokyo Institute of TechnologyMeguro-ku, TokyoJapan

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