Estimating Parameters of a Directed Weighted Graph Model with Beta-Distributed Edge-Weights

  • M. BollaEmail author
  • J. Mala
  • A. Elbanna

We introduce a directed, weighted random graph model, where the edge-weights are independent and beta distributed with parameters depending on their endpoints. We will show that the row- and column-sums of the transformed edge-weight matrix are sufficient statistics for the parameters, and use the theory of exponential families to prove that the ML estimate of the parameters exists and is unique. Then an algorithm to find this estimate is introduced together with convergence proof that uses properties of the digamma function. Simulation results and applications are also presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publishing, New York (1972).zbMATHGoogle Scholar
  2. 2.
    H. Alzer and J. Wells, “Inequalities for the polygamma functions,” SIAM J. Math. Anal., 29, No. 6, 1459–1466 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Bernardo, “Psi (digamma) function. Algorithm AS 103,” Appl. Stat., 25, 315–317 (1976).CrossRefGoogle Scholar
  4. 4.
    M. Bolla, Spectral Clustering and Biclustering, Wiley, New York (2013).CrossRefzbMATHGoogle Scholar
  5. 5.
    M. Bolla and A. Elbanna, “Estimating parameters of a probabilistic heterogeneous block model via the EM algorithm,” J. Probab. Stat., Article ID 657965 (2015).Google Scholar
  6. 6.
    S. Chatterjee, P. Diaconis, and A. Sly, “Random graphs with a given degree sequence,” Ann. Stat., 21, 1400–1435 (2010).MathSciNetzbMATHGoogle Scholar
  7. 7.
  8. 8.
    C. J. Hillar and A. Wibisono, “Maximum entropy distributions on graphs,” ArXiv, arXiv:1301.3321v2 (2013).Google Scholar
  9. 9.
    S. L. Lauritzen, Graphical Models, Oxford University Press (1995).Google Scholar
  10. 10.
    T. Yan, C. Leng, and J. Zhu, “Asymptotics in directed exponential random graph models with an increasing bi-degree sequence,” Ann. Stat., 44, 31–57 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Wainwright and M. I. Jordan, “Graphical models, exponential families, and variational inference,” Found. Trend. Mach. Learn., 1, No. 1–2, 1–305 (2008).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Institute of MathematicsELTE Eötvös Loránd UniversityBudapestHungary
  3. 3.Faculty of Science, Mathenmatics DepartmentTanta UniversityTantaEgypt

Personalised recommendations