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Dimension reduction in vertex-weighted exponential random graphs

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  • DeMuse, Ryan
  • Yin, Mei

Abstract

We investigate the behavior of vertex-weighted exponential random graphs. We show that vertex-weighted exponential random graphs with edge weights induced by products of independent vertex weights are approximate mixtures of graphs whose vertex weight vector is a near fixed point of a certain vector equation. For graphs with Hamiltonians counting cliques, it is demonstrated that, under appropriate conditions, every solution to this equation is close to a block vector with a small number of communities. We prove that for the cases of positive weights and small weights in the Hamiltonian in particular, the vector equation has a unique solution. Lastly, the behavior of vertex-weighted exponential random graphs counting triangles is studied in detail and the solution to the vector equation is shown to approach the zero vector as the weight diverges to negative infinity for sufficiently large networks.

Suggested Citation

  • DeMuse, Ryan & Yin, Mei, 2021. "Dimension reduction in vertex-weighted exponential random graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 561(C).
    • Handle: RePEc:eee:phsmap:v:561:y:2021:i:c:s0378437120306804
    DOI: 10.1016/j.physa.2020.125289
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    References listed on IDEAS

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    1. Cranmer, Skyler J. & Desmarais, Bruce A., 2011. "Inferential Network Analysis with Exponential Random Graph Models," Political Analysis, Cambridge University Press, vol. 19(1), pages 66-86, January.
    2. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
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