Learning of Weighted Multi-layer Networks via Dynamic Social Spaces, with Application to Financial Interbank Transactions

  • Chris U. CarmonaEmail author
  • Serafin Martinez-Jaramillo
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 882)


We propose a general network model suited for longitudinal data of multi-layer networks with directed and weighted edges. Our formulation built upon the latent social space representation of networks. It consists of a hierarchical formulation: deep levels of the model represent latent coordinates of agents in the social space, evolving in continuous time via Gaussian Processes; meanwhile, top levels jointly manage incidence and strength of interactions by considering a Zero-Inflated Gaussian response. Learning of the model is performed through Bayesian Inference. We develop an efficient MCMC algorithm targeting the posterior distribution of model parameters and missing data (available in GitHub). The motivation for our model lies in the context of Financial Networks, specifically the analysis of transactions between commercial banks. We evaluate the model in synthetic data, as well as our main case study: the network of inter-bank transactions in the Mexican financial system. Accurate predictions are obtained in both cases estimating out-of-sample link incidence and link strength.


  1. Battiston, S., Martinez-Jaramillo, S.: Financial networks and stress testing: challenges and new research avenues for systemic risk analysis and financial stability implications. J. Financ. Stab. 35, 6–16 (2018)CrossRefGoogle Scholar
  2. Crane, H.: Probabilistic Foundations of Statistical Network Analysis, 1st edn. Chapman and Hall/CRC (2018)Google Scholar
  3. de la Concha, A., Martinez-Jaramillo, S., Carmona, C.: Multiplex financial networks: revealing the level of interconnectedness in the banking system. In: Complex Networks & Their Applications VI, pp. 1135–1148. Springer (2018)Google Scholar
  4. Durante, D., Dunson, D.B.: Bayesian dynamic financial networks with time-varying predictors. Stat. Probab. Lett. 93, 19–26 (2014a)MathSciNetCrossRefGoogle Scholar
  5. Durante, D., Dunson, D.B.: Nonparametric Bayes dynamic modelling of relational data. Biometrika 101(4), 883–898 (2014b)MathSciNetCrossRefGoogle Scholar
  6. Durante, D., Dunson, D.B.: Locally adaptive dynamic networks. Ann. Appl. Stat. 10(4), 2203–2232 (2016)MathSciNetCrossRefGoogle Scholar
  7. Durante, D., Mukherjee, N., Steorts, R.C.: Bayesian learning of dynamic multilayer networks. J. Mach. Learn. Res. 18, 1–29 (2017)MathSciNetzbMATHGoogle Scholar
  8. Hoff, P.D.: Additive and multiplicative effects network models (2018)Google Scholar
  9. Hoff, P.D., Raftery, A.E., Handcock, M.S.: Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97(460), 1090–1098 (2002)MathSciNetCrossRefGoogle Scholar
  10. Kim, B., Lee, K.H., Xue, L., Niu, X.: A review of dynamic network models with latent variables. Stat. Surv. 12, 105–135 (2018a)MathSciNetCrossRefGoogle Scholar
  11. Kim, B., Niu, X., Hunter, D.R., Cao, X.: A dynamic additive and multiplicative effects model with application to the united nations voting behaviors (2018b)Google Scholar
  12. Linardi, F., Diks, C.G.H., van der Leij, M., Lazier, I.: Dynamic interbank network analysis using latent space models. SSRN Electron. J. (2017) Google Scholar
  13. Molina-Borboa, J., Martínez-Jaramillo, S., Lopez-Gallo, F.: A multiplex network analysis of the Mexican banking system: link persistence, overlap. J. Netw. Theory Finance 1(1), 99–138 (2015)CrossRefGoogle Scholar
  14. Poledna, S., Molina-Borboa, J.L., Martínez-Jaramillo, S., van der Leij, M., Thurner, S.: The multi-layer network nature of systemic risk and its implications for the costs of financial crises. J. Financ. Stab. 20, 70–81 (2015)CrossRefGoogle Scholar
  15. Polson, N.G., Scott, J.G., Windle, J.: Bayesian inference for logistic models using Pólya-Gamma latent variables. J. Am. Stat. Assoc. 108(504), 1339–1349 (2013)CrossRefGoogle Scholar
  16. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press (2005)Google Scholar
  17. Sewell, D.K., Chen, Y.: Latent space models for dynamic networks. J. Am. Stat. Assoc. 110(512), 1646–1657 (2015)MathSciNetCrossRefGoogle Scholar
  18. Sewell, D.K., Chen, Y.: Latent space models for dynamic networks with weighted edges. Soc. Netw. 44, 105–116 (2016)CrossRefGoogle Scholar
  19. Sewell, D.K., Chen, Y.: Latent space approaches to community detection in dynamic networks. Bayesian Anal. 12(2), 351–377 (2017)MathSciNetCrossRefGoogle Scholar
  20. Tran, D., Blei, D., Airoldi, E.M.: Copula variational inference. In: Cortes, C., Lawrence, N.D., Lee, D.D., Sugiyama, M., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 28, pp. 3564–3572. Curran Associates, Inc. (2015)Google Scholar
  21. Vehtari, A., Gelman, A., Gabry, J.: Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Stat. Comput. 27(5), 1413–1432 (2017)MathSciNetCrossRefGoogle Scholar
  22. Ward, M.D., Ahlquist, J.S., Rozenas, A.: Gravity’s rainbow: a dynamic latent space model for the world trade network. Netw. Sci. 1(01), 95–118 (2013)CrossRefGoogle Scholar
  23. Watanabe, S.: Algebraic Geometry and Statistical Learning Theory. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press (2009)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Chris U. Carmona
    • 1
    • 3
    Email author
  • Serafin Martinez-Jaramillo
    • 2
    • 3
  1. 1.Department of StatisticsUniversity of OxfordOxfordUK
  2. 2.Center for Latin American Monetary StudiesMexico CityMexico
  3. 3.Banco de MexicoMexico CityMexico

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