The Living Thing / Notebooks :

Contagion processes and their statistics

The spread of quantities of things - earthquakes/diseases/innovations/credit defaults - between different georegions/populations/vertices/banks/variates.

In my own internal taxonomy, I file contagion - growth within a single population as branching processes, and care about multivariate issues here.

For now this is a mere collection of contagion research links because I am hunting for data sets; no fancy analysis for the moment.

I’ll annotate a couple of useful models here, and hopefully talk about identifiability and noisy/incomplete data issues using a graphical model formalism.

Dirichlet Hawkes process

I don’t know anything about these, but have had them referred to me as a plausible multivariate something something. See DFAS15, YaZh13, PiCh14 and compare with the Dirichlet Poisson process.


Ahmed, E., & Elgazzar, A. S.(2007) On fractional order differential equations model for nonlocal epidemics. Physica A: Statistical Mechanics and Its Applications, 379(2), 607–614. DOI.
Amini, H., Cont, R., & Minca, A. (2013) Resilience to Contagion in Financial Networks. Mathematical Finance, n/a-n/a. DOI.
Aragón, T. J.(2012) Applied epidemiology using R. . MedEpi Publishing. http://www. medepi. net/epir/index. html. Calendar Time. Accessed
Aral, S., Muchnik, L., & Sundararajan, A. (2009) Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks. Proceedings of the National Academy of Sciences, 106(51), 21544–21549. DOI.
Azizpour, S., Giesecke, K., & others. (2008) Self-exciting corporate defaults: contagion vs frailty. . Stanford University working paper series
Bacry, E., & Muzy, J.-F. (2014) Second order statistics characterization of Hawkes processes and non-parametric estimation. arXiv:1401.0903 [Physics, Q-Fin, Stat].
Barnett, L., Barrett, A. B., & Seth, A. K.(2009) Granger causality and transfer entropy are equivalent for Gaussian variables. Physical Review Letters, 103(23), 238701.
Barrett, A. B., Barnett, L., & Seth, A. K.(2010) Multivariate Granger causality and generalized variance. Phys. Rev. E, 81(4), 041907. DOI.
Battey, H., & Sancetta, A. (2013) Conditional estimation for dependent functional data. Journal of Multivariate Analysis, 120, 1–17. DOI.
Brault, R., Lim, N., & d’Alché-Buc, F. (n.d.) Scaling up Vector Autoregressive Models With Operator-Valued Random Fourier Features.
Caccioli, F., Shrestha, M., Moore, C., & Farmer, J. D.(2012) Stability analysis of financial contagion due to overlapping portfolios. arXiv:1210.5987.
Cauchemez, S., & Ferguson, N. M.(2008) Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in London. Journal of The Royal Society Interface, 5(25), 885–897. DOI.
Centola, D., & Macy, M. W.(2007) Complex contagions and the weakness of long ties. American Journal of Sociology, 113(3), 702.
Chongsuvivatwong, V. (2008) Analysis of epidemiological data using R and Epicalc. . Book Unit, Faculty of Medicine, Prince of Songkla University Thailand
Cook, A. R., Otten, W., Marion, G., Gibson, G. J., & Gilligan, C. A.(2007) Estimation of multiple transmission rates for epidemics in heterogeneous populations. Proceedings of the National Academy of Sciences, 104(51), 20392–20397. DOI.
Dahlhaus, R., & Eichler, M. (2003) Causality and graphical models in time series analysis. Oxford Statistical Science Series, 115–137.
Daneshmand, H., Gomez-Rodriguez, M., Song, L., & Schölkopf, B. (2014) Estimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-thresholding Algorithm. In ICML.
Du, N., Dai, H., Trivedi, R., Upadhyay, U., Gomez-Rodriguez, M., & Song, L. (2016) Recurrent Marked Temporal Point Processes: Embedding Event History to Vector. In Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1555–1564). New York, NY, USA: ACM DOI.
Du, N., Farajtabar, M., Ahmed, A., Smola, A. J., & Song, L. (2015) Dirichlet-Hawkes Processes with Applications to Clustering Continuous-Time Document Streams. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 219–228). New York, NY, USA: ACM DOI.
Du, N., Song, L., Gomez-Rodriguez, M., & Zha, H. (2013) Scalable influence estimation in continuous-time diffusion networks. In Advances in neural information processing systems (pp. 3147–3155).
Du, N., Song, L., Yuan, M., & Smola, A. J.(2012) Learning networks of heterogeneous influence. In Advances in Neural Information Processing Systems (pp. 2780–2788).
Eichler, M. (2000) Graphical Models in Time Series Analysis.
Eichler, Michael. (2001) Granger-causality graphs for multivariate time series. Granger-Causality Graphs for Multivariate Time Series.
Eichler, Michael. (2007) Granger causality and path diagrams for multivariate time series. Journal of Econometrics, 137(2), 334–353. DOI.
Eichler, Michael, Dahlhaus, R., & Dueck, J. (2016) Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions. Journal of Time Series Analysis, n/a-n/a. DOI.
Ferland, R., Latour, A., & Oraichi, D. (2006) Integer-Valued GARCH Process. Journal of Time Series Analysis, 27(6), 923–942. DOI.
Gibson, G. J., & Renshaw, E. (2001) Likelihood estimation for stochastic compartmental models using Markov chain methods. Statistics and Computing, 11(4), 347–358. DOI.
Glasserman, P., & Young, H. P.(2016) Contagion in Financial Networks. Journal of Economic Literature, 54(3), 779–831. DOI.
Gomez-Rodriguez, M., Leskovec, J., Balduzzi, D., & Schölkopf, B. (2014) Uncovering the structure and temporal dynamics of information propagation. Network Science, 2(01), 26–65. DOI.
Gomez-Rodriguez, M., Leskovec, J., & Schölkopf, B. (2013) Structure and Dynamics of Information Pathways in Online Media. In Proceedings of the Sixth ACM International Conference on Web Search and Data Mining (pp. 23–32). New York, NY, USA: ACM DOI.
Granger, C. W. J.(1963) Economic processes involving feedback. Information and Control, 6(1), 28–48. DOI.
Granger, C. W. J.(2004) Time series analysis, cointegration, and applications. American Economic Review, 421–425.
Greenhill, C., Isaev, M., Kwan, M., & McKay, B. D.(2016) The average number of spanning trees in sparse graphs with given degrees. arXiv:1606.01586 [Math].
Greenland, S., Pearl, J., & Robins, J. M.(1999) Causal diagrams for epidemiologic research. Epidemiology, 10(1), 37.
Guille, A., Hacid, H., Favre, C., & Zighed, D. A.(2013) Information Diffusion in Online Social Networks: A Survey. SIGMOD Rec., 42(2), 17–28. DOI.
Hartikainen, J., & Särkkä, S. (2010) Kalman filtering and smoothing solutions to temporal Gaussian process regression models. In 2010 IEEE International Workshop on Machine Learning for Signal Processing (pp. 379–384). DOI.
Iribarren, J. L., & Moro, E. (2011) Branching dynamics of viral information spreading. Physical Review E, 84(4), 046116. DOI.
Iyengar, R., Van den Bulte, C., & Valente, T. W.(2011) Opinion leadership and social contagion in new product diffusion. Marketing Science, 30(2), 195–212. DOI.
Koopman, S. J., & Durbin, J. (2000) Fast Filtering and Smoothing for Multivariate State Space Models. Journal of Time Series Analysis, 21(3), 281–296. DOI.
Kramer, A. D. I., Guillory, J. E., & Hancock, J. T.(2014) Experimental evidence of massive-scale emotional contagion through social networks. Proceedings of the National Academy of Sciences, 111(24), 8788–8790. DOI.
Kraus, A., & Panaretos, V. M.(2014) Frequentist estimation of an epidemic’s spreading potential when observations are scarce. Biometrika, 101(1), 141–154. DOI.
Lakshmanan, K. C., Sadtler, P. T., Tyler-Kabara, E. C., Batista, A. P., & Yu, B. M.(2015) Extracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays. Neural Computation, 27(9), 1825–1856. DOI.
Li, Y., Liang, Y., & Risteski, A. (2016) Recovery Guarantee of Non-negative Matrix Factorization via Alternating Updates. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, & R. Garnett (Eds.), Advances in Neural Information Processing Systems 29 (pp. 4988–4996). Curran Associates, Inc.
Liu, K.-Y., King, M., & Bearman, P. S.(2010) Social influence and the autism epidemic. American Journal of Sociology, 115(5), 1387.
Ogata, Y., Matsu’ura, R. S., & Katsura, K. (1993) Fast likelihood computation of epidemic type aftershock-sequence model. Geophysical Research Letters, 20(19), 2143–2146. DOI.
Pinto, J. C. L., & Chahed, T. (2014) Modeling Multi-topic Information Diffusion in Social Networks Using Latent Dirichlet Allocation and Hawkes Processes. In Proceedings of the 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems (pp. 339–346). Washington, DC, USA: IEEE Computer Society DOI.
Pinto, P. C., Thiran, P., & Vetterli, M. (2012) Locating the Source of Diffusion in Large-Scale Networks. Physical Review Letters, 109(6), 068702. DOI.
Pooseh, S., Rodrigues, H. S., & Torres, D. F. M.(2011) Fractional derivatives in Dengue epidemics. arXiv:1108.1683 [Math, Q-Bio], 739–742. DOI.
Pouget-Abadie, J., & Horel, T. (2015) Inferring Graphs from Cascades: A Sparse Recovery Framework. In Proceedings of The 32nd International Conference on Machine Learning.
Rasmussen, J. G., Møller, J., Aukema, B. H., Raffa, K. F., & Zhu, J. (2007) Continuous time modelling of dynamical spatial lattice data observed at sparsely distributed times. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69(4), 701–713. DOI.
Roca, C. P., Draief, M., & Helbing, D. (2011) Percolate or die: Multi-percolation decides the struggle between competing innovations.
Saichev, A., & Sornette, D. (2011a) Generating functions and stability study of multivariate self-excited epidemic processes. arXiv:1101.5564 [Cond-Mat, Physics:physics].
Saichev, A., & Sornette, D. (2011b) Hierarchy of temporal responses of multivariate self-excited epidemic processes. arXiv:1101.1611 [Cond-Mat, Physics:physics].
Särkkä, S., Solin, A., & Hartikainen, J. (2013) Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering. IEEE Signal Processing Magazine, 30(4), 51–61. DOI.
Schoenberg, F. P., Chu, A., & Veen, A. (2010) On the relationship between lower magnitude thresholds and bias in epidemic-type aftershock sequence parameter estimates. Journal of Geophysical Research: Solid Earth, 115(B4), B04309. DOI.
Shalizi, C. R., & Thomas, A. C.(2011) Homophily and Contagion Are Generically Confounded in Observational Social Network Studies. Sociological Methods & Research, 40(2), 211–239. DOI.
Shen, Y., Baingana, B., & Giannakis, G. B.(2016) Nonlinear Structural Vector Autoregressive Models for Inferring Effective Brain Network Connectivity. arXiv:1610.06551 [Stat].
Solé-Ribalta, A., De Domenico, M., Kouvaris, N. E., Díaz-Guilera, A., Gómez, S., & Arenas, A. (2013) Spectral properties of the Laplacian of multiplex networks. Physical Review E, 88(3), 032807. DOI.
Sornette, D. (2005) Constraints on the size of the smallest triggering earthquake from the epidemic-type aftershock sequence model, Båth’s law, and observed aftershock sequences. Journal of Geophysical Research, 110(B8). DOI.
Stegehuis, C., van der Hofstad, R., & van Leeuwaarden, J. S. H.(2016) Epidemic spreading on complex networks with community structures. Scientific Reports, 6, 29748. DOI.
Stiglitz, J. E.(2010) Contagion, liberalization, and the optimal structure of globalization. Journal of Globalization and Development, 1(2), 2. DOI.
Sugihara, G., May, R., Ye, H., Hsieh, C., Deyle, E., Fogarty, M., & Munch, S. (2012) Detecting Causality in Complex Ecosystems. Science, 338(6106), 496–500. DOI.
Verma, I. M.(2014) Editorial Expression of Concern: Experimental evidence of massivescale emotional contagion through social networks. Proceedings of the National Academy of Sciences, 201412469. DOI.
Wang, Y., Xie, B., Du, N., Le Song, E. D. U., & EDU, G. (n.d.) Isotonic Hawkes Processes.
Watts, D. J., & Dodds, P. S.(2007) Influentials, Networks, and Public Opinion Formation. Journal of Consumer Research, 34(4), 441–458. DOI.
Yang, D.-P., Lin, H., Wu, C.-X., & Shuai, J. (2011) Topological conditions of scale-free networks for cooperation to evolve. arXiv:1106.5386.
Yang, S.-H., & Zha, H. (2013) Mixture of Mutually Exciting Processes for Viral Diffusion. In Proceedings of The 30th International Conference on Machine Learning (Vol. 28, pp. 1–9).
Young, H. P.(2009) Innovation diffusion in heterogeneous populations: Contagion, social influence, and social learning. The American Economic Review, 99(5), 1899–1924.
Zhou, K., Zha, H., & Song, L. (2013) Learning triggering kernels for multi-dimensional Hawkes processes. In Proceedings of the 30th International Conference on Machine Learning (ICML-13) (pp. 1301–1309).