Contagion processes and their statistics

The spread of quantities of things - earthquakes/diseases/innovations/credit defaults/cat videos - between different georegions/populations/vertices/banks/variates. For internet content virality in particular, there is much more specialized analysis and particular data sets, so I recommend checking the richer models under media virality.

In my own internal taxonomy growth in a single scalar value I woudl model using branching processes. Here I am concerned with modelling contagion between different variates. This distinction is arbitrary.

For now this is a mere collection of research links.

I’ll annotate a couple of useful models here, and hopefully talk about identifiability and noisy/incomplete data issues, which are especially interesting in this context.

Dirichlet Hawkes process

I don’t know anything about these, but have had them referred to me as a plausible multivariate something something. See (J. C. L. Pinto and Chahed 2014; S.-H. Yang and Zha 2013).

Misc incoming


Achab, Massil, Emmanuel Bacry, Stéphane Gaïffas, Iacopo Mastromatteo, and Jean-Francois Muzy. 2017. “Uncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR.
Ahmed, E., and A. S. Elgazzar. 2007. “On Fractional Order Differential Equations Model for Nonlocal Epidemics.” Physica A: Statistical Mechanics and Its Applications 379 (2): 607–14.
Amini, Hamed, Rama Cont, and Andreea Minca. 2013. “Resilience to Contagion in Financial Networks.” Mathematical Finance, October, n/a–.
Aragón, Tomás J. 2012. Applied Epidemiology Using R. MedEpi Publishing. http://www. medepi. net/epir/index. html. Calendar Time. Accessed.
Aral, Sinan, Lev Muchnik, and Arun Sundararajan. 2009. “Distinguishing Influence-Based Contagion from Homophily-Driven Diffusion in Dynamic Networks.” Proceedings of the National Academy of Sciences 106 (51): 21544–49.
Azizpour, Shariar, Kay Giesecke, and others. 2008. “Self-Exciting Corporate Defaults: Contagion Vs. Frailty.” Stanford University working paper series.
Bacry, Emmanuel, Martin Bompaire, Stéphane Gaïffas, and Jean-Francois Muzy. 2020. “Sparse and Low-Rank Multivariate Hawkes Processes.” Journal of Machine Learning Research 21 (50): 1–32.
Bacry, Emmanuel, and Jean-François Muzy. 2016. “First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory 62 (4): 2184–2202.
Barnett, Lionel, Adam B. Barrett, and Anil K. Seth. 2009. “Granger Causality and Transfer Entropy Are Equivalent for Gaussian Variables.” Physical Review Letters 103 (23): 238701.
Barrett, Adam B, Lionel Barnett, and Anil K Seth. 2010. “Multivariate Granger Causality and Generalized Variance.” Phys. Rev. E 81 (4): 041907.
Battey, Heather, and Alessio Sancetta. 2013. “Conditional Estimation for Dependent Functional Data.” Journal of Multivariate Analysis 120 (September): 1–17.
Brault, Romain, Néhémy Lim, and Florence d’Alché-Buc. n.d. “Scaling up Vector Autoregressive Models With Operator-Valued Random Fourier Features.” Accessed August 31, 2016.
Burridge, James. 2013a. “Cascade Sizes in a Branching Process with Gamma Distributed Generations.” arXiv:1304.3741 [math], April.
———. 2013b. “Crossover Behavior in Driven Cascades.” Physical Review E 88 (3): 032124.
Cauchemez, Simon, and Neil M. Ferguson. 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): 885–97.
Centola, D, and Michael W Macy. 2007. “Complex Contagions and the Weakness of Long Ties.” American Journal of Sociology 113 (3): 702.
Chen, Elynn Y., Jianqing Fan, and Xuening Zhu. 2020. “Community Network Auto-Regression for High-Dimensional Time Series.” arXiv:2007.05521 [stat], July.
Chongsuvivatwong, Virasakdi. 2008. Analysis of Epidemiological Data Using R and Epicalc. Book Unit, Faculty of Medicine, Prince of Songkla University Thailand.
Cook, Alex R., Wilfred Otten, Glenn Marion, Gavin J. Gibson, and Christopher A. Gilligan. 2007. “Estimation of Multiple Transmission Rates for Epidemics in Heterogeneous Populations.” Proceedings of the National Academy of Sciences 104 (51): 20392–97.
Dahlhaus, Rainer, and Michael Eichler. 2003. “Causality and Graphical Models in Time Series Analysis.” Oxford Statistical Science Series, 115–37.
Daneshmand, Hadi, Manuel Gomez-Rodriguez, Le Song, and Bernhard Schölkopf. 2014. “Estimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-Thresholding Algorithm.” In ICML.
Davis, Richard A., Pengfei Zang, and Tian Zheng. 2012. “Sparse Vector Autoregressive Modeling.” arXiv:1207.0520 [stat], July.
Dodds, Peter Sheridan. 2017. “Slightly Generalized Generalized Contagion: Unifying Simple Models of Biological and Social Spreading.” arXiv:1708.09697 [physics], August.
Du, Nan, Hanjun Dai, Rakshit Trivedi, Utkarsh Upadhyay, Manuel Gomez-Rodriguez, and Le Song. 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, 1555–64. KDD ’16. New York, NY, USA: ACM.
Du, Nan, Le Song, Manuel Gomez-Rodriguez, and Hongyuan Zha. 2013. “Scalable Influence Estimation in Continuous-Time Diffusion Networks.” In Advances in Neural Information Processing Systems, 3147–55.
Du, Nan, Le Song, Ming Yuan, and Alex J. Smola. 2012. “Learning Networks of Heterogeneous Influence.” In Advances in Neural Information Processing Systems, 2780–88.
Eichler, Michael. 2001. “Granger-Causality Graphs for Multivariate Time Series.” Granger-Causality Graphs for Multivariate Time Series.
———. 2007. “Granger Causality and Path Diagrams for Multivariate Time Series.” Journal of Econometrics 137 (2): 334–53.
Eichler, Michael, Rainer Dahlhaus, and Johannes Dueck. 2016. “Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis, January, n/a–.
Ferland, René, Alain Latour, and Driss Oraichi. 2006. “Integer-Valued GARCH Process.” Journal of Time Series Analysis 27 (6): 923–42.
Gelper, Sarah, Ines Wilms, and Christophe Croux. 2016. “Identifying Demand Effects in a Large Network of Product Categories.” Journal of Retailing 92 (1): 25–39.
Glasserman, Paul, and H. Peyton Young. 2016. “Contagion in Financial Networks.” Journal of Economic Literature 54 (3): 779–831.
Gomez-Rodriguez, Manuel, Jure Leskovec, David Balduzzi, and Bernhard Schölkopf. 2014. “Uncovering the Structure and Temporal Dynamics of Information Propagation.” Network Science 2 (01): 26–65.
Gomez-Rodriguez, Manuel, Jure Leskovec, and Bernhard Schölkopf. 2013. “Structure and Dynamics of Information Pathways in Online Media.” In Proceedings of the Sixth ACM International Conference on Web Search and Data Mining, 23–32. WSDM ’13. New York, NY, USA: ACM.
Granger, Clive W J. 1963. “Economic Processes Involving Feedback.” Information and Control 6 (1): 28–48.
———. 2003. “Time Series Concepts for Conditional Distributions*.” Oxford Bulletin of Economics and Statistics 65 (s1): 689–701.
———. 2004. “Time Series Analysis, Cointegration, and Applications.” American Economic Review, 421–25.
Greenhill, Catherine, Mikhail Isaev, Matthew Kwan, and Brendan D. McKay. 2016. “The Average Number of Spanning Trees in Sparse Graphs with Given Degrees.” arXiv:1606.01586 [math], June.
Greenland, Sander, Judea Pearl, and James M Robins. 1999. “Causal Diagrams for Epidemiologic Research.” Epidemiology 10 (1): 37.
Guille, Adrien, Hakim Hacid, Cecile Favre, and Djamel A. Zighed. 2013. “Information Diffusion in Online Social Networks: A Survey.” SIGMOD Rec. 42 (2): 17–28.
Hartikainen, J., and S. Särkkä. 2010. “Kalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models.” In 2010 IEEE International Workshop on Machine Learning for Signal Processing, 379–84. Kittila, Finland: IEEE.
Haufe, Stefan, Klaus-Robert Müller, Guido Nolte, and Nicole Krämer. 2008. “Sparse Causal Discovery in Multivariate Time Series.” In Proceedings of the 2008th International Conference on Causality: Objectives and Assessment - Volume 6, 97–106. COA’08. Whistler, Canada:
Hoffmann, Till, Leto Peel, Renaud Lambiotte, and Nick S. Jones. 2020. “Community Detection in Networks Without Observing Edges.” Science Advances 6 (4): eaav1478.
Hsu, Nan-Jung, Hung-Lin Hung, and Ya-Mei Chang. 2008. “Subset Selection for Vector Autoregressive Processes Using Lasso.” Computational Statistics & Data Analysis 52 (7): 3645–57.
Iribarren, José Luis, and Esteban Moro. 2011. “Branching Dynamics of Viral Information Spreading.” Physical Review E 84 (4): 046116.
Iyengar, Raghuram, Christophe Van den Bulte, and Thomas W. Valente. 2011. “Opinion Leadership and Social Contagion in New Product Diffusion.” Marketing Science 30 (2): 195–212.
Khim, Justin, Varun Jog, and Po-Ling Loh. 2016. “Computationally Efficient Influence Maximization in Stochastic and Adversarial Models: Algorithms and Analysis.” arXiv:1611.00350 [cs, Stat], November.
Kiss, István Z., Joel Miller, and Péter L. Simon. 2017. Mathematics of Epidemics on Networks: From Exact to Approximate Models. Interdisciplinary Applied Mathematics. New York, NY: Springer International Publishing.
Koopman, S. J., and J. Durbin. 2000. “Fast Filtering and Smoothing for Multivariate State Space Models.” Journal of Time Series Analysis 21 (3): 281–96.
Kramer, A. D. I., J. E. Guillory, and J. T. Hancock. 2014. “Experimental Evidence of Massive-Scale Emotional Contagion Through Social Networks.” Proceedings of the National Academy of Sciences 111 (24): 8788–90.
Kraus, Andrea, and Victor M. Panaretos. 2014. “Frequentist Estimation of an Epidemic’s Spreading Potential When Observations Are Scarce.” Biometrika 101 (1): 141–54.
Lakshmanan, Karthik C., Patrick T. Sadtler, Elizabeth C. Tyler-Kabara, Aaron P. Batista, and Byron M. Yu. 2015. “Extracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays.” Neural Computation 27 (9): 1825–56.
Lamprier, Sylvain. 2019. “A Recurrent Neural Cascade-Based Model for Continuous-Time Diffusion.” In International Conference on Machine Learning, 3632–41.
Li, Liangda, and Hongyuan Zha. 2014. “Learning Parametric Models for Social Infectivity in Multi-Dimensional Hawkes Processes.” In Twenty-Eighth AAAI Conference on Artificial Intelligence.
Li, Yuanzhi, Yingyu Liang, and Andrej Risteski. 2016. “Recovery Guarantee of Non-Negative Matrix Factorization via Alternating Updates.” In Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 4988–96. Curran Associates, Inc.
Liu, Ka-Yuet, Marissa King, and Peter S. Bearman. 2010. “Social Influence and the Autism Epidemic.” American Journal of Sociology 115 (5): 1387.
Morozova, Olga, Ted Cohen, and Forrest W. Crawford. 2018. “Risk Ratios for Contagious Outcomes.” Journal of The Royal Society Interface 15 (138): 20170696.
Ogata, Yosihiko, Ritsuko S. Matsu’ura, and Koichi Katsura. 1993. “Fast Likelihood Computation of Epidemic Type Aftershock-Sequence Model.” Geophysical Research Letters 20 (19): 2143–46.
Pinto, Julio Cesar Louzada, and Tijani Chahed. 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, 339–46. SITIS ’14. Washington, DC, USA: IEEE Computer Society.
Pinto, Pedro C., Patrick Thiran, and Martin Vetterli. 2012. “Locating the Source of Diffusion in Large-Scale Networks.” Physical Review Letters 109 (6): 068702.
Pouget-Abadie, Jean, and Thibaut Horel. 2015. “Inferring Graphs from Cascades: A Sparse Recovery Framework.” In Proceedings of The 32nd International Conference on Machine Learning.
Raissi, Maziar, and George Em Karniadakis. 2017. “Machine Learning of Linear Differential Equations Using Gaussian Processes.” arXiv:1701.02440 [cs, Math, Stat], January.
Rizoiu, Marian-Andrei, and Lexing Xie. 2017. “Online Popularity Under Promotion: Viral Potential, Forecasting, and the Economics of Time.” arXiv:1703.01012 [cs], March.
Rizoiu, Marian-Andrei, Lexing Xie, Scott Sanner, Manuel Cebrian, Honglin Yu, and Pascal Van Hentenryck. 2017. “Expecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on, 1–9. WWW ’17. Perth, Australia: International World Wide Web Conferences Steering Committee.
Roca, Carlos P, Moez Draief, and Dirk Helbing. 2011. “Percolate or Die: Multi-Percolation Decides the Struggle Between Competing Innovations.”
Saichev, A., and D. Sornette. 2011a. “Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.1611 [cond-Mat, Physics:physics], January.
———. 2011b. “Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [cond-Mat, Physics:physics], January.
Särkkä, Simo, A. Solin, and J. Hartikainen. 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.
Schein, Aaron, Hanna Wallach, and Mingyuan Zhou. 2016. “Poisson-Gamma Dynamical Systems.” In Advances In Neural Information Processing Systems, 5006–14.
Shalizi, Cosma Rohilla, and Andrew C. Thomas. 2011. “Homophily and Contagion Are Generically Confounded in Observational Social Network Studies.” Sociological Methods & Research 40 (2): 211–39.
Shen, Yanning, Brian Baingana, and Georgios B. Giannakis. 2016. “Nonlinear Structural Vector Autoregressive Models for Inferring Effective Brain Network Connectivity.” arXiv:1610.06551 [stat], October.
Shin, Minjeong, Alasdair Tran, Siqi Wu, Alexander Mathews, Rong Wang, Georgiana Lyall, and Lexing Xie. 2021. “AttentionFlow: Visualising Influence in Networks of Time Series.” In Proceedings of the 14th ACM International Conference on Web Search and Data Mining, 1085–88.
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