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 would 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).


Achab, Massil, Emmanuel Bacry, StΓ©phane GaΓ―ffas, Iacopo Mastromatteo, and Jean-Francois Muzy. 2017. β€œUncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR.
Ahmed, E., and A. S. Elgazzar. 2007. β€œOn 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. β€œResilience to Contagion in Financial Networks.” Mathematical Finance, October, n/a–.
AragΓ³n, TomΓ‘s 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. β€œDistinguishing 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, et al. 2008. β€œSelf-Exciting Corporate Defaults: Contagion Vs.Β Frailty.” Stanford University working paper series.
Bacry, Emmanuel, Martin Bompaire, StΓ©phane GaΓ―ffas, and Jean-Francois Muzy. 2020. β€œSparse and Low-Rank Multivariate Hawkes Processes.” Journal of Machine Learning Research 21 (50): 1–32.
Bacry, Emmanuel, and Jean-FranΓ§ois Muzy. 2016. β€œFirst- 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. β€œGranger 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. β€œMultivariate Granger Causality and Generalized Variance.” Phys. Rev.Β E 81 (4): 041907.
Battey, Heather, and Alessio Sancetta. 2013. β€œConditional Estimation for Dependent Functional Data.” Journal of Multivariate Analysis 120 (September): 1–17.
Brault, Romain, NΓ©hΓ©my Lim, and Florence d’AlchΓ©-Buc. n.d. β€œScaling up Vector Autoregressive Models With Operator-Valued Random Fourier Features.”
Burridge, James. 2013a. β€œCascade Sizes in a Branching Process with Gamma Distributed Generations.” arXiv:1304.3741 [Math], April.
β€”β€”β€”. 2013b. β€œCrossover Behavior in Driven Cascades.” Physical Review E 88 (3): 032124.
Cauchemez, Simon, and Neil M. Ferguson. 2008. β€œLikelihood-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. β€œComplex Contagions and the Weakness of Long Ties.” American Journal of Sociology 113 (3): 702.
Chen, Elynn Y., Jianqing Fan, and Xuening Zhu. 2020. β€œCommunity 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. β€œEstimation 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. β€œCausality and Graphical Models in Time Series Analysis.” Oxford Statistical Science Series, 115–37.
Daneshmand, Hadi, Manuel Gomez-Rodriguez, Le Song, and Bernhard SchΓΆlkopf. 2014. β€œEstimating Diffusion Network Structures: Recovery Conditions, Sample Complexity & Soft-Thresholding Algorithm.” In ICML.
Davis, Richard A., Pengfei Zang, and Tian Zheng. 2012. β€œSparse Vector Autoregressive Modeling.” arXiv:1207.0520 [Stat], July.
Dodds, Peter Sheridan. 2017. β€œSlightly 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. β€œRecurrent 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. β€œScalable 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. β€œLearning Networks of Heterogeneous Influence.” In Advances in Neural Information Processing Systems, 2780–88.
Eichler, Michael. 2001. β€œGranger-Causality Graphs for Multivariate Time Series.” Granger-Causality Graphs for Multivariate Time Series.
β€”β€”β€”. 2007. β€œGranger Causality and Path Diagrams for Multivariate Time Series.” Journal of Econometrics 137 (2): 334–53.
Eichler, Michael, Rainer Dahlhaus, and Johannes Dueck. 2016. β€œGraphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis, January, n/a–.
Ferland, RenΓ©, Alain Latour, and Driss Oraichi. 2006. β€œInteger-Valued GARCH Process.” Journal of Time Series Analysis 27 (6): 923–42.
Gelper, Sarah, Ines Wilms, and Christophe Croux. 2016. β€œIdentifying Demand Effects in a Large Network of Product Categories.” Journal of Retailing 92 (1): 25–39.
Glasserman, Paul, and H. Peyton Young. 2016. β€œContagion in Financial Networks.” Journal of Economic Literature 54 (3): 779–831.
Gomez-Rodriguez, Manuel, Jure Leskovec, David Balduzzi, and Bernhard SchΓΆlkopf. 2014. β€œUncovering the Structure and Temporal Dynamics of Information Propagation.” Network Science 2 (01): 26–65.
Gomez-Rodriguez, Manuel, Jure Leskovec, and Bernhard SchΓΆlkopf. 2013. β€œStructure 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. β€œEconomic Processes Involving Feedback.” Information and Control 6 (1): 28–48.
β€”β€”β€”. 2003. β€œTime Series Concepts for Conditional Distributions*.” Oxford Bulletin of Economics and Statistics 65 (s1): 689–701.
β€”β€”β€”. 2004. β€œTime Series Analysis, Cointegration, and Applications.” American Economic Review, 421–25.
Greenhill, Catherine, Mikhail Isaev, Matthew Kwan, and Brendan D. McKay. 2016. β€œThe 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. β€œCausal Diagrams for Epidemiologic Research.” Epidemiology 10 (1): 37.
Guille, Adrien, Hakim Hacid, Cecile Favre, and Djamel A. Zighed. 2013. β€œInformation Diffusion in Online Social Networks: A Survey.” SIGMOD Rec. 42 (2): 17–28.
Hartikainen, J., and S. SΓ€rkkΓ€. 2010. β€œKalman 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 MΓΌller, Guido Nolte, and Nicole KrΓ€mer. 2008. β€œSparse 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. β€œCommunity Detection in Networks Without Observing Edges.” Science Advances 6 (4): eaav1478.
Hsu, Nan-Jung, Hung-Lin Hung, and Ya-Mei Chang. 2008. β€œSubset Selection for Vector Autoregressive Processes Using Lasso.” Computational Statistics & Data Analysis 52 (7): 3645–57.
Iribarren, JosΓ© Luis, and Esteban Moro. 2011. β€œBranching Dynamics of Viral Information Spreading.” Physical Review E 84 (4): 046116.
Iyengar, Raghuram, Christophe Van den Bulte, and Thomas W. Valente. 2011. β€œOpinion Leadership and Social Contagion in New Product Diffusion.” Marketing Science 30 (2): 195–212.
Khim, Justin, Varun Jog, and Po-Ling Loh. 2016. β€œComputationally Efficient Influence Maximization in Stochastic and Adversarial Models: Algorithms and Analysis.” arXiv:1611.00350 [Cs, Stat], November.
Kiss, IstvΓ‘n Z., Joel Miller, and PΓ©ter 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. β€œFast 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. β€œExperimental 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. β€œFrequentist 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. β€œExtracting Low-Dimensional Latent Structure from Time Series in the Presence of Delays.” Neural Computation 27 (9): 1825–56.
Lamprier, Sylvain. 2019. β€œA Recurrent Neural Cascade-Based Model for Continuous-Time Diffusion.” In International Conference on Machine Learning, 3632–41.
Li, Liangda, and Hongyuan Zha. 2014. β€œLearning 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. β€œRecovery 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. β€œSocial Influence and the Autism Epidemic.” American Journal of Sociology 115 (5): 1387.
Morozova, Olga, Ted Cohen, and Forrest W. Crawford. 2018. β€œRisk Ratios for Contagious Outcomes.” Journal of The Royal Society Interface 15 (138): 20170696.
Ogata, Yosihiko, Ritsuko S. Matsu’ura, and Koichi Katsura. 1993. β€œFast Likelihood Computation of Epidemic Type Aftershock-Sequence Model.” Geophysical Research Letters 20 (19): 2143–46.
Pinto, Julio Cesar Louzada, and Tijani Chahed. 2014. β€œModeling 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. β€œLocating the Source of Diffusion in Large-Scale Networks.” Physical Review Letters 109 (6): 068702.
Pouget-Abadie, Jean, and Thibaut Horel. 2015. β€œInferring Graphs from Cascades: A Sparse Recovery Framework.” In Proceedings of The 32nd International Conference on Machine Learning.
Raissi, Maziar, and George Em Karniadakis. 2017. β€œMachine Learning of Linear Differential Equations Using Gaussian Processes.” arXiv:1701.02440 [Cs, Math, Stat], January.
Rizoiu, Marian-Andrei, and Lexing Xie. 2017. β€œOnline 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. β€œExpecting 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. β€œPercolate or Die: Multi-Percolation Decides the Struggle Between Competing Innovations.”
Saichev, A., and D. Sornette. 2011a. β€œHierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.1611 [Cond-Mat, Physics:physics], January.
β€”β€”β€”. 2011b. β€œGenerating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics], January.
SΓ€rkkΓ€, Simo, A. Solin, and J. Hartikainen. 2013. β€œSpatiotemporal 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. β€œPoisson-Gamma Dynamical Systems.” In Advances In Neural Information Processing Systems, 5006–14.
Shalizi, Cosma Rohilla, and Andrew C. Thomas. 2011. β€œHomophily 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. β€œNonlinear 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. β€œAttentionFlow: Visualising Influence in Networks of Time Series.” In Proceedings of the 14th ACM International Conference on Web Search and Data Mining, 1085–88.
SolΓ©-Ribalta, A., M. De Domenico, N. E. Kouvaris, A. DΓ­az-Guilera, S. GΓ³mez, and A. Arenas. 2013. β€œSpectral Properties of the Laplacian of Multiplex Networks.” Physical Review E 88 (3): 032807.
Sornette, Didier. 2005. β€œConstraints on the Size of the Smallest Triggering Earthquake from the Epidemic-Type Aftershock Sequence Model, BΓ₯th’s Law, and Observed Aftershock Sequences.” Journal of Geophysical Research 110 (B8).
Stegehuis, Clara, Remco van der Hofstad, and Johan S. H. van Leeuwaarden. 2016. β€œEpidemic Spreading on Complex Networks with Community Structures.” Scientific Reports 6 (July): 29748.
Stiglitz, Joseph E. 2010. β€œContagion, Liberalization, and the Optimal Structure of Globalization.” Journal of Globalization and Development 1 (2): 2.
Sugihara, George, Robert May, Hao Ye, Chih-hao Hsieh, Ethan Deyle, Michael Fogarty, and Stephan Munch. 2012. β€œDetecting Causality in Complex Ecosystems.” Science 338 (6106): 496–500.
Tong, Anh, and Jaesik Choi. 2019. β€œDiscovering Latent Covariance Structures for Multiple Time Series.” In International Conference on Machine Learning, 6285–94.
Tran, Alasdair, Alexander Mathews, Cheng Soon Ong, and Lexing Xie. 2021. β€œRadflow: A Recurrent, Aggregated, and Decomposable Model for Networks of Time Series.” In Proceedings of the Web Conference 2021, 730–42. Ljubljana Slovenia: ACM.
Verma, Inder M. 2014. β€œEditorial Expression of Concern: Experimental Evidence of Massivescale Emotional Contagion Through Social Networks.” Proceedings of the National Academy of Sciences, July, 201412469.
Wang, Yichen, Bo Xie, Nan Du, and Le Song. 2016. β€œIsotonic Hawkes Processes.” In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, 2226–34. ICML’16. New York, NY, USA:
Watts, Duncan J., and Peter Sheridan Dodds. 2007. β€œInfluentials, Networks, and Public Opinion Formation.” Journal of Consumer Research 34 (4): 441–58.
Wilms, Ines, Sarah Gelper, and Christophe Croux. 2016. β€œThe Predictive Power of the Business and Bank Sentiment of Firms: A High-Dimensional Granger Causality Approach.” European Journal of Operational Research 254 (1): 138–47.
Yang, Dong-Ping, Hai Lin, Chen-Xu Wu, and Jianwei Shuai. 2011. β€œTopological Conditions of Scale-Free Networks for Cooperation to Evolve.” arXiv:1106.5386, June.
Yang, Shuang-Hong, and Hongyuan Zha. 2013. β€œMixture of Mutually Exciting Processes for Viral Diffusion.” In Proceedings of The 30th International Conference on Machine Learning, 28:1–9.
Yu, Hsiang-Fu, Nikhil Rao, and Inderjit S Dhillon. 2016. β€œTemporal Regularized Matrix Factorization for High-Dimensional Time Series Prediction.” In Advances In Neural Information Processing Systems 29, edited by D. D. Lee, U. V. Luxburg, I. Guyon, and R. Garnett, 847–55. Curran Associates, Inc.
Zhou, Ke, Hongyuan Zha, and Le Song. 2013. β€œLearning Triggering Kernels for Multi-Dimensional Hawkes Processes.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13), 1301–9.

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