Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T14:13:44.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Point-process models of social network interactions: Parameter estimation and missing data recovery

Published online by Cambridge University Press:  08 October 2015

JOSEPH R. ZIPKIN ,
FREDERIC P. SCHOENBERG ,
KATHRYN CORONGES  and
ANDREA L. BERTOZZI
JOSEPH R. ZIPKIN
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USA email: [email protected]; [email protected]
FREDERIC P. SCHOENBERG
Affiliation:
Department of Statistics, University of California, Los Angeles, CA 90095, USA email: [email protected]
KATHRYN CORONGES
Affiliation:
Network Science Institute, Northeastern University, Boston, MA 02115, USA email: [email protected]
ANDREA L. BERTOZZI
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USA email: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Electronic communications, as well as other categories of interactions within social networks, exhibit bursts of activity localised in time. We adopt a self-exciting Hawkes process model for this behaviour. First we investigate parameter estimation of such processes and find that, in the parameter regime we encounter, the choice of triggering function is not as important as getting the correct parameters once a choice is made. Then we present a relaxed maximum likelihood method for filling in missing data in records of communications in social networks. Our optimisation algorithm adapts a recent curvilinear search method to handle inequality constraints and a non-vanishing derivative. Finally we demonstrate the method using a data set composed of email records from a social network based at the United States Military Academy. The method performs differently on this data and data from simulations, but the performance degrades only slightly as more information is removed. The ability to fill in large blocks of missing social network data has implications for security, surveillance, and privacy.

Keywords

Hawkes processesmaximum likelihoodmissing dataconstrained optimizationsocial networks
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 3: Mathematical Modelling of Crime and Security , June 2016 , pp. 502 - 529
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barabási, A.-L. (2005) The origin of bursts and heavy tails in human dynamics. Nature 435, 207–11.CrossRefGoogle ScholarPubMed
[2]Candès, E. J., Romberg, J. K. & Tao, T. (2006) Stable signal recovery from incomplete and inaccurate measurements. Commu. Pure Appl. Math. 59, 1207–23.CrossRefGoogle Scholar
[3]Chambolle, A., Caselles, V., Cremers, D., Novaga, M. & Pock, T. (2010) An introduction to total variation for image analysis. In: Fornasier, M. (editor), Theoretical Foundations and Numerical Methods for Sparse Recovery. De Gruyter, Berlin, pp. 263–340.CrossRefGoogle Scholar
[4]Chan, T. F. & Shen, J. (2005) Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia.CrossRefGoogle Scholar
[5]Cho, Y. S., Galstyan, A., Brantingham, P. J. & Tita, G. (2014) Latent self-exciting point process model for spatial-temporal networks. Discrete Continuous Dyn. Syst. B 19, 1335–54.CrossRefGoogle Scholar
[6]Crane, R. & Sornette, D. (2008) Robust dynamic classes revealed by measuring the response function of a social system. Proc. Natl. Acad. Sci. 105, 15649–53.CrossRefGoogle ScholarPubMed
[7]Csermely, P., London, A., Wu, L.-Y. & Uzzi, B. (2013) Structure and dynamics of core/periphery networks. J. Complex Netw. 1, 93–123.CrossRefGoogle Scholar
[8]Donoho, D. L. (2006) Compressed sensing. IEEE Trans. Inform. Theory 52, 1289–1306.CrossRefGoogle Scholar
[9]Donoho, D. L. & Tanner, J. (2005) Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. 102, 9446–51.CrossRefGoogle ScholarPubMed
[10]Egesdal, M., Fathauer, C., Louie, K., Neuman, J., Mohler, G. & Lewis, E. (2010) Statistical modeling of gang violence in Los Angeles. SIAM Undergrad. Res.Google Scholar
[11]Fox, E. W., Short, M. B., Schoenberg, F. P., Coronges, K. D. & Bertozzi, A. L. Modeling e-mail networks and inferring leadership using self-exciting point processes. Submitted to J. Am. Stat. Assoc.Google Scholar
[12]Goldfarb, D., Wen, Z. & Yin, W. (2009) A curvilinear search method for p-harmonic flows on spheres. SIAM J. Imaging Sci. 2, 84–109.CrossRefGoogle Scholar
[13]Hawkes, A. G. (1971) Spectra of self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
[14]Hawkes, A. G. (1971) Point spectra of some mutually exciting point processes. J. R. Stat. Soc. B 33, 438–43.Google Scholar
[15]Hegemann, R. A., Lewis, E. A. & Bertozzi, A. L. (2013) An “Estimate & Score Algorithm” for simultaneous parameter estimation and reconstruction of incomplete data on social networks. Secur. Inform. 2, 1.CrossRefGoogle Scholar
[16]Isella, L., Stehlé, J., Barrat, A., Cattuto, C., Pinton, J.-F. & Van den Broeck, W. (2011) What's in a crowd? Analysis of face-to-face behavioral networks. J. Theor. Biol. 271, 166–80.CrossRefGoogle Scholar
[17]Lee, N. H., Yoder, J., Tang, M. & Priebe, C. E. (2013) On latent position inference from doubly stochastic messaging activities. Multiscale Model. Simul. 11, 683–718.CrossRefGoogle Scholar
[18]Lewis, E. & Mohler, G. A nonparametric EM algorithm for multiscale Hawkes processes. Preprint.Google Scholar
[19]Lewis, E., Mohler, G., Brantingham, P. J. & Bertozzi, A. (2010) Self-exciting point process models of insurgency in Iraq. UCLA CAM Report 10–38.Google Scholar
[20]Lewis, P. A. W. & Shedler, G. S. (1979) Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Q. 26, 403–13.CrossRefGoogle Scholar
[21]Marsan, D. & Lengliné, O. (2008) Extending earthquakes' reach through cascading. Science 319, 1076–79.CrossRefGoogle ScholarPubMed
[22]Masuda, N., Takaguchi, T., Sato, N. & Yano, K. (2013) Self-exciting point process modeling of conversation event sequences. In: Holme, P. & Saramäki, J. (editors), Temporal Networks, Springer–Verlag, Berlin, pp. 245–64.CrossRefGoogle ScholarPubMed
[23]McLachlan, G. J. & Krishnan, T. (2008) The EM Algorithm and Extensions, 2nd ed.Wiley, Hoboken, New Jersey.CrossRefGoogle Scholar
[24]Miritello, G., Moro, E. & Lara, R. (2011) Dynamical strength of social ties in information spreading. Phys. Rev. E 83, 045102(R).CrossRefGoogle ScholarPubMed
[25]Mohler, G. (2013) Modeling and estimation of multi-source clustering in crime and security data. Ann. Appl. Stat. 7, 1525–39.CrossRefGoogle Scholar
[26]Ogata, Y. (1981) On Lewis' simulation method for point processes. IEEE Trans. Inform. Theory 27, 23–31.CrossRefGoogle Scholar
[27]Ogata, Y. (1998) Space-time point process models for earthquake occurrences. Ann. Inst. Stat. Math. 50, 379–402.CrossRefGoogle Scholar
[28]Ogata, Y. (1999) Seismicity analysis through point-process modeling: A review. Pure Appl. Geophys. 155, 471–501.CrossRefGoogle Scholar
[29]Ozaki, T. (1979) Maximum likelihood estimation of Hawkes' self-exciting point processes. Ann. Inst. Stat. Math. 31, 145–55.CrossRefGoogle Scholar
[30]Paxson, V. & Floyd, S. (1995) Wide area traffic: The failure of Poisson modeling. IEEE/ACM Trans. Netw. 3, 226–44.CrossRefGoogle Scholar
[31]Rubin, I. (1972) Regular point processes and their detection. IEEE Trans. Inform. Theory 18, 547–57.CrossRefGoogle Scholar
[32]Rudin, L. I., Osher, S. & Fatemi, E. (1992) Nonlinear total variation based noise removal algorithms. Physica D 60, 259–68.CrossRefGoogle Scholar
[33]Rybski, D., Buldyrev, S. V., Havlin, S., Liljeros, F. & Makse, H. A. (2009) Scaling laws of human interaction activity. Proc. Natl. Acad. Sci. 106, 12640–45.CrossRefGoogle ScholarPubMed
[34]Stomakhin, A., Short, M. B. & Bertozzi, A. L. (2011) Reconstruction of missing data in social networks based on temporal patterns of interactions. Inverse Problems 27, 115013.CrossRefGoogle Scholar
[35]Vázquez, A., Oliveira, J. G., Dezsö, Z., Goh, K.-I., Kondor, I. & Barabási, A.-L. (2006) Modeling bursts and heavy tails in human dynamics. Phys. Rev. E 73, 036127.CrossRefGoogle ScholarPubMed
[36]Veen, A. & Schoenberg, F. P. (2008) Estimation of space–time branching process models in seismology using an EM-type algorithm. J. Am. Stat. Assoc. 103, 614–24.CrossRefGoogle Scholar
[37]Vese, L. A. & Osher, S. J. (2002) Numerical methods for p-harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40, 2085–2104.CrossRefGoogle Scholar
[38]Wen, Z. & Yin, W. (2013) A feasible method for optimization with orthogonality constraints. Math. Program. A 142, 397–434.CrossRefGoogle Scholar
[39]Wu, C. F. J. (1983) On the convergence properties of the EM algorithm. Ann. Stat. 11, 95–103.CrossRefGoogle Scholar