Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T03:20:54.392Z Has data issue: false hasContentIssue false
  • English
  • Français

A functional central limit theorem for SI processes on configuration model graphs

Part of: Limit theorems

Published online by Cambridge University Press:  06 September 2022

Wasiur R. Khudabukhsh ,
Casper Woroszylo ,
Grzegorz A. Rempała  and
Heinz Koeppl
Wasiur R. Khudabukhsh*
Affiliation:
University of Nottingham
Casper Woroszylo*
Affiliation:
BHP Billiton
Grzegorz A. Rempała*
Affiliation:
The Ohio State University
Heinz Koeppl*
Affiliation:
Technische Universität Darmstadt
*
*Postal address: School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
**Postal address: 480 Queen Street, Level 12, Brisbane QLD 4000, Australia.
***Postal address: Mathematical Biosciences Institute, The Ohio State University, Jennings Hall 3rd Floor, 1735 Neil Ave., Columbus, OH 43210, United States of America.
****Postal address: Bioinspired Communication Systems, Technische Universität Darmstadt, Rundeturmstrasse 12, 64283 Darmstadt, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a stochastic compartmental susceptible–infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval. We split the population of graph vertices into two compartments, namely, S and I, denoting susceptible and infected vertices, respectively. In addition to the sizes of these two compartments, we keep track of the counts of SI-edges (those connecting a susceptible and an infected vertex) and SS-edges (those connecting two susceptible vertices). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them as the number of vertices in the random graph grows to infinity. The FCLT asserts that the counts, when appropriately scaled, converge weakly to a continuous Gaussian vector semimartingale process in the space of vector-valued càdlàg functions endowed with the Skorokhod topology. We discuss applications of the FCLT in percolation theory and in modelling the spread of computer viruses. We also provide simulation results illustrating the FCLT for some common degree distributions.

Keywords

SI processfunctional CLTconfiguration modelrandom graphsscaling limit

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 92D30: Epidemiology
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 880 - 912
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.CrossRefGoogle Scholar
Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Prob. 8, 1331–1349.CrossRefGoogle Scholar
Ball, F. and Neal, P. (2002). A general model for stochastic SIR epidemics with two levels of mixing. Math. Biosci. 180, 73–102.CrossRefGoogle Scholar
Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.CrossRefGoogle ScholarPubMed
Barato, A. C. and Hinrichsen, H. (2009). Nonequilibrium phase transition in a spreading process on a timeline. J. Statist. Mech. 2009, article no. P02020.CrossRefGoogle Scholar
Barbour, A. D. and Reinert, G. (2013). Approximating the epidemic curve. Electron. J. Prob. 18, article no. 54, 30 pp.Google Scholar
Barbour, A. D. and Röllin, A. (2019). Central limit theorems in the configuration model. Ann. Appl. Prob. 29, 1046–1069.CrossRefGoogle Scholar
Barndorff-Nielsen, O. (1963). On the limit behaviour of extreme order statistics. Ann. Math. Statist. 34, 992–1002.CrossRefGoogle Scholar
Baroni, E., van der Hofstad, R. and Komjáthy, J. (2015). First passage percolation on random graphs with infinite variance degrees. Preprint. Available at https://arxiv.org/abs/1506.01255.Google Scholar
Barrat, A., Barthélemy, M. and Vespignani, A. (2008). Dynamical Processes on Complex Networks. Cambridge University Press.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 1907–1965.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Prob. Comput. 20, 683–707.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2017). Universality for first passage percolation on sparse random graphs. Ann. Prob. 45, 2568–2630.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Bohman, T. and Picollelli, M. (2012). SIR epidemics on random graphs with a fixed degree sequence. Random Structures Algorithms 41, 179–214.CrossRefGoogle Scholar
Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge University Press.CrossRefGoogle Scholar
Brauer, F. and van den Driessche, P. (2003). Some directions for mathematical epidemiology. In Dynamical Systems and Their Applications in Biology (Cape Breton Island, NS, 2001) (Fields Inst. Commun., Vol. 36), American Mathematical Society, Providence, RI, pp. 95–112.Google Scholar
Callaway, D. S., Newman, M. E., Strogatz, S. H. and Watts, D. J. (2000). Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85, article no. 5468.CrossRefGoogle ScholarPubMed
Chatterjee, S. and Durrett, R. (2009). Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Prob. 37, 2332–2356.CrossRefGoogle Scholar
Decreusefond, L., Dhersin, J.-S., Moyal, P. and Tran, V. C. (2012). Large graph limit for an SIR process in random network with heterogeneous connectivity. Ann. Appl. Prob. 22, 541–575.CrossRefGoogle Scholar
Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.CrossRefGoogle Scholar
Durrett, R. (2010). Random Graph Dynamics. Cambridge University Press.Google Scholar
Durrett, R. (2010). Some features of the spread of epidemics and information on a random graph. Proc. Nat. Acad. Sci. USA 107, 4491–4498.CrossRefGoogle Scholar
Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. John Wiley, New York.Google Scholar
Gleeson, J. P. (2011). High-accuracy approximation of binary-state dynamics on networks. Phys. Rev. Lett. 107, article no. 068701.CrossRefGoogle ScholarPubMed
Graham, M. and House, T. (2014). Dynamics of stochastic epidemics on heterogeneous networks. J. Math. Biol. 68, 1583–1605.CrossRefGoogle ScholarPubMed
Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9, 79–94.Google Scholar
Hinrichsen, H. (2006). Non-equilibrium phase transitions. Physica A 369, 1–28.CrossRefGoogle Scholar
Jacobsen, K. A., Burch, M. G., Tien, J. H. and Rempała, G. A. (2018). The large graph limit of a stochastic epidemic model on a dynamic multilayer network. J. Biol. Dynamics 12, 746–788.CrossRefGoogle ScholarPubMed
Janson, S., Luczak, M. and Windridge, P. (2014). Law of large numbers for the SIR epidemic on a random graph with given degrees. Random Structures Algorithms 45, 726–763.CrossRefGoogle Scholar
Janson, S. and Luczak, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34, 197–216.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Kephart, J. O. and White, S. R. (1993). Measuring and modeling computer virus prevalence. In Proc. 1993 IEEE Computer Society Symposium on Research in Security and Privacy, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 2–15.CrossRefGoogle Scholar
Lelarge, M. (2012). Diffusion and cascading behavior in random networks. Games Econom. Behavior 75, 752–775.CrossRefGoogle Scholar
Meyer, P. A. (1962). A decomposition theorem for supermartingales. Illinois J. Math. 6, 193–205.CrossRefGoogle Scholar
Miller, J. C. (2011). A note on a paper by Erik Volz: SIR dynamics in random networks. J. Math. Biol. 62, 349–358.CrossRefGoogle Scholar
Miller, J. C., Slim, A. C. and Volz, E. M. (2012). Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface 9, 890–906.CrossRefGoogle ScholarPubMed
Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.CrossRefGoogle Scholar
Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combinatorics Prob. Comput. 7, 295–305.CrossRefGoogle Scholar
Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, article no. 016128, 11 pp.Google ScholarPubMed
Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63, article no. 066117.CrossRefGoogle ScholarPubMed
Rand, D. A. (1999). Correlation equations and pair approximations for spatial ecologies. In Advanced Ecological Theory: Principles and Applications, Blackwell Science, Oxford, pp. 100–142.CrossRefGoogle Scholar
Rebolledo, R. (1980). Central limit theorems for local martingales. Z. Wahrscheinlichkeitsth. 51, 269–286.CrossRefGoogle Scholar
Spruyt, V. (2014). How to draw a covariance error ellipse? Available at https://www.visiondummy.com/2014/04/draw-error-ellipse-representing-covariance-matrix.Google Scholar
Van der Hofstad, R. (2010). Percolation and random graphs. In New Perspectives in Stochastic Geometry, Oxford University Press, pp. 173–247.Google Scholar
Van der Hofstad, R. (2017). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press.Google Scholar
Volz, E. (2008). SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56, 293–310.CrossRefGoogle ScholarPubMed
Wierman, J. C. and Marchette, D. J. (2004). Modeling computer virus prevalence with a susceptible–infected–susceptible model with reintroduction. Comput. Statist. Data Anal. 45, 3–23.CrossRefGoogle Scholar