Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T13:52:01.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Replica-mean-field limits of fragmentation-interaction-aggregation processes

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  17 January 2022

François Baccelli ,
Michel Davydov  and
Thibaud Taillefumier
François Baccelli*
Affiliation:
INRIA/ENS
Michel Davydov*
Affiliation:
INRIA/ENS
Thibaud Taillefumier*
Affiliation:
University of Texas
*
*Postal address: INRIA, Paris, France and Département d’informatique de l’ENS, ENS, CNRS, PSL University, Paris, France
*Postal address: INRIA, Paris, France and Département d’informatique de l’ENS, ENS, CNRS, PSL University, Paris, France
***Postal address: Department of Mathematics and Department of Neuroscience, University of Texas, Austin, TX
Get access Rights & Permissions [Opens in a new window]

Abstract

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called ‘Poisson hypothesis’. However, in most applications this hypothesis is only conjectured. In this paper we establish the Poisson hypothesis for a general class of discrete-time, point-process-based dynamics that we propose to call fragmentation-interaction-aggregation processes, and which are introduced here. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete-time Galves–Löcherbach neural networks are used as a basic instance and illustration of our analysis.

Keywords

Point processHawkes processMarkov processmean-field theoryreplica modelPoisson hypothesisneural networkqueuing network

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60G55: Point processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 38 - 59
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

Amblard, F. and Deffuant, G. (2004). The role of network topology on extremism propagation with the relative agreement opinion dynamics. Physica A 343, 725–738.10.1016/j.physa.2004.06.102CrossRefGoogle Scholar
Baccelli, F., McDonald, D. R. and Reynier, J. (2002). A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation 49, 77–97.CrossRefGoogle Scholar
Baccelli, F. and Taillefumier, T. (2019). Replica-mean-field limits for intensity-based neural networks. SIAM J. Appl. Dynam. Syst. 18, 1756–1797.CrossRefGoogle Scholar
Baccelli, F. and Taillefumier, T. (2020). The pair-replica-mean-field limit for intensity-based neural networks. Preprint, arXiv:2004.06246.Google Scholar
Benaim, M. and Le Boudec, J.-Y. (2008). A class of mean field interaction models for computer and communication systems. Perform Evalaluation 65, 589–590.Google Scholar
Castellani, T. and Cavagna, A. (2005). Spin-glass theory for pedestrians. J. Statist. Mech. 2005, 5–12.10.1088/1742-5468/2005/05/P05012CrossRefGoogle Scholar
Cessac, B. (2007). A discrete time neural network model with spiking neurons. J. Math. Biol. 56,t 311–345.10.1007/s00285-007-0117-3CrossRefGoogle Scholar
Delattre, S., Fournier, N. and Hoffmann, M. (2016). Hawkes processes on large networks. Ann. Appl. Prob. 26, 216–261.CrossRefGoogle Scholar
Galves, A. and Löcherbach, E. (2013). Infinite systems of interacting chains with memory of variable length: A stochastic model for biological neural nets. J. Statist. Phys. 151, 896–921.10.1007/s10955-013-0733-9CrossRefGoogle Scholar
Gast, N., Latella, D. and Massink, M. (2018). A refined mean field approximation of synchronous discrete-time population models. Performance Evaluation 126, 1–21.CrossRefGoogle Scholar
Gupta, A., Briat, C. and Khammash, M. (2014). A scalable computational framework for establishing long-term behavior of stochastic reaction networks. PLoS Comput. Biol. 10, e1003669.10.1371/journal.pcbi.1003669CrossRefGoogle ScholarPubMed
Kleinrock, L. (1977). Queueing Systems Vol. 2. Wiley Interscience, New York.Google Scholar
Kleinrock, L. (2007). Communication Nets: Stochastic Message Flow and Delay. Courier Corporation, North Chelmsford, MA.Google Scholar
Le Boudec, J.-Y., McDonald, D. and Mundinger, J. (2007). A generic mean field convergence result for systems of interacting objects. In Proc. 4th Int. Conf. Quantitative Evaluation of Systems. IEEE Computer Society, USA, pp. 3–18.10.1109/QEST.2007.8CrossRefGoogle Scholar
Masi, A., Galves, A., Löcherbach, E. and Presutti, E. (2015). Hydrodynamic limit for interacting neurons. J. Statist. Phys. 158, 866–902.CrossRefGoogle Scholar
Pastor-Satorras, R., Castellano, C., Van Mieghem, P. and Vespignani, A. (2015). Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979.10.1103/RevModPhys.87.925CrossRefGoogle Scholar
Robert, P. and Touboul, J. (2016). On the dynamics of random neuronal networks. J. Statist. Phys. 165, 545–584.10.1007/s10955-016-1622-9CrossRefGoogle Scholar
Seol, Y. (2015). Limit theorems for discrete Hawkes processes. Statist. Prob. Lett. 99, 223–229.10.1016/j.spl.2015.01.023CrossRefGoogle Scholar
Shriki, O., Alstott, J., Carver, F., Holroyd, T., Henson, R. N., Smith, M. L., Coppola, R., Bullmore, E. and Plenz, D. (2013). Neuronal avalanches in the resting meg of the human brain. J. Neurosci. 33, 7079–7090.10.1523/JNEUROSCI.4286-12.2013CrossRefGoogle ScholarPubMed
Sznitman, A.-S. (1989). Topics in propagation of chaos. In Ecole d’Ete de Probabilites de Saint-Flour XIX, pp. 165–251.Google Scholar
Vladimirov, A. A., Pirogov, S. A., Rybko, A. N. and Shlosman, S. B. (2018). Propagation of chaos and Poisson hypothesis. Probl. Inf. Transm. 54, 290–299.CrossRefGoogle Scholar