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

Part of: Special processes Stochastic 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
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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
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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