Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation Processes - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Journal of Applied Probability Year : 2022

Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation Processes

François Baccelli
Michel Davydov
  • Function : Author
  • PersonId : 1069793

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. Here, 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 in the present paper. 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.
Fichier principal
Vignette du fichier
discrete case.pdf (273.57 Ko) Télécharger le fichier
Origin Files produced by the author(s)
Loading...

Dates and versions

hal-02607064 , version 1 (16-05-2020)

Identifiers

Cite

François Baccelli, Michel Davydov, Thibaud Taillefumier. Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation Processes. Journal of Applied Probability, 2022, 59 (1), pp.38-59. ⟨10.1017/jpr.2021.31⟩. ⟨hal-02607064⟩
107 View
127 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More