Mean Field Limits

Replica-Mean-Field Limits for Galves-Löcherbach dynamics and for Fragmentation-Interaction-Aggregation Processes

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”.

Here are the two papers with T. Taillefumier giving instances of practical use of the existence of such mean field limits:

However, in most applications, this hypothesis is only conjectured. In this work with Thibaud Taillefumier (UT Austin),

  • F. Baccelli, M. Davydov and T. Taillefumier. ‘Replica-mean-field limits of fragmentation-interaction-
    aggregation processes’. In: Journal of Applied Probability (17th Jan. 2022), pp. 1–22.
    https://hal.archives-ouvertes.fr/hal-03542535

the Poisson Hypothesis was proved for a general class of discrete-time, point-process-based dynamics, that the authors propose to call fragmentation-interaction-aggregation processes. 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.  Discrete time Galves-Löcherbach neural networks are used as a basic instance and illustration of our analysis.

A proof of the Poisson Hypothesis for the replica-mean-field limit of the continuous-time Galves-Löcherbach model was then proposed by M. Davydov in the article

  • M. Davydov. Propagation of chaos and Poisson Hypothesis for replica mean-field models of intensity-based neural networks. 21st Nov. 2022. To appear in Annals of Applied Probability. URL : https://hal.inria.fr/hal-03863810

A natural second direction of research was to try to generalize this result to a more general class of dynamics, analoguously to what was done in discrete time. This has led to a second preprint, currently available on ArXiV, defining a class of continuous-time fragmentation-interaction-aggregation processes, proving the Poisson Hypothesis for their replica-mean-field limits and detailing the link between discrete-time and continuous-time models.

  • M. Davydov. ‘Replica-mean-field limit of continuous-time fragmentation-interaction-aggregation processes.’ 2023. https//:arXiv.2305.03464.

A new class of problems called migration-contagion process where mean field limits are central was introduced in the following paper lately

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