We study a population of $N$ particles, which evolve according to a diffusion process and interact through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e., with $O(N)$ particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with $O(1)$ particles. We also assume that the network's dynamics is much faster than the particles' dynamics, with the time-scale of the network described by a parameter $\varepsilon>0$. We combine the averaging ($\varepsilon \rightarrow 0$) and the many-particles ($N \rightarrow \infty$) limits and prove that the evolution of the particles' empirical density is described (after taking both limits) by a nonlinear Fokker--Planck equation; moreover, we give conditions under which such limits can be taken uniformly in time, hence providing a criterion under which the limiting nonlinear Fokker--Planck equation is a good approximation of the original system uniformly in time. The heart of our proof consists of controlling precisely the dependence on $N$ of the averaging estimates.


  1. interacting particle systems
  2. Markov semigroups
  3. averaging methods
  4. sparse interaction
  5. nonlinear Fokker--Plank equation

MSC codes

  1. 60K35
  2. 47D07
  3. 60J60
  4. 35Q84
  5. 35Q82
  6. 82C31

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 937 - 972
ISSN (online): 1095-7154


Submitted: 1 April 2020
Accepted: 12 October 2020
Published online: 8 February 2021


  1. interacting particle systems
  2. Markov semigroups
  3. averaging methods
  4. sparse interaction
  5. nonlinear Fokker--Plank equation

MSC codes

  1. 60K35
  2. 47D07
  3. 60J60
  4. 35Q84
  5. 35Q82
  6. 82C31



Funding Information

International Center for Mathematical Sciences
Maxwell Institute Graduate School in Analysis and its Applications
Heriot-Watt University https://doi.org/10.13039/100009767
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/L016508/01
Scottish Funding Council https://doi.org/10.13039/501100000360
University of Edinburgh https://doi.org/10.13039/501100000848

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