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In this article we formalize the problem of modeling social networks into a measure-valued process and interacting particle system. We obtain a model that describes in continuous time each vertex of the graph at a latent spatial state as a Dirac measure. We describe the model and its formal design as a Markov process on finite and connected geometric graphs with values in path space. A careful analysis of some microscopic properties of the underlying process is provided. Moreover, we study the long time behavior of the stochastic particle system. Using a renormalization technique, which has the effect that the density of the vertices must grow to infinity, we show that the rescaled measure-valued process converges in law towards the solution of a deterministic equation. The strength of our general continuous time and measure-valued dynamical system is that their results are context-free, that is, that hold for arbitrary sequences of graphs.
We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the
We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We establish the `classical re
We consider a piecewise-deterministic Markov process (PDMP) with general conditional distribution of inter-occurrence time, which is called a general PDMP here. Our purpose is to establish the theory of measure-valued generator for general PDMPs. The
In this paper, we study the averaging principle for a class of stochastic differential equations driven by $alpha$-stable processes with slow and fast time-scales, where $alphain(1,2)$. We prove that the strong and weak convergence order are $1-1/alp
The Gamma-Dirichlet structure corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the