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.
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