Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Interplay of vertex and edge dynamics for dense random graphs
We consider a dense random graph in which the vertices can hold opinion 0 or 1 and the edges can be closed or open. The vertices update their opinion at rate η times the number of incident open edges, and do so by adopting the opinion of the vertex at the other end. The edges update their status at rate ρ, and do so by switching between closed and open
with a probability that depends on their status and on whether the vertices at their ends are concordant or discordant. Let X^n(t) denote the configuration at time t of the vertices and the edges, with n the number of vertices and n the number of edges. We show that the process X^n = (X^n(t))_{t \geq 0} converges to a limiting process X = (X(t))_{t \geq 0} as n → ∞ in the Meyer-Zheng path topology. In the limiting process X, which lives on a submanifold in the space of coloured graphons, the fraction of vertices with opinion 1 evolves according to a Fisher-Wright diffusion with a diffusion constant that is proportional to the fraction of open edges, while the fraction of open edges evolves according to a stochastic flow with a drift that is a quadratic functional of the fraction of vertices with opinion 1. Their joint evolution is an example of co-evolution. The evolution of the coloured graphon is a stochastic flow that is driven by both these fractions.
Joint work with Siva Athreya and Adrian Rollin.