Challenges in co-evolution on networks

In this talk we present an overview of what is known and is not known about random processes on dynamic random graphs in co-evolution, i.e., with mutual feedback. The literature offers plenty of heuristics, simulations and conjectures, but so far mathematical results are available for only a handful of models. We describe some of these models, with a focus on opinion dynamics. In particular, we consider random graphs in which vertices can have one of two possible opinions. Pairs of vertices connected by an edge share their opinions according to a rate that may depend on the size of the graph. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same opinion or not. We exhibit joint evolution equations and describe the occurrence of phase transitions between consensus and polarisation.

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