### Majority dynamics: Some old and new conjectures

Probability Seminar

14th February 2020, 4:30 pm – 5:30 pm

Fry Building, 2.41

Abstract:

In majority dynamics on a graph G=(V,E), every vertex initially holds an opinion in {-1,1}, and the opinion evolve in time according to the "majority" rule: When a vertex "rings" (e.g. according to a Poisson clock attached to each vertex) it adopts the opinion of the majority of its neighbors. The model turns out t exhibit interesting behavior on both finite and infinite graphs.

Among the natural questions one can ask are questions regarding fixation and convergence of the opinions (Does every vertex eventually fixate on an opinion, if so, what can be said on the final configurations, and if not, what are the stationary measures of the process),

Percolation type questions (will an infinite connected component appear or disappear with time), and questions regarding the probability of having an opinion of "1" at time t given that we started with i.i.d. Bernoulli(p) opinions at time 0.

We will survey some old a new results on these and related questions, with an emphasis on some conjectures I find particularly interesting.

We will also discuss a related model (Median dynamics/process, see https://arxiv.org/abs/1904.11625 and https://arxiv.org/abs/1911.08613 ) which offers a continuous-opinion generalization of majority dynamics, and show how it offers a good language for some of the above conjectures as well as raises new questions of its own.

The talk is based on Joint works with Rangel Baldasso and Nissan Bailin.

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