Behavior of 3-Majority and 2-Choices on expander graphs with k ≥ 3 opinions

Characterize the convergence behavior—including consensus time and plurality guarantees—of the synchronous 3-Majority and 2-Choices dynamics on expander graphs when the number of opinions satisfies k ≥ 3, for arbitrary initial configurations.

Background

Most existing results for consensus dynamics on non-complete graphs focus on the two-opinion case (k=2). Extending the analysis to k ≥ 3 on expander graphs presents additional challenges, and the authors note that knowledge in this regime is limited.

The paper suggests applying the new technical tools—multi-step concentration via the Bernstein condition and drift analysis of the ℓ2-norm—to graph classes beyond the complete graph, with expander graphs being a particularly important and archetypal case.