Model
The Hegselmann-Krause (HK) model: N agents each hold an opinion x_i ∈ [0,1]. At each step, agent i updates to the average of all agents within confidence bound ε:
x_i(t+1) = ⟨x_j : |x_j − x_i| ≤ ε⟩
Small ε → many opinion clusters (fragmentation). Large ε → consensus. The transition occurs near ε ≈ 0.27 for uniform initial conditions.
Controls
Key Results
• ε < ~0.27: fragmentation into 2–5 clusters
• ε > ~0.27: global consensus in ~10–30 steps
• "Moderate cluster" at center can mediate
• Adding noise can prevent final convergence