In the adaptive voter model, nodes hold opinions (+1 or −1) and are connected by a network. At each step: with probability p, a discordant link is rewired to a same-opinion node (homophily); with probability 1−p, an opinion is copied (voter dynamics). This drives a fragmentation transition: above a critical p*, the network splits into two like-minded communities.
p < p_c: active phase — opinions mix, one consensus
p > p_c: frozen phase — fragmentation into like-minded groups
p_c ≈ (k−2)/(2(k−1)) (mean-field)
Order param: density of active links ρ_A → 0 at fragmentation
Giant component splits at transition