Fragility of nonconvergence in preferential attachment graphs with three types

Preferential attachment networks are a type of random network where new nodes connected to existing ones at random, and more likely connect those that already have many connections. We investigate further family models introduced by Antunovi\'{c}, Mossel R\'{a}cz each vertex in preferential graph is assigned type, based on the types its neighbours. Instances this process proportions present do not converge over time seem be rare. Previous work found "rock-paper-scissors" setup node's was determined rock-paper-scissors contest between two neighbours does converge. Here, cases similar considered, one which like above but with an arbitrarily small chance picking there four perform knockout tournament determine type. These setups, despite seeming very model, fact converge, perhaps surprisingly.