The minority network majority rule with different tie-breaking strategies (prudent, risky, fixed) produces different dynamics: prudent converges to fixed points or 2-cycles, risky can produce very long cycles or complex behavior, and fixed strategy creates deterministic consensus on finite toruses

The majority rule cellular automaton on arbitrary graphs is P-complete to predict, meaning that determining whether a specific node will change state by time t is as hard as solving any problem in the complexity class P, but on planar graphs the problem remains P-complete due to the existence of a crossover gadget allowing signal routing without signal intersection