Stable leader election in population protocols requires linear time: Preliminary version

A population protocol stably elects a leader if, for all n, starting from an initial configuration with n agents each in an identical state, with probability 1 it reaches a configuration y that is correct (exactly one agent is in a special leader state `) and stable (every configuration reachable from y also has a single agent in state `). We show that any population protocol that stably elects a leader requires Ω(n) expected “parallel time” — Ω(n2) expected total pairwise interactions — to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.