Stable Leader Election in Population Protocols Requires Linear Time

Abstract
A population protocol *stably elects a leader* if, for all , starting from an initial configuration with agents each in an identical state, with probability 1 it reaches a configuration that is correct (exactly one agent is in a special leader state ) and stable (every configuration reachable from also has a single agent in state ). We show that any population protocol that stably elects a leader requires expected "parallel time" --- 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.
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