Self-stabilizing Leader Election in Population Protocols over Arbitrary Communication Graphs

This paper considers the fundamental problem of self-stabilizing leader election ( $\mathcal{SSLE}$ ) in the model of population protocols. In this model, an unknown number of asynchronous, anonymous and finite state mobile agents interact in pairs over a given communication graph. $\mathcal{SSLE}$ has been shown to be impossible in the original model. This impossibility can been circumvented by a modular technique augmenting the system with an oracle - an external module abstracting the added assumption about the system. Fischer and Jiang have proposed solutions to $\mathcal{SSLE}$ , for complete communication graphs and rings, using an oracle Ω?, called the eventual leader detector. In this work, we present a solution for arbitrary graphs, using a composition of two copies of Ω?. We also prove that the difficulty comes from the requirement of self-stabilization, by giving a solution without oracle for arbitrary graphs, when an uniform initialization is allowed. Finally, we prove that there is no self-stabilizing implementation of Ω? using $\mathcal{SSLE}$ , in a sense we define precisely.