In the stochastic population protocol model, we are given a connected graph with $n$ nodes, and in every time step, a scheduler samples an edge of the graph uniformly at random and the nodes connected by this edge interact. A fundamental task in this model is stable leader election, in which all nodes start in an identical state and the aim is to reach a configuration in which (1) exactly one node is elected as leader and (2) this node remains as the unique leader no matter what sequence of interactions follows. On cliques, the complexity of this problem has recently been settled: time-optimal protocols stabilize in $\Theta(n \log n)$ expected steps using $\Theta(\log \log n)$ states, whereas protocols that use $O(1)$ states require $\Theta(n^2)$ expected steps. In this work, we investigate the complexity of stable leader election on graphs. We provide the first non-trivial time lower bounds on general graphs, showing that, when moving beyond cliques, the complexity of stable leader election can range from $O(1)$ to $\Theta(n^3)$ expected steps. We describe a protocol that is time-optimal on many graph families, but uses polynomially-many states. In contrast, we give a near-time-optimal protocol that uses only $O(\log^2n)$ states that is at most a factor $O(\log n)$ slower. Finally, we observe that for many graphs the constant-state protocol of Beauquier et al. [OPODIS 2013] is at most a factor $O(n \log n)$ slower than the fast polynomial-state protocol, and among constant-state protocols, this protocol has near-optimal average case complexity on dense random graphs.

翻译：在随机群体协议模型中，给定一个包含$n$个节点的连通图，在每个时间步中，调度器均匀随机地抽样图的一条边，并由该边连接的两个节点进行交互。该模型中的一个基本任务是稳定领导者选举，即所有节点从相同状态开始，目标是达到一个配置，其中(1)恰好有一个节点被选为领导者，并且(2)无论后续发生何种交互序列，该节点始终保持唯一领导者的身份。在完全图上，该问题的复杂度近期已被解决：时间最优协议以$\Theta(n \log n)$期望步骤稳定，使用$\Theta(\log \log n)$个状态，而使用$O(1)$个状态的协议则需要$\Theta(n^2)$期望步骤。在本工作中，我们研究了图上稳定领导者选举的复杂度。我们首次给出了在一般图上的非平凡时间下界，表明当超越完全图时，稳定领导者选举的复杂度范围可从$O(1)$到$\Theta(n^3)$期望步骤。我们描述了一个协议，它在许多图族上是时间最优的，但使用了多项式数量的状态。相比之下，我们给出了一个近时间最优的协议，仅使用$O(\log^2 n)$个状态，且至多慢$O(\log n)$倍。最后，我们观察到对于许多图，Beauquier等人[OPODIS 2013]提出的常状态协议至多比快速多项式状态协议慢$O(n \log n)$倍，并且在常状态协议中，该协议在密集随机图上具有近最优的平均情况复杂度。