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

This paper considers the fundamental problem of \emph{self-stabilizing leader election} ($\mathcal{SSLE}$) in the model of \emph{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 \emph{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 $\Omega?$, called the \emph{eventual leader detector}. In this work, we present a solution for arbitrary graphs, using a \emph{composition} of two copies of $\Omega?$. 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 \emph{implementation} of $\Omega?$ using $\mathcal{SSLE}$, in a sense we define precisely

Uniform Bipartition in the Population Protocol Model with Arbitrary Communication Graphs

In this paper, we focus on the uniform bipartition problem in the population protocol model. This problem aims to divide a population into two groups of equal size. In particular, we consider the problem in the context of arbitrary communication graphs. As a result, we investigate the solvability of the uniform bipartition problem with arbitrary communication graphs when agents in the population have designated initial states, under various assumptions such as the existence of a base station, symmetry of the protocol, and fairness of the execution. When the problem is solvable, we present protocols for uniform bipartition. When global fairness is assumed, the space complexity of our solutions is tight

Loosely-Stabilizing Leader Election on Arbitrary Graphs in Population Protocols Without Identifiers nor Random Numbers

In the population protocol model Angluin et al. proposed in 2004, there exists no self-stabilizing leader election protocol for complete graphs, arbitrary graphs, trees, lines, degree-bounded graphs and so on unless the protocol knows the exact number of nodes. To circumvent the impossibility, we introduced the concept of loose-stabilization in 2009, which relaxes the closure requirement of self-stabilization. A loosely-stabilizing protocol guarantees that starting from any initial configuration a system reaches a safe configuration, and after that, the system keeps its specification (e.g. the unique leader) not forever, but for a sufficiently long time (e.g. exponentially large time with respect to the number of nodes). Our previous works presented two loosely-stabilizing leader election protocols for arbitrary graphs; One uses agent identifiers and the other uses random numbers to elect a unique leader. In this paper, we present a loosely-stabilizing protocol that solves leader election on arbitrary graphs without agent identifiers nor random numbers. By the combination of virus-propagation and token-circulation, the proposed protocol achieves polynomial convergence time and exponential holding time without such external entities. Specifically, given upper bounds N and Delta of the number of nodes n and the maximum degree of nodes delta respectively, it reaches a safe configuration within O(m*n^3*d + m*N*Delta^2*log(N)) expected steps, and keeps the unique leader for Omega(N*e^N) expected steps where m is the number of edges and d is the diameter of the graph. To measure the time complexity of the protocol, we assume the uniformly random scheduler which is widely used in the field of the population protocols

Population Protocols for Graph Class Identification Problems

In this paper, we focus on graph class identification problems in the population protocol model. A graph class identification problem aims to decide whether a given communication graph is in the desired class (e.g. whether the given communication graph is a ring graph). Angluin et al. proposed graph class identification protocols with directed graphs and designated initial states under global fairness [Angluin et al., DCOSS2005]. We consider graph class identification problems for undirected graphs on various assumptions such as initial states of agents, fairness of the execution, and initial knowledge of agents. In particular, we focus on lines, rings, k-regular graphs, stars, trees, and bipartite graphs. With designated initial states, we propose graph class identification protocols for k-regular graphs and trees under global fairness, and propose a graph class identification protocol for stars under weak fairness. Moreover, we show that, even if agents know the number of agents n, there is no graph class identification protocol for lines, rings, k-regular graphs, trees, or bipartite graphs under weak fairness, and no graph class identification for lines, rings, k-regular graphs, stars, trees, or bipartite graphs with arbitrary initial states

Fast Graphical Population Protocols

Let $G$ be a graph on $n$ nodes. In the stochastic population protocol model, a collection of $n$ indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other's states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when $G$ is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in $n \operatorname{polylog} n$ pairwise interactions, with high probability, using at most $\operatorname{polylog} n$ states per node. In this work, we consider the more general setting where $G$ is an arbitrary graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As a sample application, we show that, in any regular graph with conductance $\phi$, both leader election and exact majority can be solved in $\phi^{-2} \cdot n \operatorname{polylog} n$ pairwise interactions, with high probability, using at most $\phi^{-2} \cdot \operatorname{polylog} n$ states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties. We believe our results will prove generally useful, as they allow efficient technology transfer between the well-mixed (clique) case, and the under-explored spatial setting.Comment: 47 pages, 5 figure