Determining majority in networks with local interactions and very small local memory

Abstract

Abstract. We study here the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types (states). The vertices may have a few additional possible states and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority, i.e. its output function must interpret the local state of each vertex so that each vertex outputs the initial majority type. We first provide a protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler. Under the uniform probabilistic scheduler of pairwise interactions, we prove that our protocol stabilizes in expected polynomial time for any network and is quite fast on the clique. As we prove, this protocol is optimal, in the sense that there does not exist any population protocol that always computes majority with fewer than 4 states per vertex. However this does not rule out the existence of a protocol with 3 states per vertex that is correct with high probability (whp). To this end, we examine an elegant and very natural majority protocol with 3 states per vertex, introduced in [2] where its performance has been analyzed for the clique graph. In particular, it determines the correct initial majority type in the clique very fast and whp under the uniform probabilistic scheduler. We study the performance of this protocol in arbitrary networks. We prove that, when the two initial states are put uniformly at random on the vertices, the protocol of [2] converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we present an infinite family of graphs, on which the protocol of [2] can fail, i.e. it can converge to the initial minority type whp, even when the difference between the initial majority and the initial minority is n − Θ(ln n). We also present another infinite family of graphs in which the protocol of [2] takes an expected exponential time to converge. These two negative results build upon a very positive result concerning the robustness of the protocol of [2] on the clique, namely that if the initial minority is at most n 7, the protocol fails with exponentially small probability. Surprisingly, the resistance of the clique to failure causes the failure in general graphs. Our techniques use new domination and coupling arguments for suitably defined processes whose dynamics capture the antagonism between the states involved.

Optimal proportion computation with population protocols

“…Then in [5] the authors have completely characterized the computational power of the model by establishing the equivalence between predicates computable in the population model and those that can be defined in the Presburger arithmetic. Since then, there has been a lot of work on population protocols including the majority problem [2,6,12,14,17], the leader election problem [9,15], in presence of faults [11], and on variants of the model [10,13].…”

“…n A < n B ). In [12,14] the authors propose a four-state protocol that solves the majority problem with a convergence parallel time logarithmic in n but only in expectation. Moreover, the expected convergence time is infinite when n A and n B are close to each other (that is γ approaches 0).…”

“…Examples of systems whose behavior can be modeled by population protocols range from molecule interactions of a chemical process to sensor networks in which agents, which are small devices embedded on animals, interact each time two animals are in the same radio range. A considerable amount of work has been done so far to determine which properties can emerge from pairwise interactions between finite-state agents, together with the derivation of lower bounds on the time and space needed to reach such properties (e.g., [2,6,12,14,17]). Among them, is majority.…”

Synthesizing and tuning stochastic chemical reaction networks with specified behaviours

“…We next considered CRNs with four species, to explore how many reactions are required to stably compute the majority specification. There is a known CRN (or population protocol) with four species that stably computes the majority problem with three reactions and four species [ 44 – 46 ]. Indeed, it has been shown that four species are necessary [ 35 , 46 ] to stably compute majority.…”

“…There is a known CRN (or population protocol) with four species that stably computes the majority problem with three reactions and four species [ 44 – 46 ]. Indeed, it has been shown that four species are necessary [ 35 , 46 ] to stably compute majority. However, this known CRN does not satisfy Ψ Maj .…”