Brief Announcement: Lower Bounds for Asymptotic Consensus in Dynamic Networks
Résumé
In the asymptotic consensus problem a set of agents, each starting from an initial value in R d , update their values such that all agents' values converge to a common value within the convex hull of initial values. The problem is closely related to the approximate consensus problem, in which agents have to irrevocably decide on values that lie within a predefined distance ε > 0 of each other. The latter is weaker than the exact consensus problem in which agents need to decide on the same value. Both the asymptotic and the approximate consensus problems have not only a variety of applications in the design of man-made control systems like sensor fusion [1], clock synchronization [8], formation control [6], rendezvous in space [9], or load balancing [5], but also for analyzing natural systems like flocking [11], firefly synchronization [10], or opinion dynamics [7]. These problems often have to be solved under harsh-environmental restrictions: with limited computational power and local storage, under restricted communication abilities, and in presence of communication uncertainty. In this work we study asymptotic consensus in round-based computational models with a dynamic communication topology whose directed communication graphs are chosen each round from a predefined set of communication graphs, the so-called network model. In previous work [2], Charron-Bost et al. showed that asymptotic consensus is solvable precisely within rooted network models in which all communication graphs contain rooted spanning trees. These rooted spanning trees need not have any edges in common and can change from one round to the next. An interesting special case of rooted network models are network models whose graphs are non-split, that is, any two agents have a common incoming neighbor. Their prominent role is motivated by two properties: (i) They occur as communication graphs in benign classical distributed failure models. For example, in synchronous systems with crashes, in asynchronous systems with a minority of crashes, and synchronous systems with send omissions. (ii) In [2], Charron-Bost et al. showed that non-split graphs also play a central role in arbitrary rooted network models: they showed that any product of n − 1 rooted graphs with n nodes is non-split, allowing to transform asymptotic consensus algorithms for non-split network models into their amortized variants for rooted models. Interestingly, solvability in any rooted network model is already provided by deceptively simple algorithms [2]: so-called averaging or convex combination algorithms, in which agents repeatedly broadcast their current value, and update it to some weighted average of the values they received in this round. One instance, proposed by Charron-Bost et al. [3] is the midpoint algorithm, in which agents update their value to the midpoint of the set of received values, i.e., the average of the smallest and the largest of the received values. Regarding time complexity, for dimension d = 1, the midpoint algorithm was shown to *
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