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# Noidy Conmunixatipn: On the Convergence of the Averaging Population Protocol

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## Cite As

Frederik Mallmann-Trenn, Yannic Maus, and Dominik Pajak. Noidy Conmunixatipn: On the Convergence of the Averaging Population Protocol. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 148:1-148:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.148

## Abstract

We study a process of averaging in a distributed system with noisy communication. Each of the agents in the system starts with some value and the goal of each agent is to compute the average of all the initial values. In each round, one pair of agents is drawn uniformly at random from the whole population, communicates with each other and each of these two agents updates their local value based on their own value and the received message. The communication is noisy and whenever an agent sends any value v, the receiving agent receives v+N, where N is a zero-mean Gaussian random variable. The two quality measures of interest are (i) the total sum of squares TSS(t), which measures the sum of square distances from the average load to the initial average and (ii) bar{phi}(t), which measures the sum of square distances from the average load to the running average (average at time t). It is known that the simple averaging protocol - in which an agent sends its current value and sets its new value to the average of the received value and its current value - converges eventually to a state where bar{phi}(t) is small. It has been observed that TSS(t), due to the noise, eventually diverges and previous research - mostly in control theory - has focused on showing eventual convergence w.r.t. the running average. We obtain the first probabilistic bounds on the convergence time of bar{phi}(t) and precise bounds on the drift of TSS(t) that show that although TSS(t) eventually diverges, for a wide and interesting range of parameters, TSS(t) stays small for a number of rounds that is polynomial in the number of agents. Our results extend to the synchronous setting and settings where the agents are restricted to discrete values and perform rounding.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Distributed computing models
##### Keywords
• population protocols
• noisy communication
• distributed averaging

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