Consensus protocols play an important role in the study of distributed algorithms. In this paper, we study the effect of bias on two popular consensus protocols, namely, the {\em voter rule} and the {\em 2-choices rule} with binary opinions. We assume that agents with opinion $1$ update their opinion with a probability $q_1$ strictly less than the probability $q_0$ with which update occurs for agents with opinion $0$. We call opinion $1$ as the superior opinion and our interest is to study the conditions under which the network reaches consensus on this opinion. We assume that the agents are located on the vertices of a regular expander graph with $n$ vertices. We show that for the voter rule, consensus is achieved on the superior opinion in $O(\log n)$ time with high probability even if system starts with only $\Omega(\log n)$ agents having the superior opinion. This is in sharp contrast to the classical voter rule where consensus is achieved in $O(n)$ time and the probability of achieving consensus on any particular opinion is directly proportional to the initial number of agents with that opinion. For the 2-choices rule, we show that consensus is achieved on the superior opinion in $O(\log n)$ time with high probability when the initial proportion of agents with the superior opinion is above a certain threshold. We explicitly characterise this threshold as a function of the strength of the bias and the spectral properties of the graph. We show that for the biased version of the 2-choice rule this threshold can be significantly less than that for the unbiased version of the same rule. Our techniques involve using sharp probabilistic bounds on the drift to characterise the Markovian dynamics of the system.

翻译：共识协议在分布式算法研究中扮演重要角色。本文研究偏向性对两种流行共识协议（即具有二元意见的**投票规则**和**二选一规则**）的影响。我们假设持有意见$1$的智能体以概率$q_1$更新其意见，该概率严格小于持有意见$0$的智能体的更新概率$q_0$。我们将意见$1$称为优势意见，旨在研究网络就这种意见达成共识的条件。假设智能体位于具有$n$个顶点的正则扩展图的顶点上。研究表明：对于投票规则，即使系统初始时仅有$\Omega(\log n)$个智能体持有优势意见，系统也能以高概率在$O(\log n)$时间内就该意见达成共识。这与经典投票规则形成鲜明对比——经典规则下共识需$O(n)$时间达成，且就某一特定意见达成共识的概率直接正比于初始持有该意见的智能体数量。对于二选一规则，当初始持有优势意见的智能体比例超过特定阈值时，系统能以高概率在$O(\log n)$时间内就该意见达成共识。我们明确刻画了该阈值作为偏向强度和图谱特性的函数，并证明在有偏版本的二选一规则中，该阈值可显著低于无偏版本的阈值。我们的技术方法利用关于漂移的尖锐概率界来表征系统的马尔可夫动力学特性。