Biased Consensus Dynamics on Regular Expander Graphs

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.