Efficient and Noise-Resilient rumor-spreading in Large, Anonymous Populations

The study in this paper aims to further our understanding of communication in large anonymous populations composed of simple agents which interact through short and highly unreliable messages. We focus on the rumor-spreading problem, and initiate the study of reliable rumor-spreading under communication noise. Specifically, the noisy rumor-spreading problem considers a population of n anonymous agents, with one distinguished source agent representing the environment associated with an opinion (a bit B in {0,1}). Few agents obtain noisy samples of this bit and all agents use noisy interactions to guarantee that eventually, all agents hold the correct opinion B with high probability. Our model for communication is extremely weak and follows the push gossip communication paradigm: In each synchronous round each agent that wishes to send information delivers a message to a randomly chosen anonymous agent. Since our objective is to study restricted communication channels, we assume that each message can contain only one bit (essentially representing an opinion). The system is furthermore assumed to be so noisy that the bit in each message sent is flipped independently with probability $1/2-\eps$, for small $\eps$, that is, it carries only $H(\eps)$ amount of entropy. Even in this severely restricted, stochastic and noisy setting we give natural protocols that solve the noisy rumor-spreading problem efficiently. In particular, our protocol run in $O(\frac{1}{\eps^2}\log n)$ rounds and use $O(\frac{1}{\eps^2} n \log n)$ messages (or bits) in total. These bounds are both asymptotically optimal. Our efficient, robust, and simple algorithms suggest balancing between silence and transmission, synchronization, and majority-based decisions as important ingredients towards understanding collective communication schemes in anonymous and noisy natural populations.