Breathe before Speaking: Efficient Information Dissemination Despite Noisy, Limited and Anonymous Communication

Distributed computing models typically assume reliable communication between processors. While such assumptions often hold for engineered networks, their relevance to biological systems, wherein messages are often distorted before reaching their destination, is quite limited. In this study we aim at bridging this gap by rigorously analyzing a model 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 the majority-consensus problem, two fundamental tasks in distributed computing, and initiate their study under communication noise. 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 random anonymous agent. This communication is further restricted to contain only one bit (essentially representing an opinion). Lastly, the system is assumed to be so noisy that the bit in each message sent is flipped independently with probability $1/2-\epsilon$, for some small $\epsilon >0$. Even in this severely restricted, stochastic and noisy setting we give natural protocols that solve the noisy rumor-spreading and the noisy majority-consensus problems efficiently. Our protocols run in $O(\log n / \epsilon^2)$ rounds and use $O(n \log n / \epsilon^2)$ messages/bits in total, where $n$ is the number of agents. These bounds are asymptotically optimal and, in fact, are as fast and message efficient as if each agent would have been simultaneously informed directly by the source. 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 populations.