Reliable communication over highly connected noisy networks

We consider the task of multiparty computation performed over networks in the presence of random noise. Given an n-party protocol that takes R rounds assuming noiseless communication, the goal is to find a coding scheme that takes $$R'$$R′ rounds and computes the same function with high probability even when the communication is noisy, while maintaining a constant asymptotic rate, i.e., while keeping $$\liminf _{n,R\rightarrow \infty } R/R'$$lim infn,R→∞R/R′ positive. Rajagopalan and Schulman (STOC ’94) were the first to consider this question, and provided a coding scheme with rate $$O(1/\log (d+1))$$O(1/log(d+1)), where d is the maximal degree in the network. While that scheme provides a constant rate coding for many practical situations, in the worst case, e.g., when the network is a complete graph, the rate is $$O(1/\log n)$$O(1/logn), which tends to 0 as n tends to infinity. We revisit this question and provide an efficient coding scheme with a constant rate for the interesting case of fully connected networks. We furthermore extend the result and show that if a (d-regular) network has mixing time m, then there exists an efficient coding scheme with rate $$O(1/m^3\log m)$$O(1/m3logm). This implies a constant rate coding scheme for any n-party protocol over a d-regular network with a constant mixing time, and in particular for random graphs with n vertices and degrees $$n^{\varOmega (1)}$$nΩ(1).