How to Distribute Computation in Networks

In network function computation is as a means to reduce the required communication flow in terms of number of bits transmitted per source symbol. However, the rate region for the function computation problem in general topologies is an open problem, and has only been considered under certain restrictive assumptions (e.g. tree networks, linear functions, etc.). In this paper, we propose a new perspective for distributing computation, and formulate a flow-based delay cost minimization problem that jointly captures the costs of communications and computation. We introduce the notion of entropic surjectivity as a measure to determine how sparse the function is and to understand the limits of computation. Exploiting Little's law for stationary systems, we provide a connection between this new notion and the computation processing factor that reflects the proportion of flow that requires communications. This connection gives us an understanding of how much a node (in isolation) should compute to communicate the desired function within the network without putting any assumptions on the topology. Our analysis characterizes the functions only via their entropic surjectivity, and provides insight into how to distribute computation. We numerically test our technique for search, MapReduce, and classification tasks, and infer for each task how sensitive the processing factor to the entropic surjectivity is.