How to Distribute Computation in Networks
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We study the function computation problem in a communications network. The rate region for the function computation problem in general topologies is an open problem, and has been considered under certain restrictive assumptions (e.g. tree networks, linear functions, etc.). We are motivated by the fact that in network computation can be as a means to reduce the required communication flow in terms of number of bits transmitted per source symbol and provide a sparse representation or labeling. To understand the limits of computation, we introduce the notion of entropic surjectivity as a measure to determine how surjective the function is. Exploiting Little's law for stationary systems, we later provide a connection between this notion and the proportion of flow (which we call computation processing factor) that requires communications. This connection gives us an understanding of how much a node (in isolation) should compute (or compress) in order to communicate the desired function within the network. Our analysis does not put any assumptions on the network topology and characterizes the functions only via their entropic surjectivity, and provides insight into how to distribute computation depending on the entropic surjectivity of the computation task.
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