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We study the communication cost of collecting correlated data at a sink over a network. To do so, we introduce Distance Entropy, an intrinsic quantity that characterizes the data gathering limit of networked sources. We demonstrate that, for any network embedded with any set of sources and a cost function [cost]=[data rate]×[link weight], distance entropy is a lower bound on the optimal communication cost. This is true for the most general data collection schemes that allow arbitrary routing and coding operations, including network coding and source coding. This lower bound can be matched using optimal rate Slepian-Wolf encoding plus shortest path routing. For more general communication cost functions, we show that the optimal scheme among schemes using Slepian-Wolf codes is also optimal over all of the possible schemes. Our results imply that for collecting data from correlated sources at a single sink, Network Coding does not help in the sense of lowering the optimal communication cost. We then extend our results to the case that includes broadcast links in the network. We show that the same optimal cost holds even if we allow broadcasting. In other words, neither broadcasting nor Network Coding improves the total cost of collecting data from correlated sources at a single sink.