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Systems that employ network coding for content distribution convey to the receivers linear combinations of the source packets. If we assume randomized network coding, during this process, the network nodes collect random subspaces of the space spanned by the source packets. We establish several fundamental properties of the random subspaces induced in such a system and show that these subspaces implicitly carry topological information about the network and its state that can be passively collected and inferred. We leverage this information toward a number of applications that are interesting in their own right, such as topology inference, bottleneck discovery in peer-to-peer systems, and locating Byzantine attackers. We thus argue that randomized network coding, apart from its better known properties for improving information delivery rate, can additionally facilitate network management and control.