This paper considers the problem of minimum cost communication of correlated sources over a network with multiple sinks, which consists of distributed source coding followed by routing. We introduce a new routing paradigm called dispersive information routing, wherein the intermediate nodes are allowed to `split' a packet and forward subsets of the received bits on each of the forward paths. This paradigm opens up a rich class of research problems which focus on the interplay between encoding and routing in a network. Unlike conventional routing methods such as in [1], dispersive information routing ensures that each sink receives just the information needed to reconstruct the sources it is required to reproduce. We demonstrate using simple examples that our approach offers better asymptotic performance than conventional routing techniques. This paradigm leads to a new information theoretic setup, which has not been studied earlier. We propose a new coding scheme, using principles from multiple descriptions encoding [2] and Han and Kobayashi decoding [3]. We show that this coding scheme achieves the complete rate region for certain special cases of the general setup and thereby achieves the minimum communication cost under this routing paradigm.