—In this paper we face the following problem: how to provide each peer local access to the full information (not just a summary) that is distributed over all edges of an overlay network? How can this be done if local access is performed at a given rate? We focus on large and sparse information and we propose to exploit the compressive sensing (CS) theory to efficiently collect and pro-actively disseminate this information across a large overlay network. We devise an approach based on random walks (RW) to spread CS random combinations to participants in a random peer-topeer (P2P) overlay network. CS allows the peer to compress the RW payload in a distributed fashion: given a constraint on the RW size, e.g., the maximum UDP packet payload size, this amounts to being able to distribute larger information and to guarantee that a large fraction of the global information is obtained by each peer. We analyze the performance of the proposed method by means of a simple (yet accurate) analyti...