Fast gossiping with short unreliable messages

Each of n nodes of a communication network has a piece of information (gossip) which should be made known to all other nodes. Gossiping is done by sending letters. In a unit of time each node can either send one letter to a neighbor or receive one such letter, containing one gossip currently known to the sender. Letters reach their destinations with constant probability 0 < q < 1, independently of one another. For a large class of networks, including rings, grids, hypercubes and complete graphs, we construct gossip schemes working in linear time and successfully performing gossiping with probability converging to 1, as the number of nodes grows.