  In this paper, we investigate the topology convergence problem for the
gossip-based Gradient overlay network. In an overlay network where each node
has a local utility value, a Gradient overlay network is characterized by the
properties that each node has a set of neighbors with the same utility value (a
similar view) and a set of neighbors containing higher utility values (gradient
neighbor set), such that paths of increasing utilities emerge in the network
topology. The Gradient overlay network is built using gossiping and a
preference function that samples from nodes using a uniform random peer
sampling service. We analyze it using tools from matrix analysis, and we prove
both the necessary and sufficient conditions for convergence to a complete
gradient structure, as well as estimating the convergence time and providing
bounds on worst-case convergence time. Finally, we show in simulations the
potential of the Gradient overlay, by building a more efficient live-streaming
peer-to-peer (P2P) system than one built using uniform random peer sampling.
