A polynomial expansion network of a plurality of nodes, each node capable of communicating with the other nodes through a communications system. The network has an originating node that sends a message for ultimate receipt by a subset of nodes in a polynomial expansion manner. The network also has a first and a second receiving node, each of which receives the message from the originating node. The first and second receiving nodes also receive a unique list of other nodes to which the message has not yet been sent. The polynomial expansion manner can be optimized such that each node sends the message to no more than a predetermined number of other nodes to which the message has not yet been sent, or such that, before each node receives the message, the message passes through no more than a predetermined number of other nodes.