it is known both scientifically and anecdotally that typical large-scale social networks exhibit short paths between pairs of nodes, hence the common phrase ``six degrees of separation''. in this talk, mathematical background for this phenomenon is given, together with a study of the algorithmic question of how to search such large-scale networks using only local information. in particular, one could imagine that each person in the network is allowed to pass a message to one of their acquaintances, until a particular target person in the network receives the message. remarkably, for many networks with $n$ nodes, there is a simple algorithm which does this extremely efficiently -- a ``decentralized search'' algorithm which runs in time polylogarithmic in the number of nodes. the mathematics in this talk involves only elementary combinatorics and probability.