The Forgiving Graph: A distributed data structure for low stretch under adversarial attack

Abstract

We consider the problem of self-healing in peer-to-peer networks that are under repeated attack by an omniscient adversary. We assume that, over a sequence of rounds, an adversary either inserts a node with arbitrary connections or deletes an arbitrary node from the network. The network responds to each such change by quick "repairs," which consist of adding or deleting a small number of edges. These repairs essentially preserve closeness of nodes after adversarial deletions, without increasing node degrees by too much, in the following sense. At any point in the algorithm, nodes $v$ and $w$ whose distance would have been $\ell$ in the graph formed by considering only the adversarial insertions (not the adversarial deletions), will be at distance at most $\ell \log n$ in the actual graph, where