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We consider the problem of self-healing in peer-to-peer networks that areunder repeated attack by an omniscient adversary. We assume that, over asequence of rounds, an adversary either inserts a node with arbitraryconnections or deletes an arbitrary node from the network. The network respondsto each such change by quick "repairs," which consist of adding or deleting asmall number of edges.  These repairs essentially preserve closeness of nodes after adversarialdeletions, 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 (notthe adversarial deletions), will be at distance at most $\ell \log n$ in theactual graph, where $n$ is the total number of vertices seen so far. Similarly,at any point, a node $v$ whose degree would have been $d$ in the graph withadversarial insertions only, will have degree at most 3d in the actual graph.Our algorithm is completely distributed and has low latency and bandwidthrequirements.

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