Cascade attacks
Related link: http://xxx.lanl.gov/PS_cache/cond-mat/pdf/0301/0301086.pdf
You attack and kill a node. Its load is redistributed elsewhere. The nodes that take on this new work are now more likely to be overloaded, which means they're weaker. You attack these overloaded nodes. When they go down their responsibilities, which include those of the first victim, are also redistributed. Nodes that take on this new load have even more new work than after the first attack, so are even more overloaded, and are more vulnerable.
"Cascade-based attacks on complex networks" analyzes such attacks. The abstract is:
We live in a modern world supported by large, complex networks. Examples range from financial markets to communication and transportation systems. In many realistic situations the flow of physical quantities in the network, as characterized by the loads on nodes, is important. We show that for such networks where loads can redistribute among the nodes, intentional attacks can lead to a cascade of overload failures, which can in turn cause the entire or a substantial part of the network to collapse. This is relevant for real-world networks that possess a highly heterogeneous distribution of loads, such as the Internet and power grids. We demonstrate that the heterogeneity of these networks makes them particularly vulnerable to attacks in that a large-scale cascade may be triggered by disabling a single key node. This brings obvious concerns on the security of such systems.
Their key finding is that self-healing networks have pretty good resistance to natural failures, but not to deliberate attacks on high-degree nodes.
This doesn't apply to all peer-to-peer apps. BitTorrent, for example, doesn't do load balancing in a way that would make it susceptible. A distributed hash table, on the other hand, would be very susceptible.
Thanks to Anatole Shaw for the link.
Update: Gojomo points out that I'm dead wrong for DHTs with each real node at multiple places in the virtual space. He's right, and not just when each real node is at multiple virtual places. This attack only applies to networks with uneven distributions, which is exactly what the original paper said.
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DHTs
Most DHT actually arrange each real node to be at many places in the virtual space -- so that when any one node fails, rather than hitting one or a small number of "neighbors" with extra traffic, some large number 'n' of other nodes each get 1/n of the resulting traffic -- with n perhaps approaching the total number of nodes in the system, if such a level of smooth redistribution is desired.
It's hard to imagine any system doing better...