Distributed function calculation via linear iterations in the presence of malicious agents — Part I: Attacking the network

We consider the problem of distributed function calculation in the presence of faulty or malicious agents. In particular, we consider a setup where each node has an initial value and the goal is for (a subset of) the nodes to calculate a function of these values in a distributed manner. We focus on linear iterative strategies for function calculation, where each node updates its value at each time-step to be a weighted average of its own previous value and those of its neighbors; after a sufficiently large number of time-steps, each node is expected to have enough information to calculate the desired function of the initial node values. We study the susceptibility of such strategies to misbehavior by some nodes in the network; specifically, we consider a node to be malicious if it updates its value arbitrarily at each time-step, instead of following the predefined linear iterative strategy. If the connectivity of the network topology is 2f or less, we show that it is possible for a set of f malicious nodes to conspire in a way that makes it impossible for a subset of the other nodes in the network to correctly calculate an arbitrary function of all node values. Our analysis is constructive, in that it provides a specific scheme for the malicious nodes to follow in order to obfuscate the network in this fashion.