Fundamental limitations for anonymous distributed systems with broadcast communications

We consider deterministic anonymous distributed systems with broadcast communications where each node has some initial value, and the goal is to compute a function of all these values. We show that only a very restricted set of functions can be computed if the nodes do not know (and cannot use) the number of their out-neighbors. Our results remain valid even if nodes know the precise structure of the network but do not know where they lie within the structure. They also remain valid if nodes know their out-degree up to an uncertainty of 1. These results are a variation of those obtained by Boldi and Vigna (1997) for a weaker computation model. As a consequence, computing more complex functions in the context of broadcast communications requires the explicit or implicit knowledge or use of either (a) the out-degree of each node, (b) global node identifiers, (c) randomization, or (d) asynchronous updates with specific properties.