Distributed function computation is the problem, for a networked system of $n$ autonomous agents, to collectively compute the value $f(v_1, \ldots, v_n)$ of some input values, each initially private to one agent in the network. Here, we study and organize results pertaining to distributed function computation in anonymous networks, both for the static and the dynamic case, under a communication model of directed and synchronous message exchanges, but with varying assumptions in the degree of awareness or control that a single agent has over its outneighbors. Our main argument is three-fold. First, in the "blind broadcast" model, where in each round an agent merely casts out a unique message without any knowledge or control over its addressees, the computable functions are those that only depend on the set of the input values, but not on their multiplicities or relative frequencies in the input. Second, in contrast, when we assume either that a) in each round, the agents know how many outneighbors they have; b) all communications links in the network are bidirectional; or c) the agents may address each of their outneighbors individually, then the set of computable functions grows to contain all functions that depend on the relative frequencies of each value in the input - such as the average - but not on their multiplicities - thus, not the sum. Third, however, if one or several agents are distinguished as leaders, or if the cardinality of the network is known, then under any of the above three assumptions it becomes possible to recover the complete multiset of the input values, and thus compute any function of the distributed input as long as it is invariant under permutation of its arguments. In the case of dynamic networks, we also discuss the impact of multiple connectivity assumptions.

翻译：分布式函数计算问题是指：一个由$n$个自主智能体组成的网络系统，需共同计算某个输入值（每个输入初始由网络中一个智能体私有）的函数$f(v_1, \ldots, v_n)$。本文针对匿名网络中的分布式函数计算，系统梳理了静态与动态两种情况下的相关结果。研究基于有向同步消息交换的通信模型，但针对单个智能体对其出向邻居的感知或控制程度，采用了不同假设。我们的核心论点包含三层：第一，在"盲广播"模型中（每轮智能体仅发送唯一消息，无法知晓或控制接收对象），可计算的函数仅限于仅依赖输入值集合、而不依赖输入值多重性或相对频率的函数。第二，相反，若假设：a) 每轮智能体知道自身出向邻居数量；b) 网络中所有通信链路为双向；或c) 智能体可分别寻址每个出向邻居，则可计算函数集扩展至包含所有依赖输入中各值相对频率的函数（如平均值），但排除依赖多重性的函数（因此不包括求和）。第三，若存在一个或多个被区分的领导智能体，或网络基数已知，则在上述任一假设下均可恢复输入值的完整多重集，从而计算分布式输入的任何函数（只要该函数在参数置换下保持不变）。针对动态网络，我们还讨论了多种连通性假设的影响。