We study how the strongly sublinear MPC model relates to the classic, graph-centric distributed models, focusing on the Node-Capacitated Clique (NCC), a bandwidth-parametrized generalization of the Congested Clique. In MPC, $M$ machines with per-machine memory $S$ hold a partition of the input graph, in NCC, each node knows its full neighborhood but can send/receive only a bounded number of $C$ words per round. We are particularly interested in the strongly sublinear regime where $S=C=n^δ$ for some constant $0 < δ<1$. Our goal is determine when round-preserving simulations between these models are possible and when they are not, when total memory and total bandwidth $SM=nC$ in both models are matched, for different problem families and graph classes. On the positive side, we provide techniques that allow us to replicate the specific behavior regarding input representation, number of machines and local memory from one model to the other to obtain simulations with only constant overhead. On the negative side, we prove simulation impossibility results, which show that the limitations of our simulations are necessary.

翻译：我们研究了强亚线性MPC模型如何与经典的、以图为中心的分布式模型相关联，重点关注节点容量Clique（NCC）——一种带宽参数化的拥塞Clique推广模型。在MPC中，$M$台机器每台具有内存$S$，共同持有输入图的一个划分；在NCC中，每个节点知晓其全部邻居，但每轮只能发送/接收有限数量的$C$个单词。我们特别关注强亚线性机制，其中$S=C=n^δ$，$0 < δ<1$为常数。我们的目标是确定在总内存和总带宽$SM=nC$在两个模型中匹配的情况下，对于不同问题族和图类，这些模型之间保持轮数不变的模拟何时可行、何时不可行。在积极方面，我们提供了能够将一个模型中关于输入表示、机器数量和本地内存的具体行为复制到另一个模型的技术，从而仅以常数开销获得模拟。在消极方面，我们证明了模拟不可能性结果，这些结果表明我们模拟的局限性是必然存在的。