Bonne and Censor-Hillel (ICALP 2019) initiated the study of distributed subgraph finding in dynamic networks of limited bandwidth. For the case where the target subgraph is a clique, they determined the tight bandwidth complexity bounds in nearly all settings. However, several open questions remain, and very little is known about finding subgraphs beyond cliques. In this work, we consider these questions and explore subgraphs beyond cliques. For finding cliques, we establish an $\Omega(\log \log n)$ bandwidth lower bound for one-round membership-detection under edge insertions only and an $\Omega(\log \log \log n)$ bandwidth lower bound for one-round detection under both edge insertions and node insertions. Moreover, we demonstrate new algorithms to show that our lower bounds are tight in bounded-degree networks when the target subgraph is a triangle. Prior to our work, no lower bounds were known for these problems. For finding subgraphs beyond cliques, we present a complete characterization of the bandwidth complexity of the membership-listing problem for every target subgraph, every number of rounds, and every type of topological change: node insertions, node deletions, edge insertions, and edge deletions. We also show partial characterizations for one-round membership-detection and listing.

翻译：Bonne与Censor-Hillel（ICALP 2019）开创了有限带宽动态网络中分布式子图查找的研究。针对目标子图为团的情况，他们在几乎所有设定下确定了紧致的带宽复杂度边界。然而，若干开放问题依然存在，且关于团之外子图查找的研究甚少。本文针对这些问题展开研究，并探索团之外的子图查找。在团查找方面，我们建立了仅边插入场景下单轮成员检测的$\Omega(\log \log n)$带宽下界，以及边插入与节点插入共存场景下单轮检测的$\Omega(\log \log \log n)$带宽下界。此外，我们提出新算法证明：当目标子图为三角形且网络度数有界时，所得下界是紧致的。此前研究尚未给出这些问题的任何下界。针对团之外的子图查找，我们完整刻画了所有目标子图、任意轮次及各类拓扑变化（节点插入、节点删除、边插入、边删除）场景下成员列举问题的带宽复杂度。同时，我们还给出了单轮成员检测与列举问题的部分特征刻画。