Distributed detection of cliques in dynamic networks

This paper provides an in-depth study of the fundamental problems of finding small subgraphs in distributed dynamic networks. While some problems are trivially easy to handle, such as detecting a triangle that emerges after an edge insertion, we show that, perhaps somewhat surprisingly, other problems exhibit a wide range of complexities in terms of the trade-offs between their round and bandwidth complexities. In the case of triangles, which are only affected by the topology of the immediate neighborhood, some end results are: The bandwidth complexity of 1-round dynamic triangle detection or listing is Θ(1). The bandwidth complexity of 1-round dynamic triangle membership listing is Θ(1) for node/edge deletions, Θ(n¹/²) for edge insertions, and Θ(n) for node insertions. The bandwidth complexity of 1-round dynamic triangle membership detection is Θ(1) for node/edge deletions, O(log n) for edge insertions, and Θ(n) for node insertions. Most of our upper and lower bounds are tight. Additionally, we provide almost always tight upper and lower bounds for larger cliques.