# Distributed Detection of Cliques in Dynamic Networks

## TL;DR

This paper investigates the complexities of detecting small subgraphs, like triangles and larger cliques, in distributed dynamic networks, revealing tight bounds and trade-offs between round and bandwidth complexities.

## Contribution

It provides tight bounds and complexity trade-offs for detecting and listing small subgraphs, especially triangles, in dynamic distributed networks.

## Key findings

- Bandwidth complexity of 1-round triangle detection is Θ(1).
- Complexity varies with node/edge insertions and deletions.
- Almost tight bounds are established for larger cliques.

## Abstract

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: \begin{itemize} \item The bandwidth complexity of $1$-round dynamic triangle detection or listing is $\Theta(1)$. \item The bandwidth complexity of $1$-round dynamic triangle membership listing is $\Theta(1)$ for node/edge deletions, $\Theta(n^{1/2})$ for edge insertions, and $\Theta(n)$ for node insertions. \item The bandwidth complexity of $1$-round dynamic triangle membership detection is $\Theta(1)$ for node/edge deletions, $O(\log n)$ for edge insertions, and $\Theta(n)$ for node insertions. \end{itemize} Most of our upper and lower bounds are \emph{tight}. Additionally, we provide almost always tight upper and lower bounds for larger cliques.