Link streams offer a good model for representing interactions over time. They consist of links $(b,e,u,v)$, where $u$ and $v$ are vertices interacting during the whole time interval $[b,e]$. In this paper, we deal with the problem of enumerating maximal cliques in link streams. A clique is a pair $(C,[t_0,t_1])$, where $C$ is a set of vertices that all interact pairwise during the full interval $[t_0,t_1]$. It is maximal when neither its set of vertices nor its time interval can be increased. Some of the main works solving this problem are based on the famous Bron-Kerbosch algorithm for enumerating maximal cliques in graphs. We take this idea as a starting point to propose a new algorithm which matches the cliques of the instantaneous graphs formed by links existing at a given time $t$ to the maximal cliques of the link stream. We prove its validity and compute its complexity, which is better than the state-of-the art ones in many cases of interest. We also study the output-sensitive complexity, which is close to the output size, thereby showing that our algorithm is efficient. To confirm this, we perform experiments on link streams used in the state of the art, and on massive link streams, up to 100 million links. In all cases our algorithm is faster, mostly by a factor of at least 10 and up to a factor of $10^4$. Moreover, it scales to massive link streams for which the existing algorithms are not able to provide the solution.

翻译：链接流为随时间进行的交互提供了一种良好的建模方式。它们由链接$(b,e,u,v)$构成，其中$u$和$v$是在整个时间区间$[b,e]$内进行交互的顶点。本文研究链接流中极大团的枚举问题。一个团是一对$(C,[t_0,t_1])$，其中$C$是一个顶点集合，这些顶点在整个区间$[t_0,t_1]$内两两交互。当该团的顶点集合或时间区间无法进一步扩大时，称其为极大团。现有解决此问题的主要工作大多基于著名的Bron-Kerbosch算法（一种用于枚举图中极大团的算法）。我们以此思想为出发点，提出一种新算法，该算法将给定时间$t$上由现有链接形成的瞬时图的团与链接流的极大团进行匹配。我们证明了该算法的正确性并计算了其复杂度，在众多感兴趣的情况下优于现有方法。我们还研究了输出敏感复杂度，该复杂度接近输出规模，从而表明我们的算法是高效的。为验证这一点，我们在现有研究使用的链接流以及高达1亿条链接的大规模链接流上进行了实验。在所有情况下，我们的算法均更快，多数情况下快至少10倍，最高可达$10^4$倍。此外，该算法能够扩展到现有算法无法求解的大规模链接流。