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：链路流为随时间推移的交互提供了一种良好的表示模型。它由链接$(b,e,u,v)$组成，其中$u$和$v$是在整个时间区间$[b,e]$内交互的顶点。本文研究链路流中极大团的枚举问题。一个团定义为二元组$(C,[t_0,t_1])$，其中$C$是在完整区间$[t_0,t_1]$内两两交互的顶点集合。当顶点集和时间区间均无法扩展时，该团称为极大团。现有解决该问题的主要方法基于经典的Bron-Kerbosch图极大团枚举算法。以此为出发点，我们提出一种新算法，该算法将在给定时间$t$存在的链接所构成的瞬时图中的团与链路流的极大团进行匹配。我们证明了算法的正确性并计算了其复杂度，该复杂度在多数实际应用场景中优于现有最优算法。我们还研究了输出敏感复杂度，其值接近输出规模，从而证明算法具有高效性。为验证这一点，我们在现有研究使用的链路流以及大规模链路流（包含高达1亿条链接）上进行了实验。在所有情况下，我们的算法均表现出更快的速度，通常至少提升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路流。