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：链接流为表示随时间演化的交互关系提供了良好模型，其由形如$(b,e,u,v)$的链接构成，其中$u$和$v$是在整个时间区间$[b,e]$内持续交互的顶点。本文研究链接流中最大团的枚举问题。团定义为二元组$(C,[t_0,t_1])$，其中$C$是在完整区间$[t_0,t_1]$内所有顶点两两交互的顶点集合。当无法通过扩展顶点集或时间区间生成更大团时，该团即为最大团。现有解决该问题的主要工作大多基于图论中枚举最大团的经典Bron-Kerbosch算法。我们以此为出发点提出新算法，将给定时刻$t$的瞬时图（由当前存在的链接构成）中的团与链接流中的最大团建立对应关系。我们证明了算法的正确性并计算其复杂度，在多种重要场景下优于现有方法。同时研究了输出敏感复杂度，其值接近输出规模，表明算法的高效性。为验证这一点，我们在现有文献使用的链接流及高达一亿条链接的大规模链接流上开展实验。在所有测试案例中，本算法的运行速度均更快——多数情况下提速至少10倍，最高可达$10^4$倍。此外，该算法能处理现有方法无法求解的大规模链接流。