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:链接流为代表长期互动提供了一个很好的模式。 它们由链接 $( b, e, u, v) 美元组成, 美元和 $v) 美元, 其中美元和 $v 美元是整个时间间隔 $[ b, e, e] 美元 。 在本文中, 我们处理在链接流中计算最大 cliques 的问题。 一个 cloique 是一对 $( C, [ t_ 0, t_ 1) 美元, 美元是一组在全部时间间隔 $[ t_ 0, t_ 1] 中所有自动互动的顶端。 当它的顶端或时间间隔都无法增加时, 美元和 美元是顶尖的顶尖的顶端点 。 解决此问题的主要工作是以著名的 Bron- Kerbosch 算法来计算最大 cliquequencle 。 我们以这个想法作为起点的新的算法, 在一个特定时间段段里, $tal legal lex 中, 我们以最快速的递解算算算算出它的精度 。 。 在 10 ral 中, 我们以 lix lix lix 的精度中, 的精度中, 我们的精度的精度的精度中, 的精度能到 。