A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. Introduced by Fox, Roughgarden, Seshadhri, Wei and Wein [ICALP 2018, SICOMP 2020], this definition is an abstraction of the triadic closure property exhibited by many real-world social networks, namely, friends of friends tend to be friends themselves. Social networks, however, are often temporal rather than static -- the connections change over a period of time. And hence temporal graphs, rather than static graphs, are often better suited to model social networks. Motivated by this, we introduce a definition of temporal c-closed graphs, in which if two vertices u and v have at least c common neighbors during a short interval of time, then u and v are adjacent to each other around that time. Our pilot experiments show that several real-world temporal networks are c-closed for rather small values of c. We also study the computational problems of enumerating maximal cliques and similar dense subgraphs in temporal c-closed graphs; a clique in a temporal graph is a subgraph that lasts for a certain period of time, during which every possible edge in the subgraph becomes active often enough, and other dense subgraphs are defined similarly. We bound the number of such maximal dense subgraphs in a temporal c-closed graph that evolves slowly, and thus show that the corresponding enumeration problems admit efficient algorithms; by slow evolution, we mean that between consecutive time-steps, the local change in adjacencies remains small. Our work also adds to a growing body of literature on defining suitable structural parameters for temporal graphs that can be leveraged to design efficient algorithms.

翻译：若图G中任意两个拥有至少c个共同邻居的顶点均彼此邻接，则称该图是c-闭的。该定义由Fox、Roughgarden、Seshadhri、Wei和Wein[ICALP 2018, SICOMP 2020]提出，是对现实社交网络中三元闭包性质的抽象——即朋友的朋友往往也是朋友。然而，社交网络通常具有时序性而非静态——连接关系随时间变化。因此，时序图比静态图更适合建模社交网络。受此启发，我们提出了时序c-闭图的定义：若两个顶点u和v在短时间区间内拥有至少c个共同邻居，则u和v在该时间段前后彼此邻接。初步实验表明，多个现实世界时序网络在较小c值下即呈现c-闭特性。我们还研究了时序c-闭图中极大团及类似稠密子图的计算枚举问题：时序图中的团是指持续特定时间段的子图，在此期间子图内所有可能边均频繁激活，其他稠密子图的定义类似。通过约束缓慢演化时序c-闭图中此类极大稠密子图的数量，我们证明了相应枚举问题存在高效算法；所谓缓慢演化，是指相邻时间步之间邻接关系的局部变化保持较小。本研究也为时序图结构参数定义的学术积累作出贡献，这类参数可用于设计高效算法。