Temporal Triadic Closure: Finding Dense Substructures in Social Networks That Evolve over Time

A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. 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 other 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; 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.