Temporal Triadic Closure: Finding Dense Structures in Social Networks That Evolve

TL;DR

This paper introduces the concept of temporal c-closed graphs to model evolving social networks and demonstrates that dense subgraphs like maximal cliques can be efficiently enumerated in such structures.

Contribution

It extends the static notion of c-closure to temporal graphs, providing bounds and algorithms for enumerating dense subgraphs in evolving social networks.

Findings

Real-world temporal networks are often c-closed for small c.

Efficient algorithms exist for enumerating maximal dense subgraphs in slowly evolving temporal graphs.

The study advances structural parameterization for temporal graph algorithms.

Abstract

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.