The iterated local directed transitivity model for social networks

Abstract

We introduce a new, deterministic directed graph model for social networks, based on the transitivity of triads. In the Iterated Local Directed Transitivity (ILDT) model, new nodes are born over discrete time-steps and inherit the link structure of their parent nodes. The ILDT model may be viewed as a directed graph analog of the ILT model for undirected graphs introduced in [4]. We investigate network science and graph-theoretical properties of ILDT digraphs. We prove that the ILDT model exhibits a densification power law, so that the digraphs generated by the models densify over time. The number of directed triads are investigated, and counts are given of the number of directed 3-cycles and transitive 3-cycles. A higher number of transitive 3-cycles are generated by the ILDT model, as found in real-world, on-line social networks that have orientations on their edges. We discuss the eigenvalues of the adjacency matrices of ILDT digraphs. We finish by showing that in many instances of the chosen initial digraph, the model eventually generates digraphs with Hamiltonian directed cycles.