The Iterated Local Directed Transitivity Model for Social Networks

TL;DR

This paper introduces the ILDT model for social networks, capturing directed transitivity and densification, and analyzes its properties, including cycle counts and spectral characteristics, to better understand real-world online social network structures.

Contribution

The paper presents the ILDT model as a directed analogue of the ILT model, demonstrating its properties like densification, cycle counts, and Hamiltonian cycles, with relevance to real social networks.

Findings

ILDT graphs densify over time.

ILDT generates more transitive 3-cycles, similar to real social networks.

Many initial conditions lead to graphs with Hamiltonian directed cycles.

Abstract

We introduce a new 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 analogue of the ILT model for undirected graphs introduced in \cite{ilt}. 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. In many instances of the chosen initial digraph, the model eventually generates graphs with Hamiltonian directed cycles. We finish with a discussion of the eigenvalues of the adjacency matrices of ILDT directed graphs, and provide further directions.