The Iterated Local Transitivity Model for oriented graphs

Abstract

Transitivity in social and other complex networks describes the tendency for two nodes with a common neighbor to become adjacent. We propose a new model for complex networks based on transitivity, called the Iterated Local Transitivity Model for Oriented Graphs (ILTO). The model generalizes the Iterated Local Model for Tournaments (ILMT). As in ILMT, oriented graphs are generated iteratively by cloning nodes and their adjacencies, with arcs between clones oriented either as the corresponding arcs between their parents or in the reverse direction. This process produces dense oriented graphs and, under suitable conditions, graphs with high connectivity and small diameter. We analyze the universality properties of ILTO, showing that, for generating sequences with infinite support, the base oriented graph determines which oriented graphs appear as induced subgraphs, unlike in ILMT. We study motifs in ILTO and show that when the base graph is a tournament and the generating sequence has infinite support, the model generates quasirandom tournament sequences. Finally, we study domination numbers, the cop number, and the dichromatic number. We bound the cop number in terms of the independence number of the base graph and show that, for each fixed base graph, the dichromatic number is $Θ(2^t)$ after $2t$ consecutive 0-steps.