Directed cycles and related structures in random graphs: II--Dynamic properties

TL;DR

This paper investigates the dynamic evolution of directed random graphs, analyzing how different growth strategies influence the formation of cycles and strong components through analytic and simulation methods.

Contribution

It introduces and compares two evolution strategies for directed graphs, providing new insights into their degree distributions and cycle formation over time.

Findings

Near acyclicity is maintained under one strategy.

Directed cycles emerge under the alternative strategy.

Strong component behavior is characterized during graph evolution.

Abstract

We study directed random graphs (random graphs whose edges are directed) as they evolve in discrete time by the addition of nodes and edges. For two distinct evolution strategies, one that forces the graph to a condition of near acyclicity at all times and another that allows the appearance of nontrivial directed cycles, we provide analytic and simulation results related to the distributions of degrees. Within the latter strategy, in particular, we investigate the appearance and behavior of the strong components that were our subject in the first part of this study.