Hierarchy in directed random networks

TL;DR

This paper explores the hierarchical structures in directed random networks, analyzing how correlations influence hierarchy, with findings indicating non-monotonic effects and an optimal correlation level for maximum hierarchy.

Contribution

It provides qualitative analytic results on hierarchy in zero-correlation networks and numerical insights into how correlations affect hierarchical properties.

Findings

Hierarchy varies with the presence of giant components.

One-point correlations can significantly alter hierarchical structure.

Hierarchy peaks at an optimal non-zero correlation level.

Abstract

In recent years, the theory and application of complex networks have been quickly developing in a markable way due to the increasing amount of data from real systems and to the fruitful application of powerful methods used in statistical physics. Many important characteristics of social or biological systems can be described by the study of their underlying structure of interactions. Hierarchy is one of these features that can be formulated in the language of networks. In this paper we present some (qualitative) analytic results on the hierarchical properties of random network models with zero correlations and also investigate, mainly numerically, the effects of different type of correlations. The behavior of hierarchy is different in the absence and the presence of the giant components. We show that the hierarchical structure can be drastically different if there are one-point correlations in the network. We also show numerical results suggesting that hierarchy does not change monotonously with the correlations and there is an optimal level of non-zero correlations maximizing the level of hierarchy.