Simple models of small world networks with directed links

TL;DR

This paper models small world networks with directed links, analyzing how shortcuts and unfavorable bonds influence spreading processes and network accessibility through exact analytical methods.

Contribution

It introduces a simple directed small world network model and derives exact analytical expressions for spreading dynamics and accessibility.

Findings

Unfavorable bonds hinder spreading and reduce accessible network size.

Shortcuts facilitate spreading but are affected by bond directionality.

The model reveals the interplay between shortcuts and bonds in small world properties.

Abstract

We investigate the effect of directed short and long range connections in a simple model of small world network. Our model is such that we can determine many quantities of interest by an exact analytical method. We calculate the function $V(T)$, defined as the number of sites affected up to time $T$ when a naive spreading process starts in the network. As opposed to shortcuts, the presence of un-favorable bonds has a negative effect on this quantity. Hence the spreading process may not be able to affect all the network. We define and calculate a quantity named the average size of accessible world in our model. The interplay of shortcuts, and un-favorable bonds on the small world properties is studied.