How likely is a random graph shift-enabled?

TL;DR

This paper investigates the conditions under which random graphs are shift-enabled, revealing that unweighted graphs are typically shift-enabled with moderate density, while weighted graphs are almost always shift-enabled except when very sparse.

Contribution

It provides a comprehensive analysis of shift-enabled properties in various random graph models, highlighting differences between weighted and unweighted graphs.

Findings

Unweighted connected graphs are shift-enabled with moderate edges.

Very dense or fully connected unweighted graphs are not shift-enabled.

Weighted connected graphs are generally shift-enabled unless very sparse.

Abstract

The shift-enabled property of an underlying graph is essential in designing distributed filters. This article discusses when a random graph is shift-enabled. In particular, popular graph models ER, WS, BA random graph are used, weighted and unweighted, as well as signed graphs. Our results show that the considered unweighted connected random graphs are shift-enabled with high probability when the number of edges is moderately high. However, very dense graphs, as well as fully connected graphs, are not shift-enabled. Interestingly, this behaviour is not observed for weighted connected graphs, which are always shift-enabled unless the number of edges in the graph is very low.