Investigating Randomly Generated Adjacency Matrices For Their Use In Modeling Wireless Topologies

TL;DR

This paper analyzes the limitations of randomly generated adjacency matrices in accurately modeling wireless network topologies, showing most such matrices do not conform to real-world constraints, especially with more than three nodes.

Contribution

It demonstrates that random adjacency matrices often violate geometric constraints of wireless topologies and provides a probability model for their failure, aiding in better topology generation algorithms.

Findings

Over 90% of random matrices with more than 3 nodes do not conform to wireless topologies.

The number of invalid node triplets grows as O(base^3), where base is the distance granularity.

A probability estimate for random matrices failing to realize valid topologies is provided.

Abstract

Generation of realistic topologies plays an important role in determining the accuracy and validity of simulation studies. This study presents a discussion to justify why, and how often randomly generated adjacency matrices may not not conform to wireless topologies in the physical world. Specifically, it shows through analysis and random trials that, more than 90% of times, a randomly generated adjacency matrix will not conform to a valid wireless topology, when it has more than 3 nodes. By showing that node triplets in the adjacency graph need to adhere to rules of a geometric vector space, the study shows that the number of randomly chosen node triplets failing consistency checks grow at the order of O(base^3), where base is the granularity of the distance metric. Further, the study models and presents a probability estimate with which any randomly generated adjacency matrix would fail realization. This information could be used to design simpler algorithms for generating k-connected wireless topologies.