  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.
