TABLE OF CONTENTS All the results given for the random connection model also hold in the special case of the boolean model. It is interesting, however, to restate the phase transition theorem, and emphasise the scaling properties of the Poisson process. We define the node degree of the random boolean network as the average number of connections of a point of the Poisson process, given by ? = 4 ?r 2 ?, and we give the following three equivalent formulations of the phase transition.
Theorem 2.6.1
In a boolean random network of radius r, there exists a critical density 0 < ? c < ? such that ?( ?) = 0 for ? < ? c , and ?( ?) > 0 for > ? c .
In a boolean random network of density ?, there exists a critical radius 0 < r c < ? such that ?(r) = 0 for r < r c , and ?(r) > 0 for r > r c .
In a boolean random network, there exists a critical node degree 0 < ? c < ? such that ?( ?) = 0 for ? < ? c , and ?( ?) > 0 for ? > ? c .
Although exact values of the critical quantities in Theorem 2.6.1 are not known, analytic...
