An interesting feature of circle patterns is the distribution of "holes" where future circles could be placed. If you look at such a pattern you readily see them. The task of the random placement algorithm is to search out such holes and place new circles in them.
I have been working on ways to construct the pattern of "holes" in a circle pattern, taking as my examples cases with fairly high c values (circles rather tightly packed). The results are in this .pdf document click here.
For computational reasons I have placed all the circles within a larger bounding circle. This works fine.
If you have placed n circles, I believe the combinatorial rules say you have n!/3! = n!/6 circle triplets, each of which could (in principle) result in a hole. Most of these triplets have "touching" circles which are too big to fit inside the bounding circle or overlap previous circles; only a small minority will create "holes".
The pattern has strong constraints arising from the space-filling power-law size sequence and the "not overlap" condition and this presumably accounts for the observed trend.
If you think about the placement of a new circle in a hole you can see it as going into a hole with 3 nearest neighbors and creating 3 new holes while annihilating 1 hole, for a net gain of 2. This says that the count of holes will go up by 2 as n is incremented by 1, which is what is seen. Why the number of "edge" holes should be the same as the "interior" holes isn't obvious to me. The nonrandom Apollonian circle packing fractals show the same property -- each new circle creates three new holes and eliminates one hole.
John