Wireless networks are typically modeled as random geometric graphs. The number of nodes $N$ in the network is drawn from a Poisson distribution with intensity $\lambda$

$$P(N = n) = \frac{\lambda^n e^{-n}}{n!}.$$

Once this number has been chosen, the $N$ nodes are placed uniformly at random over a circular (or squared) area of radius $R$. Two nodes are then connected by an edge if their euclidean distance is less than some predefined range, say $r_0$, where $r_0 << R$.

As a result of **Penrose's Theorem** to ensure that the graph is $k$-connected, it is sufficient to show that the minimum degree is at least $k$, i.e., $d_{\rm min} \geq k$. This holds true asymptotically for a geometric random graph.

I came across two papers dealing with connectivity in (wireless) networks (this is **one**, and here is **another one**, but there are others by different Authors in the literature). When posing the condition that the minimum degree is at least $k$, I often find this approximation:

$$ P(G{\rm ~is~k~conn}) \cong P (d_{\rm min} \geq k) \cong P(d \geq k)^n$$

where $P(d \geq k)$ is the probability that the degree of a node (any node) is at least $k$. The first approximation is essentially true when the number of nodes is at least of a few hundreds. The second approximation is not true since the nodes degrees are correlated. What people say when using it, is that they assume *almost* statistical independence of the nodes degrees.

In practice there is an excellent match between simulation results and the approximation above. Since I need to use the very same approximation, I was wondering if someone had a better argument to justify its use, rather than assuming its validity from the very beginning.