I have adjacency matrices which have nearly connected components. That is partitions with a dense number of edges between nodes in the same group and few edges acting as bridges between these groups. I have clustered them so that they form a band matrix. 

What ways exist to identity these nodes linking groups? These nodes that act as bridges between these nearly connected components.

It appears that each method I have seen has some element of it being a heuristic in some way.

Here is a heuristic argument that there is nothing to explain:

The probability that $p$ divides the sum of the preceding primes is $1/p$. So the expected number of primes less than $10^9$ with this property is $\sum_{p \leq 10^9} \frac{1}{p}$. Using Mertens' second theorem,
$$\sum_{p \leq 10^9} \frac{1}{p} \approx \log \log 10^9 + M \approx 3.3$$

Here $\log$ is natural log and $M \approx 0.26149$ is Mertens' constant. 

This is an example of the motto "$\log \log x$ goes to infinity but has never been observed to do so". It is quite common for people to look at primes $p$ which divide some quantity $a_p$ and conclude that they are surprisingly rare when, in fact, they are simply growing as $\log \log N$ for the reason above.
