  Considering all possible paths that a natural number can take following the
rules of the algorithm proposed in the Collatz conjecture we construct a graph
that can be interpreted as an infinite network that contemplates all possible
paths within the conjecture. This allows us to understand why the minimal
element of the Collatz orbit $x$ is equal to 1. Subsequently we define the
extended Collatz conjecture equal to $o x+1$ when $x$ is odd and $x/2$ when $x$
is even with $o$ an odd number greater than $3$ and we show that there are
infinite orbits in the extended Collatz conjecture that diverges. Finally, we
find interesting theorems relating the Fibonacci sequence to prime numbers.
