Perfect Communication Networks

In a perfect communication model, every vertex can communicate with every other, receiving and sending messages, The network must be connected, so that there are no isolated vertices or isolated part of the network. If the netwok is drawn as a digraph - directed graph - then lines must enter and leave every vertex. It is not necessary that every vertex be directly connected to every other, only that messages can pass both ways between them, possibly through intermediate vertices

 A B C D A 0 1 1 1 B 1 0 1 0 C 1 1 0 1 D 1 0 1 0
This is a perfect communication network. In particular every vertx can obtain feedback.
$M= \left( \begin{array}{cccc} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{array} \right)$
is the matrix representing the connectedness of the netweok.
$M^2= \left( \begin{array}{cccc} 3 & 1 & 2 & 1 \\ 1 & 2 & 1 & 2 \\ 2 & 1 & 3 & 1 \\ 1 & 2 & 1 & 2 \end{array} \right), \; M^3 = \left( \begin{array}{cccc} 6 & 5 & 5 & 5 \\ 2 & 4 & 2 & 4 \\ 5 & 5 & 4 & 5 \\ 5 & 2 & 5 & 2 \end{array} \right)$

$M^2$
and
$M^3$
give the number of feedback routes - along the main diagonal - with 1 and 2 intermediat vertices repectively. The entry in the 1st row, 1st column of
$M$
is 3, so there are three feedback routes with 1 intermediate vertex. These are A-B-A, A-C-A and A-D-A.