Representing a network with two types of connections: A fishing application

I want to represent a fishing network using a graph representation. My question surrounds how I can write the adjacency matrix if there are two types of connections, which I want to capture together.

Here's the most basic setup. Say there are two people represented by nodes.

1. Person one owns a fishing right and also acts as an authority who can land fish.
2. Person two is a fisherman.

Person one sells the signle fishing right to person two. Person two catches the fish and sells the fish back to person one. Intuitively, I want to write a directed adjacency matrix, in which positive i,j values represent the fishing rights transaction and negative j,i values represent the physical fish transaction, which leads to the following adjacency matrix:

\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}

My question is: Can I use signs like this to represent related but distinct connections? The basic logis follows from the fact the total fish caught = the total fishing rights. By summing the elements, we get zero representing a closed system. Here is a more complicated example:

\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & -1 & 0 \\ \end{bmatrix}

In this example person one sells the fishing right to person two as before. Now there is a person three, however, who lands the fish. Again the sum of all the elements is zero as total fishing rights = total fish caught. The network captures a different reality, however. In this second network, all the activities are separated as in an open market. In the original network, all the activities were through a relationship between person one and person two.

• I don't understand your question. You seem to be asking, "Can I do this thing that I've just done?" The answer is "yes", right? You just did it. – David Richerby Aug 18 '18 at 9:00