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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.

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    $\begingroup$ 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. $\endgroup$ – David Richerby Aug 18 '18 at 9:00
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Sure, of course. You can define a matrix to contain whatever numbers you want it to contain. There's nothing that prevents you. The real question is whether the result has the properties you want it to have, but since you haven't listed any properties, there's nothing to answer here.

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Assuming for the moment that the two types of person are distinct, your graph is (directed) bipartite, so it makes more sense to store it as a matrix whose rows correspond to people with fishing rights, and whose columns correspond to fishermen.

If a person can function in both roles, you can think of their two personas as distinct, i.e., have a row and a column for each.

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