When using GraphPlot with an adjacency matrix, how can I make Mathematica draw exactly one self loop for any non-zero weight?

Question

I have tried using the option MultiedgeStyle->None, but apparently Mathematica doesn't treat self loops in the same way it treats edges. For example, I find

GraphPlot[{{1,2},{2,0}}, SelfLoopStyle->All, MultiedgeStyle->None]

while the same command, but with twice the weight on the self loop gives

GraphPlot[{{2,2},{2,0}}, SelfLoopStyle->All, MultiedgeStyle->None]

I would like the output from both commands to be the same. I know I could write a function to replace all non-zero values on the diagonal with 1, but I'd prefer not to for two reasons. One is that it seems as if this should be possible without resorting to that, and the other is that I'm actually using this inside another routine to draw graphs with edge weights displayed, to get output such as

where the label inside the self loop should list the correct weight, taken from the diagonal of the adjacency matrix. Removing the diagonals for later labeling then changing them in the adjacency matrix just seems like a hack.

Answer 1

A simple hack of a way to obtain the behaviour that you want is to negate the adjacency matrix. Now, don't ask me what that means or represents, but it works!

Modifying the example from the accepted answer in the question that CHM shared to include multiple self loops:

You can see that the second one displays the behaviour you want.