Depends what you mean with "network diagram". What is your biological question?
There is a lot of way you can formulate networks, not all of them relevant for every biological question, and the linked paper doesn't seem to mention any.
As you pointed out, you have a rectangular, not square, matrix. You could make a bipartite graph, where a water environmental factor can be connected to several dominant orders (and reversely), but orders can't be connected with each other (nor factors with each other). You could also want to make a graph of most similar dominant orders (or a graph of most similar environmental factors), computing distances between columns (or rows).
I think this chapter could be useful (see e.g. figure 5.3 which makes use of distances between cancer samples computed from a rectangular matrix). You might find other ideas in the trees and networks section of the same book, which also looks at bacterial abundance (though probably not with the same biological question, what they're doing there may be more advanced than what you need).