Generally speaking, for these sorts of complex reactions, you want to go to more than two dimensions. (At least for the initial bit.)
For each potential binding reaction you have a different dimension. That is, for $L$ binding to $A_1$ on site 1, you go horizontally. For $L$ binding to $A_2$ on site 1, you go vertically. For $L$ binding to $A_1$ on site 2, you go in the Z dimension, and then for $L$ binding to $A_2$ on site 2, you go in the 4th dimension.
Obviously, you can't put an actual 4D representation on paper. In fact, you can't put a 3D representation on paper. But if you're lucky, you can do a bit of topological rearrangement of the network and then do a projection of your network onto a 2D plane.
For example, if you have three potential binding interactions, each independent, then your system can be represented as the corners and edges of a cube. You can then represent this system by any of the standard projections of a 3D cube into 2D space.
For four potential binding interactions, it's the the corners and edges of a tesseract. Projecting a tesseract onto 2D space is much harder, in that there's no way to do it without lines crossing. But depending on what you want to represent, you might be able to adequately mark the crossings, or add "jumps" between two sub-graphs. (That is, have two more sub-graphs and have a systematic way of going from nodes on one graph to the node on the other graph with implicit reaction edges.)
For an arbitrary reaction graph, there's no guarantee that the crossing number of the graph is going to be zero -- that is, there's no guarantee that you'll be able to represent the network on a 2D page without crossings or jumps. Often the best you can do is attempt to be systematic about things, and then add crossings/jumps as appropriate.