Any Graph is a Model (! or ?)
If you look at things from a model-theoretic perspective, the basic ingredients you have are atoms, relations, and functions. Atoms could be modelled using nodes of the graph, and binary relations and single argument functions could be modelled using graphs. If you are willing to allow hyper-edges in your graph, then the remainder of the relations and functions can be modelled. 

I don't think that there is anything deep to this observation.