Graph-Rewriting Automata (GRA) are an extension of Cellular Automata to a dynamic structure using local graph-rewriting rules. This work introduces linear algebra based tools that allow for a practical investigation of their behavior in deeply extended time scales. A natural subset of GRA is explored in different ways thereby demonstrating the benefits of this method. Some elements of the subset were discovered to create chaotic patterns of growth and others to generate organic-looking graph structures. These phenomena suggest a strong relevance of GRA in the modeling natural complex systems. The approach presented here can be easily adapted to a wide range of GRA beyond the chosen subset.