Test problems and representations for graph evolution

Graph evolution - evolving a graph or network to fit specific criteria - is a recent enterprise because of the difficulty of representing a graph in an easily evolvable form. Simple, obvious representations such as adjacency matrices can prove to be very hard to evolve and some easy-to-evolve representations place severe limits on the space of graphs that is explored. This study fills in a gap in the literature by presenting two scalable families of benchmark functions. These functions are tested on a number of representations. The first family of benchmark functions is matching the eccentricity sequences of graphs, the second is locating graphs that are relatively easy to color non-optimally. One hundred examples of the eccentricity sequence matching problem are tested. The examples have a difficulty, measured in time to solution, that varies through four orders of magnitude, demonstrating that this test problem exhibits scalability even within a particular size of problem. The ordering by problem hardness, for different representations, varies significantly from representation to representation. For the difficult coloring problem, a parameter study is presented demonstrating that the problem exhibits very different results for different algorithm parameters, demonstrating its effectiveness as a benchmark problem.