Colored Gaussian graphical models are linear concentration models arising from undirected graphs with a coloring in its vertices and edges. We focus on maximum likelihood (ML) thresholds for these models: the minimum number of observations that ensure existence (with either probability one or strictly positive probability) of the maximum likelihood estimator. We extend results for ML thresholds to the colored setting, implement algorithms that exploit the underlying algebraic geometry and compute the thresholds for certain families graphs. Finally, we present a novel computational method using topological data analysis. Joint work with Olga Kuznetsova and Bernadette J. Stolz.