Understanding and modeling go hand in hand – we develop models not only to make predictions but also to see where the models fail and there is more to do. Large-scale networks are immensely challenging to model mathematically. In this talk, we present our arguments for what features are important to measure and reproduce. In the undirected case, we show that graphs with high clustering coefficients (i.e., many triangles) must have dense Erdȍs-Rényi subgraphs. This is a key theoretical finding that may yield clues in understanding network structure. Following this line, we propose the Block Two-level Erdȍs-Rényi (BTER) model because it reproduces a given degree distribution and clustering coefficient profile (i.e., the triangle distribution), scales linearly in the number of edges, and is easily parallelized. We also consider the extension of this work to bipartite graphs, where we consider bipartite four-cycles, and propose a bipartite BTER (biBTER) model. These models can be used to generate artificial graphs that capture salient features of real graphs. We compare the artificial and real-world graphs so that we can understand where the models are accurate or not. Time permitting, we also explain how these models can be specified with very few parameters, which is useful for benchmarking purposes. We close with open questions for future investigations. This is joint work with S. Aksoy, A. Pinar, T. Plantenga, and C. Seshadhri.