Review Materials for Networks Qual

1.1. Graphs. Graphs, or abstract collections of nodes and edges, form the main objects studied in the theory of complex networks. These models can be classified according to many different types of topological properties, but the underlying structures are quite simple. The general procedure for modeling with networks is to first identify the objects of interest (nodes), the crucial relations between then (edges), and then add more structure (such as weights or embeddings) depending on what is available in the original data. Although there are many textbooks focusing on graph theory from a combinatorial perspective, these resources tend to be mostly irrelevant for the study of complex networks. The problem is that in real world settings many of the combinatorial properties and statistics are nonsensical. As an example, consider these basic graph theory questions: how many nodes does the internet have? does your router have an even or odd number of neighbors? what is a minimal independent set on Facebook? These are clearly ill–posed questions, but they highlights the disparity between the two fields. It is not possible to do much precise counting when the objects that you are studying are at best a noisy approximation to a fixed real world scenario, and more importantly analysis of one particular instance of the network has only small practical significance.