Exploring complex networks
https://www.nature.com/articles/35065725
This article discusses some ways to reason about the overall structure of networks, and why this type of reasoning can be so important. It highlights the way these networks form a link between different fields of science that are seemingly unrelated – from dynamical physics, to power grids, to food webs, and many others. The article describes some of the same properties of networks we discussed in class. The structural complexity of a graph relates to the pattern of edges across nodes. Network evolution relates to a graph changing over time. Connection diversity describes the ability to attach additional information to edges or nodes in graphs. For example, we’ve reasoned about Structural Balance in class by attaching positive or negative weights to each edge. Node diversity refers to different nodes representing distinct types of objects. All these network properties allow us to take the simple yet abstract nature of a graph and apply it to countless real-world problems across different fields of study. The article dives deeply into an example involving dynamical physics and oscillators reaching an equilibrium.
The article also describes regular and random network architectures, and how they can even be combined and reasoned about. As we’ve learned in class, a regular network follows a strict set of rules governing its overall structure. For example, a fully-connected network (where every node has an edge connecting it to every other node) is a regular network. On the other end, we could have completely random networks, where a given set of nodes gains edges by randomly connecting pairs of nodes. The article then describes a very interesting bridge between regular and random called a small-world network. It gives the following two formulations. First, a regular graph where each link can be randomly replaced by a random link with some probability. Or, a regular ring graph where shortcut edges are randomly added across the center of the ring. I found it especially interesting that, using this second ring-formulation, the average path length of the entire system could be mathematically calculated based on the number of random shortcuts added. Overall, this article was a great read that not only granted insight into some specific utilizations of networks in science, but also expanded upon the abstract nature of graphs themselves.