Using random matrices to represent interactions in large-scale systems

In Chapter 10, we encountered the adjacency matrix method for representing a network. A network also represents a set of components, the nodes, and the network edges can be used to represent the pairwise interactions between the nodes. Figure 15.1 gives an example of a network and its adjacency matrix representation.

Figure 15.1: Network interactions represented as a matrix

It is natural to use a matrix to represent pairwise interactions between elements of any interacting system and not just between nodes in a network. Some of these systems may be physical, such as particles interacting via some force. They may also be non-physical, where the components of the system do not directly exert forces on each other but may influence each other, such as correlations between the share prices of different companies listed on a stock market.

Many of the systems that we study can be large...