Rather than actually doing my math assignments - I thought it might be interesting to show some applications of graph theory to economic models.
First a basic overview of graph theory.
Graph Theory is a very simple way to show (visually) how different objects are related. It primarily consists of sets of edges and vertices that connect to make something like this:
Where the vertices are the numbered circles, and the edges are the lines connecting them. Of course, this example shows a very simple graph. Before you know it, they can start to look like this:
I'll be borrowing a large chunk of this post from a paper by Michael Konig and Stefano Battiston. I won't have time to cover the complex parts of the paper, but this will be a good preview of some of analysis that graph theory can help with.
The paper uses graph theory to analyze economic networks, which are just economic actors (firms, individuals, groups, etc.) that are organized to behave some way. Network economics differs from most neoclassical models, which use the perfect price competition models. The assumptions that people act individually and rationally are relaxed and analyzed more closely in economic network analysis.
In this post, we'll just look at network diffusion, and how knowledge can move from one person to another. Let's assume that vertices represent an individual, and the edges represent some sort of personal connection. (Note: Different lengths in the edges represent varying degrees of social connection. Small edges show that two people are closer than two people connected by a longer edge.) Therefore, closer edges are better ways of diffusing information than long edges.
The following equation is the average distance of all the edges. Where d is the distance between vertices i and j.
As you could assume, the lower average of the edge distance means that a network is more efficient at diffusing information. Konig and Battison cite a paper that shows that nearly 50% of all employment is facilitated through personal social circles. Using this kind of analysis is important in understanding how networks change and the impact of various inputs. (In this case, the diffusion of knowledge among a group of people.)
I would argue that network economics is becoming increasingly important as individuals become more interconnected through online social mediums. Similarly, this type of economic analysis could certainly be applied to urban environments, where interactions are becoming increasingly interrelated.