Network graphs were first developed in the 18th century in order to solve a problem about 7 bridges in a town called Konigsburg. The problem was developing a route to cross over every bridge at least once, and it had many issues to solve. Leonard Euler envisioned this problem as a network with 4 nodes that are all connected to each other. This link has more details regarding the initial discovery of networks:
http://www.jcu.edu/math/vignettes/bridges.htm

The original network problem looked similar to this:

There are a lot of interesting aspects to a network graph including odd and even vertices. An odd vertex has an odd number of links. An even vertex has an even number of links. In addition to vertices, graphs also have edges. Edges connect nodes together. There is a lot of mathematics behind the network graphs including the edge and vertex concepts.

Network graphs can be also used to understand sequences. One interesting fact that we learned was the idea of 6 degrees of separation. A degree can apply to individual nodes or groups. I was skeptical at first when I heard this theory. The theory basically states that you can connect to anyone in the world with an average of 6 degrees of separation. In an interesting article, Microsoft proves that the six degrees of separation is valid. Source: http://www.guardian.co.uk/technology/2008/aug/03/internet.email

Microsoft decided to test this by using 30 billion electronic messages. It seems that using a social media network can also prove this topic. We are continuously becoming more interconnected with each other regardless of physical location. Facebook, twitter, and LinkedIn are allowing us to maintain connections that we would not have otherwise. It is important to be able to visualize these networks to make more sense. Below is an example of a LinkedIn network graph:

This image helps to paint a picture of how one person is connected with all of their connections. LinkedIn is able to do this by using the data when making connections with other people. I think it is a very useful tool.