Authors in a Markov matrix: Which author do people find most inspiring? (4)

Introduction to graph theory

In mathematics and computer science, graph theory is the study of graphs. A graph represents relationship between objects. I am going to use this theory to represent relationship between authors. It is straightforward to use graph theory to understand the relationships between objects. In that sense, this is nothing new, but I am curious about the application of this method to understand an aspect of literature --- the relationship between authors.

Fundamental objects of graph

Graph theory has two fundamental objects: a node (also called a vertex) and an edge. A graph, then, is a set of nodes connected by edges. If two nodes are connected by an edge, these two nodes are defined to have a relationship.

Figure 1 shows these fundamental objects. Graph theory only thinks about these two objects. You might think that this is simple, since there are only two types of objects. However, we will see that complexity is still possible.

Figure 1. Node and Edge.

Usually, there is more than one node in a graph. To distinguish multiple nodes, we typically put numbers on them (as shown in Figure 2). However, since the purpose of using number is to distinguish the nodes, we don't have to use numbers. However, numbering nodes is the easiest way to distinguish them, so many of the examples will use numbers.

Figure 2. To distinguish different nodes, put the numbers on them.

There are two types of edges shown in Figure 3: a directed edge and an undirected edge. The difference between them is that one has a direction and the other has no direction, which usually means it represents both directions. These types reflect the two type of relations: a one-directional relation and a bi-directional relation. For example, a person in the past can effect a person today with their writing but someone today cannot afect someone in the past. This relation is one-directional.

Figure 3. Directed and undirected edge.

Today, I show you who are the dramatis personae in graph theory. They are node and edge. So far, so simple. Next time I would like to talk about what is graph.

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

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