## What is a graph?

A graph is a set of nodes connected by edges. When there is a relation between two nodes, these nodes are connected by an edge. Figure 4 shows a graph example that has four nodes. In the figure, the node 1 ($$N_1$$) and the node 2 ($$N_2$$) is connected by the edge $$E_{1,2}$$. A graph is only concerned with the connections between nodes, so how the nodes are arranged in a figure doesn't matter. Figure 5 has three graphs, but all the graphs are considered to be the same in graph theory.

 Figure 4. Edges connect nodes.

 Figure 5. The same graphs. Graph only cares the connections between nodes.

I will show you some examples next time.

## 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.

## 2012-09-24

### Can you say equal to''?

What I first asked to children is
Kannst du ,,gleich'' sagen?
(Can you say equal to''?)
Because many children say
Eins plus zwei ist drei.
(One plus two is three.)
Of course it is no problem if the children know the difference. But when a child use ist (is)'' I am not sure they know what the 1 + 2 = 3 means. Therefore, I asked often children to say,
Eins plus zwei (ist) gleich drei.
(One plus two equals to three.)
But if I force this, children might hate mathematics, so I only ask this sometimes. I am afraid they hate mathematics. If they hate, it doesn't matter what is correct.

By the way, I just thought that does usual grown up know the difference? I asked some of colleagues, though, they earn money by doing mathematics, so they knew it. (Maybe wrong samples.)

The problem is how to explain this to 8-11 years old. I explain as the following. All the children said I see''. But still some children forget to say equal to'', so I am not yet sure.
7 is equal to 2 + 5.
2 + 5 is equal to 7. These are both correct.
Hitoshi is a Japanese.
A Japanese is Hitoshi? There are other Japanese names.
Therefore, Hitoshi is equal to Japanese.'' is wrong. A Japanese is equal to Hitoshi.'' is also wrong.
Another example.
Daniel is 160 cm (tall).
160 cm is Daniel? 160 cm is a length, a length is not Daniel.
Therefore, Daniel is equal to 160cm.'' is not correct.
Equal things can be exchange-able.
Can you see ist (is)'' is not equal to gleich (equal to)''?
So far, I explain a concept gleich (equal to)'' as austauschbar (exchange-able)''. This works numbers, money, length, and so on. But when a quartic object (e.g., area) is shown up, I have a problem. In fact, in ancient Greek (if my memory is correct), x*x = x can not be a computable equation. Ancient Greek people thought the left hand side is an area and the right hand side is a length. How can you exchange area with length? The concept of equal to'' is not so simple.  But I think this is fine for the first step.

In my case, I even can not speak the correct German, so I hope my students think I am not always correct. I wish my students think me as: the teacher is usually correct, but sometimes makes mistakes.

### How can I help the children's learning? My current method.

I think mathematics and computer science are fun. I can explain some example cases, but, why do I think these are fun in general?  The same as why am I interested in novels, movies, climbing, or teaching? I don't have a clear answer. How do children think about mathematics? If I could understand a part of that, maybe I could help them. Therefore, I would like to learn how the children think rather than teaching. I learn how the people think about mathematics.

I am interested in artificial intelligence. Therefore, I am also interested in how the people's intelligence develop. Children learn speaking, counting numbers, logical thinking, then develop their intelligence further. I should have experienced these process, but, I totally forgot how to learn my native language, how to learn counting numbers.

Because of that, I would like to ask children to teach me them. Though, I found it is not easy. Children hardly teach me how they think. I am not sure, but maybe children themselves don't know that.

Therefore, I first observe children. How children reacts the inputs. I would like to know what kind of model they have. It is a bit similar to debug a program. For example, a child can not process specific inputs, another child always makes a specific mistake. Then, I think, how this happens? What kind of mental model they have? This is very hard problem, but it is quite interesting. I think if I could understand their model (how they think), and if I could see the problem of the model, I might help them. This is just my hypothesis, but this is usually my case. I hope this works. So, what I do in the class is the following loop: observe a child, learn their model, think it, try to teach based on my model, fail, observe the child again, ...

I would like to record my those experiences.

## 2012-09-23

### Ich bin ein Mathe-Helfer bei Hasenschule

I started leaning how to teach mathematics at Hasenschule (http://www.hasenschule.de/). I attend twice to four times a week, one hour lesson each. This is a voluntary activity and I don't get any money. (Well, sometimes I got sweets or a piece of cake.) I would like to write about this activity.

Briefly speaking, Hasenschule is a special school for children who could not catch up with their school. This school teach how to read, write, and calculate. I am interested in teaching math and physics to children and want to learn how to teach that. With help with my friends, I found the following web pages. Gute-tat is a kind of portal organization, they introduced me Hasenschule. There are many activity of this kind in Berlin, if you are interested in, these pages are good start. My lessen is taught in German, but my German is not well. Hasenschule provides a rechtschreiben (reading/writing) course for 8-10 years old children, I maybe should take the course.

## Problem statement: Analyzing author relationships (2)

I would like to suggest we start with a simpler problem. Let's forget about the contents of a book and how much it influences others. Our model only concerns the existence of an influence.

Fortunately, there is a mathematical method based on the existence of a relationship, called graph theory.  One of the tools used by graph theory is linear algebra. Graph theory has been widely used for a long time.  When the World Wide Web era came, this method was also applied to the principles of web search to implement a basic idea.  If you can find the most influential web page, you should list these results on top of the search results. There is also a lot of research to find a method to automatically compute the influence between web pages.

Probably the most frequently used method for such a web page ranking only considers the linking relationship between web pages. Since this method only uses link information, it doesn't need to know the contents of individual web pages. This method was a breakthrough in web search engine technology. The inventors of this method started a company called Google, using a computed influence rank for listing the search results[bib:pagerank]. Because the method doesn't need to understand the contents of a web page to evaluate that page's ranking, it can used for web pages in any language with any type of content. Furthermore, the ranking can be fully automated. The search engines of some other companies evaluated the rank by hand, but these companies were easily outperformed by Google. Ranking by hand may be more accurate than automated methods but the automated method is much better at updating speed and handling large numbers of web pages.

The method used for web page ranking by Google is called PageRank [bib:pagerank], though this method is based on the traditional technique of eigenanalysis in linear algebra.

In this article, I first provide an overview of graph theory.  I then apply eigenanalysis to rank the influence of authors.

Therefore, my next blog entry will be introduction to graph theory.''

## Problem statement: Analyzing author relationships (1)

Let's start thinking about what is the problem.

One of my friends who studies literature asked me how I would analyze the relationships between authors. For instance, How can we measure the influence of Shakespeare in English literature compared to other writers?''

The discussion of which literary works are canonical and the attempts to define the boundaries of the canon are endless (http://en.wikipedia.org/wiki/Western_canon).  Typically, a specific individual or group has defined the boundaries of the canon. If we consider creating a ranking system using numbers, I wouldn't even know how to begin such a discussion. It is easy to imagine such discussions raising an even more generic question: What is art?'' Answering that question is clearly beyond my ability!

If nobody can agree on such a ranking system for authors, I could at least start by creating an incomplete definition, hoping to improve it later. However, my definition would only reflect my personal taste. We could extend the result of one person's opinion to many people's opinions by asking a lot of people. We could start by assuming that there is a correlation between the most frequently printed book and the most preferred book. If this is true, we can just check the number of printings of a particular book. Simple web search tells us that the two most printed books are IKEA's catalog (http://en.wikipedia.org/wiki/Ikea_catalogue) and the Bible. I doubt that the most influential book for English literature is IKEA's catalog. My intuition says maybe not. What is literature?'' is also not a simple question. We could replace the number of printings with the total sales, but then magazines and newspapers would have quite a large influence. But again, I am not so sure that this is what my friend and I are looking for.

It seems problem is not so simple, where can I start?

## Introduction

One of my friends who studies literature asked me how I would analyze the relationships between authors. I answered that if you could represent these relationships by a graph, you could use a mathematical method called eigenanalysis'' for that.  Google determines the relationship between web pages using this method in a software program called PageRank.'' I am interested in this method as a problem in a new approach to literary analysis called computational literature.'' In this paper I describe the results of my research.

I will describe this in the following blog entries.