Abstract:

A question that inevitably we face, when teaching new mathematical concepts to Undergraduate mathematics students, is the following: is it going to help me later? One way to answer that question is to try to relate the new theoretical concepts to some real life problems.

The main goal of this paper is to introduce the concepts of graph, distance, probabilistic distribution, and in general, the importance of the mathematical modeling, through what is known as “modeling of real networks”.

In a broad variety of systems, we find what is known as “structure of real networks”. The term real network makes reference to the fact that these networks are defined in the mathematical sense but they represent a process or a system that is part of our life. For instance, in Physics a great deal of research consists of studying a large number of particles (as the molecules that form the atmosphere) and the way they interact. In Biology, the cell can be described as a network of chemical components connected by chemical reactions. Also, we can find real social networks, as the one formed by the scientific collaboration around the world, where the vertices are the scientists, and two vertices are connected by a line, if both wrote a scientific paper together. The telephone calls are other example, where the vertices are the telephone numbers and each finished call represents a connection between the two telephone numbers. In the same way, we can talk about ecological networks, commercial networks, etc.

The World Wide Web or internet is a huge network of computers and servers, connected by a physical wire or wireless. The internet affects many aspects of real life such as the way we store and recover information, the way we do shopping, bank transactions, and in general, how we communicate. For instance, today the information can be found not only in written form, but on-line, through a complex network of web pages connected between them.

In this paper, we will focus in the study of what we called the web graph, that will be denoted by W. First of all, as its name indicates, W is a graph from the mathematical point of view. W consists of vertices (points), that represent the web pages, and of edges (line segments) that join the vertices that correspond to links between them.

Recently the web graph has been a very active field of study, theoretically and experimentally. The web is not only fascinating by itself, but it also gives us a better understanding about the networks in general and its evolution. The complexity of W lies in the fact that is an evolutive structure with pages and links that appear and disappear continuously with time, so the first question arise: How big is the web?

We know that there exist a huge number of web pages and links between them, but it is difficult to obtain the exact number. In 2005, a study by Hirate (see [H]), found 53.7 billions of web pages, although this number is constantly changing, but how is this change occurring? Has W some properties that characterize it? Are there good mathematical models that described these properties? Is it helpful the graph structure to obtain information about W?

Keywords:

Graph theory, probability, education.