Question

Three-Click Rule An unofficial, and often contested, guideline for website design is to make all website content available to a user within three clicks. The webpage adjacency matrix for a certain website is given by $$ A=\left[\begin{array}{lllll} 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \end{array}\right] $$ (a) Find \(B_{3}\). Does this website satisfy the Three-Click Rule? (b) Which page can be reached the most number of ways from page 1 within three clicks?

Short answer

a) Yes, it satisfies the Three-Click Rule. b) Page 2.

Step-by-step solution

Understanding the Problem

The given website's structure is represented by an adjacency matrix. We need to find if any page on this website can be reached within three clicks from any other page by calculating the matrix to the third power, i.e., finding matrix B3. Then, we need to determine which page can be reached the most number of ways from page 1 within three clicks.

Matrix Multiplication

Calculate the square of matrix A, denoted as A^2. This will be achieved by multiplying matrix A by itself: \[ A^2 = A \times A \] Conduct matrix multiplication using rules of linear algebra.

Second Multiplication to Find B3

Next, multiply the result from A^2 by matrix A again to get A^3, also referred to as B3: \[ B_3 = A \times A^2 \] This step gives us the number of ways to get from one page to another in exactly three clicks.

Interpreting Matrix B3

Number of ways each page can be reached from other pages within three clicks are described by matrix B3. A non-zero entry at position (i,j) of B3 signifies that page j can be reached from page i within three clicks. Verify if all entries in B3 are non-zero to check compliance with the Three-Click Rule.

Identifying Maximum Reach from Page 1

Look at the first row of matrix B3 to find the page that has the maximum number of ways to be reached from page 1. Identify the column in the first row of B3 with the highest value.

Key concepts

adjacency matrix

An adjacency matrix is a way to represent the connections between nodes in a graph. In our case, the nodes are web pages and the connections are the links between them. The matrix is a square grid where each element represents a connection. If we have a connection from page i to page j, we put a 1 in the matrix at position (i,j). Otherwise, we put a 0.
For instance, consider the matrix for our website:
$$ A=\begin{pmatrix} 0 & 1 & 1 & 0 & 0 \ 1 & 0 & 0 & 1 & 1 \ 1 & 0 & 0 & 1 & 0 \ 0 & 0 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 0 d{pmatrix} $$
This means page 1 has links to pages 2 and 3, page 2 has links to pages 1, 4, and 5, and so on. Understanding this matrix is crucial for finding the number of clicks required to reach from one page to another.

three-click rule

The Three-Click Rule is a guideline for website design, which suggests that a user should be able to find any information on a website within three clicks. In practice, this means designing a website so that users can easily navigate through the pages.
To check if our website follows this rule, we must see if every page can be reached within three clicks from any other page. We do this by computing the cube of the adjacency matrix. The entries of the resulting matrix will tell us the number of ways pages can be reached within three clicks.
This involves matrix multiplication, an important concept within linear algebra, where each element in the resulting matrix is a sum of products.

linear algebra

Linear algebra is a branch of mathematics focused on vectors, matrices, and the operations that can be performed on them. In this scenario, we utilize matrix multiplication to analyze the web page connections.
Here’s a quick rundown on matrix multiplication: when two matrices are multiplied, an element in the resulting matrix (let’s say at position (i, j)) is found by taking the ith row of the first matrix and the jth column of the second matrix, and computing the dot product.
For our problem: $$ A^2 = A \times A $$ $$ B_3 = A \times A^2 $$
These new matrices tell us about 2-click and 3-click connections, respectively. By analyzing these matrices, we can determine if our website meets the Three-Click Rule and identify the page with maximum connections from page 1 within three clicks.