HUB:

• These are the pages that link to authorities.
• For example: list of newspapers.

Authority:

• These are the page which contain useful information.
• For example: newspaper home page.

HITS Algorithm:

Link matrix L = $\left[\begin{matrix} A & 0 & 1 & 0 \\ B & 0 & 0 & 1 \\ C & 1 & 1 & 0 \\ \end{matrix}\right]$

$L^T$ = $\left[\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix}\right]$

Initial vector h = $\left[\begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix}\right]$

Iteration 1: $a^1 = L^T h$ = $\left[\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix}\right]$ = $\left[\begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right]$ = $\left[\begin{matrix} 1 \\ 2 \\ 1 \\ \end{matrix}\right]$

Scaling down a’ = $\left[\begin{matrix} 1/2 \\ 1 \\ 1/2 \\ \end{matrix}\right]$

H’ = L a’ = $\left[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{matrix}\right]$ = $\left[\begin{matrix} 1/2 \\ 1 \\ 1/2 \\ \end{matrix}\right]$ = $\left[\begin{matrix} 1/2 \\ 1 \\ 1/2 \\ \end{matrix}\right]$

Scaling down h’ = $\left[\begin{matrix} 1/6 \\ 1/3 \\ 1/2 \\ \end{matrix}\right]$

Iteration 2:

$a^2 = L^Th’$ = $\left[\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix}\right]$ = $\left[\begin{matrix} 1/6 \\ 1/3 \\ 1/2 \\ \end{matrix}\right]$

$a^2$ = $\left[\begin{matrix} 1/2 \\ 2/3 \\ 1/3 \\ \end{matrix}\right]$

No need to scale as all values less than 1

$h^2 = La^2$

= $\left[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{matrix}\right]$ = $\left[\begin{matrix} 1/2 \\ 2/3 \\ 1/3 \\ \end{matrix}\right]$

After scaling:

$h^2$ = $\left[\begin{matrix} 2/21 \\ 1/21 \\ 1/6 \\ \end{matrix}\right]$

Thus $a^2$ and $b^2$ represent the hub and authority score after two iterations.