Google PageRank

CS problem: Ranking nodes

• We are given a directed graph $G=(V, E)$ (WWW pages, scientific papers, people relations)
• How do we rank nodes in the graph

Example: Internet search (page order on search page)

Graph adjacency matrix

• A directed graph of four vertices: $v_1, v_2, v_3, v_4$.
• Five edges: $e_{1,2}, e_{2,1}, e_{1, 3}, e_{1, 4}, e_{2, 4}$.

Adjacency matrix: $$A=\begin{pmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ $a_{ij}\ne0 \Leftrightarrow$ there is an edge from $v_i$ to $v_j$.

Edge weights

Edges may have weights (e.g. edge length):

$$A=\begin{pmatrix} 0 & 0.5 & 4 & 1\\ 0.25 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Undirected graph

$A$ of undirected graph is symmetric:

$$A=\begin{pmatrix} 0 & 0.5 & 4 & 1\\ 0.5 & 0 & 0 & 4 \\ 4 & 0 & 0 & 0 \\ 1 & 4 & 0 & 0 \end{pmatrix}$$
$a_{ij}=a_{ji}\,\forall i, j$

Vectors and matrices

• vectors are 1-dimensional: $\pmb{v}=\begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$
• matrices are 2-dimensional: $\pmb{A}=\begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$

A scalar is a vector (or a matrix) with 1 element.

Matrix multiplication

$\begin{pmatrix}2\end{pmatrix}\begin{pmatrix}3\end{pmatrix}=\begin{pmatrix}6\end{pmatrix}$

$\begin{pmatrix}2 & 1 \\-1 & 2\end{pmatrix}\begin{pmatrix}1 \\ -1\end{pmatrix}=\begin{pmatrix}1\\-3\end{pmatrix}$

$\begin{pmatrix}2 & 1 \\-1 & 2\end{pmatrix}\begin{pmatrix}1&2 \\ -1&1\end{pmatrix}=\begin{pmatrix}1&5\\-3&0\end{pmatrix}$

Matrix inverse

$I_{2\times2}=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}$

$\begin{pmatrix}2 & 1 \\-1 & 2\end{pmatrix}\begin{pmatrix}0.4&-0.2 \\ 0.2&0.4\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}$

Eigenvalues and eigenvectors

• Vectors are scaled and rotated by matrices. $$\begin{pmatrix}1 & -1 \\-1 & 1\end{pmatrix}\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}1\\-1\end{pmatrix}$$
• Some vectors are only scaled. $$\begin{pmatrix}1 & -1 \\-1 & 1\end{pmatrix}\begin{pmatrix}-1\\1\end{pmatrix}=\begin{pmatrix}-2\\2\end{pmatrix}$$

  They are eigenvectors.

Eigenvalues and eigenvectors

If $Av=\lambda v$ then

• $v$ — eigenvector (וקטור עצמי)
• $\lambda$ — eigenvalue (ערך עצמי)

Google PageRank

• A search engine finds some pages.
• Pages must be sorted.
• How to assign page importance? PageRank ($R$)
• Measures centrality

Simplified PageRank

Simplified PageRank

$$a_{ij}=$$

• $\frac 1 {N_i}$ if edge $(j, i)$ in the graph,
• otherwise 0.

$R$ is an eigenvector with eigenvalue $1$!

PageRank

Users stay with probability $d$ --- the damping factor

$$a_{ij}=$$

• $\frac 1 {N_i}$ if edge $(j, i)$ in the graph,
• otherwise 0.

$R$ is an eigenvector with eigenvalue $d < \pmb{\lambda} < 1$!

PageRank computation

Iterative (power iteration):

1. Set: $R_i = \frac 1 N\, \forall i$
2. At each step: $R' = \frac {1-d} N\mathbf{1} + dAR$
3. Stop when: $|R'-R| < \varepsilon$
4. $R \gets R'$

Algebraic: