Sparse matrix data structure one example

1
Sparse matrix data structure (one example)

• Full
• 2-dimensional array of real or complex numbers
• (nrowsncols) memory

• Sparse
• compressed column storage (CSC)
• about (2nzs ncols) memory

2
Graphs and Matrices
Matrix
Graph

• Starting with the matrix
• One graph vertex for each row (or column) of the
  matrix
• One graph edge (i,j) for each nonzero A(j,i) in
  the matrix
• (Some people direct the edges the opposite way,
  from rows to columns either way is ok as long as
  its consistent.)
• Or, starting with the graph
• The adjacency matrix has A(j,i)1 if (i,j) is an
  edge.

3
Google and the Random Surfer
How does Google figure out which web pages are
most important?

• An important page is one that lots of important
  pages point to.
• Start at any web page and follow links at random.
  Forever.
• Youll see important pages more often than
  unimportant ones.

4
Analyzing the Web with graphs and matrices
Matrix
Graph

• Graph nodes are web pages
• Arrows between nodes are links between web pages
• Matrix entries are links from column pages to
  row pages
• The Page Rank comes from algebra on the matrix
• The matrix has 8,058,044,651 rows columns (in
  March 2005)

5
Web graph PageRank (Google)
An important page is one that many important
pages point to.

• Markov process follow a random link most of the
  time otherwise, go to any page at random.
• Importance stationary distribution of Markov
  process.
• Transition matrix is pG (1-p)ones(size(G)),
  scaled so each column sums to 1.
• Importance of page i is the i-th entry in the
  principal eigenvector of the transition matrix.

6
A Page Rank Matrix

• Importance ranking of web pages
• Stationary distribution of a Markov chain
• Power method matvec and vector arithmetic
• MatlabP page ranking demo (from SC03) on
  a web crawl of mit.edu (170,000 pages)