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In the subfield of numerical analysis, a sparse matrix is a matrix populated primarily with zeros (Stoer & Bulirsch 2002, p. 619).

Huge sparse matrices often appear in science or engineering when solving partial differential equations (wikipedia).

Tim Davis from the University of Florida did a great job collecting a myriad of datasets, known as the The University of Florida Sparse Matrix Collection. As the fastcodesign guys say, http://www.fastcodesign.com/1662136/geeky-science-problems-double-as-works-of-art

Timothy Alden Davis, a computer, information science, and engineering professor at the University of Florida, is a sort of Eli Broad with a pocket protector. He’s got the largest open-source collection of sparse matrices in the world [...].

Sparse matrices can be represented as graphs, and this is what they look like using MATLAB/Octave to obtain static visualizations:

How do we map sparse matrices to weighted graphs? the algorithm for square matrices is straightforward: Given a square sparse matrix M = n x n, its corresponding graph G = (N, E) contains n nodes and the edge between nodes n1 and n2 is weighted as M[n1,n2]. If M[n1,n2] = 0, then there is no edge between nodes n1 and n2.

The interested reader may find more details at this link: Visualizing Sparse Matrices. In addition, there are a few academic references at the end of this post.

You can explore this data interactively using GraphInsight with 2D and 3D layouts. To export square sparse matrices from MATLAB/Octave to the DIMACS format (supported by GraphInsight), you can use the following function:

function sparse2dimacs(A,filename)
fid = fopen(filename,'wt');
n=length(A);
for i=1:n
 klen = length( nonzeros(A(i,[1:(i-1) (i+1):end])) );
 fprintf(fid,'%d ( 0.0, 0.0 ) %d ',i, klen );
 % find non zero elements of row i
 [p]=find(A(i,:));
 for m=p
  if (i~=m )
   fprintf(fid,'%d ',m);
   end
  end
  fprintf(fid,'\n');
end

In the following, we show some screenshots obtained from GraphInsight. We observe that these are screenshots from interactive 3D visualizations.

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Bibliographic notes