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6 Sparse Matrices The find Function and Sparse Matrices For any matrix, full or sparse, the find function returns the indices and values of nonzero elements. Its syntax is [i,j,s] = find(S) find returns the row indices of nonzero values in vector i, the column indices in vector j, and the nonzero values themselves in the vector s. The example below uses find to locate the indices and values of the nonzeros in a sparse matrix. The sparse function uses the find output, together with the size of the matrix, to recreate the matrix. [i,j,s] = find(S) [m,n] = size(S) S = sparse(i,j,s,m,n) 6-16
Adjacency Matrices and Graphs Adjacency Matrices and Graphs This section includes: • An introduction to adjacency matrices • Instructions for graphing adjacency matrices with gplot • A Bucky ball example, including information about using spy plots to illustrate fill-in and distance • An airflow model example Introduction to Adjacency Matrices The formal mathematical definition of a graph is a set of points, or nodes, with specified connections between them. An economic model, for example, is a graph with different industries as the nodes and direct economic ties as the connections. The computer software industry is connected to the computer hardware industry, which, in turn, is connected to the semiconductor industry, and so on. This definition of a graph lends itself to matrix representation. The adjacency matrix of an undirected graph is a matrix whose (i,j)th and (j,i)th entries are 1 if node i is connected to node j, and 0 otherwise. For example, the adjacency matrix for a diamond-shaped graph looks like A = 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 2 4 Since most graphs have relatively few connections per node, most adjacency matrices are sparse. The actual locations of the nonzero elements depend on how the nodes are numbered. A change in the numbering leads to permutation 3 6-17
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