This is known as an adjacency matrix.
There is some information in the R FAQ on plotting social networks expressed as matrices:
Plotting social network data can be easily done with the igraph
package in R. [...] In order for the igraph package to recognize this table as a network, we can first convert it to a matrix. Then, if we wish to calculate graph-related statistics on it (betweenness, closeness, degree), we can use the matrix to create a graph object.
The FAQ gives some examples, resulting in sociogram diagrams, for example:

The igraph
library itself is documented here - it can produce much more complex examples - see the screenshots page and the example below:
