A network graph, when the term is used by mathematicians, means a structure comprised of nodes and edges. For example, a law department could represent the law firms it retained during the previous year by means of such a graph. The department would be the central node on the graph and each law firm would be at the end of an edge.
The visual depiction would be more informative if the edges were of different thicknesses to indicate the amount paid the law firm during the year. Likewise, the length of the edge could be proportional to the number of matters handled by the law firm.
If the data from a law department were represented this way, you could also show the size of the law firms by the area of their node shape (a circle). One further use of this graph would be to color or shape the node of each law firm according to some other factor, such as the number of timekeepers used by the firm or the number of areas of law serviced.