Fruchterman and Rheingold (Fruchterman 1991) have created an efficient algorithm using different forces (attraction Fa = d2/k and repulsion Fr = -k2/d, with k adjusting the scaling of the graph). Moreover, recent software like Gephi draw the visualization at each iteration, thus providing real-time feedback to users. When layouts are implemented with no stopping condition, users can tweak the layout parameters in real-time until they decide to stop its execution. Interaction while calculating layout is usually made technically possible by using multi-threading processing, and by using the GPU for rendering the visualization. The goal is to avoid the layout algorithm being perceived as a “black box” by the analyst (although no scientific study has been performed yet to verify this belief), and to accelerate the testing of the layout parameters to obtain an aesthetically good visualization.
The targeted visualization of force-directed layouts is a rough mapping between the distances in the projection space and the distances in the graph topology. The goal is to enable a visual interpretation of the topology using the spatial positions of nodes. When a “good” layout is applied, the resulting image hastens the understanding of the network structure by revealing visual patterns. The readability of graphical representations can be defined by the relative ease with which users finds the information they are looking for. Alternative definitions include the potential to make sense of the data, the familiarity to users, and aesthetic criteria; readability is subjective because the result should be visually appealing and depend on the analysis task. However, some metrics are available to compare layouts, such as the number of occlusions, the uniformity of edge lengths, and the number of edge crossings. A more detailed introduction to this topic can be found in (Tamassia 2013).
Other kinds of representation exist, but you should be able to cover most of your needs using matrix-based representations and nodelink diagrams. As choosing a representation may also depend on the analysis task to perform, (Henry 2007) provides the following comparison guide, see the table below.