A GALLERY OF SOCIAL STRUCTURES

A closer look at the gravity iterations ...

The result of the embedding iterations is a low energy solution under a given set of constraints (i.e. the geometric shape the nodes of a network have been fitted to).

Using gravity to put strain on this fitted solution will move the nodes to new locations depending if the empirical links contain additional information which is not contained in the embedding solution yet.

An ideal morphing procedure provides us with a series of solutions which gradually adds more and more of the information contained in the empirical data and will finally end in an equilibrium solution where forces derived from the links are balanced. This lets us choose to pick any of these solutions if they are informative for answering our theoretical questions. Three information components are available for use in the gravity approach

• the locations of the embedding solution
• the empirical links among the nodes
• small repelling forces among all nodes that can prevent structurally equivalent nodes from taking the same location

Using the iterated solution and the forces of the links only ...

Taking the original locations of the embedding procedure into account can thus produce a series of solutions, which transforms the iteration solution stepwise into an structural equivalence solution. This makes it possible to choose among solutions which gradually deviate from the a priori constrained start configuration.

This can be illustrated with an iteration history diagram which contains the location of the nodes for each iteration step, and where we look from a z-axis onto the xy- plane.

• Paths of Iteration - a look onto the xy plane from above (the z-axis)
  (60 Kb);
  Yellow nodes mark the input configuration from the minimization procedure, red nodes the equilibrium solution, grey nodes show intermediary locations

  
  

Adding small repelling forces ..

• Gravity with repell component
  (56 Kb);
  As above: yellow nodes give the start locations, red ones the equilibrium solution, grey ones show intermediary steps.
  
  

The repelling forces prevent all nodes from overlapping. Compare especially the locations of the two equilibrium solution for the lowest hierachy level, where nodes now stay apart and are not overlapping.