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

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

A gravity approach which makes only use of the ties among the nodes will converge to a structural equivalence solution: any two nodes linked to a same set of other nodes will occupy the same location in the final solution.

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.

The members of the lowest level hierarchy (the small yellow nodes) move quickly to the same location after just a few iteration steps and overlap since there is no information in the links, which would enable us to differentiate among them: the nodes are structurally equivalent.

Adding small repelling forces ..

Adding small repelling forces between any locations helps to meet the demand for a solution where there is no overlap between any nodes. This is often a preferred solution to a visually untrained audience.

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.

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Lothar Krempel, MPI für Gesellschaftsforschung, Lothringerstr.78, 50677 Köln, Germany
email: krempel@mpi-fg-koeln.mpg.de