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
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
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
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
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.
Paths of Iteration - a look onto the xy plane from above (the z-axis)
Yellow nodes mark the input configuration from the minimization procedure,
red nodes the equilibrium solution, grey nodes
show intermediary locations
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.
Gravity with repell component
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.
Lothar Krempel, MPI für Gesellschaftsforschung, Lothringerstr.78, 50677 Köln, Germany