A Well-Known Example

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A Well-Known Example

A more complex and well-studied dataset is that of Doreian (1989), which describes the structure between politicians engaged in decision-making in a county in the American Midwest. This dataset has been extensively studied using various analytic procedures (Doreian (1988), (1989), Doreian and Albert (1989)), so that the scope and the subtleties of its solutions are quite well known. A reanalysis with our algorithm is therefore a kind of validation for our procedure. The comparison with the outcomes of various other analytical procedures also helps us to understand, how our algorithm works.

Doreian's network is made up of the fourteen most prominent political actors of a Midwestern County engaged in making decisions about the construction of a new jail. Seven of these make up the County Council which is the legislative and taxing authority of the county. Members serve for four years, with one of them as the Council President, the other six being labeled Council 1 through Council 6. Two of these lost their reelection bids: the Former Council President and the Former Co President. Other actors are the County Executive as the chief officer of the county, the County Auditor responsible for the county's administration, the elected Sheriff as the law enforcement officer and the County Prosecutor, the chief legal officer of the county.

Doreian and Albert (1986) hypothesised that the political actors would be partionable into two camps. One would be associated with the County Executive, while the other would be associated with the County Auditor. Their second hypothesis was that the distribution of votes would be conditioned by the partitioned network of strong ties among the political actors. Both hypotheses were confirmed, leading them to conclude that the structure of the network of strong political ties was responsible for the prolonged inaction of the County Council.

.. on the right the diagram is a clustered set of five points containing the County Auditor (B), while the left contains a clustered set of seven points containing the County Executor (A). ... In the County Auditor's alliance, Council 5 (I) and Council 6 (J) occupy the same location. The Council President H is located close to B and the City Mayor (M) can be grouped with the other four points. In the left hand cluster, the Sheriff (C) and Council 4 (G) are close, as are Council 1 (D) and Council 3 (F). The County Auditor (A) can be grouped with C, G, D and F. Finally Council 2 (E) and the County Prosecutor (N) join the cluster but at a greater distance. At the center of the Euclidian space is the Former Council President (L) with the Former Council member (K) at the periphery of the diagram. p. 285 f.

In Figure gif we present our solution for Doreian's political actors gif , yielded by applying our algorithm to the solution space of 14 equally spaced locations on a circle, which corresponds to the number of actors in the network.

We used the degree of centrality to specify the priority sequence of how nodes enter into the computations. This makes possible errors likely to occur only for peripheral actors.

There are the same two cliques as identified by Doreian: the A clique { A, C, D, E, F, G, N } and the B clique { B, H, I, J, M }. The Former Co President { L }, as the most central actor in the network, is located between the two subsystems A and B, with the Former Council {K} attached to {L}.

Figure: Doreian politicians: solution based on geodesics

Figure: Doreian politicians: solution based on direct links only

What is different, however, in Figure gif is the ordering of nodes in each of these cliques. This is not completely surprising, since Doreian computed his MDS solution on the basic of geodesics (using the shortest graph-theoretic distances between all pairs), while our solution is based on the direct links only. In our case, it is evident that the boundary spanning actors {M,H} and { A, D, F } in each of the cliques are located closer to the center of the subsystems, while completely peripheral actors 'buried with the alliances' { G, N, E, K }, are found on the subsystem's periphery, with exception of {E}.

A further test of our algorithm is to use the same graphtheoretic distance information that was used by Doreian. As this information accounts additionally for the shortest indirect links between any actors, it will raise those actors in the rank-order of centrality (closeness), who occupy boundary spanning positions.gif

If we use the rank-order in 'closeness' to specify the priorities in which nodes are taken into account, 'boundary spanners' should move towards the intersection of both subsystems. The results of this enhanced solution are shown in Figure gif.

If we view the ordering in the second solution from the position of {L}, {B,H,J,M,I} is changed to {B,H,J,I,M) whereas {N,F,E,C,D,A,G} is changed to {D,E,F,N,G,C,A} . With the exception of {F}, who is located in between {D} and {A}, the results of this analysis have become similar to Doreian's MDS solution and mimic even the internal organization of Doreian's MDS clusters to some degree.

The primary focus in this section has been to demonstrate that the algorithm yields acceptable results when compared to more powerful analytic procedures. We have shown, that our algorithm is able to find cliques or clusters of cohesion. We have demonstrated that it can use quite extensive information, and, that the results in such cases mimic the results obtained with more powerful analytic procedures, despite the given constraints of a strongly restricted solution space.

We have argued that the real virtue of the proposed procedure is that the algorithm works on an a priori constrained solution space, giving us the means to control the complexity and the design of a solution. While this may not have become evident in this section, we will try to demonstrate exactly this in the next section, when we use the proposed strategy to visualize more complex datasets and use the outlined procedure as a building block for more complex designs of the solution space.

next up previous contents
Next: Complex Examples Up: Examples Previous: Toy Examples

Lothar Krempel
Fri Mar 31 13:14:02 MET DST 1995