Partitioning Political Actor Networks: Some Quantitative Tools for Analyzing Qualitative Networks
Journal of Quantitative Anthropology 1: 279-291, 1989
@ 1989 KluwerAcademic Publishers. Printed in the Netherlands.
PATRICK DORELAN*
Department of Sociology, University of Pittsburgh, Pittsburgh, PA 15260 U.S.A.LOUIS H. ALBERT
Graduate School of Public and International Affairs University of PittsburghThis paper outlines ways in which qualitative and quantitative tools can be used in a complementary fashion to delineate the structure of social networks. The network used for this demonstration is made up of prominent political actors in a Standard Metropolitan Statistical Area. Mutually consistent representations of the network structure in terms of group distance, partitions. centrality and first order stars are obtained. Moreover, these representations provide a perfect prediction of the voting behaviour of the members of the County Council. ABSTRACT
KEY WORDS: networks, research methods, equivalence.
A methodology is a system of rules guiding scientific inquiry and has four components - substantive, epistemological, technical, and purposive (Doreian and Hummon 1976). While the technical component is made up of techniques, instruments, and procedures applied in specific contexts, the other components make it clear that a methodology is far more than a set of data gathering and data analytic techniques. The evaluation of specific tools can only be considered in an overall methodology and in specific applications. Without these, debates over the appropriateness of technique are fruitless.
Within the technical component of a methodology, it is conventional to make a distinction between qualitative and quantitative methods. Although a sharp distinction can be made between the two approaches, ideally, both can be used in mutually enriching ways for the study of social phenomena. The goal of this paper is a demonstration of the complementary roles that can played by qualitative and quantitative techniques.
1. QUALITATIVE AND QUANTITATIVE METHODS
The discussion of qualitative and quantitative methods hinges on two specific ideas: measurement and mathematics. More precisely, it is necessary to consider measurement practices and the role of mathematics in the study of social phenomena.
Measurement
Measurement is a process whereby, given a set of units of analysis and values of a variable, each unit of analysis is mapped to a single value among the set of values that make up the variable. Variables can be categorized in many ways. One scheme goes back to Stevens (1966) where variables can be nominal, ordinal, interval or ratio. Koopmans (1987) distinguishes only categorical (nominal) and measurement (ordinal, interval, and ratio) variables. Categorical variables have values where only qualitative distinctions can be made among the units of analysis according to a specific criterion. Measurement variables are quantified in the sense of having order and a metric. Borgatta and Bohmstedt (1981) point out that most social scientific variables are quantitative in this sense and that ordinal variables can be viewed as imperfect interval variables. Given this claim, we will use Koopmans' distinction between categorical (qualitative)
and measurement (quantitative) variables.Mathematics
Discussions of the role of mathematics in social -science has been plagued by the confusion of mathematics and quantification. Many practitioners of quantitative social science see the following progression for social science. First, social scientists observe some regularity in social phenomena. Second, measurement is attempted in some crude fashion. Next, as our conceptualization of the phenomenon becomes more sophisticated and our measurement more precise, there is the attempt to quantify variables. Finally, there is mathernatization in the sense of having quantified variables linked in particular mathematical formulae. (See, for example Coleman (1964) and Golembiewski et aL (1969) for early statements and Afifi and Clark (1984) for current recipes in the final phase of the sequence. See also most issues of the major social journals where regression and related methods are used.)While it is clear that the use of mathematics in the social sciences does include quantified variables and formal equations as a mode of expression, there are many areas of qualitative mathematics that are useful for representing and analysing social phenomena. 'These are found loosely within the rubric of 'discrete' mathematics. Of value here is modem algebra, topology, and (as a special case of these) graph theory (Roberts 1976). Discrete mathematics provides a vast array of qualitative models that provide the foundation for social network analysis. Although many quantitative methods have been used in network analysis, its distinctive flavour comes from using qualitative tools.
When qualitative tools are discussed (see for example, Taylor and Bogdan 1984) the arguments in favour of their use are couched in terms of a broad methodology (as defined above) as they come from the substantive and epistemological components. Claims that qualitative researchers use induction and "emphasize validity in their research" (Taylor and Bogdan 1984) carry with them the implicit connotation that quantitative researchers do not or that, on epistemological grounds, qualitative methods are better. The accuracy of these claims are not at issue here. What is at issue is the extent to which qualitative claims apply for qualitative mathematics.
"In qualitative methodology, the researcher looks at settings and people holistically; people, settings, or groups are not reduced to variables, but are viewed as whole" (Taylor and Bogdan 1984: 6). By locating actors in a network structure the network is considered in holistic terms and the structural location of the actor only has meaning in the context of the network as a whole. The use of graph theory as a qualitative model is completely consistent with the qualitative declaration of intent. Given the graph theoretic representation of a social network, it is clear that the qualitative analysis can be used. Less clear is the utility of coupling some quantitative procedures to the qualitative models.
2. NETWORK ANALYSIS: QUALITATIVE AND QUANTITATIVE TOOLS
Among the exciting aspects of contemporary social network analysis is the potential to combine quantitative and qualitative techniques in a complementary fashion. Networks represented in graph theoretical terms are qualitative while the construction of indexes measuring structural properties are quantitative. The specific combination considered here is the construction of a graph and the delineation of subgraphs (both qualitative), together with the measurement of centrality and the computation of graph distances for an MDS analysis (all quantitative).
Central to network analysis is the assumption that the structure of human relations making up a social network is critical in the distribution of behavioural outcomes over the network. Both local structure (at the nodes) and global structure (of the network as a whole) are seen as conditioning behavioural outcomes. We provide a partial examination of these assumptions by considering a network of strong ties among a set of prominent local politicians and the voting outcomes of an elected council made up of some of these political actors.
3. EMPIRICAL BACKGROUND
The network consists of the most prominent political actors of a Midwestern County. The county is part of a Standard Metropolitan Statistical Area (SMSA) and contains a city, the population of which is slightly over 250,000. The substantive issue for which the network data are relevant is coalition formation among the political actors, and the County Council voting patterns concerning the construction of a correctional facility.
The current jail has never met the minimum standards for a facility of its type nor has it ever passed a fire or electrical inspection. In 1976 and again in 1984 the County was taken to Federal Court on charges that conditions in the facility were unconstitutional. In December 1984, the County signed a consent decree to change the conditions in the jail. Under terms of the consent decree the jail had to average 162 inmates or less. The County also was required to release inmates to keep the inmate population from exceeding a limit of 201. Even with the jail population as low as it had ever been, the population exceeded the 201 limit on the first day of the order and the jail population has never come close to the 162 figure. The inadequacy of the jail and the constraints of the consent decree makes it a serious problem. The County Council has taken no definitive course of action and their behaviour, as a collectivity, and a component of a wider political network, merits attention.
The Data
Our specific focus is the structure of the network of strong political ties among the 14 most prominent political actors (see below) of a Midwestern County. The County Council, the legislative and taxing authority of the county, contains 7 of the political elite. Members serve for four years, with one of them as the Council President. The other six members are labelled Council 1 through Council 6. Under a Home Rule Charter, the Council was elected in 1980. In November, 1984 there was a two person change in the council when the Former Council President and the Former Council Member lost re-election bids. The Former Council President remained a powerful
leadership figure.The County Executive is responsible for the administration of county business except where specified otherwise by law. The current County Executive was elected in 1980 and re-elected in 1984. The current Sheriff, appointed in 1980 and has been re-elected in both 1980 and 1984. He is the chief law enforcement officer of the county and custodian of the county jail. Another county position is the County Prosecutor, the chief legal officer of the county and a member of the county budget committee. The fourteenth political figure is the City Mayor.
The network of the central political actors can be observed in some contexts, but, in the main, their interaction is not open to the public in
any systematic way. As an elite group they are beyond the public and, perhaps to a lesser extent, the researcher. Of course, an attempt can be made to solicit information from each actor separately concerning their interaction. The drawbacks of this strategy are found in the possibility of refusals, so that the network data would be incomplete, and the generation of invalid data given the political nature of the actors and their network(s). In the context of field research, a preferable alternative is to use key informants who can be a major source of useful information (Orenstein and Phillips 1978). Using a member of the network as such an informant runs into all of the problems of attempting to get information directly from the group members. Instead. we sought individuals who could perform many of the same functions. These individuals must possess enough information about the political actors to be of value and also be, at most, on the margins of the group under study. Such individuals can be found among news reporters (Orenstein and Phillips 1978). We used the expertise
of the reporters of the local daily paper whose beats included political affairs.Our options were to interview all of the reporters separately or to approach one of their senior members who could provide a summary response based on the views and perceptions of the group of reporters. Interviewing all reporters separately would leave us the problem of combining and reconciling any differing responses (without us having the knowledge needed for reconciling different views). As there were few reporters, it is not clear that statistical tools could be useful by themselves for combining different responses. Instead we gave the list of major political actors to one of the reporters and requested verification of the list and information on the existence of strong ties.' He provided them after the reporters had an extensive discussion about the actors involved and the political ties linking them. Actors inked by a strong tie can expect public and private support from each other and they are also friends. Public support can be discerned in a straightforward fashion (although it can be fabricated for a public display). Detection of misleading public support and assessing private support demands knowledgeable observers.
Another advantage of having the tie data reported by a key person is that any differences between the observers can be ironed out by the observers themselves. They are, after all, in the ideal situation for this. Yet another advantage is that the definition of 'strong political tie' is
uniform across the entire network. Table I contains the network data.These data are reported by one individual but should not be viewed as 'just one response'. It is a set of responses that have been generated by group discussion and reported for the group. Such validated data are more reliable than having less knowledgeable people
people aggregate distinct responses, if any, at some remove from the phenomenon under study.TABLE I
Sociomatrix for strong ties between major political actors
County Executive
County Auditor
Sheriff
Council Member I
Council Member 2
Council Member 3
Council Member 4
Council President
Council Member 5
Council Member 6
Former Council Memb
Former Council Pres
City Mayor
County Prosecutor(A) 0 0 1 1 0 0 1 0 0 0 0 0 1 0
(B) 0 0 0 0 0 0 0 1 1 1 0 1 0 0
(C) 1 0 0 1 0 1 1 0 0 0 0 0 0 1
(D) 1 0 1 0 1 1 1 0 0 0 0 1 0 0
(E) 0 0 0 1 0 1 0 0 0 0 0 0 0 0
(F) 0 0 1 1 1 0 0 0 0 0 0 1 0 1
(G) 1 0 1 1 0 0 0 0 0 0 0 0 0 0
(H) 0 1 0 0 0 0 0 0 1 1 0 1 1 0
(1) 0 1 0 0 0 0 0 1 0 1 0 0 1 0
(J) 0 1 0 0 0 0 0 1 1 0 0 0 1 0
(K) 0 0 0 0 0 0 0 0 0 0 0 1 0 0
(L) 0 1 0 1 0 1 0 1 0 0 1 0 1 0
(M) 1 0 0 0 0 0 0 1 1 1 0 1 0 0
(N) 0 0 1 0 0 1 0 0 0 0 0 0 0 0
In 1985, the County Executive, seeking to consolidate his power, proposed eliminating some elected positions, including that of the County Auditor, and replacing them with appointed positions. He was in favour of constructing a new jail. The Auditor brought himself into the jail issue although he had no legal authority to do so. He publicly opposes the construction of a new jail.Our initial intuition was that the political actor network would contain two coalitions - one associated with the County Auditor, the other with the County Executive. If correct, a variety of analyses should point to a partition of the actors in the network, in which one subset contains the County Executive with the other containing the County Auditor. Given such a partition, our hypothesis is that the distribution of votes on the issue of the county jail in the County Council is conditioned by the partitioned network of strong ties among the political actors. In particular, the Council voting in December 1984 and the voting of the Public Safety Committee in November 1984 were distributed as follows: for construction of the jail: Council 1, Council 2. Council 3 and Council 4 and against construction: Council President, Council 5 and Council 6. If our hypothesis is correct, the set of actors voting for the construction of the jail will be contained in one political camp while the opponents to the construction of the jail will be contained in the other.
It is important to note that this hypothesis is not expressed in a quantitative form linking two variables in an explicit equation. It is a
qualitative hypothesis. Further, several obvious variables that might account in an equational form for the voting distribution have no variance. As the network is overwhelmingly white (13 of 14), male (13 of 14) and Democratic (12 of 14), the variables of race, gender and political affiliation are of no value in understanding the political dynamics of the network. We turn to consider its qualitative structure.4. DATA ANALYSIS
Corresponding to the adjacency matrix of Table I is a graph (shown in Figure 2) where the points represent the political actors and the edges represent strong political ties. There are paths between all pairs of points. The length of a path is the number of edges in it and the graph theoretical distance between two points is the length of the shortest path between them2. A fundamental property of a point is its proximity to other points in the structure and is captured by its graph distances to other points. The matrix of the graph theoretic distances formed the immediate input for a multidimensional scaling analysis in order to obtain a representation of the structure of strong political ties.
Multidimensional scaling (MDS) algorithms seek co-ordinates for a set of points in some Euclidean space where the distances between pairs of objects in the space correspond as closely as possible to the dissimilarities that provide the input. If the initial intuition is correct then there should be two distinct sets of points in the spatial representation returned by MDS.3 Figure I contains the two-dimensional MDS representation for the graph theoretic distance input. The stress for this representation is 0.0004, a very small value indicating a very good fit (without being degenerate'). On the right of the diagram is a clustered set of five points containing the County Auditor (B), while
Fig. 1. MDS representation of graph theoretic distancesthe left contains a clustered set of seven points containing the County Executive (A). ne initial intuition is confirmed. In the County Auditor's alliance, Council 5 (1) and Council 6 (J) occupy the same location.5 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 I (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 centre of the Euclidean space is the Former Council President (L) with the Former Council member (K) at the periphery of the diagram.Figure 2 shows the graph of the imposed on the Euclidean representation where the lines between points represent the edges of the graph. The cluster containing A is Alliance A while that containing B is Alliance B. The remaining two points are the Unaligned. The structure contained in this figure is striking. A pictorial representation of the information contained in Table I is not seen easily from the matrix alone. The set of quantitative co-ordinates provided by MDS provide straightforward locations for the nodes, over which the sociograrn can be drawn.
Internally, each alliance has far greater density of ties compared to the structure as a whole and those actors forming bridges between the major alliances occupy critical structural positions. Given this pattern, it is reasonable to expect that the alliances approximate structural equivalence classes. This expectation is confirmed by use of CONCOR and STRUCTURE (Dorelan 1988b). The partition rests on methods using correlation's or computed euclidean distances which are quantitative.
In addition, given the partition into alliances (as structural equivalence classes), the nodes in the graph, for each alliance. can be partitioned into boundary spanning nodes and nodes that lie within an alliance. It is a short step from this partition to thinking in terms of regular
Fig. 2. Political actor sociograrn overlaid on MDS representationequivalence (White & Reitz 1983) for the integration of the network. Indeed when the extent of regular equivalence is measured (by use of REGE 7 ) and clustered, the partition that results is (Doreian 1988a):
Partition Structural Role
L Central core of the network
M, H, B, D, F Boundary spanning nodes linked to the core
C, A, 1, J Provide some integration within alliances
G,N,E,K Completely peripheral and buried with the alliances.This partition, capturing clearly the integrative roles of the nodes is qualitative. However, it is reached by the systematic use of quantitative
procedures.When the membership of the two alliances is considered it is clear that alliance A contains all the council members voting one way, while alliance B contains the council members voting in the opposite way. The match between the voting patterns and the alliance is exact and provides strong support for the voting hypothesis.
From the structure shown in Figure 2, it is clear that L is the most central point, followed by the bridging members of the two alliances. Table II contains the centrality scores computed for the network of strong ties. The first column contains Freeman's (1979) betweenness centrality while the second panel contains the closeness centrality scores. These scores are consistent with the integrative scores are
TABLE II
Centrality scores for political actor network
