Network Analysis and Sociometry

Overview The term network analysis has largely supplanted the earlier term sociometry, but both refer to the analysis of social networks in part utilizing graphical methods. While some use the term "sociometry" to refer to all research using quantitative scales, it is used here in its narrower meaning, sometimes called sociography, which is a method of presenting data about complex individual relationships and networks in graph form. Sociometry was developed by Jacob L. Moreno in the 1930s and became closely associated with small group research and a focus on interpersonal choices. As a largely subjective but empirical, phenomenological approach to the study of group structure, sociometry can serve as a contrast or complement to the formal study of group structure through archival records. In addition to its social scientific purposes, discussed below, sociometric assessment of interpersonal choices also plays a role in therapy by helping facilitate constructive change in individuals and groups through greater interpersonal awareness. For this reasons, in some circles the term sociometry refers to a form of therapy related to psychodrama. See also: network theory, actor-network theory.

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography

Key Concepts and Terms

• Sociometric tests are simply surveys which are administered to subjects (typically all subjects in a small group) to determine the direction of certain relationships, although sometimes this can be assessed through simple observation. For instance, a sociometric test might ask members of a political group which other members they are most/least likely to consult on political questions; whom they hold in highest/lowest esteem; or with whom they most/least likely to see face-to-face during the previous week.

• Sociometric representation refers to various graphical methods of data presentation. Some common sociometric representations are illustrated in the figure below. In each representation, individuals are depicted as points. The star representation is made up of several lines pointing toward or away from a central point, which represents an individual who is involved in realationships with each of the other points, and where the arrows represent relatinships such as reciprocation, ignoring, or rejection. Other common representations include the chain (a series of relationships); the pair (a mutually reciprocated relationship); the power (short for "power behind the throne," who is the object of attraction of a few very sociometrically attractive individuals; and the isolate (a subject not chosen by anyone).
  
  • Inclusiveness is the percentage of non-isolated points in a sociometric diagram. A 25-point graph with five isolated points has an inclusiveness of 0.80.

  • Density is the number of lines as a percentage of total lines when all points are connected. Total lines, based on probability, is _nC₂ = n!/(n - 2)!2!. For example, the number of lines connecting 6 points taken 2 at a time is 6!/[(6-2)!2!] = 720/(24*2) = 15. If a sociometric diagram had 10 lines, out of the possible total of 15, its density would be .67. This assumes a non-directed graph in which the lines are bi-directional or non-directional. For a directed graph in which all lines are one-directional, the total possible number of lines would be twice the calculation above (30) and thus the density would be half as much (.33).

• Sociometric diagrams ("sociograms") map the relationships revealed by sociometric tests, as illustrated in the figure below While sociometric relationships for a five-person group such as depicted below could easily be described verbally, as the size of the group increases to dozens of subjects, sociometric diagrams are increasingly helpful in conveying the relationship structure. While sociometric studies are often static measurements at one point in time, there is nothing inherent in sociometry to prevent the recording of panel data over time. Moreover, sociometric representation can now be animated on the web to provide dynamic data diagrams.
  

• Sociometric matrices are an alternative form of data representation, used primarily when group size makes sociometric diagrams too complex. The matrix is an n-b-n square table representing subjects both across the top and down the side. For each possible pair, the range of the criterion is shown. For instance, for the criterion "With whom do you most like to work?", the ranges might be attraction = +1, indifference=0, dislike = -1. This sociometric information may be used to create an index of popularity by group by comparing the proportion of members chosen as desirable work partners in one group divided by group size, compared to a similar index in another group. An adjacency matrix is essentially similar, but cell entries are 1's and 0's, depending on whether the pair of subjects are adjacent by some criterion.

• Network data diagrams exist to provide a variety of graphic representations of networks alternative to classic sociograms (for illustration, see Brandes et al., 1999):
  • Path diagrams, not necessarily based on actual path analysis, represent variables or groups as circles, relationships (which may be correlations, communications, formal associations, or other interactions) as arrows, and, often, magnitude of relationship by thickness of the arrow.

  • Cluster diagrams represent variables or groups as points on one or more two-dimensional scatterplots or polar plots, with the proximity of points representing their similarity on the dimensions, and clusters of points may be highlighted by perimeter lines around each cluster (including the possibility of intersecting perimeters where a point may belong to two or more clusters).
    • Cliques are identified by UCINET software as one of its options. A clique by clique co-membership matrix can also be output.

  • Factor plots similarly represent variables or groups as points on one or more two-dimensional scatterplots, where the dimensions are factors (see factor analysis); optionally, factor space may be divided into non-intersecting quadrants to highlight similarities among points.

  • Centrality measures the relative importance of nodes in a network. ICINET outputs four centrality measures: degree, closeness, betweenness, and eigenvalue.
    • Centrality degree. Degree for a node is highest when the node has the maximum possible number of direct connections to other nodes. Degree is thus the number of direct ties to other nodes. Nodes of high centrality degree have more opportunity of giving and receiving information or other phenomena under study.

    • Centrality closeness: Closeness for a node is highest when a node can reach all other nodes in the network. Mathematically, closeness is the graph-theoretic distance of a node to all other nodes. In diffusion theory, nodes with high centrality closeness are ones most likely to receive and transmit innovations.

    • Centrality betweenness: Betweenness for a node is highest when that node is maximally utilized by nodes connecting to other nodes. That is, betweenness measures how many paths pass through a node. Nodes high on betweenness have high opportunity to play gatekeeper, liaison, or broker roles.

    • Centrality eigenvector: The node with the highest centrality eigenvector is the node for which its directly connected nodes have the highest centrality. Nodes high on centrality eigenvector are high on the possibility of receiving/transmitting to the most nodes on the network quickly in one path step.

    • Centrality plots are polar plots in which the heavier the loading of the variable or group on the dimension, the closer it is located to the center of the plot. Optionally, concentric circles may highlight which points share a similar degree of centrality on the depicted dimension. Loadings may reflect factor loadings, path distances, or an index of the author's devising. Centrality index numbers, if assigned to points, are usually coded such that heavier loadings are represented as lower numbers. In centrality plots, direction of location with respect to the center (up/down, left/right) often has no meaning other than aesthetics of placement, but direction can be used to depict a second and third dimension.

  • Spatial network diagrams. In the context of geographic information systems, software such as ArcView implement network analysis modules which generate map graphics depicting such things as shortest route between two objects, optimal route passing through a series of objects, or service areas (by best time or shortest distance) associated with multiple points.

Assumptions

• Measurement inerrancy. Sociometric mapping assumes, of course, that individuals respond accurately to sociometric surveys or can be assessed accurately through observation. However, sociometry is open to several common problems. It tends not to record subconscious or illicit relationships. It may be biased toward recording attractions rather than dislikes because subjects more easily reveal the former. It is best when subjective sociometric responses can be validated through external objective measures.

• Small group size. Sociometric diagramming becomes unwieldy and difficult for readers to interpret for very large groups. Also, sociometric techniques may be biased in larger groups since subjects tend to confine their choices to their own class range.

Frequently Asked Questions

• What computer programs exist to generate sociograms or similar representations?
  Probably the leading software for network analysis at present is UCI-Net, which is a comprehensive program s a comprehensive program supporting centrality measures, dyadic cohesion measures, positional analysis algorithms, clique finders, stochastic dyad models (P1), network hypothesis testing procedures (including QAP matrix correlation/regression and categorical and continuous attribute autocorrelation tests), plus general statistical and multi­variate analysis tools such as multi­dimensional scaling, correspondence analysis, factor analysis, cluster analysis, multiple regression, etc. In addition, UCINET provides a host of data management and transformation tools ranging from graph-theoretic procedures to a full-featured matrix algebra language. A free evaluation version is available.

  A public administration example is Kapucu, Naim (2003).Coordinating without hierarchy: Public-nonprofit partnerships. International Association of Schools and Institutes of Administration, Conference on Public Administration: Challenges of Inequality and Exclusion, Miami (USA), 14-18 September 2003.

  Other packages include:

  • NetDraw. See http://www.analytictech.com/netdraw.htm. NetDraw is a complement to UCI-Net, from the same publisher.

  • Pajek. A recent academic package capable of analyzing large networks. Pajek is described in the text by Nooy, Wouter “de” ; Andrej Mrvar, & Vladimir Batagelj (2005).Exploratory Social Network Analysis with Pajek. Cambridge, UK: Cambridge University Press. Pajek software and datasets for all examples are freely available at http://vlado.fmf.uni-lj.si/pub/networks/pajek/.. A brief tutorial is available at http://www.ccsr.ac.uk/methods/publications/snacourse/pajek.html.

  • ArcView is leading geographic information systems software from ESRI. It has a "Network Analysis" module, which is for calculating service areas around geographic points, or calculating shortest routes. This is a different meaning of network analysis from sociometry.

  • FatCat. See http://www.sfu.ca/~richards/. A sociometric matrix manipulation program. It makes graphic representations of two-dimensional crosstabulation tables, called "panigrams", which make the information contained in the "who" rows and "to whom" columns.

  • MultiNet. See http://www.sfu.ca/~richards/Multinet/Pages/multinet.htm. It gives histograms with frequency distributions, panigrams with crosstabs, 3-d ribbon plots with ANOVAs, scatterplots with correlations, network displays with graph spectra. Graph theoretic spectra (eigenvalues and eigenvectors allows the researcher to rotate any axis, change any axis, color nodes by categories, and examine the network in 1, 2 and 3-dimensions, and to define new node variables based on eigenvectors and induced partitions, which may be used to examine the network for group structures.

  • NEGOPY. See http://www.sfu.ca/~richards/Pages/negopy.htm. One of the oldest network analysis programs, NEGOPY finds cliques, liaisons, and isolates in networks having up to 1,000 members and 20,000 links. In use at over 100 universities and research centers around the world.

  • Walsh's Classroom Sociometrics. Designed to identify classroom cliques, this package prints out a sociometric survey for classroom distribution. Output includes sociograms, barcharts, and scatterplots, as well as tab-delimited data files SPSS and other packages can import. Quickly indicates rejected, isolated, popular, and controversial students. Demo version available. Site has some links to other sociometry resources.

  • Viewnet. See http://www.datashare.com.au/.%5CViewnet.htm. Works with DataShare software and is geared toward electric and other physical networks.

• What is role analysis in sociometry?
  One can cluster subjects into roles, such as functional work roles in an organization, then average data for the many incumbents of each of several roles so as to use the averages rather than the individual responses when constructing sociometric diagrams and matrices. This general procedure may be used to relate occupational categories, social classes, age cohorts, and so on.

Example

• Uncloaking Terrorist Networks, by Valdes E. Krebs

Bibliography

• Brandes, Ulrik, Patrick Kenis, Jorg Raab, Volker Schneider, and Dorothea Wagner (1999). Explorations into the visualization of policy networks. Journal of Theoretical Politics 11(1): 75-106. Broader than sociometry but using it as a focus, this article places sociometry in the context of network analysis for policy research, with numerous alternative modes of graphical representation.
• Brandes, Ulrik & Thomas Erlebach, eds. (2005). Network analysis: Methodological foundations. Berlin, Heidelberg: Springer-Verlag.
• Breiger, Ronald L. (2004). The analysis of social networks. Pp. 505-526 in Melissa Hardy & Alan Bryman, eds., Handbook of Data Analysis. London: Sage Publications.
• Carrington, Peter J., John Scott and Stanley Wasserman, eds. (2005). Models and methods in social network analysis. New York: Cambridge University Press.
• Degenne, Alain and Michel Forse (1999). Introducing social networks. Thousand Oaks, CA: Sage Publications. Reviews the literature and provides an introductory text. using the paradigm of structural analysis. Covers graph theory and includes guides for usage of software for network analysis.
• Freeman, Linton C. (2004). The development of social network analysis: A study in the sociology of science. Vancouver: Empirical Press.
• Geer, John P. van der (1971). Introduction to multivariate analysis for the social sciences. San Francisco, CA: Freeman. Chapter 2 discusses use of matrix algebra to process sociometric data.
• Hanneman, Robert A. (1999). Introduction to Social Network Methods.
• Haythornthwaite, Caroline (2001). Exploring multiplexity: social network analysis in a computer-supported distance learning class. The Information Society 17(3): 211-226. Good example of sociogram use.
• Moreno, Jacob L. (1934). Who shall survive? Washington, DC: Nervous and Mental Disorders Publishing Co. This was the seminal study popularizing sociometry. It was reprinted in 1953 by Beacon House, Beacon, NY.
• Moreno, Jacob L. (1960). The sociometry reader. Glencoe, IL: Free Press.
• Northway, Mary L. (1967). A primer of sociometry. Toronto, Canada: University of Toronto Press.
• Scott, John (1992). Social network analysis : A handbook. Thousand Oaks, CA: Sage Publications. Covers network theory.
• Wasserman, Stanley and Katherine Faust (1994). Social network analysis : Methods and applications (Structural Analysis in the Social Sciences, 8). Cambridge, UK: Cambridge University Press. A current leading textbook on network analysis.
• Wellman, Barry and Berkowitz, S.D. (1988). Social structures: A network approach. Cambridge: Cambridge University Press.