Network Analysis

• Data matrices, sometimes called sociometric matrices, can be used to represent network relationships. The matrix typically is an n-by-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.

• Adjacency matrices are essentially similar, but cell entries are 1's and 0's, depending on whether the pair of subjects are adjacent by some criterion.

• Two-mode networks. Most network analyses are one-mode using a data matrix in which both rows and columns are the same actors or objects. However, UCINET and some other packages support two-mode networks using a data matrix in which the rows are of one type (ex., authors) and the columns are of another type (ex., articles published by authors).

  In the two-mode network illustration below, Tom, Dick, and Harry form a work group. They are asked to rate 1 - 10 how much they like, trust, and respect other members of the group. The UCINET inset (invoked by the Data, Spreadsheet, Matrix commands) shows the responses of each worker. The NetDraw graph shows how the network is visualized after being input as a two-mode model (with the File, Open, Ucinet Dataset, 2-Mode Network commands in NetDraw) in a common layout (with the Layout, Graph-theoretic layout, Spring embedding commands) and then sizing the edges according the the response (with the Properties, Lines, Size, Tie Strength commands). Lines appear proportionate to the responses, showing graphically the higher responses of Harry compared to Tom, for instance.

  

• Inclusiveness is the percentage of non-isolated nodes (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).

• Centrality measures the relative importance of nodes in a network. UCINET outputs four centrality measures: degree, closeness, betweenness, and eigenvalue. See the example section below for further discussion.
  • Degree centrality. 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. Degree centrality is sometimes interpreted as a measure of "direct influence". Past research has shown a correlation of degree centrality with such things as job satisfaction, knowledge, and leadership. Beware that an individual may have high degree centrality within a component that is relatively unconnected to the overall network. Degree centrality is based only on direct connections.

  • Closeness centrality: 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. Closeness centrality can be interpreted as a measure of the mean time it takes information or flow to arrive at a given node, or to pass from the node to other nodes. In a gossip network, nodes with highest closeness hear things first.

  • Betweenness centrality: 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. Betweenness is sometimes interpreted as "influence" or as degree of "control of information".

  • Eigenvector centrality: The node with the highest eigenvector centrality is the node for which its directly connected nodes have the highest centrality. That is, high eigenvector centrality indicates being connected to the well-connected. Nodes high on eigenvector centrality have high possibility of receiving/transmitting to the most nodes on the network quickly in one path step. This implies high exposure or risk vis-a-vis whatever is flowing through the network. Eigenvector centrality can be interpreted as measuring "popularity" or "indirect influence" as a "power behind the throne". A node with high eigenvecgtor centrality tends to be near the center of a clique (cliques are discussed below).

  • 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.

Network Analysis with UCINET

• UCINET, developed by Borgatti, Everett, & Freeman (2002), is commonly cited as the leading social network analysis package. A 60-day free implementation of the current version may be downloaded from the Analytic Technologies website. UCINET is designed to analyze network or other proximity data. It supports computation of centrality measures, dyadic cohesion measures, positional analysis algorithms, clique finders, stochastic dyad models (P1), network hypothesis testing procedures (including QAP quadratic assignment procedure matrix correlation/regression and categorical and continuous attribute autocorrelation tests), graph-theoretic procedures, as well as general statistical tools including multidimensional scaling, correspondence analysis, factor analysis, cluster analysis, mmultiple regression, and more.

  • Data. Data input for network analysis is typically in the format of a data matrix in which the rows and columns are the same objects and cell entries express a relationship between the row object and the column object. For instance, rows/columns may be people and cell entries a 1-to-10 self-rating of how much one person trusts another person. Or rows/columns may be cities and cell entries may be the distances between them. Note that the people example is asymmetric (person 1 may trust person 2 at level 3, but person 2 may trust person 1 at level 9), whereas the cities example is symmetric (the distance from NYC to LA is the same as from LA to NYC). For symmetric data, only one triangle of the matrix is needed since the other is redundant.

    For either asymmetric (a.k.a., "directed") or symmetric (a.k.a., "bonded" or "simple") input matrices, the cell data may be binary (ex., 0=doesn't trust, 1=trusts), multinomial (ex, "signed data", where -1 = dislikes, 0 = indifferent, 1= likes), ordinal (ranked), or interval (ex., inter-city distances). Graphically, asymmetric ties are indicated by lines with arrowheads (double-headed arrows if mutual choice), symmetric ties by lines without arrowheads.

    Cells in the data matrix may represent similarity or dissimilarity coding (high may represent similarity or objects or dissimilarity). The researcher must be careful to specify which type of coding is used.

    The figure below shows a symmetric data matrix with interval dissimilarity data. The example is the simple one of distances between cities, an example file provided with UCINET (CITIES.##H). Output below was generated by selecting Data, Display, from the UCINET menu.

    

    The next figure below shows an asymmetric data matrix with ordinal data. These data are also a UCINET example file, "Camp92.##h". Data were collected by placing each respondent's name on a card and asking each respondent to sort the cards by amount of interaction with that person. This resulted in rank order data with "1" indicating the most and "17" indicating the least interaction. Note the directed nature of the data: the number of interactions Holly perceives with Brazey is different from the number of interactions Brazey perceives with Holly.

    

    In summary, for conventional statistical analysis, objects of analysis are data rows. For network analysis, objects are both columns and rows. This implies that for conventional statistical analysis, objects are described by their column attributes. For network analysis, objects are described by their interrelationships with other objects. Indeed, that is the uniqueness of network analysis.

    In spite of the foregoing, note that it is possible for the rows and columns to represent different objects. For instance, one UCINET sample data file, "supremecourt judge attributes.##h", has the nine Supreme Court justices as rows and selected years as columns, with cell entries being number of times a given judge voted with the majority in a given year. However, in that case, while judges are connected to years, the usual network analysis questions about relationships among judges cannot be answered.

  • Visualization. Network visualization in UCINET is accomplished through the companion program, NetDraw, invoked from within UCINET by selecting Visualization, NetDraw, from the UCINET menus.

    The diagram below was created in NetDraw using the NetDraw menu commands File, Open, Ucinet dataset, Network; then in the ensuing "Open Data File" dialog, browse to the example file "CITIES.##h". A diagram will appear, but for the default assumption that the data are similarities. As inter-city distances are dissimilarity data, the diagram must be rescaled accordingly by selecting Layout, Scaling/Ordination, from the NetDraw menus, then checking the radio button indicating dissimilarity data.

    After rescaling, a map-like network diagram is generated based not on map coordinates but simply from iteratively positioning the city nodes to reflect the paired dissimilarity measures (distances). The diagram employs double-headed arrows by default, but arrowheads could by suppressed by selecting Options, Display, and unchecking arrowheads. The statistical iteration process used is a type of multidimensional scaline, and, indeed, multidimensional scaling in this case could achieve the same results.

    

    
    

  • Statistical analysis

    
    

  • Example. In the example below, UCINET is used to explore a network of 13 individuals. Data are binary, coded 0 or 1 according to whether there is a relationship. The relationship might be liking, trusting, communicating with, or any other relation which can be coded absent/present.
    • Data. Data are entered in an Excel spreadsheet. Cell entries indicate "from" rows and "to" columns. Thus, for instance, if the relationship is "trusts", then the "1" in the Brett row, Jane column indicates that Brett trusts Jane. The "0" in the Jane row, Brett column indicates that Jane does not trust Brett. Some relations are reciprocal, some are not. Alice is not trusted by anyone.
      

    • Importing Excel to UCINET. After saving the Excel file above in .xls format, the file can be imported as depicted below by selecting Data, Import via spreadsheet, Full matrix w/ multiple sheets from the UCINET menu. (It does not matter that this example has a single sheet). In the ensuing dialog box, also depicted below, row and column labels' presence checkboxes are checked and the filename given (click on the ellipses icon to browse to the file). The importation process will cause the file to be saved in UCINET format to files ending in .##d and .##h, which are the data and data description file formats for UCINET.
      

    • Visualization. After the importation process has saved files in UCINET's formats, the researcher can switch to the Netdraw component and open the .##h file, in this example networkgroup.##h. In Netdraw menus, select File, Open, Ucinet dataset, Network, and browse to the .##h file wanted.
      

      The network will be graphed automatically, though the researcher may wish to use Netdraw's Layout, Move/Rotate menu choice, which brings up the dialog shown below, to redraw the network in the desired orientation.

      

      Under the Analysis menu choice, the researcher can color code isolates, components, blocks & cutpoints (shown in the diagram above), K-cores, or subgroups. Various statistical output may also be selected, discussed further below.

      
      1. Isolates are nodes with no relations (edges, geodesics) with any other node. Alice is an isolate in this network.

      2. Components are sets of nodes connected by relation arrows or edges. Components are not connected to each other. In this network, there are two components: the isolate Alice and all the rest. To take another example, in a network study of faculty in a college with eight departments, there might be eight components if there were no cross-departmental relations among facult. In directed graphs, components are strong if connected by bidirectional arrows, weak if unidirectional arrows.

      3. Blocks & cutpoints. A cutpoint is a node which, if removed, would increase the number of components. In the example network, Jane, Joe, Pete, and Grace are cutpoints, shown in blue in the network diagram above. If a network has no cutpoints, it is said to be "non-separable," not the case in this example.

         In this example there are 10 blocks. The Network, Regions, Bi-component menu choice in UCINET yields a list of blocks:

         BI-CONNECTED COMPONENTS (BLOCKS)
         --------------------------------------------------- 
         Input dataset:                networkgroup 
         Dataset symmetrized via maximum method.
         10 blocks found.
         
         BLOCKS:
         Block    1:  Jane Macy
         Block    2:  Jane Brett
         Block    3:  Jane Sam
         Block    4:  Pete Lynn
         Block    5:  Joe Pete
         Block    6:  Grace Larry
         Block    7:  Grace Marcia
         Block    8:  Grace Greg
         Block    9:  Joe Grace
         Block   10:  Jane John Joe
         --------------------------------------------------- 
         

      4. K-cores and degrees. The degrees for a node is the number of relations (edges) directly related to it. In the example, there are 0 degrees associated with Alice, 1 with Macy, 2 with Peter, 4 with Joe or Grace, and 5 with Jane (the most), for instance.

         K-cores identify clique-like structures, though these are not necessarily cohesive subsets. Alice has 0 relations with anyone else and has a K-core value of 0. Jane, Joe, and John each have relations with two others in this core group and have a K-core value of 2. All others have a K-core value of 1.

      5. Cliques. Cliques are groups of 2 or more (in most analyses, 3 or more) all connected to each other by strong (double-headed arrow) reciprocal relationships in a directed network (arrows have directional heads) or simply by relationships (lines, not arrows are edges) in non-directional networks. In network analysis jargon, a clique is a maximally complete subgraph. Cliques are calculated in UCINET (not Netdraw) using the menu choice Network, Subgroups, Cliques. In this example there are only cliques of size 2:

         CLIQUES
         ---------------------------------------------------------
         Minimum Set Size:             2
         Input dataset:                networkgroup 
         
         NOTE: Directed graph. You may prefer to symmetrize first.
         6 cliques found.
         
            1:  Jane Macy
            2:  Jane John
            3:  Jane Joe
            4:  Joe Pete
            5:  Grace Marcia
            6:  Grace Greg
         ---------------------------------------------------------
         

         • N-cliques relax the definition of cliques to allow clique members to belong as long as they are not more than n relations away from each other clique member. If the relationship is friendship, for example, in a 2-clique every member is either a friend of each other member or at least a friend of a friend. Choose Network,Subgroups,N-Cliques from the UCINET (not Netdraw) menu.

         • N-clans are like n-cliques but require any indirect connections (ex., friend of a friend) to pass only through another clique member. (N-cliques can have the undesirable property of having indirect relations involve paths through network members who do not qualify as clique members). Choose Network, Subgroups, N-Clan from the UCINET (not Netdraw) menu.

         • K-plexes are a different type of relaxation of the definition of clique. Given a k-plex of size n, a node can be a member as long as it has a relation to at least (n -k) other nodes in the k-plex. If k=2, for instance, a node can be a member as long as it has a relation with all but two other nodes in the k-plex. Choose Network, Subgroups, K-plex from the UCINET (not Netdraw) menu.

         • K-cores are an even more relaxed definition of clique. A node can be a member of a k-core as long as it has a relation to at least k other nodes in the k-core. If k=5, for instance, a node can be a member as long as it has a relation with five other nodes in the k-core, regardless of how many it does not have relations with. Choose Network, Regions, K-core from the UCINET menu. Alternatively, choose Analysis, K-cores, from the Netdraw menu.

      6. Subgroups. In Netdraw, the menu choice Analysis, Subgroups supports a variety of different methods of partitioning the network into clusters. The Newman-Girvan algorithm, results of which are shown below, is a form of block modeling. It tries to group nodes into various numbers of clusters, giving a goodness-of-fit measure for each. The researcher may need to make the trade-off between goodness of fit and parsimony. NetDraw draws the graph for the smallest number of clusters (here, 2), but using the "Choose node colors" icon on the NetDraw toolbar allows any of the partitions to be displayed. In the diagram below, four clusters are displayed.
         

         An alternative to Newman-Girvan is the Analysis, Subgroups, Factions choice. Faction analysis attempts to create the researcher-specified number of clusters such that within-cluster notes are more tightly connected to each other than to nodes in other factions. Using a different algorithm, factions often yields similar results. It too computes a goodness of fit measure. The diagram below shows the example network partitioned into four factions.

         

         Other Analysis, Subgroups choices in NetDraw are the hierarchical clustering (see the separate Statnotes section on hierarchical clustering) and block-based approaches.

    • Centrality measures. Often a central research purpose is to generate centrality coefficients for the network under study. These coefficients become added as additional columns to the nodes spreadsheet built into NetDraw. This spreadsheet starts with a single column, ID, listing the nodes (here the 13 individuals, starting with Alice). As the researcher selects choices from the Analysis menu in NetDraw, additional columns of coefficients are added. These may be viewed at any time in NetDraw by selecting Transform, Node attribute editor. One can cut-and-paste coefficients as additional data columns to other programs for statistical analysis using external packages. Below is the example output for a directed network (relationships are directional and hence can be asymmetric) copied from the NetDraw node attribute editor into an Excel spreadsheet:
      

      See related discussion of centrality measures earlier in this section. The coefficients below are generated when a directed graph is specified. For non-directed graphs, a simple coefficient without "In" and "Out" variants is computed.

      1. InDegree: number of incoming relationship arrows for the node. Jane has the most, with 5. A high in-degree score may indicate prominence or prestige.

      2. OutDegree: number of outgoing relationship arrows for the node. Jane and Joe have the most, with 3. For information flows, a high out-degree score may indicate ability to communicate with others, make them aware of one's views, and hence be influential.

      3. Betweenness: The higher the betweenness, the more the given node is along the shortest path between pairs of other nodes. Betweenness is the number of times a node is on the shortest path between two other nodes, adjusting for the number of alternative shortest paths. For directed networks, direction of edges is taken into account in the betweenness-dir coefficient. In this example, Joe has the highest betweenness, 32.

      4. Closeness. Closeness is interpreted as measuring how long information or other flow takes on the average to (or from) go from a given node to (or to) all other reachable nodes in the network. Closeness is the mean geodesic distance between a given node and all other nodes with paths from the given node to the other node. This is close to being the mean shortest path, but geodesic distances give higher values for more central nodes. For directed networks, InCloseness and OutCloseness are computed, taking account of inbound or outbound paths.

      5. HarmonicCloseness: Harmonic closeness, also called information closeness, is an alternative measure of closeness based on an algorithm that uses the harmonic mean length of paths ending at the given node. Lower numbers are associated with a nodee connected to other nodes by many short paths. For directed networks, InHarmonicCloseness and OutHarmonicCloseness are computed, taking account of inbound and outbound paths.

      6. Eigenvector: Interpreted as measuring a node's network importance, eigenvector centrality gives greater weight to a node the more it is connected to other highly connected nodes. A node connected to five high-scoring nodes will have higher eigenvector centrality than a node connected to five low-scoring nodes. In directed networks, there is an "In" and an "Out" version. In information flow studies, for instance, InEigenvector scores would reflect which nodes are high on receiving informationj while OutEigenvector scores would reflect which nodes are high on broadcasting information.

      • Many additional network coefficients may be generated through UCINET and NetDraw. In addition, any coefficient can be visualized in the network diagram by selecting the NetDraw menu choice, Properties, Nodes, Symbols. This choice allows keying of symbol size, color, and/or shape to any coefficient, as illustrated below. It is also possible to assign properties to lines (edges) in a similar manner, not illustrated here.
        

    • Statistical analysis. UCINET supports a variety of built-in statistical procedures, including cluster analysis, multi-dimensional scaling, regression, and many more. For instance, a given centrality measure may be regressed against demographic and other possible causal variables to determine the possible causes or at least correlates of that measure. These related statistical procedures are accessed from the UCINET "Tools" menu choice illustrated below
      

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.

• Correlated data. In most forms of network analysis, one is dealing with correlated data but network analysis packages are designed to handle data which are not statistically independent. While the groups studied may be selected at random from some universe (ex., post offices from a list of all federal post offices), the responses of the individuals within the group are not independent (ex., how much employee 1 likes employees 2 and 3 may affect how much employee 2 likes employee 3). The researcher must take care when network data are exported to conventional statistical programs whose procedures may assume data independence.

• 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. On the other hand, statistical network a analysis packages (ex., Pajek) can handle large datasets.

• Model specification. If aspects of group structure, such as centrality of one actor compared to another, are not causes of the dependent variable(s) of interest to the researcher, network analysis may not be the appropriate tool. Examples of research questions appropriate for network analysis are (1) in an organized crime network, arresting which individual(s) would cause the most disruption to its work?; (2) to diffuse public health information in a small rural community where only three individuals could be hired, which three individuals would make the most influential advocates?; or (3) do the trust structures of male and female groups differ for groups of the same size performing the same function?

• Ecological fallacy. Network coefficients by node are individual centered, but what is true at the individual level may not be true at the group level. This is the "ensemble problem". Forming a team of peer influencers (ensemble) by picking, say, the five individuals with the highest betweenness centrality may not be optimal. for instance, the top-scoring five may tend to be between many of the same individuals as other top five scorers. Picking an ensemble with more diversity may be optimal.

Frequently Asked Questions

• Where can I find out more about network analysis?
  A good start is the website of the International Network for Social Network Analysis.

• 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/Software/ViewNET/viewnet.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

• Borgatti, S.P.; Everett, M.G.; & Freeman, L.C. (2002). Ucinet for Windows: Software for social network analysis. Harvard, MA: Analytic Technologies.
• 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. Uses UCINET.
• Hanneman, Robert A. & Mark Riddle. (2005). Introduction to social network methods. Riverside, CA: University of California, Riverside (published in digital form at http://faculty.ucr.edu/~hanneman/nettext/). Uses UCINET.
• 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.
• Newman, M. E. J. (2010). Networks: An introduction. NY: Oxford University Press.
• 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.