Scott, John. (2000) “Social Network analysis”
Course: Inter organisational relationships
Author: Michiel Westra ([email protected]
Abstract: This article provides an introduction to the theory and practice of network analysis in the social sciences. It explains the general framework, the basic concepts, technical measures and it reviews the available computer programs. Definitions of points,
lines and paths are given and their use in clarifying such measures as density, fragmentation and centralization. The author identifies the various cliques, components and circles
into which networks are formed, and outlines an approach to the study of socially structured positions. Finally, the use of multi dimensional methods for investigating social
networks is discussed.
CHAPTER 3:
Social network analysis emerged as a set of methods for the analysis of social
HANDLING RE-
structures, methods that specifically allow an investigation of the relational as-
LATIONAL DA-
pects of these structures. All social research data, once collected, must be held in
TA
some kind of data matrix (Galtung 1967), a framework in which the raw or coded
data can be organised in a more or less efficient way. The logical structure of the
data matrix is always that of a table.
Variables
Sex
Age
Income etc...
CASE-BYCases: 1
AFFILIATION
MATRIX
2
attributes
3
4
…
figure 3.1 a data matrix for variable analysis
This data matrix (case by variable) cannot be used for relational data. This kind of
data must be seen in terms of a case-by-affiliation matrix in which affiliations are
organisations, events etc, in which the cases (agents) are involved. This is called a
two-mode or rectangular, because the rows and columns refer to different sets of
data.
Figure 3.2. show this kind of matrix. Three people are involved (labelled 1,2,3)
and three events (labelled A,B,C). Where a specific individual participates in a
particular event, there’s a ‘1’ in the corresponding cell of the matrix. Non1/10
participation is shown by a ‘0’. The sociogram next to the matrix can be read as
saying that each person meets the other two at a particular event.
CASE-BY-CASE
It may be quitte impossible, using conventional manual methods of drawing, to
MATRIX
construct a sociogram for a large network. Alternative ways of recording the connections are made. The solution that has been most widely adopted has been to
construct a case-by-case matrix in which each agent is listed twice – once in the
rows and once in the columns. In this case, both the rows and the columns will
represent the cases, and the individual cells will show whether or not pairs of
individuals are related through common affiliation. The presence or absence of
connections between pairs of agents is represented by ‘1’ or ‘0’ in the appropriate
cells of the matrix. This matrix shows the actual relations or ties among the
agents.
AFFILIATION-
This matrix shows affiliations in both rows and its columns, with the individual
BY-
cells showing whether particular pairs of affiliations are linked through common
AFFILIATION
agents. This matrix is extremely important in network analysis and can often
MATRIX
throw light on important aspects of the social structure that are not apparent from
the case-by-case matrix.
INCIDENCE
The rectangular matrix (case-by-affiliation) is generally termed an incidence ma-
AND ADJACEN-
trix, while the square matrices (case-by-case, or affiliation-by-affiliation) are
CY MATRIX
termed adjacency matrices.
(See also appendix, figure 3.3)
Most techniques of network analysis involve the direct manipulation of adjacency
matrices, and so involve a prior conversion of the original incidence matrix into its
two constituent adjacency matrices. It is critically important therefore that researchers understand the for of their data (incidence or adjacency) and the assumptions that underpin particular procedures of network analysis.
For example, if a researcher collects two-mode data on cases and their affiliations,
then, it will generally be most appropriate to organize this info into an incidence
matrix form which the adjacency matrices used in network analysis can later be
derived. However in some situations it will be possible for a researcher to collect
relational data in a direct case-by-case form, for example friendship choices within
a small group. In this situation, the information can be immediately organized in
an adjacency matrix. (there are no corresponding incidence matrix and no complementary adjacency matrix of affiliations. All agents have merely a single affiliation in common – the fact of having chosen one another as friends).
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The choice of which set of agents to treat as the cases for the purpose of network
analysis will depend simply on which is seen as being the most significant in terms
of the research design. (For example, if the organizations are assumed to be of
the greatest importance, then a sample of organizations will be selected for study
and the only people who will figure in the analysis will be those who happen to be
members of these organizations). It makes no difference which of the two (organizations or members) are regarded as cases.
The distinction between cases and affiliations may generally be regarded as a
purely conventional feature of research designs for network analysis.
SOME CONVEN- A further aspect of this convention is to place the cases on the rows of the inciTIONS
dence matrix and the affiliations on its columns. It is also conventional to refer to
the rows before the columns when describing the contents of any particular cell,
and to use the letter ‘a’ to refer to the actual value contained in that cell. a(i,j) is
the general form for the content which refers to the value contained in the cell
corresponding to the intersection of row ‘i’ with column ‘j’.
ADJACENCY
The diagonal cells with this matrix are different from all other cells in the matrix.
MATRIX
In a square matrix, the diagonal cells shows the relations between any particular
case and itself. Therefore, these cells contain no values and should be ignored in
the analysis. Appendix 3.5 also shows that the adjacency matrix are symmetrical
around their diagonals; the top half of each matrix is an identical, mirror image of
its bottom half. Therefore, many analytical procedures in network analysis require
only the bottom half of this matrix and not the full matrix.
LEVEL OF
One of the most important considerations in variable analysis is the level of meas-
MEASUREMENT
urement that is appropriate for a variable. (nominal, ordinal, ratio, interval). Appendix 3.6 uses two dimensions to classify the four levels of measurement in
relational data;
1. Numeration (binary or valued)
2. Directionality (undirected or directed).
For example, the simplest type of relational data (type 1) is that which is both
undirected and binary.
STORAGE OF
If relational data is properly stored, they can be managed and manipulated more
RELATIONAL
efficiently. The need to use computers for network analysis, then, means that it is
DATA
important to consider how the logical structure of the data matrix can be translated into a computer file. For example the research on interlocking directorships
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invloves (1) generating a list of directors in the target compagnies (2) sorting this
into alphabetical order, (3) and then identifying those names that appear two or
more times. (using a word processor to create a data file..)
THE SELECTION The selection problems concern the boundedness of social relations and the posOF RELATIONAL sibility of drawing relational data from samples. Also, social relations are social
DATA
constructs, produced on the basis of the definitions of the situation made
by
group members. Therefore, ‘ close friendship’ for example, may mean different
things to different people. This issue is important, as researchers often have unrealistic views about the boundaries of relational systems (Laumann et al., 1989)
Researchers are involved in a process of conceptual elaboration and model building, not a simple process of collecting pre-formed data.
Two general approaches to this task have been identified: (1) positional approach
and (2) the relation approach.
POSITIONAL
In this approach, the researcher samples from among the occupants of particular
APPROACH
formally defined positions or group memberships. First, the positions or groups
that are of interest are identified, and then their occupants or members are sampled.
RELATIONAL
This approach can be used where there are no relevant positions , where there is
APPROACH
no comprehensive listing available, or where the knowledge of the agents themselves is crucial in determining the boundaries of the population. (for example,
informants are asked to nominate powerful members of the community and these
nominations are combined into a target population). The researcher must have
good reasons to believe that the informants will have a good knowledge of the
target population and are able to report this accurately.
A particular variant of this approach is the ‘snowballing technique’ in which a small
numbers of informants are studied and each is asked to nominate others for
study. In this method, the social relation itself is used as a chain of connection for
building the group.
SAMPLING
There are different sample problems, especially when focusing on large scale social systems. For example, in the case of a fairly small village with a population of
5,000 people, the adjacency matrix would contain 25,000,000 cells, which is beyond the capacity of most available computers and software. It might be assumed, therefore, that sampling from large populations would provide a similar
workable solution for social network analysis. (appendix 3.8 gives a schematic
account of the ideal sampling process in social) But, a representative sample of
agents, does not, in itself, give a useful sample of relations (Alba, 1982:44).
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There seem to be three different responses to these sampling problems;
1. to abandon any attempt to measure the global properties of social networks and to restrict attention to personal ego-centric networks;
2. to use a form of snowballing (which is not a random sample);
3. a way moving on to some of the more qualitative features of social networks (Burt, 1983). Burt, in particular, is concerned with the identification
of ‘positions’ or structural locations such as roles. (obtaining info from
each respondent about their social attributes and the attributes of those
to whom they are connected with. Then grouping of agents into sets with
commonly combinations of attributes, and then arrange these into a setby-set square matrix that shows the frequencies of relations between
members of the various categories.
CHAPTER 4:
If the sociogram is one way of representing relational matrix data (see previous
POINTS, LINES
chapter), the language of graph theory is another, and more general, way of do-
AND DENSITY
ing this. It is a starting point for many of the most fundamental ideas of social
network analysis.
Data in matrix form can be read by the programs, and suitable graph theoretical
concepts can be explored without the researcher needing to know anything about
the mechanism of the theory or of the matrix algebra.
SOCIOGRAMS
Graph theory concerns sets of elements and the relations among these; elements
AND GRAPH
being termed as points and the relations lines.Thus a matrix describing the rela-
THEORY
tions among a group a people can be converted into a graph of points connected
by lines.
The graphs of graph theory express the qualitative patterns of connection among
points. In a graph, it is the pattern of connections that is important, and not the
actual positioning of the points on the page. This theory does involve concepts of
length and location, for example, but these do not correspond to these concepts
of physical length and location which we are most familiar with.
There are different graphs, depending on the element which is focused on;
1. directed graph = if the relations are directed from one agent to another.
2. valued graph (example: appendix 3.5)= if the intensity of the relation is
an important consideration and can be represented by a numerical value.
A matrix for a directed graph will not usually be symmetrical, as relations will not
normally be reciprocated.
IMPORTANT
Two points that are connected by a line are said to be a adjacent to one anoth-
TERMS
er. It means that two agents represented by the points are directly related or
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connected with one another.
Those points to which a particular point is adjacent are termed its neighbourhood, and the total number of other points in its neighbourhood is termed its
degree. Thus, the degree of points is a numerical measure of the size of its
neighbourhood.
A sequence of lines in a graph is called a walk, and a walk in which each point
and each line are district is called a path.
The length of the path is the number of lines it contains. The distance between two points is the length of the shortest path that connects them.
The indegree of a point is the total number of other points that have lines directed
towards it; and its outdegree is the total number of points to which it direct
lines.
The more points that are connected to one another, the more dense will the
graph be. Density depends upon two other parameters of network structure: (1)
inclusiveness – the total number of points minus the number of isolated pointsand (2) the sum of the degree of its points.
Density of a graph is defined as the number of lines in a graph, expressed as a
proportion of the maximum possible number of lines.
The formula for this is:
L : n (n-1) / 2
The simplest and most straightforward way to measure the density of a large
network from a sample data would be to estimate it from the mean degree of
cases included in the sample. The degree sum – the sum of the degrees of all
points in the graph – is equal to the estimated mean degree multiplied by the
total number of case in the population. As the maximum possible number of lines
can always be calculated directly from the total number of points (it is always
equal to n (n-1) / 2 in an undirected graph), the density of the graph can be estimated by calculating:
(n x mean degree) /2 : n (n-1) / 2
When reduced, the formula = (n x mean degree) / n (n-1).
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APPENDIX
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