Comparing Social Networks:
Comparing Multiple Social Networks using MultiDimensional Scaling.
John Stevens
Abstract
Exploring the comparison of social networks and giving a description of the method I
developed for comparing many social networks. I develop a new technique by using
multidimensional scaling to differentiate between several social networks. Others have used
various methods and factors for comparing these social Networks, such as comparing
degree of node, density of the social network, path length and the number of triads in a
network. This technique works better if more than THREE social Networks are compared.
Keywords: Network comparison, Social Network Analysis,
Multi Dimensional Scaling.
1 Introduction
In this methodological paper I explore using multi-dimension scaling to compare
social networks. I have discovered from my own experience that most British
trained sociologists do not like numbers and prefer either text or graphics. For this
reason I have developed this graphical technique for comparing social networks.
In this paper I describe the relevant areas of social network analysis to my research
before investigating other academics’ work on the comparison of social networks,
following on with a description on the methods I developed for completing many
social networks. I then explore a new technique by using multidimensional scaling to
differentiate between several social networks. Following on from this I then give an
example of using the multidimensional scaling techniques to express graphically the
comparison of seven social Networks. In the final section of this paper I highlight
and summarize my findings.
2 What is Network Analysis
The following sub-section provides formal network definitions that are used in the
rest of the paper. If the reader is familiar with the subject area of social networks
please feel free to skip this sub-section.
2.1 The Basics
Social Network analysis as defined by Elizabeth Bott (1957) and Linton Freeman
(1968) is the symbol of a friendship network between individuals represented as
points and friendships link between them represented by lines. An example of this is
shown in figure 1 with nodes A, B and C and links X and Y. In this example the
nodes represent individuals and the links in this case, are friendship links. It should
also be noted from figure 1 that line X is a directed line while line Y is an undirected
line.
Figure 1 – An Example of a small Social Network
A realistic image of a social network is reproduced bellow in figure 2
Figure 2 – Plot of friendships on the rural estate
(Source: Output from graph UCINET and NetDRAW)
2.3 Density
One of the first network measures to be investigated by researchers was the
network density. This measure was used to provide a rough measure of the
connectedness of the network. A point’s density counts the number of complete
triangles emanating from a given point. An example of this is shown in the following
diagram.
Figure 3 – An Example of point density
Network density is then calculated by averaging all the individual point densities
across the whole network. The denominator in this equation is the number of nodes
in the network.
2.2 Shortest Path length
Path length is calculated as the shortest non-circular number of edges between 2
nodes. An example of this n=4 is shown bellow in figure 4. The average path length
is the average path length of all the pairs of nodes in a network.
Figure 4 – An example of a path of length (size = 3)
2.5 Global Centrality
Global centrality is similar to local centrality that was also initially developed by
Freeman (1979, 1980). Global centrality is calculated for each node as the distance
to all other nodes from that nodes point in the network. An example of this is given
below in Figure 5 and Table 1.
Figure 5 – An example of global centrality
Node
Global
centrality
A,C
19
B
16
All other points
26
Table 1 – An example of global centrality
2.6 Betweeness
The concept of betweeness was also initially developed by Freeman (1979) and is
related to global centrality. This subject describes the detection of brokers or
gatekeepers in social network analysis or locating gateways in computer networks as
described in electronics or computing. In figure 6, shown below node B fulfils this
role, as it is a gatekeeper between the networks centred round nodes A and C
Figure 6 – An Example of betweeness
3 Others Work Comparing Social Networks
Using exponential random graph models as explored by Wasserman and Patterson,
(1996), in which they compare the direction and magnitudes of parameters
characterizing local Comparing Social Networks: Size, Density, and Local Structure in
graphs, allows for calculated measures of dissimilarity between graphs for a variety of
social Networks. Results have shown the differences in the “structural signatures” of
different kinds of relations, notably antagonistic relations such as fighting and
dominance on the one hand, and relations of affection (friendship, liking) and
affiliation on the other. Differences between species became apparent only for the
first kind of relation, where humans showed tendencies toward mutuality and instars and away from transitivity, compared to non-human primates who showed
tendencies in the opposite direction on these properties Skvoretz and Faust, (2002).
The current work continues the line of inquiry initiated by Skvoretz and Faust. In
particular, it uses the triples or triad census [reference original triad reference: Davis,
Holland, Leinhardt] as a vehicle for comparisons to investigate local structural
similarities among a collection of 51 Networks of different relational contents and
measured on different species. The method of triad census was also used in Faust, K
& J. Skvoretz. (2007) in “Comparing Networks Across Space and Time, Sizes and
Species.” They managed to study 42 Networks from 4 types of species, human,
nonhuman primates, non primate mammals, and birds. Their aim was to assess
whether two, three… or many differing Networks were similarly structured despite
their surface differences.
One of the pieces of work on comparing social networks is the work reported in a
recent paper by Katherine Faust (2006) based on the fact of where she also uses a
“Triad census “to compare different social Networks of, amongst other species,
chimpanzees she comes to the conclusion that “caution should be taken in
interpreting higher order structural properties when they are explained by local
network features”. The majority of social network studies are case studies of a
single group or setting. Relatively less attention has been paid to comparisons using
Networks from multiple settings or longitudinal comparisons. Studies employing
multiple Networks focus on one of two distinct general questions. The first asks
whether a network of a specific relational content, in aggregate, exhibit common
structural tendencies. The second enquires as to what structural features are
distinguished among different kinds of social relations. In approaching the first sort of
question, some studies have examined the same relation measured in multiple
settings. Empirical examples include friendships in schools or classrooms see
Bearman, Jones, and Udry, (1997); Hallinan, (1974) and Leinhardt, (1972);
Snijders. T & Baerveldt. C (2003) uses a Multilevel Network Study of the Effects of
Delinquent Behaviour on Friendship Evolution. They use a similar multi level
approach to propose the study of the evolution of multiple Networks. Actor
orientated models of 10 school classes, looking at delinquency and friendship
Networks and network evolution within those classes (Snijders and Baerveldt, 2003),
social interactions in workplaces, Johnson villages Laumann and Pappi, (1976);
Rindfuss et al., (2004); and so on. Wasserman (1987) and Patterson and Wasserman
(1999) describe methodology for these comparisons. In addition, there are studies in
which roughly similar relations are compared across different settings. Killworth &
Bernard, conducted studies on Sailor’s of informant accuracy, by using observations
and verbal reports of interactions. A good example of such applications is by Bernard
et al., (1984) as is Freeman’s (1992) study of group structure of social interactions in
different settings (Freeman 1972). This review focuses on problems of informant
accuracy in reporting past events, behaviour, and circumstances. This paper is
concerned with validity and accuracy in what we call their native senses, the
literature on informant accuracy, Childcare, Health care, communication and social
interactions. Many questions regarding accuracy could be asked of the seven data
sets.
4 Methods
In order to help me investigate social Networks, I will examine several different
network variables including the number of nodes in a network, the degree of node,
the density of Networks, the average shortest path length, and network
centralization, the number of triads and finally the number and size of cliques. I
produced this data using UCINET to compare results and to find, among other
things if there is a relationship between density of the network, the degree of nodes,
number of triads and the number of cliques. I will then analyse this data using
techniques described in Marsh Catherine, (1988), and in Upton, Graham and Cook,
Ian, (1996) which are implemented in SPSS. In anticipation of the failures to find a
single variable leads me to investigate the following methods.
Secondly to compare several different social Networks, I produced a metadata table
detailing five network variables across all 7 datasets for the number of nodes in a
given social Networks, the density of social network, network centralization and the
average shortest path length between nodes. This data will then be analysed using
multi-dimensional scaling techniques that are implemented in SPSS and also described
in Coxon (1982). The metadata table is an input to the following procedures
implemented in SPSS.
The ALSCAL multidimensional scaling procedure is
implemented as part of SPSS. I discounted that it has 2 main problems those of which
are firstly, it allows for the plotting of rows and columns in the same model space
which is confusing for the viewer, and secondly this procedure has its faults with
small value differences in the data identified in Ramsey (1982). I similarly discounted
the next two multidimensional scaling procedures Proxscal and Minissa as they are
designed to work only on square matrix and not rectangle matrix data. I finally
chose to use the multidimensional scaling techniques of Prefmap detailed by Frank
Busing, M.T.A, (2006) and Hiclus as these are implemented in SPSS for ease of use.
The scaling procedure Prefmap, although designed for preference data, gives a
virtually identical output to the multidimensional scaling procedure Minirsa detailed
in Katrijn Van Deun, Willem J. Heiser, Luc Delbeke, (2007). In the past I have found
it to be more preferable as it meant for a more general use but not implemented in
SPSS. I will then integrate the respective outputs using techniques described in
Coxon (1982).
5 Variables that effect comparing social Networks
I investigated the comparison of social Networks by concentrating on which social
network variables that would be most useful to differentiate between differing social
Networks and their structures. The social network variables I compare are the
number of nodes, the degree of a node, the density of a social network, the shortest
path length between nodes, network centralisation and betweeness
5.1 Number of nodes
The first and most obvious variable to investigate when comparing social Networks
is the size of the social network (number of nodes). The following table gives the
number of nodes in each of the social Networks used in the paper.
Data set
Rural housing social
network
Urban housing social
network
Student social network
wave 1
Student social network
wave 2
Student social network
wave 3
Surname data
No of nodes
24
27
60
65
64
95
Student email social
324
network
Staff email social
833
network
Mean
186.5
Table 2 degree of node within social Networks
(Source: Figures given by UCINET for all the papers data sets)
Displaying the data contained in Table 2, as a figure would be pointless as there is
only one dimension to the data. This output is only of very limited use for
comparing the structure of social Networks.
5.2 Density
The density of a social network is a commonly used concept within social network
analysis. There are many researchers such as for example Favavo and Sunshine
(1968), Anderson B. et al. (1999) and Lin et al (1999) have all been interested in the
density of social Networks to see how tight or weak a community is. I am going to
compare and contrast several different social Networks of differing types in differing
location in this paper. A commonly used measure of the density of a friendship
social network is the clustering coefficient. I will be using the clustering coefficient
interchangeably with density in this paper.
In plain language, these are
representations of a person but shaped as triangles, X friends called Y and Z who
know each other as well. The clustering coefficient is used regularly when describing
friendship and social Networks within literature with whole sections dedicated to it
in books by John Scott (1991:73-84), Stanley Wassermann and Katherine Frast
(1994:101-103)
To add to the confusion, some academics call this simple concept the clustering
coefficient or reciprocity amongst friends. This is confused further by Claude Fischer
(1982) introducing the concept on page 139 to 143 of multistrandedness that is by
my reading very similar if not identical to density. In Scott (1991: pg 77) the
scalability of the clustering coefficient is analysed expressing an opinion that the
clustering coefficient is unable to scale for differing sizes of social Networks. I take
the view that the clustering coefficient can be used for comparing social Networks of
similar sizes (+- 10%). To examine this, the table below has listed the number of
nodes in each of my real world social Networks with there is corresponding social
network density.
Data set
No of nodes
The density of social
Networks
0.0451
Rural housing social
24
network
Urban housing social
27
0.0116
network
Student social network
60
0.0370
wave 1
Student social network
65
0.0938
wave 2
Student social network
64
0.1796
wave 3
Student email social
324
0.0118
network
Staff email social
833
0.0051
network
Mean
199.5
0.048
Table 3– Comparing social network size and density
(Source: Calculated using UCINET using all the papers datasets)
It should be noted from the above table that the mean density does not increase
linearly with the number of nodes in a network. However an experimenter could
compare two social Networks of similar sizes that have been collected using the
same data collection technique and methodology. It can also be seen from this figure
that the average degree of a node is not relative to the size of the social network.
0.2
0.18
0.16
0.14
Density
0.12
0.1
0.08
0.06
0.04
0.02
0
0
100
200
300
400
500
600
700
800
900
No of nodes
Figure 7 – Number of nodes against social network density
(Source: Output from MS EXCEL XP when plotting no nodes against social network
density)
5.3 Path Length (small world social Networks)
The next variable is the shortest path length that is unrelated to network density
between 2 nodes. This is also a popular area for social network studies. The
research area for this section is also called Small World Social network Studies that
was first detailed in Milgram (1967). The following table details some of these
studies
Type of
experiment
Number of
targets
Size of
community
Initial
sample size
Completed
chains
Mean links
in chains
Experiment
al
Theoretical
2
1967
(Milgram)
Post out /
post back
1 per sample
group
180 million
1
128
Not applicable
500
1
18
1
1
5.5
2.53
50
approximate
4
1
1989
(Tjaden)
Database
2002
(Dobbs)
Internet
Not applicable
2000
(Wiseman)
Post out /
post back
1
35,000
56 million
1 billion
(estimate)
24,163
Table 4 – Summary Details of Selected Small World Studies
18
384
6
600000000
500000000
400000000
300000000
200000000
Population
100000000
0
-100000000
0
1
2
3
4
5
6
7
Number of steps
Figure 8- Plot of number steps against population
(Source: Table 7)
The small world literature contains some significant gaps because I have discovered
there is no published work on the sampling for small world studies that can address
the question of how many cases you need to collect to make a reasonable model
from the data. Likewise, I found no published work that examines the effects of
response rates of data or resulting model quality. Duncan Watts and Steven Strogatz
(1998) show that the addition of a handful of “random links” can turn a disconnected
social network into a highly connected one. These “links” can generate social
networks with significant social consequences (as happened through the spread of
infectious disease such as AIDS and SARS). Similarly, key people may facilitate
constructive links, such as the extensive fundraising achieved by the political outsider
Howard Dean through Internet social networks rather than conventional advertising
in the 2004 US presidential election. Gladwell (2000) argues that the six-degree
phenomenon is dependent on a few extraordinary people (connectors) with large
social networks of contacts and friends. These people mediate the connections
between the vast majorities of otherwise weakly-connected individuals. To this
extent, these “connectors” can mediate social network interactions- a process
Gladwell calls “funnelling”. The opposite of the shortest path between 2 nodes is
the less used average longest path as referred to in Alon, N., Yuster, R., and Zwick,
U. (1994). This measurement gives a good approximation of the maximum distance
across a social network by tacking an average to remove the return path. I used
both measurements in the following table.
Data set
Rural housing social
network
Urban housing social
No of nodes
24
Shortest Path
3.160
27
1.547
network
Student social network
60
2.924
wave 1
Student social network
65
2.535
wave 2
Student social network
64
2.053
wave 3
Student email social
324
4.442
network
Staff email social
833
3.618
network
Mean
199.5
2.534
Table 5– number of nodes and path length
(Source: Calculated using UCINET using all the papers datasets)
It can also be seen from this figure that the average degree of a node is not relative
to the shortest or longest path length in the social network.
5
4.5
Shortest Path Length
4
3.5
3
2.5
2
1.5
1
0.5
0
0
100
200
300
400
500
600
700
800
900
No of Nodes
Figure 9 – Number of nodes against path lengths
(Source: Output from MS EXCEL XP when plotting no nodes against average
shortest path length)
5.4 Centrality
The next variable to be investigated was the network known as centrality. The
following table, as above gives the network centrality for the 7 datasets I collected.
Data set
No of
nodes
Mean
centralization
Network %
centralization
Rural housing
24
7.737
19.84
social network
Urban housing
27
3.250
5.78
social network
Student social
60
3.157
14.80
network wave 1
Student social
65
6.514
20.12
network wave 2
Student social
64
12.000
32.80
network wave 3
Student email
324
3.834
18.06
social network
Staff email social
833
4.223
4.88
network
Mean
199.5
5.852
16.60
Table 6- Number of nodes and network centrality
(Source: Calculated using UCINET using all the papers datasets)
The data in this table is represented in figure 10 as a graph plotting the number of
nodes against the network centrality.
35
Network % Centralization
30
25
20
15
10
5
0
0
200
400
600
800
1000
No of Nodes
Figure 10 – Number of nodes against path lengths
(Source: Output from MS EXCEL XP when plotting no nodes against centrality)
It can also be seen from this figure that similar to the previous comparisons the
degree of a node is not relative to the centrality of the social network.
5.5 Betweeness
It has appeared to me that the next natural step to be investigated is the network
variable of betweeness, which once again also appears to have been overlooked by
researchers working on comparing Networks. The following table has provided the
values for the average betweeness for the 7 data sets.
Data set
No of nodes
Betweeness
Rural housing social
24
20.404
network
Urban housing social
27
0.604
Network
Student social network
60
75.458
wave 1
Student social network
65
91.200
wave 2
Student social network
64
63.373
wave 3
Student email social
324
221.775
Network
Staff email social
833
1072.264
Network
Mean
199.5
220.725
Table 7- Number of nodes and betweeness
(Source: Calculated using UCINET using all the papers dataset)
This data is represented differently in figure 11. It can be seen from this figure that
the average degree of a node is not relative to betweeness in the social network.
Betweeness is affected by at least 2 variables of the age of the social network and the
physical location of its nodes. This variability is highlighted in the following figure,
which plots the number of nodes against betweeness.
1200
1000
Betweeness
800
600
400
200
0
24
27
60
65
64
324
833
No of Nodes
Figure 11 – Number of nodes plotted against betweeness
(Source: Output from MS EXCEL XP when plotting no nodes against betweeness)
5.6 Summary
As seen in the previous sections, there are many possible factors to think about
when comparing social Networks. These include the average degree of node, the
social network density, the number of triads. It should be noted that these plots are
very similar as the variables chosen are all related to the density of the social
network to a greater or lesser extent. A second point that should also be noted is
that the graphs describing the network variables network centralization and the
shortest path length is quite different and is not related to network density. The
variable that I have chosen, as the above sections highlight, are the most relevant
variable when comparing these networks are the number of nodes, density of the
graph, the shortest path length and network centrality. I will use these network
variables in the following section of this paper. I am certain this is not a complete list
of variables but it should be considered from the above work that no one network
variable is the best for comparing social Networks.
6 Comparing many social Networks
As I previously stated, I have found no single variable that adequately defines a social
network completely therefore comparing social Networks is a difficult task.
Traditionally the most commonly used quantities way of comparing social Networks
was to compare the density of different social Networks but this is a very broadbrush approach. As a way forward K, Faust and J. Skvoretz (2007) compared the
number of triads between different social Networks (triads being cliques of 3 nodes).
I chose to ignore Cliques as I anticipated using the Multidimensional scaling technique
as described by Prof A P M Coxon (1982) because it is a way of analysing several
variables of the social network together and not just the number of cliques.
6.1 Metadata table
To compare many social Networks, I have used Multidimensional-scaling procedures
on a metadata table of variables, which I have selected and described in section 5.6. I
then followed on from this by using the multidimensional scaling tools of Prefmap
and Hiclus on the metadata table. This produced a graphical output to compare the
social Networks when the procedures outputs are combined together.
Data
set
No of
nodes
Rural
housing
social
network
Urban
housing
social
network
Student
social
network
wave 1
Student
social
network
wave 2
Student
social
network
wave 3
Student
email
social
network
Staff
email
social
network
24
Density Average
Network %
the of
Shortest centralization
social
Path
network
0.0451
3.160
19.84
Betweeness
????
27
0.0116
1.547
5.78
????
60
0.0370
2.924
14.80
75.458
65
0.0938
2.535
20.12
91.200
64
0.1796
2.053
32.80
63.373
324
0.0118
4.442
18.06
221.775
833
0.0051
3.618
4.88
????
Table 8 - Metadata table
(Source: Values calculated using UCINET using all the papers dataset)
As can easily be seen from this table, which was has also been mentioned in the
previous section, I have found there is no single variable that can be used for
comparing social Networks.
6.2 Alscal, Proxscal and Minissa
Firstly I investigate the metadata table as input for the Alscal multidimensional scaling
procedure that is implemented as part of SPSS. A fault with this procedure is that it
dose analyses both rows and column in the same model space and the dis-advantage
that it dose not handle small value differences correctly as explored in Ramsay. J. 0.
(1982). Because of this feature I chose not to proceed with the procedure any
further. The next two multidimensional scaling procedures I investigate will be
Proxscal and Manissa. However I have decided to discount these procedures
because they only use square data matrix data as input and they are similar to Allscal
in analysing both rows and column in the same model space but with the advantage
they handle small value differences correctly.
6.3 Prefmap
The fourth multidimensional scaling technique I will explore is the Prefmap
procedure that is also implemented in SPSS14 (and up). The table 13 below shows
the output for this procedure. This procedure has the notable advantage that the
user can select to output rows or columns in the model space and use a rectangular
input data matrix.
Figure 13 - Prefmap output
(Final Stress: 0.0000: Penalty 4.5677)
(Source: Scaling output of Prefmap as implemented in SPSS when computing
metadata Table 8)
The output of Prefmap is similar to that of Alscal with the notable exceptions that
the graph has no major grouping.
6.4 Hierarchical clustering
Next I am going to use the Hiclus procedure also implemented in SPSS on the
metadata table above to find a dendrogram of the hierarchical clustering of the table.
I found this graph has the advantage over other graphs as the output, being graphical
was simple and easy to understand. It has worked well for the computer program
that I used to create the output being implemented in SPSS.
****HIERARCHICAL CLUSTER ANALYSIS**
Dendrogram using Average Linkage (Between Groups)
Rescalde Distancie Cluster Combine
CASE
0
5
10
15
20
25
Label Num +---------+---------+---------+---------+---------+
Urban
wave1
rural
wave2
wave3
Student
staff
2
3
1
4
5
6
7
─┐
─┤
─┼───┐
─┤
├───────────────────────────────────────────┐
─┘
│
│
─────┘
│
────────────────────────────────────────────────┘
Figure 14 - Hiclus output
(Source: Scaling output of Hiclus as implemented in SPSS when computing metadata
Table 8)
The output of the Hiclus procedure gives the similar results as the Prefscal
procedure grouping the 5 data-set waves 1, 2 and 3, the urban and the rural
Networks together while differentiating them from the other 2 data sets.
6.5 Interpreting results
Multidimensional scaling by its nature does not easily allow for a scale to be applied
to its output as the procedure folds multi dimensions into a 2 dimensional work
space. To interpret the output I am going to combine the results of the Prefscal and
Hierarchical clustering producers shown in figure 13 and figure 14 using the
techniques described in Coxon, T (1982). This is shown below in figure 15
Figure 15 - Combined output
(Final Stress: 0.0000: Penalty 4.5677)
(Source: Combined output of Prefscal and Hiclus as implemented in SPSS when
computing metadata Table 8)
When it comes to comparing many social Networks, the above techniques are
successful for showing the differences and similarities for comparing many social
Networks. A minimum of 3 social Networks should be compared using the above
method. An area for further investigation would be to access the accuracy of the
minimum number of social Networks to be compared. To summarize comparing
social Networks using multidimensional scaling on a meta-data table is one way I
have discovered that works well when many social Networks need to be compared.
7 Summary
As I have already stated in the previous section, there are many factors to consider
when comparing social Networks. Others have used various methods and factors for
comparing these social Networks, such as comparing degree of node, density of the
social network, path length and the number of triads in a network. The technique I
developed, which was far more successful for comparing several social Networks
was achieved by creating a Metadata table of many social network variables and using
multidimensional scaling techniques such as Prefscal and Hiclus. When the output of
these procedures was combined and analysed, a graphical representation of the
differences between several social Networks is shown. This technique works better
if more than 3 social Networks are compared. To summarize, this technique is not
perfect but it does provide a more comprehensive and systematic method of
comparing social Networks.
Bibliography.
Alon, N., Yuster, R., and Zwick, U. (1994), ``Colour-coding: a new method for
finding simple paths, cycles and other small subgraphs within large graphs'', Proc.
26th Ann. ACM Symp. On Theory of Comp., ACM, 326-335.
Anderson B. et al. (1999). The Interactions of size and density. Social Networks.
Bearman, PS., J. Jones, and J. R. Udry. (1997), “Connections Count: Adolescent
Health and the Design of the National Longitudinal Study of Adolescent
Health.”
Busing, Frank M.T.A, (2006) Avoding degeneracy in metric unfolding by penalizing the
intercept, pp. 419-427(9)
Coxon (1982) The Users Guide to Multidensional Scaling. Exeter, NH: Heinemann
Davis, J.A. and Leinhardt, S. (1972). ``The Structure of Positive Interpersonal
Relations in Small Groups.'' In J. Berger (Ed.), Sociological Theories in Progress,
Volume 2, 218-251.
Faust, K & J. Skvoretz. (2002) “Comparing Networks across Space and Time, Sizes
and species. Sociological Methodology.
Faust, Katherine (2006) “Comparing social Networks: Size, density and local
structure.” Metodološki Zvezki Advances in Methodology and Statistics 3(2):185216.
Faust, K & J. Skvoretz. (2007) in “Comparing Networks Across Space and Time,
Sizes and Species
Fararo, TJ & Sunshine, M H. (1964) A Study of a biased friendship Net. Syracuse,
New York: Syracuse University Press.
Fischer, Claude (1982) To Dwell Among Friends: Personal Networks in Town and
city. Chicago: University of Chicago Press
Freeman, Linton .C. (1992). The Sociological Concept of `Group': An Empirical Test
of Two Models.' American Journal of Sociology, 98, 1992, 55-79
Gladwell, M. (2000) The Tipping Point. Abacus. London.
Hallinan, M. T. (1974).A structural model of sentiment relations. American Journal of
Sociology.80, 364-378 Hallinan, M, T, and R, A Wilson. (1989), Interracial
Friendship Choices in Secondary Schools. American Sociological Review 54:67-78.
Hallinan, M, T, and R, A Wilson. (1989), Interracial Friendship Choices in Secondary
Schools. American Sociological Review 54:67-78.
Laumann, E, O and Pappi, F (1976). Network of Collective Action: A Perspective on
Community Influence System. New York: Academy Press.
Leinhardt. S. (1972) Developmental change in the sentiment structure of children’s
groups. American Sociological Review.370202-212
Lin, N. (1999) Building a network theory of social capital
Marsh, Catherine, (1988), Exploring Data, Polity Press.
Millgam Stanley. (1967) ‘Small world problem. Psychology Today 2, pp.66-67.
Patterson and Wasserman (1999), Advances in Exponential Random Graph (p*)
Models,
Social
Networks,
vol
29
issue
2,
Pages
169-172
Ramsay. J. 0. (1982). Royal Statistical Society, A. vol. 145. 285-312. (Foundations for one
aspect of the current state of the art. Introduces hypothesis testing into the
MDS framework, providing statistical tests to help decide on the appropriate
dimensionality and model.)
Rindfuss et al., 2004; the Power Lies in the Structure: Economic Policy Forum
Networks in Israel, the British Journal of Sociology, Vol. 48, No. 2. (Jun., 1997), pp.
267-285.
Scott, John (1991:73-84), Social Network Analysis. Sage.
Skovoretz. S. & Faust. K. (2002) Relations. Species and Network Structure
Snijders. T. & Baerveldt. C (2003) a Multilevel Network Study of the Effects of
Delinquent Behaviour on Friendship Evolution. Taylor & Francis.
Van Deun, Katrijn; Heiser, Willem J.; Delbeke, Luc. (2007). Multivariate Behavioral
Research, British Journal of Mathematical and Statistical Psychology, Volume
59, Number 2, November 2006, pp. 419-427(9)
Wasserman, S. (1987). Conformity of Two sociometric relations Psychometrika. 52, 3-18
Wasserman and Faust (1994), social network analysis methods and applications,
Cambridge University press.
Wasserman and Patterson, (1996), Comparing Social Networks: Size, Density, and Local
Structure 1987
Wasserman, S. (1987). Conformity of Two sociometric relations Psychometrika.
52,3-18
Watts, D. and Strogatz S. (1998), "Collective dynamics of small-world Networks",
Nature 393V
Upton, Graham and Cook, Ian, (1996), Understanding Statistics, Oxford University
Press.