The basic network hypothesis is that the structure of a network affects the likelihood that goods wi

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Complete Network Analysis Network Connections
Large-Scale network structure
The basic network hypothesis is that the
structure of a network affects the likelihood
that goods will flow through the network. While
direct measures are fine for smaller networks, we
often want to make generalizations to very
large-scale network structure. The next
section covers large-scale network topography and
bridges us to generalized images of the network
structure captured by cohesive groups and
blockmodels. We focus on 3 such factors
today 1) Basic structure of large-scale
networks 2) Cohesive Peer Groups 3) Identifying
Role positions (blockmodels)
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Based on Milgrams (1967) famous work, the
substantive point is that networks are structured
such that even when most of our connections are
local, any pair of people can be connected by a
fairly small number of relational steps.
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Watts says there are 4 conditions that make the
small world phenomenon interesting
1) The network is large - O(Billions) 2) The
network is sparse - people are connected to a
small fraction of the total network 3) The
network is decentralized -- no single (or small
) of stars 4) The network is highly clustered --
most friendship circles are overlapping
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Formally, we can characterize a graph through 2
statistics.

• 1) The characteristic path length, L
• The average length of the shortest paths
  connecting any two actors.
• (note this only works for connected graphs)
• 2) The clustering coefficient, C
• Version 1 the average local density. That is,
  Cv ego-network density, and C Cv/n
• Version 2 transitivity ratio. Number of closed
  triads divided by the number of closed and open
  triads.
• A small world graph is any graph with a
  relatively small L and a relatively large C.

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The most clustered graph is Watts Caveman
graph
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C and L as functions of k for a Caveman graph of
n1000
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Compared to random graphs, C is large and L is
long. The intuition, then, is that clustered
graphs tend to have (relatively) long
characteristic path lengths. But the small world
phenomenon rests on just the opposite high
clustering and short path distances. How is this
so?
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A model for pair formation, as a function of
mutual contacts.
Using this equation, a produces networks that
range from completely ordered (caveman-like) to
random.
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CLarge, L is Small SW Graphs
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Why does this work? Key is fraction of shortcuts
in the network
In a highly clustered, ordered network, a single
random connection will create a shortcut that
lowers L dramatically
Watts demonstrates that Small world graphs occur
in graphs with a small number of shortcuts
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1) Movie network Actors through Movies Lo/Lr
1.22 Co/Cr 2925 2) Western Power Grid
Lo/Lr 1.50 Co/Cr 16 3) C. elegans Lo/Lr
1.17 Co/Cr 5.6
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What are the substantive implications? Return
to the initial interest in connectivity disease
diffusion
1) Diseases move more slowly in highly clustered
graphs (fig. 11) - not a new finding. 2) The
dynamics are very non-linear -- with no clear
pattern based on local connectivity.
Implication small local changes (shortcuts) can
have dramatic global outcomes (disease diffusion)
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How do we know if an observed graph fits the SW
model?
Random expectations For basic one-mode networks
(such as acquaintance nets), we can get
approximate random values for L and C
as Lrandom ln(n) / ln(k) Crandom k /
n As k and n get large. Note that C essentially
approaches zero as N increases, and K is assumed
fixed. This formula uses the density-based
measure of C, but the substantive implications
are similar for the triad formula.
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How do we know if an observed graph fits the SW
model?
One problem with using the simple formulas for
most extant data on large graphs is that, because
the data result from people overlapping in
groups/movies/publications, necessary clustering
results from the assignment to groups.
G1 G2 G3 G4 G5 Amy 1 0 1 0
0 Billy 0 1 0 1 0 Charlie 0 1 0 1
0 Debbie 1 0 0 0 0 Elaine 1 0 1 0
1 Frank 0 1 0 1 0 George 0 1 0 1 0
. . . . LINES CUT . . . . . William 0 1 0
0 0 Xavier 0 1 0 1 0 Yolanda 1 0 1 0
0 Zanfir 0 1 1 1 1 12 14 9 14 5
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How do we know if an observed graph fits the SW
model?
Newman, M. E. J. Strogatz, S. J., and Watts, D.
J. Random Graphs with arbitrary degree
distributions and their applications Phys. Rev.
E. 2001
This paper extends the formulas for expected
clustering and path length using a generating
functions approach, making it possible to
calculate E(C,L) for graphs with any degree
distribution. Importantly, this procedure also
makes it possible to account for clustering in a
two-mode graph caused by the distribution of
assignment to groups.
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How do we know if an observed graph fits the SW
model?
Newman, M. E. J. Strogatz, S. J., and Watts, D.
J. Random Graphs with arbitrary degree
distributions and their applications Phys. Rev.
E. 2001
Where N is the size of the graph, Z1 is the
average number of people 1 step away (degree) and
Z2 is the average number of people 2 steps away.
Theoretically, these formulas can be used to
calculate many properties of the network
including largest component size, based on degree
distributions. A word of warning The math in
these papers is not simple, sharpen your calculus
pencil before reading the paper
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How do we know if an observed graph fits the SW
model?
Since C is just the transitivity ratio, there are
a number of good formulas for calculating the
expected value. Using the ratio of complete to
(incomplete complete) triads, we can use the
expected values from the triad distribution in
PAJEK for a simple graph or we can use the
expected value conditional on the dyad types (if
we have directed data) using the formulas in SPAN
and Wasserman and Faust (1994).
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• Other extensions of the SW model?
• Searchability on small worlds (Klienberg)
  suggests that the ability to walk through the
  graph requires a structure based on some knowable
  distance features. That is, people cant
  search a simple random SW graph.
• Graph Dynamics. The distance-shortening effects
  of shortcut ties are much less effective when the
  graph itself changes over time. To shorten the
  distance, structurally shortcut ties must also
  be temporally sequenced. This means that when
  relations are changing quickly, the rapid returns
  to shortcuts drops significantly.

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Standard result on a static graph
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Small World Mechanisms on Dynamic Graphs
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Across a large number of substantive settings,
Barabási points out that the distribution of
network involvement (degree) is highly and
characteristically skewed.
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Many large networks are characterized by a highly
skewed distribution of the number of partners
(degree)
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Complete Network Analysis Network Connections
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Many large networks are characterized by a highly
skewed distribution of the number of partners
(degree)
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The scale-free model focuses on the
distance-reducing capacity of high-degree nodes
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Complete Network Analysis Network Connections
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The scale-free model focuses on the
distance-reducing capacity of high-degree nodes,
as hubs create shortcuts that carry network
flow.
The diffusion implications of mathematical models
based on the preferential attachment model are
dim, because the carrying capacity of the network
comes to depend entirely on a vanishingly small
number of stars, who are statistically hard to
find. Thus, random treatment to the network does
no good, but targeted treatment does.
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The scale-free model focuses on the
distance-reducing capacity of high-degree nodes,
as hubs create shortcuts that carry network
flow.
The primary mechanism hypothesized to drive a
power-law degree distribution is the
preferential attachment model. This model
suggests that new nodes enter the population and
connect to current nodes with probability
proportional to the current nodes degree.
This implies that The rich get richer and the
graph takes on a decidedly star-like shape.
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• Critiques of the Scale-free model
• The insights are not particularly new, having
  been anticipated in the epidemiology of STDs for
  some time.
• Many of the empirical claims are over-stated.
• The most common test for a scale free network
  is to plot the degree histogram on a log-log
  scale and fit a regression line to it. This is
  poor statistical practice, and better models for
  fitting distributions show that most of the
  sexual networks are not, in fact, scale free (see
  Jones and Handcock, "Sexual contacts and epidemic
  thresholds" Nature, 423, 6940, 605-606)
• Theoretically, any degree-based metric has no
  necessary relation to the arrangement of ties
  within the network. That is, there are many
  graphs with identical degree distributions but
  very different topologies.
• Preferential attachment ? scale free, but not
  vice versa
• Finding a power-law degree distribution is really
  not that useful if there is any kind of blocking
  structure (focal aspects) to the network.

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Colorado Springs High-Risk (Sexual contact only)

• Network is approximately scale-free, with l
  -1.3
• But connectivity does not depend on the hubs.

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Social Subgroups
A primary interest in Social Network Analysis is
the identification of significant social
subgroups some smaller collection of nodes in
the graph that can be considered, at least in
some senses, as a unit based on the pattern,
strength, or frequency of ties. There are many
ways to identify groups. They all insist on a
group being in a connected component, but other
than that the variation is wide.
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• A) Graph theoretical methods Cliques and
  extensions of cliques
• Cliques
• k-cores
• k-plexes
• Freeman (1992) Models
• K-components
• B) Algorithmic methods search through a network
  trying to maximize for a particular pattern.
  Adjust assignment of actors to groups until a
  particular pattern of ties (block diagonal,
  usually) is identified.
• Standard models
• - Factions (UCI-NET)
• - NEGOPY (Richards)
• - KliqueFinder (Frank)
• - RNM, JIGGLE (Moody)
• - Betweeness Centrality (Newman)
• - General Distance Clustering Methods

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Graph Theoretical Models.
Start with a clique. A clique is defined as a
maximal subgraph in which every member of the
graph is connected to every other member of the
graph. Cliques are collections of nodes where
density 1.0.

• Properties of cliques
• Density 1.0
• Everyone connected to n-1 alters
• Distance between every pair is 1
• Ratio of within group ties to between group ties
  is infinite
• All triads are transitive

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Graph Theoretical Models.
In practice, complete cliques are not very
useful. They tend to overlap heavily and are
limited in their size.
Graph theorists have thus relaxed the complete
connectivity requirement (with varying degrees of
success). See the Moody White (2003) for a
discussion of these attempts.
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Graph Theoretical Models.
k-cores Every person connected to at least k
other people.
Ideally, they would look something like this
(here two 3-cores). However, adding a single
tie from A to B would make the whole graph a
3-core
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Extensions of this idea include
K-Core Every person has ties to at least k other
people in the set. K-plex Every member
connected to at least n-k other people in the
graph (recall in a clique everyone is connected
to n-1, so this relaxes that condition. n-clique
Every person is connected by a path of N or less
(recall a clique is with distance 1).
N-clan same as an n-clique, but all paths must
be inside the group. Ive never had much luck
with any of these methods empirically. Real
data is usually too messy to work well. Since
many of the graph-theoretic options seem not to
work well, authors have used optimization
techniques, that attempt to identify groups
iteratively.
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Algorithmic Approaches to Identifying Primary
Groups 1) Measures of fit To identify a
primary group, we need some measure of how
clustered the network is. Usually, this is a
function of the number of ties that fall within
group to the number of ties that fall between
group. 2.1) Processes designed to maximize
(1) Once we have such an index, we need a method
for searching through the network to maximize the
fit. 2.2) Generalized cluster analysis In
addition to maximizing a group function such as
(1) we can use the relational distance directly,
and look for clusters in the data.
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Segregation Index (Freeman, L. C. 1972.
"Segregation in Social Networks." Sociological
Methods and Research 6411-30.)
Freeman asked how we could identify segregation
in a social network. Theoretically, he argues,
if a given attribute (group label) does not
matter for social relations, then relations
should be distributed randomly with respect to
the attribute. Thus, the difference between the
number of cross-group ties expected by chance and
the number observed measures segregation.
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Consider the (hypothetical) network below. There
are two attributes in this network people with
Blue eyes and Brown eyes and people who are
square or not (they must be hip).
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Segregation Index
Mixing Matrix
Seg -0.25
Seg 0.78
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Complete Network Analysis Network Connections
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Segregation Index
To calculate the number of expected, we use the
standard formula for a contingency table Row
marginal column Marginal / Total
observed
Expected
Blue Brown Blue 6 17 23 Brown
17 16 33 23
33 56
Blue Brown Blue 9.45
13.55 23 Brown 13.55 19.45 33
23 33 56
In matrix form
E(X) RC/T
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Segregation Index
observed
Expected
Blue Brown Blue 6 17 23 Brown
17 16 33 23
33 56
Blue Brown Blue 9.45
13.55 23 Brown 13.55 19.45 33
23 33 56
E(X) (13.5513.55) X (1717) Seg
27.1 - 34 / 27.1 -6.9 / 27.1 -0.25
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Segregation Index
Observed
Expected
Hip Square Hip 20
3 23 Square 3 30 33 23 33
56
Hip Square Hip 9.45
13.55 23 Square 13.55 19.45 33
23 33 56
E(X) (13.5513.55) X (33) Seg
27.1 - 6 / 27.1 21.1 / 27.1 0.78
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Segregation Index
One problem with the segregation index is that it
is not margin free. That is, if you were to
change the distribution of the category of
interest (say race) by a constant but not the
core association between race and friendship
choice, you can get a different segregation
level. One antidote to this problem is to use
odds ratios. In this case, and odds ratio tells
us the relative likelihood that two people in the
same category will choose each other as friends.
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Odds Ratios
The odds ratio tells us how much more likely
people in the same group are to nominate each
other. You calculate the odds ratio based on the
number of ties in a group and their relative
size, based on the following table
Member of Same Group Different
Group Friends A
B Not Friends C D
OR AD/ BC
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Odds Ratios
There are 6 hip people and 9 square people in
this network. This implies that there are the
following number of possible ties in the network
Observed
Hip Square Hip 20
3 23 Square 3 30 33 23 33
56
Hip Square Hip 30
54 Square 54 72 Diagonal
ni(ni-1) off diagonal ni2
Group Same
Dif Yes 50 6 Friend
No 52 102
OR (50)102 / 52(6) 16.35
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Segregation index compared to the odds ratio
Friendship Segregation Index
r.95
Log(Same-Sex Odds Ratio)
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The second problem is that the Segregation index
has no clear maximum if every node is assigned
to a single group the value can be higher than if
everyone is assigned to the right group. This
means you cant just keep adjusting nodes until
you see a best fit, but instead have to look for
changes in fit. The modularity score solves this
problem by re-organizing the expectation in a way
that forces the value to 0 if everyone is in a
single group.
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We can also measure the extent that ties fall
within clusters with the modularity score
Where s indexes clusters in the network ls is
the number of lines in cluster s ds is the sum
of the degrees of s L is the total number of
lines
M has the advantage of going to 0 if there is
only 1 group, which means maximizing the score is
sensible
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Modularity Scores Comparison to Segregation Index
comparing values for known solutions
Modularity Score Plotted against Segregation
Index for various nets
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Modularity Scores Comparison to Segregation Index
comparing value of multiple solutions on the
same network
Number of groups ?
In-group Density ?
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Other groupedness indices include a) The ratio
of in-group to out-group ties (Negopy, UCINET
Factions) b) Maximizing the probability of
in-group contact (CliqueFinder) c) The
Segregation Matrix Index (SMI) d) The dyadic
factor loadings for overlapping groups (akin to a
latent class model) e) Minimize the within-group
distance Once a metric has been chosen, some
algorithm is needed to search through the graph
to identify clusters. These algorithms range
from very sophisticated graph-intelligent
algorithms, such as NEGOPY, to simple cluster
analysis of distance matrices. In most cases,
you have to pre-set the number of groups to use
(the exceptions are NEGOPY and CliqueFinder.
Moodys JIGGLE algorithm also has automatic
stopping criteria, but you have to give it
starting values.
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In practice, the different algorithms will give
different results. Here, I compare the NEGOPY
results to the RNM results. NEGOPY returned one
large group, RNM found many smaller, denser
groups. Its usually a good idea to explore
multiple solutions and algorithms.
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Gangon Prison Network
In practice, the different algorithms will give
different results. Here, I compare NEGOPY,
FACTIONS and RNM. Groups A and B are identical,
C is close. F, E and D differ. Its usually a
good idea to explore multiple solutions and
algorithms.
(all solutions constrained to 6 groups)
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Section of a School Friendship Network
In practice, the different algorithms will give
different results. Here, I compare NEGOPY,
FACTIONS and RNM with the k-connectivity
levels. Its usually a good idea to explore
multiple solutions and algorithms.
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Cluster analysis
In addition to tools like FACTIONS, we can use
the distance information contained in a network
to cluster observations that are close to each
other. In general, cluster analysis is a set of
techniques that allows you to identify
collections of objects that are simmilar to each
other in some degree. A very good reference is
the SAS/STAT manual section called, Introduction
to clustering procedures. (http//wks.uts.ohio-s
tate.edu/sasdoc/8/sashtml/stat/chap8/index.htm)
(See also Wasserman and Faust, though the
coverage is spotty). We are going to start with
the general problem of hierarchical clustering
applied to any set of analytic objects based on
similarity, and then transfer that to clustering
nodes in a network.
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Cluster analysis
Imagine a set of objects (say people) arrayed in
a two dimensional space. You want to identify
groups of people based on their position in that
space. How do you do it?
How Smart you are
How Cool you are
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Cluster analysis
Start by choosing a pair of people who are very
close to each other (such as 15 16) and now
treat that pair as one point, with a value equal
to the mean position of the two nodes.
x
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Cluster analysis
Now repeat that process for as long as possible.
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Cluster analysis
This process is captured in the cluster tree
(called a dendrogram)
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Cluster analysis

• As with the network cluster algorithms, there are
  many options for clustering. The three that I
  use most are
• Wards Minimum Variance -- the one I use almost
  95 of the time
• Average Distance -- the one used in the example
  above
• Median Distance -- very similar
• The SAS manual is the best single place Ive
  found for information on each of these
  techniques.
• Some things to keep in mind
• Units matter. The example above draws together
  pairs horizontally because the range there is
  smaller. Get around this by standardizing your
  data.
• This is an inductive technique. You can find
  clusters in a purely random distribution of
  points. Consider the following example.

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Cluster analysis
The data in this scatter plot are produced using
this code
data random do i1 to 20 xrannor(0)
yrannor(0) output end run
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Resulting dendrogram
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Resulting cluster solution
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Cluster analysis works by building a distance
matrix between each pair of points. In the
example above, it used the Euclidean distance
which in two dimensions is simply the physical
distance between the points in a plot. Can
work on any number of dimensions. To use
cluster analysis in a network, we base the
distance on the path-distance between pairs of
people in the network. Consider again the
blue-eye hip example
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Distance Matrix 0 1 3 2 3 3 4 3 3 2 3 2 2 1 1 1 0
2 2 2 3 3 3 2 1 2 2 1 2 1 3 2 0 3 2 4 3 3 2 1 1 1
2 2 3 2 2 3 0 1 1 2 1 1 2 3 3 3 2 1 3 2 2 1 0 2 1
1 1 1 2 2 3 3 2 3 3 4 1 2 0 1 1 2 3 4 4 4 3 2 4 3
3 2 1 1 0 2 2 2 3 3 4 4 3 3 3 3 1 1 1 2 0 1 2 3 3
4 3 2 3 2 2 1 1 2 2 1 0 1 2 2 3 3 2 2 1 1 2 1 3 2
2 1 0 1 1 2 2 2 3 2 1 3 2 4 3 3 2 1 0 1 2 2 3 2 2
1 3 2 4 3 3 2 1 1 0 1 1 2 2 1 2 3 3 4 4 4 3 2 2 1
0 2 2 1 2 2 2 3 3 4 3 3 2 2 1 2 0 1 1 1 3 1 2 2 3
2 2 2 3 2 2 1 0
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The distance matrix implies a space that nodes
are embedded within. Using something like MDS,
we can represent the space implied by the
distance matrix in two dimensions. This is the
image of the network you would get if you did
that.
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When you use variables, the cluster analysis
program generates a distance matrix. We can,
instead use the network distance matrix directly.
If we do that with this example network, we get
the following
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The one other program you should know about is
NEGOPY. Negopy is a program that combines
elements of the density based approach and the
graph theoretic approach to find groups and
positions. Like CROWDS, NEGOPY assigns people
both to groups and to outsider or between
group positions. Negopy also determines how
many groups are in the network, though in my
experience it often finds a single large group.
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The Recursive Neighborhood Means Jiggle
algorithms (Moody) create the variables that are
then used in the cluster analysis to identify
groups based on a simulated peer influence
process.

• Start by randomly assigning every node a random
  value on k variables
• Then calculate the average for each variable for
  the people each person is tied to
• Repeat this process many times
• ? This results in people who have many ties to
  each other having similar values on the k random
  variables. This similarity then gets picked up
  in a cluster analysis.

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Example of the RNM procedure
Time 3
Time 2
Time 1
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As an example, consider the process active on a
known-to-be clustered networks, starting with 2
random k variables. You get something like this,
where the nodes are now placed according to their
resulting values on the 2 variables.
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All of these techniques are inductive procedures.
It is possible to specify a deductive, test for
group membership, if you have a prior reasons to
assume that a particular set of people are a
group. The simplest way would be to specify a
dyadic model on the adjacency matrix, then model
the probability (strength) of a tie as a function
of dyadic characteristics and your indicator for
being in the same group. If this parameter is
large and significant, then you have evidence for
the group. This is, in fact, what KliqueFinder
does inductively.
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Role Positions

• Overview
• Social life can be described (at least in part)
  through social roles.
• To the extent that roles can be characterized by
  regular interaction patterns, we can summarize
  roles through common relational patterns.
• Identifying these sets is the goal of block-model
  analyses.
• Nadel The Coherence of Role Systems
• Background ideas for White, Boorman and Brieger.
  Social life as interconnected system of roles
• Important feature thinking of roles as connected
  in a role system social structure
• White, Harrison C. Boorman, Scott A., and
  Breiger, Ronald L. Social Structure from Multiple
  Networks I. American Journal of Sociology. 1976
  81730-780.
• The key article describing the theoretical and
  technical elements of block-modeling

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• Elements of a Role
• Rights and obligations with respect to other
  people or classes of people
• Roles require a role compliment another person
  who the role-occupant acts with respect to
• Examples
• Parent child
• Teacher student
• Lover lover
• Friend Friend
• Husband - Wife
• Nadel (Following functional anthropologists and
  sociologists) defines logical types of roles,
  and then examines how they can be linked together.

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Role Positions
White et al From logical role systems to
empirical social structures
Start with some basic ideas of what a role is
An exchange of something (support, ideas,
commands, etc) between actors. Thus, we might
represent a family as
H
W
C
C
C
Provides food for
(and there are, of course, many other relations
inside a family)
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The key idea, is that we can express a role
through a relation (or set of relations) and thus
a social system by the inventory of roles. If
roles equate to positions in an exchange system,
then we need only identify particular aspects of
a position. But what aspect? Structural
Equivalence
Two actors are structurally equivalent if they
have the same types of ties to the same people.
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Structural Equivalence
A single relation
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Structural Equivalence
Graph reduced to positions
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Blockmodeling basic steps
In any positional analysis, there are 4 basic
steps 1) Identify a definition of
equivalence 2) Measure the degree to which pairs
of actors are equivalent 3) Develop a
representation of the equivalencies 4) Assess
the adequacy of the representation
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1) Identify a definition of equivalence
Structural Equivalence Two actors are
equivalent if they have the same type of ties to
the same people.
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Automorphic Equivalence Actors occupy
indistinguishable structural locations in the
network. That is, that they are in isomorphic
positions in the network. Automorphically
equivalent nodes are equivalent with respect to
all graph theoretic properties (I.e. degree,
number of people reachable, centrality,
etc.) (Which suggests a simple way of using
cluster analyses to find these groups)
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Automorphic Equivalence
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Regular Equivalence Regular equivalence does not
require actors to have identical ties to
identical actors or to be structurally
indistinguishable. Actors who are regularly
equivalent have identical ties to and from
equivalent actors. If actors i and j are
regularly equivalent, then for all relations and
for all actors, if i k, then there exists
some actor l such that j l and k is regularly
equivalent to l.
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Regular Equivalence
There may be multiple regular equivalence
partitions in a network, and thus we tend to want
to find the maximal regular equivalence position,
the one with the fewest positions.
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Role or Local Equivalence While most
equivalence measures focus on position within the
full network, some measures focus only on the
patters within the local tie neighborhood. These
have been called local role equivalence.
Note that Structurally equivalent actors are
automorphically equivalent, Automorphically
equivalent actors are regularly
equivalent. Structurally equivalent and
automorphically equivalent actors are role
equivalent In practice, we tend to ignore some
of these distinctions, as they get blurred
quickly once we have to operationalize them in
real-world graphs. It turns out that few people
are ever exactly equivalent, and thus we
approximate the links between the types. In
all cases, the procedure can work over multiple
relations simultaneously. The process of
identifying positions is called blockmodeling,
and requires identifying a measure of similarity
among nodes.
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Once you identify equivalent actors, block them
in the matrix and reduce it, based on the number
of ties in the cell of interest. The key values
are a zero block (no ties) and a one-block (all
ties present)
1 2 3 4 5 6 1 0 1 1 0 0 0 2 1 0 0 1 0 0 3 1 0 1
0 1 0 4 0 1 0 1 0 1 5 0 0 1 0 0 0 6 0 0 0 1 0 0
1
2
3
4
5
6
1
. 1 1 1 0 0 0 0 0 0 0 0 0 0 1 . 0 0 1 1 0 0 0 0 0
0 0 0 1 0 . 1 0 0 1 1 1 1 0 0 0 0 1 0 1 . 0 0 1 1
1 1 0 0 0 0 0 1 0 0 . 1 0 0 0 0 1 1 1 1 0 1 0 0 1
. 0 0 0 0 1 1 1 1 0 0 1 1 0 0 . 0 0 0 0 0 0 0 0 0
1 1 0 0 0 . 0 0 0 0 0 0 0 0 1 1 0 0 0 0 . 0 0 0 0
0 0 0 1 1 0 0 0 0 0 . 0 0 0 0 0 0 0 0 1 1 0 0 0 0
. 0 0 0 0 0 0 0 1 1 0 0 0 0 0 . 0 0 0 0 0 0 1 1 0
0 0 0 0 0 . 0 0 0 0 0 1 1 0 0 0 0 0 0 0 .
2
3
4
5
6
Structural equivalence thus generates 6 positions
in the network
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Once you partition the matrix, reduce it
. 1 1 1 0 0 0 0 0 0 0 0 0 0 1 . 0 0 1 1 0 0 0 0 0
0 0 0 1 0 . 1 0 0 1 1 1 1 0 0 0 0 1 0 1 . 0 0 1 1
1 1 0 0 0 0 0 1 0 0 . 1 0 0 0 0 1 1 1 1 0 1 0 0 1
. 0 0 0 0 1 1 1 1 0 0 1 1 0 0 . 0 0 0 0 0 0 0 0 0
1 1 0 0 0 . 0 0 0 0 0 0 0 0 1 1 0 0 0 0 . 0 0 0 0
0 0 0 1 1 0 0 0 0 0 . 0 0 0 0 0 0 0 0 1 1 0 0 0 0
. 0 0 0 0 0 0 0 1 1 0 0 0 0 0 . 0 0 0 0 0 0 1 1 0
0 0 0 0 0 . 0 0 0 0 0 1 1 0 0 0 0 0 0 0 .
1 2 3 1 1 1 0 2 1 1 1 3 0 1 0
1
2
3
Regular equivalence
(here I placed a one in the image matrix if there
were any ties in the ij block)
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Operationally, you have to measure the similarity
between actors. If two actors are structurally
equivalent, then they will have identical ties to
other people. Consider the example again
C and D match on all 12 other people, and are
thus structurally equivalent.
1
2
3
4
5
6
C D Match 1 1 1 0 0 1 . 1 . 1 . . 0
0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1
1 0 0 1 0 0 1 0 0 1 0 0 1 Sum 12
1
. 1 1 1 0 0 0 0 0 0 0 0 0 0 1 . 0 0 1 1 0 0 0 0 0
0 0 0 1 0 . 1 0 0 1 1 1 1 0 0 0 0 1 0 1 . 0 0 1 1
1 1 0 0 0 0 0 1 0 0 . 1 0 0 0 0 1 1 1 1 0 1 0 0 1
. 0 0 0 0 1 1 1 1 0 0 1 1 0 0 . 0 0 0 0 0 0 0 0 0
1 1 0 0 0 . 0 0 0 0 0 0 0 0 1 1 0 0 0 0 . 0 0 0 0
0 0 0 1 1 0 0 0 0 0 . 0 0 0 0 0 0 0 0 1 1 0 0 0 0
. 0 0 0 0 0 0 0 1 1 0 0 0 0 0 . 0 0 0 0 0 0 1 1 0
0 0 0 0 0 . 0 0 0 0 0 1 1 0 0 0 0 0 0 0 .
2
3
4
5
6
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If the model is going to be based on asymmetric
or multiple relations, you simply stack the
various relations, usually including both
directions of asymmetric relations
Stacked
Romance 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
H
W
0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Feeds 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
C
C
C
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0
Romantic Love
Provides food for
Bicker 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0
0 1 1 0
Bickers with
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0
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The metric used to measure structural equivalence
by White, Boorman and Brieger is the correlation
between each nodes set of ties. For the
example, this would be
1.00 -0.20 0.08 0.08 -0.19 -0.19 0.77 0.77
0.77 0.77 -0.26 -0.26 -0.26 -0.26 -0.20 1.00
-0.19 -0.19 0.08 0.08 -0.26 -0.26 -0.26 -0.26
0.77 0.77 0.77 0.77 0.08 -0.19 1.00 1.00
-1.00 -1.00 0.36 0.36 0.36 0.36 -0.45 -0.45
-0.45 -0.45 0.08 -0.19 1.00 1.00 -1.00 -1.00
0.36 0.36 0.36 0.36 -0.45 -0.45 -0.45
-0.45 -0.19 0.08 -1.00 -1.00 1.00 1.00 -0.45
-0.45 -0.45 -0.45 0.36 0.36 0.36 0.36 -0.19
0.08 -1.00 -1.00 1.00 1.00 -0.45 -0.45 -0.45
-0.45 0.36 0.36 0.36 0.36 0.77 -0.26 0.36
0.36 -0.45 -0.45 1.00 1.00 1.00 1.00 -0.20
-0.20 -0.20 -0.20 0.77 -0.26 0.36 0.36 -0.45
-0.45 1.00 1.00 1.00 1.00 -0.20 -0.20 -0.20
-0.20 0.77 -0.26 0.36 0.36 -0.45 -0.45 1.00
1.00 1.00 1.00 -0.20 -0.20 -0.20 -0.20 0.77
-0.26 0.36 0.36 -0.45 -0.45 1.00 1.00 1.00
1.00 -0.20 -0.20 -0.20 -0.20 -0.26 0.77 -0.45
-0.45 0.36 0.36 -0.20 -0.20 -0.20 -0.20 1.00
1.00 1.00 1.00 -0.26 0.77 -0.45 -0.45 0.36
0.36 -0.20 -0.20 -0.20 -0.20 1.00 1.00 1.00
1.00 -0.26 0.77 -0.45 -0.45 0.36 0.36 -0.20
-0.20 -0.20 -0.20 1.00 1.00 1.00 1.00 -0.26
0.77 -0.45 -0.45 0.36 0.36 -0.20 -0.20 -0.20
-0.20 1.00 1.00 1.00 1.00
Another common metric is the Euclidean distance
between pairs of actors, which you then use in a
standard cluster analysis.
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The initial method for finding structurally
equivalent positions was CONCOR, the CONvergence
of iterated CORrelations.
Concor iteration 1
1.00 -.77 0.55 0.55 -.57 -.57 0.95 0.95 0.95 0.95
-.75 -.75 -.75 -.75 -.77 1.00 -.57 -.57 0.55 0.55
-.75 -.75 -.75 -.75 0.95 0.95 0.95 0.95 0.55 -.57
1.00 1.00 -1.0 -1.0 0.73 0.73 0.73 0.73 -.75 -.75
-.75 -.75 0.55 -.57 1.00 1.00 -1.0 -1.0 0.73 0.73
0.73 0.73 -.75 -.75 -.75 -.75 -.57 0.55 -1.0 -1.0
1.00 1.00 -.75 -.75 -.75 -.75 0.73 0.73 0.73
0.73 -.57 0.55 -1.0 -1.0 1.00 1.00 -.75 -.75 -.75
-.75 0.73 0.73 0.73 0.73 0.95 -.75 0.73 0.73 -.75
-.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77 0.95
-.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77
-.77 -.77 -.77 0.95 -.75 0.73 0.73 -.75 -.75 1.00
1.00 1.00 1.00 -.77 -.77 -.77 -.77 0.95 -.75 0.73
0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77
-.77 -.75 0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77
-.77 1.00 1.00 1.00 1.00 -.75 0.95 -.75 -.75 0.73
0.73 -.77 -.77 -.77 -.77 1.00 1.00 1.00 1.00 -.75
0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77 -.77 1.00
1.00 1.00 1.00 -.75 0.95 -.75 -.75 0.73 0.73 -.77
-.77 -.77 -.77 1.00 1.00 1.00 1.00
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The initial method for finding structurally
equivalent positions was CONCOR, the CONvergence
of iterated CORrelations.
Concor iteration 2
1.00 -.99 0.94 0.94 -.94 -.94 0.99 0.99 0.99 0.99
-.99 -.99 -.99 -.99 -.99 1.00 -.94 -.94 0.94 0.94
-.99 -.99 -.99 -.99 0.99 0.99 0.99 0.99 0.94 -.94
1.00 1.00 -1.0 -1.0 0.97 0.97 0.97 0.97 -.97 -.97
-.97 -.97 0.94 -.94 1.00 1.00 -1.0 -1.0 0.97 0.97
0.97 0.97 -.97 -.97 -.97 -.97 -.94 0.94 -1.0 -1.0
1.00 1.00 -.97 -.97 -.97 -.97 0.97 0.97 0.97
0.97 -.94 0.94 -1.0 -1.0 1.00 1.00 -.97 -.97 -.97
-.97 0.97 0.97 0.97 0.97 0.99 -.99 0.97 0.97 -.97
-.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99 0.99
-.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99
-.99 -.99 -.99 0.99 -.99 0.97 0.97 -.97 -.97 1.00
1.00 1.00 1.00 -.99 -.99 -.99 -.99 0.99 -.99 0.97
0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99
-.99 -.99 0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99
-.99 1.00 1.00 1.00 1.00 -.99 0.99 -.97 -.97 0.97
0.97 -.99 -.99 -.99 -.99 1.00 1.00 1.00 1.00 -.99
0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99 -.99 1.00
1.00 1.00 1.00 -.99 0.99 -.97 -.97 0.97 0.97 -.99
-.99 -.99 -.99 1.00 1.00 1.00 1.00
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The initial method for finding structurally
equivalent positions was CONCOR, the CONvergence
of iterated CORrelations.
Concor iteration 3
1.00 -1.0 1.00 1.00 -1.0 -1.0 1.00 1.00 1.00 1.00
-1.0 -1.0 -1.0 -1.0 -1.0 1.00 -1.0 -1.0 1.00 1.00
-1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00 1.00 -1.0
1.00 1.00 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0 -1.0
-1.0 -1.0 1.00 -1.0 1.00 1.00 -1.0 -1.0 1.00 1.00
1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 -1.0 -1.0
1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00
1.00 -1.0 1.00 -1.0 -1.0 1.00 1.00 -1.0 -1.0 -1.0
-1.0 1.00 1.00 1.00 1.00 1.00 -1.0 1.00 1.00 -1.0
-1.0 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00
-1.0 1.00 1.00 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0
-1.0 -1.0 -1.0 1.00 -1.0 1.00 1.00 -1.0 -1.0 1.00
1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00 -1.0 1.00
1.00 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0
-1.0 -1.0 1.00 -1.0 -1.0 1.00 1.00 -1.0 -1.0 -1.0
-1.0 1.00 1.00 1.00 1.00 -1.0 1.00 -1.0 -1.0 1.00
1.00 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0
1.00 -1.0 -1.0 1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00
1.00 1.00 1.00 -1.0 1.00 -1.0 -1.0 1.00 1.00 -1.0
-1.0 -1.0 -1.0 1.00 1.00 1.00 1.00
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The initial method for finding structurally
equivalent positions was CONCOR, the CONvergence
of iterated CORrelations.
Concor iteration 3 (permuted)
1.00 1.00 1.00 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0
-1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00 1.00 1.00
1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00
1.00 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0
-1.0 -1.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 -1.0
-1.0 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00
1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0
-1.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 -1.0 -1.0
-1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00 1.00
1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0
-1.0 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00
1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0
1.00 1.00 1.00 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0
-1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00 1.00 1.00
1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00
1.00 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0
-1.0 -1.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 -1.0
-1.0 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00
1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0
1.00 1.00 1.00 1.00 1.00 1.00 1.00
1 3 4 7 8 9 10 2 5 6 11 12 13 14
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Repeat the process on the resulting 1-blocks
until you have reached structural equivalent
blocks
Because CONCOR splits every sub-group into two
groups, you get a partition tree that looks
something like this
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Automorphic and Regular equivalence are more
difficult to find, and require iteratively
searching over possible class assignments for
sets that have the same graph theoretic patterns.
Usually start with a set of nodes defined as
similar on a number of network measures, then
look within these classes for automorphic
equivalence classes. A theoretically appealing
method for finding structures that are very
similar to regular equivalence, role equivalence,
uses the triad census. Each node is involved in
(n-1)(n-2)/2 triads, and occupies a particular
position in each of these triads.
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Triadic Position Census 36 Positions within 16
Directed Triads
Indicates the position.
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36 36 10 10 10 10 43 43 43 43 43 43 43 43 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 20 20 41 41 41 41 14 14 14 14
14 14 14 14 9 9 11 11 11 11 12 12 12 12 12 12
12 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 10 10 1 1 1 1 8
8 8 8 8 8 8 8 2 2 10 10 10 10 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 5 5
5 5 1 1 1 1 1 1 1 1
Triad position vectors for the example network,
resulting in 3 positions
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Correlating each persons triad position vector
with each other persons results in the following
table, which clearly shows the positions that are
equivalent
1.00 1.00 0.64 0.64 0.64 0.64 0.98 0.98 0.98 0.98
0.98 0.98 0.98 0.98 1.00 1.00 0.64 0.64 0.64 0.64
0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.64 0.64
1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.50 0.50
0.50 0.50 0.64 0.64 1.00 1.00 1.00 1.00 0.50 0.50
0.50 0.50 0.50 0.50 0.50 0.50 0.64 0.64 1.00 1.00
1.00 1.00 0.50 0.50 0.50 0.50 0.50 0.50 0.50
0.50 0.64 0.64 1.00 1.00 1.00 1.00 0.50 0.50 0.50
0.50 0.50 0.50 0.50 0.50 0.98 0.98 0.50 0.50 0.50
0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.98
0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 0.98 0.98 0.50 0.50 0.50 0.50 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.98 0.98 0.50
0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 0.98 0.98 0.50 0.50 0.50
0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.98
0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 0.98 0.98 0.50 0.50 0.50 0.50 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00
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Triadic Position Census 40 Positions within all
on two types of mutual ties
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Moving from a similarity/distance matrix to a
blockmodel number of groups and determining
blocks An important decision in an analysis
using CONCOR is how fine the partition should be
in other words, when should one stop splitting
positions? Theory and the interpretability of
the solution are the primary consideration in
deciding how many positions to produce. (WF,
p.378) In defining positions of actors, the
trick is to choose the point along the series
that gives a useful and interpretable partition
of the actors into equivalence classes. (WF
p.383)
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Once you have decided on a number of blocks, you
need to determine what counts as a one block or
a zero block. Usually this is a some
function of the density of the resulting
block. General rules Fat Fit Only put a one
in blocks with all ones in the adjacency
matrix Lean Fit Put a zero if all the cells
are zero, else put a one Density fit If the
average value of the cell is above a certain
cutoff. White, Boorman and Breiger used a lean
fit (zeroblock) rule for the examples in their
paper
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Most common block structures identified in Add
Health
Based on CONCOR, imposing a 5-block fit
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An example Padgett, J. F. and Ansell, C. K.
Robust action and the rise of the Medici,
1400-1434. American Journal of Sociology. 1993
981259-1319. Political Groups in the attribute
sense do not seem to exist, so PA turn to the
pattern of network relations among
families. This is the block reduction of the
full 92 family network.
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An example based on regular equivalence using the
Add Health data.
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Jefferson High School
Sunshine High School
School provides a good boundary for social
relations
School does not provide a good boundary for
social relations
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64
1 Outsiders
Jefferson High School
276
Semi- periphery
212
2 Insiders
544
3 Outsiders
147
Periphery
268
4 Insiders
121
5 Aloof
80
6 Most Popular
87
Core
288
7 2nd String
121
Wards minimum variance clustering on the 36
triad types, plus number of ties sent
out-of-school
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253
Sunshine High School
Receiving Periphery
432
179
903
Sending Periphery
421
471
50
Semi- Periphery
255
742
Lieutenants
332
818
487
102
155
53
76
Core
Wards minimum variance clustering on the 36
triad types, plus number of ties sent
out-of-school
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Jefferson High School
Sunshine High School
4
34
43
32
52
33
Image networks. Width of tie is proportional to
the ratio of cell density to mean cell density.
110
 
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Jefferson High School
Sunshine High School
 
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Jefferson High School
Sunshine High School
112
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Role Positions
Jefferson High School

• Being in the same block significantly increases
  the likelihood of being the same behavioral
  cluster
• Locally defined OR 1.13
• Globally defined OR 1.12
• The effect is differential across blocks
• Being adjacent in the network has a consistent
  positive effect
• Local OR 1.21
• Global OR 1.35

Coefficients based on a dyad-level logistic
regression model. Models control for grade,
gender and SES.
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Sunshine High School

• Being in the same block barely increases the
  likelihood of being the same behavioral cluster
• Locally defined OR 1.03
• Globally define OR 1.02
• The effect is differential across blocks
• Being adjacent in the network has a weaker, but
  still positive effect
• Local 1.13
• Global 1.08

Coefficients based on a dyad-level logistic
regression model. Models control for grade,
race, gender SES.
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Role Positions
Compound Relations
One of the most powerful tools in role analysis
involves looking at role systems through compound
relations. A compound relation is formed by
combining relations in single dimensions. The
best example of compound relations come from
kinship.
Nephew/Niece 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
Sibling 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 0
Child of 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0

x
Sibling
Child of
S?C SC
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Complete Network Analysis Network Connections
Role Positions
An example of compound relations can be found in
WF. This role table catalogues the compounds
for two relations Is boss of and Is on the
same level as
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• The newest work in block modeling comes from
  Doreian, Batagelj, and Ferligoj, who have
  proposed a system for generalized block models
• Instead of having blocks composed of zeros or
  ones, you can specify the type of relation within
  and b