Complete Network Analysis

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Complete Network Analysis Exploratory Analysis
Social Networks capture the relations between
people. These relations form a system that can
be thought of as a social space. The advantage
of the space analogy is that it captures the
topography of social networks classes,
clusters, distance, centrality, etc. The
disadvantage is that spaces and fields are
notoriously difficult to study, because key
features are simultaneously active. Current
calls for relational sociology make this point
clearly (See Martin 2003, Abbott 2001). Field
serves as some sort of representation for those
overarching social regularities that may also be
visualized as quasi-organisms, systems or
structures J. L. Martin AJS 2003. Examples of
fields range from abstract notions of status
spaces to concrete examples such as the French
academic system.
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Complete Network Analysis Exploratory Analysis
Bourdieu Social Space and Symbolic Space
Sociologists often use spatial analogies, such as
MDS or correspondence analysis, based on patterns
of actor attributes. Social Network Analysis
lets you explore the relational space directly,
by mapping relations directly. The first step in
this exploration is often visualizing the network.
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Complete Network Analysis Exploratory Analysis
Network visualization
Network visualization helps build intuition, but
you have to keep the drawing algorithm in mind
Spring-embeder layouts
Tree-Based layouts
Most effective for very sparse, regular graphs.
Very useful when relations are strongly directed,
such as organization charts, internet connections,
Most effective with graphs that have a strong
community structure (clustering, etc). Provides
a very clear correspondence between social
distance and plotted distance
Two images of the same network
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Complete Network Analysis Exploratory Analysis
Network visualization
Network visualization helps build intuition, but
you have to keep the drawing algorithm in mind
Spring-embeder layouts
Tree-Based layouts
Two images of the same network
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Complete Network Analysis Exploratory Analysis
Network visualization
Network visualization helps build intuition, but
you have to keep the drawing algorithm in
mind. Hierarchy Tree models Use optimization
routines to add meaning to the Y-axis of the
plot. This makes it possible to easily see who
is most central because of who is on the top of
the figure. Usually includes some routine for
minimizing line-crossing. Spring Embedder
layouts Work on an analogy to a physical system
ties connecting a pair have springs that pull
them together. Unconnected nodes have springs
that push them apart. The resulting image
reflects the balance of these two features. This
usually creates a correspondence between physical
closeness and network distance.
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Complete Network Analysis Exploratory Analysis
Network visualization
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Complete Network Analysis Exploratory Analysis
Network visualization
Using colors to code attributes makes it simpler
to compare attributes to relations. Here we can
assess the effectiveness of two different
clustering routines on a school friendship
network.
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Complete Network Analysis Exploratory Analysis
Network visualization
Using colors to code attributes makes it simpler
to compare attributes to relations. Here color
size are used to express node characteristics.
Trade among OECD Countries in 1981
http//www.mpi-fg-koeln.mpg.de/lk/netvis/trade/W
orldTrade.html
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Complete Network Analysis Exploratory Analysis
Network visualization
Using colors to code attributes makes it simpler
to compare attributes to relations. Here color
size are used to express node characteristics.
Trade among OECD Countries in 1981
http//www.mpi-fg-koeln.mpg.de/lk/netvis/trade/W
orldTrade.html
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Complete Network Analysis Exploratory Analysis
Network visualization
Using colors to code attributes makes it simpler
to compare attributes to relations. Here
clusters are abstracted from the nodes then
colored based on functional elements.
NATURE VOL 433 24 FEBRUARY 2005
www.nature.com/nature
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Complete Network Analysis Exploratory Analysis
Network visualization
As networks increase in size, the effectiveness
of a point-and-line display diminishes, because
you simply run out of plotting dimensions. Ive
found that you can still get some insight by
using the overlap that results in from a
space-based layout as information. Here you see
the clustering evident in movie co-staring for
about 8000 actors.
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Complete Network Analysis Exploratory Analysis
Network visualization
As networks increase in size, the effectiveness
of a point-and-line display diminishes, because
you simply run out of plotting dimensions. Ive
found that you can still get some insight by
using the overlap that results in from a
space-based layout as information. This figure
contains over 29,000 social science authors. The
two dense regions reflect different topics.
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Where does sociology fit?
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Where does sociology fit?
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Where does sociology fit?
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Complete Network Analysis Exploratory Analysis
Network visualization
Adding time to social networks is also
complicated, as you run out of space to put time
in most network figures. One solution is to
animate the network. Here we see streaming
interaction in a classroom, where the teacher
(yellow square) has trouble maintaining
order. The SONIA software program (McFarland and
Bender-deMoll) will produce these figures.
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Complete Network Analysis Exploratory Analysis
Network visualization
Adding time to social networks is also
complicated, as you run out of space to put time
in most network figures. One solution is to
animate the network. When the network is very
sparse, sometimes it makes more sense to build
layers in a flipbooks style
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Complete Network Analysis Exploratory Analysis
Network visualization
Data on drug users in Colorado Springs, over 5
years
Drug Relations, Colorado Springs, Year 1
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Complete Network Analysis Exploratory Analysis
Network visualization
Data on drug users in Colorado Springs, over 5
years
Drug Relations, Colorado Springs, Year 2 Current
year in red, past relations in gray
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Complete Network Analysis Exploratory Analysis
Network visualization
Data on drug users in Colorado Springs, over 5
years
Drug Relations, Colorado Springs, Year 3 Current
year in red, past relations in gray
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Complete Network Analysis Exploratory Analysis
Network visualization
Data on drug users in Colorado Springs, over 5
years
Drug Relations, Colorado Springs, Year 4 Current
year in red, past relations in gray
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Complete Network Analysis Exploratory Analysis
Network visualization
Data on drug users in Colorado Springs, over 5
years In general, adding time to networks
changes many of our notions of structure
which well go through in detail later.
Drug Relations, Colorado Springs, Year 5 Current
year in red, past relations in gray
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Complete Network Analysis Exploratory Analysis
Network visualization
Visualization is a tool, but networks are complex
and our visualization tools can sometimes
confound. The strong advantage is that you get a
complete overview of multiple features at once.
The difficulty comes with trying to map a complex
multi-dimensional object in low-dimensional
space. Here we use a hierarchy to trace
diffusion from 10 seed nodes, but display in two
formats.
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Complete Network Analysis Network Connections
Goods flow through networks
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Complete Network Analysis Network Connections

• We often care about networks because of how
  goods travel through the network.
• In addition to the simple pairwise probability
  that one actor passes information on to another
  (pij), two factors affect flow through a network
• Topology
• the shape, or form, of the network
• - Example one actor cannot pass information to
  another unless they are either directly or
  indirectly connected
• Time
• - the timing of contact matters
• - Example an actor cannot pass information he
  has not receive yet

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Complete Network Analysis Network Connections
Topology
Two features of the networks topology are known
to be important connectivity and centrality

• Connectivity refers to how actors in one part of
  the network are connected to actors in another
  part of the network.
• Reachability Is it possible for actor i to
  reach actor j? This can only be true if there is
  a chain of contact from one actor to another.
• Distance Given they can be reached, how many
  steps are they from each other?
• Number of paths How many different paths
  connect each pair?

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Complete Network Analysis Network Connections
Topology
Without full network data, you cant distinguish
actors with limited flow potential from those
more deeply embedded in a setting.
c
b
a
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Complete Network Analysis Network Connections
Connectivity
Indirect connections are what make networks
systems. One actor can reach another if there is
a path in the graph connecting them.
a
b
d
a
c
e
f
Paths can be directed, leading to a distinction
between strong and weak components
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Complete Network Analysis Network Connections
Connectivity

• Basic elements in connectivity
• A path is a sequence of nodes and edges starting
  with one node and ending with another, tracing
  the indirect connection between the two. On a
  path, you never go backwards or revisit the same
  node twice.
• Example a ? b ? c?d
• A walk is any sequence of nodes and edges, and
  may go backwards. Example a ? b ? c ? b ?c ?d
• A cycle is a path that starts and ends with the
  same node. Example a ? b ? c ? a

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Complete Network Analysis Network Connections
Connectivity
Reachability
If you can trace a sequence of relations from one
actor to another, then the two are reachable. If
there is at least one path connecting every pair
of actors in the graph, the graph is connected
and is called a component. Intuitively, a
component is the set of people who are all
connected by a chain of relations.
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Complete Network Analysis Network Connections
Connectivity
This example contains many components.
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Complete Network Analysis Network Connections
Connectivity
Because relations can be directed or undirected,
components come in two flavors For a graph with
any directed edges, there are two types of
components Strong components consist of the
set(s) of all nodes that are mutually
reachable Weak components consist of the set(s)
of all nodes where at least one node can reach
the other.
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Complete Network Analysis Network Connections
Connectivity
There are only 2 strong components with more than
1 person in this network. Components are the
minimum requirement for social groups. As we
will see later, they are necessary but not
sufficient
All of the major network analysis software
identifies strong and weak components
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Complete Network Analysis Network Connections
Distance
Geodesic distance is measured by the smallest
(weighted) number of relations separating a pair
Actor a is 1 step from 4 2 steps from 5
3 steps from 4 4 steps from 3 5 steps from 1
a
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Complete Network Analysis Network Connections
Distance
When the graph is directed, distance is also
directed (distance to vs distance from),
following the direction of the tie.
a b c d e f g h i j k l
m ------------------------------------------ a.
. 1 2 . . . . . . . . 2 1 b. 3 .
1 . . . . . . . . 1 2 c. . . . .
. . . . . . . . . d. 4 3 1 . 1 2
1 . 2 . . 2 3 e. 3 2 2 1 . 1 2 .
1 . . 1 2 f. 4 3 3 2 1 . 3 . 2 .
. 2 3 g. 5 4 4 3 2 1 . . 3 . . 3
4 h. . . . . . . . . 1 . . . . i.
. . . . . . . . . . . . . j. . .
. . . . . . 1 . . . . k. . . . .
. . . . 1 . . . . l. 2 1 2 . . .
. . . . . . 1 m. 1 2 3 . . . . .
. . . 1 .
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Complete Network Analysis Network Connections
Distance

• High-risk actors over 4 years
• 695 people represented
• Longest path is 17 steps
• Average distance is about 5 steps
• Average person is within 3 steps of 75 other
  people
• 137 people connected through 2 independent paths,
  core of 30 people connected through 4 independent
  paths

Reachability in Colorado Springs (Sexual contact
only)
(Node size log of degree)
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Complete Network Analysis Network Connections
Distance
Calculating distance in global networks Powers
of the adjacency matrix
Calculate reachability through matrix
multiplication. (see p.162 of WF)
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Complete Network Analysis Network Connections
Distance
Calculating distance in global networks
Breadth-First Search
In large networks, matrix multiplication is just
too slow. A breadth-first search algorithm works
by walking through the graph, reaching all nodes
from a particular start node.
Distance is calculated directly in most SNA
software packages.
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Complete Network Analysis Network Connections
Distance
As a graph statistic, the distribution of
distance can tell you a good deal about how close
people are to each other (well see this more
fully when we get to closeness centrality). The
diameter of a graph is the longest geodesic,
giving the maximum distance. We often use the l,
or the mean distance between every pair to
characterize the entire graph. For example, all
else equal, we would expect rumors to travel
faster through settings where the average
distance is small.
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Complete Network Analysis Network Connections
Distance
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Complete Network Analysis Network Connections
Distance
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Complete Network Analysis Network Connections
Distance
Travers and Milgrams work on the small world is
responsible for the standard belief that
everyone is connected by a chain of about 6
steps.
Two questions Given what we know about networks,
what is the longest path (defined by handshakes)
that separates any two people? Is 6 steps a
long distance or a short distance?
43
What if everyone maximized structural holes?
Associates do not know each other Results in an
exponential growth curve. Reach entire planet
quickly.
Complete Network Analysis Network Connections
Distance
44
What if people know each other randomly? Random
graph theory shows that we could reach people
quite quickly if ties were random
Complete Network Analysis Network Connections
Distance
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Complete Network Analysis Network Connections
Distance
Random Reachability By number of close friends
100
Degree 4
Degree 3
80
Degree 2
60
Percent Contacted
40
20
0
0
1
2
3
4
5
6
7
8
9
10
11
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Remove
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Complete Network Analysis Network Connections
Distance
Distance-Reach Distribution for a large Jr. High
School (Add Health data)
Random graph
Observed
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Complete Network Analysis Network Connections
Distance
Milgrams test Send a packet from sets of
randomly selected people to a stockbroker in
Boston.
Experimental Setup
Arbitrarily select people from 3 pools a)
People in Boston b) Random in Nebraska c)
Stockholders in Nebraska
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Complete Network Analysis Network Connections
Distance
Milgrams Findings Distance to target person,
by sending group.
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Complete Network Analysis Network Connections
Distance
Most chains found their way through a small
number of intermediaries.
Understanding why this is true has been called
the Small-World Problem, which has since been
generalized to a much more formal understanding
of tie patterns in large networks (see
below) For purposes of flow through graphs,
distance is a primary concern so long as pij lt 1.
Most measures of position in a network account
for some aspect of distance.
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Complete Network Analysis Network Connections
Connectivity
We can extend our conception of component to
increase the structural cohesion of the
definition. Multiple connectivity Two paths
with the same start and end point, but that have
no other nodes in common are called node
independent. In every component, the paths
linking actors i and j must pass through a set of
nodes, S, that if removed would disconnect the
graph. The number of nodes in the smallest S is
equal to the number of independent paths
connecting i and j.
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Complete Network Analysis Network Connections
Connectivity
Simple component
Every path from 1 to 8 must go through 4. S(1,8)
4, and N(1,8)1. That is, the graph is a
component.
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Complete Network Analysis Network Connections
Connectivity
In this graph, there are multiple paths
connecting nodes 1 and 8.
Multiple connectivity
1
But only 2 of them are independent.
5
2
3
4
8
1
1
6
6
2
5
4
7
7
3
8
6
5
8
8
N(1,8) 2.
7
8
8
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Complete Network Analysis Network Connections
Connectivity
A bicomponent is the set of all nodes connected
by at least 2 node-independent paths.
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Complete Network Analysis Network Connections
Connectivity
Bicomponents can overlap by at most 1 person.
These nodes are cutpoints in the graph. If that
node is removed, the graph would be disconnected.
1
4 is a cutpoint
1 is a cutpoint
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Complete Network Analysis Network Connections
Connectivity
White, D. R. and F. Harary. 2001. "The
Cohesiveness of Blocks in Social Networks Node
Connectivity and Conditional Density."
Sociological Methodology 31305-59. Moody,
James and Douglas R. White. 2003. Structural
Cohesion and Embeddedness A hierarchical
Conception of Social Groups American
Sociological Review 68103-127 White, Douglas
R., Jason Owen-Smith, James Moody, Walter W.
Powell (2004) "Networks, Fields, and
Organizations Scale, Topology and Cohesive
Embeddings."  Computational and Mathematical
Organization Theory. 1095-117 Moody, James "The
Structure of a Social Science Collaboration
Network Disciplinary Cohesion from 1963 to 1999"
American Sociological Review. 69213-238

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Complete Network Analysis Network Connections
Connectivity
Analytically, most of work on connectivity has
focused on summaries of completely local
properties (degree distributions or clustering).
We turn the argument around and ask what features
of a network are essential for holding the whole
structure together?
Def. 1 A collectivity is cohesive to the
extent that the social relations of its members
hold it together.
What network pattern embodies all the elements of
this intuitive definition?
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Complete Network Analysis Network Connections
Connectivity

• This definition contains 5 essential elements
• Focuses on what holds the group together
• Expressed as a group level property
• The conception is continuous
• Rests on observable social relations
• Applies to groups of any size

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Complete Network Analysis Network Connections
Connectivity
1) Actors must be connected a collection of
isolates is not cohesive.
Minimally cohesive a single path connects
everyone
Not cohesive
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Complete Network Analysis Network Connections
Connectivity
1) Reachability is an essential element of
relational cohesion. As more paths re-link
actors in the group, the ability to hold
together increases.
The important feature is not the density of
relations, but the pattern.
Cohesion increases as of paths connecting
people increases
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Complete Network Analysis Network Connections
Connectivity
Consider the minimally cohesive group
Moving a line keeps density constant, but changes
reachability.
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Complete Network Analysis Network Connections
Connectivity
What if density increases, but through a single
person?
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Complete Network Analysis Network Connections
Connectivity
Cohesion increases as the number of independent
paths in the network increases. Ties through a
single person are minimally cohesive.
D . 39 More cohesive
D . 39 Minimal cohesion
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Complete Network Analysis Network Connections
Connectivity
Substantive differences between networks
connected through a single actor and those
connected through many.
Minimally Cohesive Strongly Cohesive Power is
centralized Power is decentralized Information
is concentrated Information is
distributed Expect actor inequality Actor
equality Vulnerable to unilateral action Robust
to unilateral action Segmented structure Even
structure
Def 2. A group is structurally cohesive to the
extent that multiple independent relational paths
among all pairs of members hold it together.
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Complete Network Analysis Network Connections
Connectivity
Def 2. A group is structurally cohesive to the
extent that multiple independent relational paths
among all pairs of members hold it together.
0
1
2
3
Node Connectivity
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Complete Network Analysis Network Connections
Connectivity
Formalize the argument
If there is a path between every node in a graph,
the graph is connected, and called a
component. In every component, the paths linking
actors i and j must pass through a set of nodes,
S, that if removed would disconnect the graph.
The number of nodes in the smallest S is equal
to the number of independent paths connecting i
and j.
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Complete Network Analysis Network Connections
Connectivity
The relation between cut-set size and number of
paths (recall our discussion of bicomponents)
leads to the two versions of our final
definition
Def 3a A groups structural cohesion is equal
to the minimum number of actors who, if removed
from the group, would disconnect the group. Def
3b A groups structural cohesion is equal to
the minimum number of independent paths linking
each pair of actors in the group.
These two definitions are equivalent.
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Complete Network Analysis Network Connections
Connectivity
Some graph theoretic properties of k-components
1) Every member of a k-components must have at
least k-ties. If a person has less than k ties,
then there would be fewer than k paths connecting
them to the rest of the network. 2) A graph
where every person has k-ties is not necessarily
a k-component. That is, (1) does not work in
reverse. Structures can have high degree, but
low connectivity. 3) Two k-components can only
overlap by k-1 members. If the k-components
overlap by more than k-1 members, then there
would be at least k paths connecting the two
components, and they would be a single
k-component. 4) A clique is n-1 connected. 5)
k-components can be nested, such that a kl
component is contained within a k-component.
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Complete Network Analysis Network Connections
Connectivity
Nested connectivity sets An operationalization
of embeddedness.
2
3
1
9
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11
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14
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Complete Network Analysis Network Connections
Connectivity
Embeddedness refers to the fact that economic
action and outcomes, like all social action and
outcomes, are affected by actors dyadic
(pairwise) relations and by the structure of the
overall network of relations. As a shorthand, I
will refer to these as the relational and the
structural aspects of embeddedness. The
structural aspect is especially crucial to keep
in mind because it is easy to slip into dyadic
atomization, a type of reductionism. (Granovetter
199233, italics in original)
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Complete Network Analysis Network Connections
Connectivity
G
7,8,9,10,11 12,13,14,15,16
1, 2, 3, 4, 5, 6, 7, 17, 18, 19, 20, 21,
22, 23
7, 8, 11, 14
1,2,3,4, 5,6,7
17, 18, 19, 20, 21, 22, 23
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Complete Network Analysis Network Connections
Connectivity
Empirical Examples a) Embeddedness and School
Attachment b) Political similarity among Large
American Firms
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Complete Network Analysis Network Connections
Connectivity
School Attachment
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Complete Network Analysis Network Connections
Connectivity
Business Political Action
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Complete Network Analysis Network Connections
Connectivity

• Theoretical Implications
• Resource and Risk Flow
• Structural cohesion increases the probability of
  diffusion in a network, particularly if flow
  depends on individual behavior (as opposed to
  edge capacity).

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Complete Network Analysis Network Connections
Connectivity
Structural Cohesion also provides a new way of
thinking about STD cores
Project 90, Sex-only network (n695)
3-Component (n58)
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Complete Network Analysis Network Connections
Connectivity
Connected Bicomponents
IV Drug Sharing Largest BC 247 k gt 4 318 Max k
12 Structural Cohesion simultaneously gives us a
positional and subgroup analysis.
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Complete Network Analysis Network Connections
Connectivity
Development of STD cores in low-degree networks
rapid transition without stars.
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Complete Network Analysis Network Connections
Connectivity
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
Probability of transfer
by distance and number of paths, assume a
constant pij of 0.6
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
Clustering and diffusion
Arcs 11 Largest component 12, Clustering 0
Arcs 11 Largest component 8, Clustering 0.205
Clustering turns network paths back on already
identified nodes. This has been well known since
at least Rappaport, and is a key feature of the
Biased Network models in sociology.
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
Define as a general measure of the diffusion
susceptibility of a graph as the ratio of the
area under the observed curve to the area under
the random curve. As this gets smaller than 1.0,
you get effectively slower median transmission.
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
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Complete Network Analysis Network Connections
Distance, cohesion Diffusion
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Complete Network Analysis Network Connections
Centrality

• Distance Connectivity measures locate a node
  based on particular features of the path
  strcutre, but there are many other ways of
  locating nodes in networks.
• Centrality refers to (one dimension of) location,
  identifying where an actor resides in a network.
  
• For example, we can compare actors at the edge
  of the network to actors at the center.
• In general, this is a way to formalize intuitive
  notions about the distinction between insiders
  and outsiders.
• As a terminology point, some authors distinguish
  centrality from prestige based on the
  directionality of the tie. Since the formulas
  are the same in every other respect, I stick with
  centrality for simplicity.

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Complete Network Analysis Network Connections
Centrality

• Conceptually, centrality is fairly straight
  forward we want to identify which nodes are in
  the center of the network. In practice,
  identifying exactly what we mean by center is
  somewhat complicated, but substantively we often
  have reason to believe that people at the center
  are very important.
• The standard centrality measures capture a wide
  range of importance in a network
• Degree
• Closeness
• Betweenness
• Eigenvector / Power measures
• After discussing these, I will describe measures
  that combine features of each of them.

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Complete Network Analysis Network Connections
Centrality
The most intuitive notion of centrality focuses
on degree. Degree is the number of direct
contacts a person has. The ideas is that the
actor with the most ties is the most important
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Complete Network Analysis Network Connections
Centrality
In a simple random graph (Gn,p), degree will have
a Poisson distribution, and the nodes with high
degree are likely to be at the intuitive center.
Deviations from a Poisson distribution suggest
non-random processes, which is at the heart of
current scale-free work on networks (see below).
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Complete Network Analysis Network Connections
Centrality
Degree centrality, however, can be deceiving,
because it is a purely local measure.
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Complete Network Analysis Network Connections
Centrality
If we want to measure the degree to which the
graph as a whole is centralized, we look at the
dispersion of centrality
Simple variance of the individual centrality
scores.
Or, using Freemans general formula for
centralization (which ranges from 0 to 1)
UCINET, SPAN, PAJEK and most other network
software will calculate these measures.
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Complete Network Analysis Network Connections
Centrality
Degree Centralization Scores
Freeman 0.0 Variance 0.0
Freeman 1.0 Variance 3.9
Freeman .02 Variance .17
Freeman .07 Variance .20
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Complete Network Analysis Network Connections
Centrality
A second measure of centrality is closeness
centrality. An actor is considered important if
he/she is relatively close to all other actors.
Closeness is based on the inverse of the distance
of each actor to every other actor in the network.
Closeness Centrality
Normalized Closeness Centrality
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Complete Network Analysis Network Connections
Centrality
Closeness Centrality in the examples
Distance Closeness normalized
0 1 1 1 1 1 1 1 .143 1.00 1 0 2
2 2 2 2 2 .077 .538 1 2 0 2 2 2 2
2 .077 .538 1 2 2 0 2 2 2 2
.077 .538 1 2 2 2 0 2 2 2 .077
.538 1 2 2 2 2 0 2 2 .077 .538
1 2 2 2 2 2 0 2 .077 .538 1 2
2 2 2 2 2 0 .077 .538
Distance Closeness normalized
0 1 2 3 4 4 3 2 1 .050 .400 1
0 1 2 3 4 4 3 2 .050 .400 2 1 0 1 2
3 4 4 3 .050 .400 3 2 1 0 1 2 3 4 4
.050 .400 4 3 2 1 0 1 2 3 4 .050
.400 4 4 3 2 1 0 1 2 3 .050
.400 3 4 4 3 2 1 0 1 2 .050 .400
2 3 4 4 3 2 1 0 1 .050 .400 1 2
3 4 4 3 2 1 0 .050 .400
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Complete Network Analysis Network Connections
Centrality
Closeness Centrality in the examples
Distance Closeness normalized
0 1 2 3 4 5 6 .048 .286 1 0 1 2 3 4
5 .063 .375 2 1 0 1 2 3 4 .077
.462 3 2 1 0 1 2 3 .083 .500
4 3 2 1 0 1 2 .077 .462 5 4 3 2 1
0 1 .063 .375 6 5 4 3 2 1 0 .048
.286
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Complete Network Analysis Network Connections
Centrality
Closeness Centrality in the examples
Distance Closeness normalized 0 1 1
2 3 4 4 5 5 6 5 5 6 .021 .255 1 0 1 1
2 3 3 4 4 5 4 4 5 .027 .324 1 1 0 1 2
3 3 4 4 5 4 4 5 .027 .324 2 1 1 0 1 2
2 3 3 4 3 3 4 .034 .414 3 2 2 1 0 1 1
2 2 3 2 2 3 .042 .500 4 3 3 2 1 0 2 3
3 4 1 1 2 .034 .414 4 3 3 2 1 2 0 1 1
2 3 3 4 .034 .414 5 4 4 3 2 3 1 0 1 1
4 4 5 .027 .324 5 4 4 3 2 3 1 1 0 1 4
4 5 .027 .324 6 5 5 4 3 4 2 1 1 0 5 5
6 .021 .255 5 4 4 3 2 1 3 4 4 5 0 1 1
.027 .324 5 4 4 3 2 1 3 4 4 5 1 0 1
.027 .324 6 5 5 4 3 2 4 5 5 6 1 1 0
.021 .255
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Centrality
Betweenness Centrality Model based on
communication flow A person who lies on
communication paths can control communication
flow, and is thus important. Betweenness
centrality counts the number of shortest paths
between i and k that actor j resides on.
b
a
C d e f g h
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Betweenness Centrality
Where gjk the number of geodesics connecting
jk, and gjk(ni) the number that actor i is on.
Usually normalized by
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Centrality
Betweenness Centrality
Centralization 1.0
Centralization 0
Centralization .59
Centralization .31
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Betweenness Centrality
Centralization .183
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Information Centrality It is quite likely that
information can flow through paths other than the
geodesic. The Information Centrality score uses
all paths in the network, and weights them based
on their length.
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Graph Theoretic Center (Barry or Jordan Center).
Identify the points with the smallest, maximum
distance to all other points.
Value longest distance to any other node.
The graph theoretic center is 3, but you might
also consider a continuous measure as the inverse
of the maximum geodesic
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Information Centrality
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• Comparing across these 3 centrality values
• Generally, the 3 centrality types will be
  positively correlated
• When they are not (low) correlated, it probably
  tells you something interesting about the network.

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Centrality
Bonacich Power Centrality Actors centrality
(prestige) is equal to a function of the prestige
of those they are connected to. Thus, actors who
are tied to very central actors should have
higher prestige/ centrality than those who are
not.

• a is a scaling vector, which is set to normalize
  the score.
• b reflects the extent to which you weight the
  centrality of people ego is tied to.
• R is the adjacency matrix (can be valued)
• I is the identity matrix (1s down the diagonal)
• 1 is a matrix of all ones.

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Bonacich Power Centrality
The magnitude of b reflects the radius of power.
Small values of b weight local structure, larger
values weight global structure. If b is
positive, then ego has higher centrality when
tied to people who are central. If b is
negative, then ego has higher centrality when
tied to people who are not central. As b
approaches zero, you get degree centrality.
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Bonacich Power Centrality
b 0.23
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Centrality
Bonacich Power Centrality
b-.35
b.35
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Bonacich Power Centrality
b.23
b-.23
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Centrality
In recent work, Borgatti (2003 2005) discusses
centrality in terms of two key dimensions
Medial
Radial
Frequency
Degree Centrality Bon. Power centrality
Betweenness
(empty but would be an interruption measure
based on distance)
Distance
Closeness Centrality
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In recent work, Borgatti (2003 2005) discusses
centrality in terms of two key dimensions

• Substantively, the key question for centrality is
  knowing what is flowing through the network. The
  key features are
• Whether the actor retains the good to pass to
  others (Information, Diseases) or whether they
  pass the good and then loose it (physical
  objects)
• Whether the key factor for spread is distance
  (disease with low pij) or multiple sources
  (information)
• The off-the-shelf measures do not always match
  the social process of interest, so researchers
  need to be mindful of this.

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Centrality
Actors that appear very different when seen
individually, are comparable in the global
network.
Graph is 27 centralized
(Node size proportional to betweenness centrality
)
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Centrality example Add Health
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Centrality
Node size proportional to betweenness centrality
Graph is 45 centralized
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Network Topology Centrality and Centralization
Measures research

• Rothenberg, et al. 1995. "Choosing a Centrality
  Measure Epidemiologic Correlates in the Colorado
  Springs Study of Social Networks." Social
  Networks Special Edition on Social Networks and
  Infectious Disease HIV/AIDS 17273-97.
• Found that the HIV positive actors were not
  central to the overall network
• Bell, D. C., J. S. Atkinson, and J. W. Carlson.
  1999. "Centrality Measures for Disease
  Transmission Networks." Social Networks 211-21.
• Using a data-based simulation on 22 people, found
  that simple degree measures were adequate,
  relative to complexity
• Poulin, R., M.-C. Boily, and B. R. Masse. 2000.
  "Dynamical Systems to Define Centrality in Social
  Networks." Social Networks 22187-220
• Method that allows one to compare across
  non-connected portions of a network, applied to a
  network of 40 people w. AIDS

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Network Dynamics
Two factors that affect network
flows Topology - the shape, or form, of the
network - simple example one actor cannot pass
information to another unless they are either
directly or indirectly connected Time - the
timing of contacts matters - simple example an
actor cannot pass information he has not yet
received.
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The Cocktail Party Problem

• Imagine a typical mixer party, where one of the
  guests knows a bit of gossip that everyone would
  like to know.
• Assuming that people tell this gossip to the
  people they meet at the party
• How many people would eventually hear the gossip?
• How long would it take to spread through the
  group?

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The Cocktail Party Problem

• Some specifics to narrow down the problem.
• 30 people invited, party lasts an hour.
• At any given moment in time, you can only carry
  on a conversation with 3 other people
• Guests mingle well they spend a short time
  talking to most people, but a long time to a
  small number (such as their date).
• Mingling is somewhat space-based you talk to
  the people you bump into, then move on to someone
  else after a short time.
• The bit of gossip moves instantaneously across
  connected sets (so time-to-diffuse0).

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• Some specifics to narrow down the problem.
• A (seemingly) simple network problem record who
  talks to who, and map the network.

Mean distance 1.99 Diameter 4 steps
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• But such an image conflates many temporally
  distinct events. A more accurate image is
  something like this

• In general, the graphs over which diffusion
  happens often
• Have timed edges
• Nodes enter and leave
• Edges can re-occur multiple times
• Edges can be concurrent
• These features break transmission paths,
  generally lowering diffusion potential and
  opening a host of interesting questions about the
  intersection of structure and time in networks.

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Network Evolution
Timing in networks

• A focus on contact structure has often slighted
  the importance of network dynamics,though a
  number of recent pieces are addressing this.
• Time affects networks in two important ways
• The structure itself evolves, in ways that will
  affect the topology an thus flow.
• Wasserheit and Aral, 1996. The dynamic topology
  of Sexually Transmitted Disease Epidemics The
  Journal of Infectious Diseases 74S201-13
• Rothenberg, et al. 1997 Using Social Network
  and Ethnographic Tools to Evaluate Syphilis
  Transmission Sexually Transmitted Diseases 25
  154-160
• 2) The timing of contact constrains flow
• Moody 2002, Social Forces,
• Morris and Kretchmar, 1995

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Network Evolution
Sexual Relations among A syphilis outbreak
Rothenberg et al map the pattern of sexual
contact among youth involved in a Syphilis
outbreak in Atlanta over a one year period.
(Syphilis cases in red)
Jan - June, 1995
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Sexual Relations among A syphilis outbreak
July-Dec, 1995
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Sexual Relations among A syphilis outbreak
July-Dec, 1995
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Network Evolution
Drug Relations, Colorado Springs, Year 1
Data on drug users in Colorado Springs, over 5
years
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Network Evolution
Drug Relations, Colorado Springs, Year 2 Current
year in red, past relations in gray
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Network Evolution
Drug Relations, Colorado Springs, Year 3 Current
year in red, past relations in gray
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Network Evolution
Drug Relations, Colorado Springs, Year 4 Current
year in red, past relations in gray
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Network Evolution
Drug Relations, Colorado Springs, Year 5 Current
year in red, past relations in gray
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Network Evolution
How do we analyze change in networks over
time? a) Descriptive techniques (change in
measures over time) b) Visualization c) Network
statistical models (Sienna, see below under
models)
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Social Balance

• One of the best theoretical approaches to
  understanding change in networks over time is to
  ask how the current relational patterns are
  likely to affect future relations. That is, make
  relational change endogenous.
• There are many models that do this, but the most
  famous for affective relations is social balance.
  
• Other models include
• Preferential attachment the rich get richer
  (Barabasi)
• Avoiding asymmetry (Gould)
• Avoiding close past relations (cycles of 4)
  (Bearman, Moody Stovel)
• Development of Hierarchy (Ivan Chase)

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Social Balance
Social Balance Transitivity
We determine balance based on the product of the
edges
A friend of a friend is a friend
()()() ()
Balanced
An enemy of my enemy is a friend
-
-
(-)()(-) (-)
Balanced

An enemy of my enemy is an enemy
-
-
(-)(-)(-) (-)
Unbalanced
-
A Friend of a Friend is an enemy


()(-)() (-)
Unbalanced
-
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Social Balance
Heider argued that unbalanced triads would be
unstable They should transform toward balance


Become Friends

-



Become Enemies
-
-

-
Become Enemies
-
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Social Balance
IF such a balancing process were active
throughout the graph, all intransitive triads
would be eliminated from the network. This would
result in one of two possible graphs (Balance
Theorem)
Complete Clique
Balanced Opposition
Friends with
Enemies with
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Empirically, we often find that graphs break up
into more than two groups. What does this imply
for balance theory?
It turns out, that if you allow all negative
triads, you can get a graph with many clusters.
That is, instead of treating (-)(-)(-) as an
forbidden triad, treat it as allowed. This
implies that the micro rule is different
negative ties among enemies are not as motivating
as positive ties.
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Empirically, we also rarely have symmetric
relations (at least on affect) thus we need to
identify balance in undirected relations.
Directed dyads can be in one of three
states 1) Mutual 2) Asymmetric 3) Null
Every triad is composed of 3 dyads, and we can
identify triads based on the number of each type,
called the MAN label system
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Social Balance
Balance in directed relations Actors seek
out transitive relations, and avoid intransitive
relations. A triple is transitive

If
then

• A property of triples within triads
• Assumes directed relations
• The saliency of a triad may differ for each
  actor, depending on their position within the
  triad.

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Once we admit directed relations, we need to
decompose triads into their constituent triples.
Ordered Triples
a
b
c
a
c
Transitive
b
a
c
b
a
b
Vacuous
a
c
b
c
b
Vacuous
a
c
b
c
a
b
a
Intransitive
120C
a
b
c
b
c
Intransitive
c
b
a
c
a
Vacuous
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Network Sub-Structure Triads
(0)
(1)
(2)
(3)
(4)
(5)
(6)
003
012
102
111D
201
210
300
021D
111U
120D
Intransitive
Transitive
021U
030T
120U
Mixed
021C
030C
120C
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An Example of the triad census
Type Number of
triads ---------------------------------------
1 - 003 21 ----------------------
----------------- 2 - 012 26
3 - 102 11 4 - 021D
1 5 - 021U 5 6 -
021C 3 7 - 111D
2 8 - 111U 5 9 - 030T
3 10 - 030C 1
11 - 201 1 12 - 120D
1 13 - 120U 1 14 -
120C 1 15 - 210
1 16 - 300
1 --------------------------------------- Sum (2
- 16) 63
Pajek SPAN will give you the triad census
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As with undirected graphs, you can use the type
of triads allowed to characterize the total
graph. But now the potential patterns are much
more diverse
1) All triads are 030T
A perfect linear hierarchy.
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Social Balance
Triads allowed 300, 102
N
M
M
1
0
1
0
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Cluster Structure, allows triads 003, 300, 102
N
Eugene Johnsen (1985, 1986) specifies a number of
structures that result from various triad
configurations
M
M
N
N
N
N
M
M
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PRC300,102, 003, 120D, 120U, 030T, 021D, 021U
Ranked Cluster
1
0
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
1
1
0
1
1
1
1
0
And many more...
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Substantively, specifying a set of triads defines
a behavioral mechanism, and we can use the
distribution of triads in a network to test
whether the hypothesized mechanism is active. We
do this by (1) counting the number of each triad
type in a given network and (2) comparing it to
the expected number, given some random
distribution of ties in the network. See
Wasserman and Faust, Chapter 14 for computation
details, and the SPAN manual for SAS code that
will generate these distributions, if you so
choose.
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Structural Indices based on the distribution of
triads
The observed distribution of triads can be fit to
the hypothesized structures using weighting
vectors for each type of triad.
Where l 16 element weighting vector for the
triad types T the observed triad census mT
the expected value of T ST the
variance-covariance matrix for T
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For the Add Health data, the observed
distribution of the tau statistic for various
models was
Indicating that a ranked-cluster model fits the
best.
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So far, the structural features of a network
focus on the graph at equilibrium. That is,
we have hypothesized structures once people have
made all the choices they are going to make.
What we have not done, is really look closely at
the implication of changing relations. That
is, we might say that triad 030C should not
occur, but what would a change in this triad
imply from the standpoint of the actor making a
relational change?
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030C
120C
102
111U
021C
201
012
111D
300
003
210
021D
120U
030T
021U
120D
(some transitions will both increase transitivity
decrease intransitivity the effects are
independent they are colored here for net
balance)
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Observed triad transition patterns, from Sorensen
and Hallinan (1976)
150
Comp