Foundations of Network Analysis

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Foundations of Network Analysis
Overview

• Theory A structural Approach to Sociology
• Wellman
• Emirbayer

• Methods
• Points and Lines
• Data formats
• Matrices
• Adjacency Lists
• Edge Lists
• Basic Graph Theory

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Homework Results JWMs 3-step kinship
neighborhood (plus in-laws for fun)
N70
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Foundations Theory
A manifesto for Relational Sociology

• Substantialism vs Relationalism
• Theoretical Domains
• Power, equality, freedom, agency
• Substantive domains (research)
• Social Structure
• Network analysis
• Culture
• Social Psychology
• Problems
• Boundary specification
• Network dynamics
• Causality
• Normative implication

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Foundations Theory
Structural Analysis from method and metaphor to
theory and substance.

• Five elements
• Structural constraint on activity (as opposed to
  inner forces)
• focus on relations among units (as opposed to
  categories)
• relationships among multiple alters affect people
  behavior
• structure is a network of networks
• analytic methods deal with this structure
  directly

• Historical roots
• Social anthropology (Barnes 1954 Bott 1957).
  Moved from normative relations to observed
  relations.
• Early sociologists Social psychologists start
  using sociograms (Moreno, Coleman). Focused on
  details of sociometric structure.
• Group around white really pushed the theoretical
  development of a network perspective as the basis
  for sociology (late 60s, early 70s)

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Foundations Theory
Structural Analysis from method and metaphor to
theory and substance.
H. White The presently existing, largely
categorical descriptions of social structure have
no solid theoretical grounding furthermore,
network concepts may provide the only way to
construct a theory of social structure. (p.25)
Integration of large-scale social systems
Form Vs. Content
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Foundations Theory
Structural Analysis from method and metaphor to
theory and substance.
Major Claims

• Structured social relationships are a more
  powerful source of sociological explanation than
  personal attributes of system members.
• Norms emerge from location in structured systems
  of social relationships
• Social Structures determine the operation of
  dyadic relationships
• The world is composed of networks, not groups
• Structural methods supplant and supplement
  individualistic methods

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Foundations Theory
Structural Analysis from method and metaphor to
theory and substance.
Analytic Principles

• Ties are usually asymmetrically reciprocal,
  differing in content and intensity
• Ties link network members indirectly as well as
  directly. Hence, they must be defined within the
  context of larger network structures.
• Ties are structured, and thus networks are not
  random, but instead clusters, boundaries and
  cross-linkages
• Cross-linkages connected clusters as well as
  individuals
• Asymmetric ties and complex networks
  differentially distribute scares resources
• Networks structure collaborative and competitive
  activities to secure scarce resources

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Foundations Key Questions

• Social Network analysis lets us answer questions
  about social interdependence. These include
• Networks as Variables approaches
• Are kids with smoking peers more likely to smoke
  themselves?
• Do unpopular kids get in more trouble than
  popular kids?
• Are people with many weak ties more likely to
  find a job?
• Do central actors control resources?
• Networks as Structures approaches
• What generates hierarchy in social relations?
• What network patterns spread diseases most
  quickly?
• How do role sets evolve out of consistent
  relational activity?
• We dont want to draw this line too sharply
  emergent role positions can affect individual
  outcomes in a variable way, and variable
  approaches constrain relational activity.

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Foundations Data
The unit of interest in a network are the
combined sets of actors and their relations. We
represent actors with points and relations with
lines. Actors are referred to variously
as Nodes, vertices, actors or
points Relations are referred to variously
as Edges, Arcs, Lines, Ties
Example
b
d
a
c
e
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Foundations Data

• Social Network data consists of two linked
  classes of data
• Nodes Information on the individuals (actors,
  nodes, points, vertices)
• Network nodes are most often people, but can be
  any other unit capable of being linked to another
  (schools, countries, organizations,
  personalities, etc.)
• The information about nodes is what we usually
  collect in standard social science research
  demographics, attitudes, behaviors, etc.
• Often includes dynamic information about when the
  node is active
• b) Edges Information on the relations among
  individuals (lines, edges, arcs)
• Records a connection between the nodes in the
  network
• Can be valued, directed (arcs), binary or
  undirected (edges)
• One-mode (direct ties between actors) or two-mode
  (actors share membership in an organization)
• Includes the times when the relation is active
• Graph theory notation G(V,E)

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Foundations Data
In general, a relation can be (1) Binary or
Valued (2) Directed or Undirected
The social process of interest will often
determine what form your data take. Almost all
of the techniques and measures we describe can be
generalized across data format.
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Foundations Data
Global-Net
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Foundations Data
We can examine networks across multiple levels
1) Ego-network - Have data on a respondent (ego)
and the people they are connected to (alters).
Example 1985 GSS module - May include estimates
of connections among alters
2) Partial network - Ego networks plus some
amount of tracing to reach contacts of contacts
- Something less than full account of
connections among all pairs of actors in the
relevant population - Example CDC Contact
tracing data for STDs
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Foundations Data
We can examine networks across multiple levels

• 3) Complete or Global data
• - Data on all actors within a particular
  (relevant) boundary
• - Never exactly complete (due to missing data),
  but boundaries are set
• Example Coauthorship data among all writers in
  the social sciences, friendships among all
  students in a classroom

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Foundations Graphs
Working with pictures. No standard way to draw a
sociogram each of these are equal
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Foundations Graphs
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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Foundations Graphs
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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Foundations Graphs
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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Foundations Graphs
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Foundations Graphs
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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Foundations Graphs
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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Foundations Graphs
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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Foundations Graphs
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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Foundations Graphs
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.
http//www.sociology.ohio-state.edu/jwm/NetMovies/
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Foundations Methods
Analytically, graphs are cumbersome to work with
analytically, though there is a great deal of
good work to be done on using visualization to
build network intuition. I recommend using
layouts that optimize on the feature you are most
interested in. The two I use most are a
hierarchical layout or a force-directed layout
are best.
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Foundations Methods
From pictures to matrices
Undirected, binary
Directed, binary
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Foundations Methods
From matrices to lists
Arc List
Adjacency List
a b b a b c c b c d c e d c d e e c e d
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Foundations Basic Measures
Basic Measures A little graph theory For
greater detail, see http//www.analytictech.com/
networks/graphtheory.htm
Volume
The first measure of interest is the simple
volume of relations in the system, known as
density, which is the average relational value
over all dyads. Under most circumstances, it is
calculated as
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Foundations Basic Measures
Basic Measures A little graph theory
Volume
At the individual level, volume is the number of
relations, sent or received, equal to the row and
column sums of the adjacency matrix.
Node In-Degree Out-Degree a
1 1 b 2 1 c
1 3 d 2 0 e
1 2 Mean 7/5 7/5
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Foundations Data
Basic Measures A little graph theory
Reachability
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
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Foundations Basic Matrix Operations
One of the key advantages to storing networks as
matrices is that we can use all of the tools from
linear algebra on the socio-matrix. Some of the
basics matrix manipulations that we use are as
follows

• Definition
• A matrix is any rectangular array of numbers. We
  refer to the matrix dimension as the number of
  rows and columns

(5 x 5)
(5x2)
(5x1)
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Foundations Basic Matrix Operations
Matrix operations work on the elements of the
matrix in particular ways. To do so, the
matrices must be conformable. That means the
sizes allow the operation. For addition (),
subtraction (-), or elementwise multiplication
(), both matrices must have the same number of
rows and columns. For these operations, the
matrix value is the operation applied to the
corresponding cell values.
-1 0 -3 6 2 1
3 6 11 8 2 9
1 3 4 7 2 5
2 3 7 1 0 4
A-B
AB
A
B
2 9 28 7 0 20
3 9 12 21 6 15
AB
Multiplication by a scalar 3A
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Foundations Basic Matrix Operations
The transpose ( or T) of a matrix reverses the
row and column dimensions. AtijAji So a M x
N matrix becomes an N x M matrix.
T
a b c d e f
a c e b d f

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Foundations Basic Matrix Operations
The matrix multiplication (x) of two matrices
involves all elements of the matrix, and will
often result in a matrix of new dimensions. In
general, to be conformable, the inner dimension
of both matrices must match. So A3x2 x B2x3
C3 x 3 But A3x3 x B2x3 is not
defined Substantively, adding names to the
dimensions will help us keep track of what the
resulting multiplications mean So multiplying
(send x receive)x (send x receive) (send x
receive), giving us the two-step distances (the
senders recipient's receivers).
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Foundations Basic Matrix Operations
The multiplication of two matrices Amxn and Bnxq
results in Cmxq
a b c d
e f g h
aebg afbh cedg cfdh

a b c d e f
agbj ahbk aibl cgdj chdk cidl egfg
ehfk eifl
g h i j k l

(3x2) (2x3)
(3x3)
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Foundations Basic Matrix Operations
The powers (square, cube, etc) of a matrix are
just the matrix times itself that many
times. A2 AA or A3 AAA We often use
matrix multiplication to find types of people one
is tied to, since the 1 in the adjacency matrix
effectively captures just the people each row is
connected to.
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Foundations Data
Basic Measures A little graph theory
Reachability
The distance from one actor to another is the
shortest path between them, known as the geodesic
distance. If there is at least one path
connecting every pair of actors in the graph, the
graph is connected and is called a component.
Two paths are independent if they only have the
two end-nodes in common. If a graph has two
independent paths between every pair, it is
biconnected, and called a bicomponent. Similarly
for three paths, four, etc.
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Foundations Data
Basic Measures A little graph theory
Calculate reachability through matrix
multiplication. (see p.162 of WF)
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Foundations Data
Basic Measures A little graph theory
Mixing patterns
Matrices make it easy to look at mixing patterns
connections among types of nodes. Simply
multiply an indicator of category by the
adjacency matrix.
e
d
c
f
b
a
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Foundations Data
Basic Measures A little graph theory
Matrix manipulations allow you to look at
direction of ties, and distinguish symmetric
from asymmetric ties.
To transform an asymmetric graph to a symmetric
graph, add it to its transpose.
X 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0
0 1 1 0
XT 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1
0 0
Max Sym MIN Sym 0 1 0 0 0 0 1 0 0
0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0
0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0
1 0 0
0 2 0 0 0 2 0 1 0 0 0 1 0 1 2 0 0 1 0 1 0 0 2 1 0
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Foundations Software

• UCINET
• The Standard network analysis program, runs in
  Windows
• Good for computing measures of network topography
  for single nets
• Input-Output of data is a special 2-file format,
  but is now able to read PAJEK files directly.
• Not optimal for large networks
• Available from
• Analytic Technologies

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Foundations Software

• PAJEK
• Program for analyzing and plotting very large
  networks
• Intuitive windows interface
• Used for most of the real data plots in this
  presentation
• Started mainly a graphics program, but has
  expanded to a wide range of analytic capabilities
• Can link to the R statistical package
• Free
• Available from

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Foundations Software

• Cyram Netminer for Windows
• Newest Product, not yet widely used
• Price range depends on application
• Limited to smaller networks O(100)

http//www.netminer.com/NetMiner/home_01.jsp
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Foundations Software

• NetDraw
• Also very new, but by one of the best known names
  in network analysis software.
• Free
• Limited to smaller networks O(100)

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Foundations Software

• NEGOPY
• Program designed to identify cohesive sub-groups
  in a network, based on the relative density of
  ties.
• DOS based program, need to have data in arc-list
  format
• Moving the results back into an analysis program
  is difficult.
• Available from
• William D. Richards
• http//www.sfu.ca/richards/Pages/negopy.htm
• SPAN - Sas Programs for Analyzing Networks
  (Moody, ongoing)
• is a collection of IML and Macro programs that
  allow one to
• a) create network data structures from nomination
  data
• b) import/export data to/from the other network
  programs
• c) calculate measures of network pattern and
  composition
• d) analyze network models
• Allows one to work with multiple, large networks
• Easy to move from creating measures to analyzing
  data
• Available by sending an email to
• Moody.77_at_sociology.osu.edu