Transcript No Slide Title

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
Homework Results
JWM’s 3-step kinship neighborhood (plus in-laws for fun)
N=70+
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
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)
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)
Form Vs. Content
Integration of large-scale social
systems
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
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
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 don’t want to draw this line too sharply: emergent role positions can
affect individual outcomes in a ‘variable’ way, and variable approaches
constrain relational activity.
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
a
d
c
e
Foundations
Data
Social Network data consists of two linked classes of data:
a) 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)
Foundations
Data
In general, a relation can be: (1) Binary or Valued (2) Directed or Undirected
b
b
d
a
c
a
e
c
1
a
b
d
1
3
c
e
Directed, binary
Undirected, binary
b
d
d
2
4
Undirected, Valued
e
a
c
e
Directed, Valued
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.
Foundations
Data
Primary
Group
Global-Net
Ego-Net
Best Friend
Dyad
2-step
Partial network
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
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
Foundations
Graphs
Working with pictures.
No standard way to draw a sociogram: each of these are equal:
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
Foundations
Graphs
Network visualization helps build intuition, but you have to keep the drawing
algorithm in mind:
Tree-Based layouts
Spring-embeder layouts
Two images of the same network
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 linecrossing.
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.
Foundations
Graphs
2
12
9
63
Male
Female
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.
Foundations
Graphs
As networks increase in size, the
effectiveness of a point-and-line
display diminishes, because you
simply run out of plotting
dimensions.
I’ve 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.
Foundations
Graphs
As networks increase in size, the
effectiveness of a point-and-line
display diminishes, because you
simply run out of plotting
dimensions.
I’ve 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.
Foundations
Graphs
As networks increase in size, the
effectiveness of a point-and-line
display diminishes, because you
simply run out of plotting
dimensions.
I’ve 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.
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/
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.
Foundations
Methods
From pictures to matrices
b
b
d
a
c
e
Undirected, binary
a
b
1
a
b 1
c
1
d
e
c
d
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1
c
e
a
1
a
b 1
c
1
d
e
1
1
a
e
Directed, binary
1
1
d
b
1
c
1
d
e
1
1
1
Foundations
Methods
From matrices to lists
a
a
b 1
c
d
e
b
1
c
d
e
1
1
1
1
1
1
1
1
Adjacency List
ab
bac
cbde
dce
ecd
Arc List
ab
ba
bc
cb
cd
ce
dc
de
ec
ed
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:
D=
X
N ( N - 1)
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.
a
a
b 1
c
d
e
b
1
c
1
1
d
e
1
1
1
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
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.
b
a
a
d
c
b
e
f
c
f
d
e
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:
1)
Definition
A matrix is any rectangular array of numbers. We refer to the matrix dimension as
the number of rows and columns
a b c d e
a 0 1 0 0 0
b 1 0 0 0 0
c 0 1 0 1 1
d 0
e 0
0 0 0 0
0 1 1 0
(5 x 5)
W B
1 0
1 0
0 1
0 1
0 1
1 0
(5x2)
Age
13
10
7
8
16
11
(5x1)
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 3
A= 4 7
2 5
2 3
B= 7 1
0 4
3 6
A+B = 11 8
2 9
3 9
Multiplication by a scalar: 3A = 12 21
6 15
A-B =
-1 0
-3 6
2 1
2 9
A#B = 28 7
0 20
Foundations
Basic Matrix Operations
The transpose (` or T) of a matrix reverses the row and column
dimensions.
Atij=Aji
So a M x N matrix becomes an N x M matrix.
a b
c d
e f
T
=
a c e
b d f
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 sender’s recipient's receivers).
Foundations
Basic Matrix Operations
The multiplication of two matrices Amxn and Bnxq results in Cmxq
n
Cmq =  amk bkq
k =1
a b
c d
e f
g h
a b
c d
e f
g h i
j k l
(3x2)
(2x3)
=
=
ae+bg
ce+dg
ag+bj
cg+dj
eg+fg
af+bh
cf+dh
ah+bk
ch+dk
eh+fk
(3x3)
ai+bl
ci+dl
ei+fl
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.
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 endnodes 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.
Foundations
Data
Basic Measures & A little graph theory
Calculate reachability through matrix multiplication.
(see p.162 of W&F)
0
1
0
0
0
1
e
d
c
b
a
f
1
0
1
0
0
0
X
0 0
1 0
0 1
1 0
1 1
1 0
0
0
1
1
0
0
1
0
1
0
0
0
2
0
2
0
0
0
0
2
0
1
1
2
X2
2 0
0 1
4 1
1 2
1 1
0 1
Distance
. 1 2 0
1 . 1 2
2 1 . 1
0 2 1 .
0 2 1 1
1 2 1 2
0
1
1
1
2
1
0
2
1
1
.
2
4
0
6
1
1
0
X3
0 2
6 1
2 5
5 2
5 3
6 1
0
2
0
1
1
2
0
4
0
2
2
4
2
1
5
3
2
1
4
0
6
1
1
0
1
2
1
2
2
.
Distance
. 1 2 3 3
1 . 1 2 2
2 1 . 1 1
3 2 1 . 1
3 2 1 1 .
1 2 1 2 2
1
2
1
2
2
.
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
b
a
f
0
1
0
0
0
1
1
0
1
0
0
0
X
0 0
1 0
0 1
1 0
1 1
1 0
0
0
1
1
0
0
1
0
1
0
0
0
Race
1 0
1 0
0 1
0 1
0 1
1 0
R G
R 4 2
Race`(X)Race=
G 2 6
X(Race)
2 0
1 1
2 2
0 2
0 2
1 1
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.
0
1
0
0
0
1
0
1
0
0
X
0
0
0
0
1
0
0
1
0
1
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
XT
0 0
1 0
0 0
1 0
1 0
0
0
1
1
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
Max Sym
0 1 0 0 0
1 0 1 0 0
0 1 0 1 1
0 0 1 0 1
0 0 1 1 0
MIN Sym
0 1 0 0 0
1 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 1 0 0
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
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:
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
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)
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:
[email protected]