SSRI_pt1.ppt

Introduction to
Social Network Analysis
Duke University
May 2012
James Moody
Duke University
http://www.soc.duke.edu/~jmoody77/rwj/wsfiles.htm

1. Introduction
2. Social Network data
a. Basic data elements
b. Network data sources
3. Local (ego) Network Analysis
a. Introduction
b. Network Composition
c. Network Structure
d. Local Network Models
4. Complete Network Analysis
a. Exploratory Analysis
b. Network Connections
c. Network Macro Structure
d. Stochastic Network Analyses
5. Social Network Software Review
6. Work through examples
Introduction

Introduction
These patterns of connection form a social space, that can be seen in multiple
contexts:
We live in a connected world:
“To speak of social life is to speak of the association between people –
their associating in work and in play, in love and in war, to trade or to
worship, to help or to hinder. It is in the social relations men establish that
their interests find expression and their desires become realized.”
Peter M. Blau
Exchange and Power in Social Life, 1964
"If we ever get to the point of charting a whole city or a whole nation, we
would have … a picture of a vast solar system of intangible structures,
powerfully influencing conduct, as gravitation does in space. Such an
invisible structure underlies society and has its influence in determining the
conduct of society as a whole."
J.L. Moreno, New York Times, April 13, 1933

Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000, 32-42.
Introduction

High Schools as Networks
Introduction

And yet, standard social science analysis methods do not take this space
into account.
“For the last thirty years, empirical social research has been
dominated by the sample survey. But as usually practiced, …, the
survey is a sociological meat grinder, tearing the individual from his
social context and guaranteeing that nobody in the study interacts
with anyone else in it.”
Allen Barton, 1968 (Quoted in Freeman 2004)
Moreover, the complexity of the relational world makes it impossible to
identify social connectivity using only our intuition.
Social Network Analysis (SNA) provides a set of tools to empirically
extend our theoretical intuition of the patterns that compose social
structure.
Introduction

Why do Networks Matter? Local vision
Introduction

Social network analysis is:
•a set of relational methods for systematically understanding
and identifying connections among actors. SNA
•is motivated by a structural intuition based on ties linking
social actors
•is grounded in systematic empirical data
•draws heavily on graphic imagery
•relies on the use of mathematical and/or computational
models.
•Social Network Analysis embodies a range of theories
relating types of observable social spaces and their relation to
individual and group behavior.
Introduction

Introduction
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?
•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?
Both: Connectionist vs. Positional features of the network
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.

1. Introduction and Background
Why networks matter:
• Intuitive: information travels through contacts
between actors, which can reflect a power
distribution or influence attitudes and behaviors. Our
understanding of social life improves if we account
for this social space.
• Less intuitive: patterns of inter-actor contact can
have effects on the spread of “goods” or power
dynamics that could not be seen focusing only on
individual behavior.

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:
a
b
c e
d
Social Network 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)
Social Network Data
Basic Data Elements

In general, a relation can be: (1) Binary or Valued (2) Directed or Undirected
a
b
c e
d
Undirected, binary Directed, binary
a
b
c e
d
a
b
c e
d
Undirected, Valued Directed, Valued
a
b
c e
d
1 3
4
2
1
Social Network Data
Basic Data Elements
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.

In general, a relation can be: (1) Binary or Valued (2) Directed or Undirected
Social Network Data
Basic Data Elements
The social process of interest will often determine what form your data take. Conceptually, almost
all of the techniques and measures we describe can be generalized across data format, but you may
have to do some of the coding work yourself….
a
b
c e
d
Directed,
Multiplex categorical edges

Ego-Net
Global-Net
Best Friend
Dyad
Primary
Group
Social Network Data
Basic Data Elements: Levels of analysis
2-step
Partial network

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
Social Network Data

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
We can examine networks across multiple levels:
Social Network Data

A good network drawing allows viewers to come away from the image with an almost
immediate intuition about the underlying structure of the network being displayed.
However, because there are multiple ways to display the same information, and standards
for doing so are few, the information content of a network display can be quite variable.
Now trace the actual pattern of ties.
You will see that these 4 graphs are
exactly the same.
Consider the 4 graphs drawn at right.
After asking yourself what intuition
you gain from each graph, click on
the screen.
Social Network Data
Graph Layout (teaser)

In general, 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.
We’ll see some examples of “best practice” after getting a little more familiar
with data structure.
Basic Data Structures
Social Network Data

From pictures to matrices
a
b
c e
d
Undirected, binary
a b c d e
a
b
c
d
e
1
1 1
1 1 1
1 1
1 1
An undirected graph and the
corresponding matrix is symmetric.
Because network images are hard to work
with, we often use an adjacency matrix to
represent the network.
The matrix (X) at right represents an
undirected binary network. Each node (a-e)
is listed on both the row and the column.
The ith row and the jth column (Xij) records the
value of a tie from node i to node j. For
example, the line between nodes a and b is
represented as an entry in the first row and
second column (red at right).
Because the graph is undirected the ties sent
are the same as the ties receive, so every entry
above the diagonal equals the entries below
the diagonal.
Social Network Data

Directed, binary
a
b
c e
d
a b c d e
a
b
c
d
e
1
1
1 1 1
1 1
A directed graph and the
corresponding matrix is asymmetrical.
Directed graphs, on the other hand,
are asymmetrical.
We can see that Xab =1 and Xba =1,
therefore a “sends” to b and b “sends” to a.
However, Xbc=0 while Xcb=1; therefore,
c “sends” to b, but b does not reciprocate.
Social Network Data

From matrices to lists (binary)
a b c d e
a
b
c
d
e
1
1 1
1 1 1
1 1
1 1
a b
b a c
c b d e
d c e
e c d
a b
b a
b c
c b
c d
c e
d c
d e
e c
e d
Adjacency List
Arc List
Social network analysts also use adjacency lists and arc lists
to more efficiently store network data.
a
b
c e
d
Social Network Data

From matrices to lists (valued)
a b c d e
a
b
c
d
e
1
1 2
2 3 5
3 1
5 1
a b
b a c
c b d e
d c e
e c d
a b 1
b a 1
b c 2
c b 2
c d 3
c e 5
d c 3
d e 1
e c 5
e d 1
Adjacency List
Arc List
Social network analysts also use adjacency lists and arc lists
to more efficiently store network data.
a
b
c e
d
Social Network Data
1 2
5
1
3 a 1
b 1 2
c 2 3 1
d 3 1
e 5 1
contact value

Social network data are substantively divided by the number of
modes in the data.
1-mode data represents edges based on direct contact between
actors in the network. All the nodes are of the same type (people,
organization, ideas, etc). Examples:
Communication, friendship, giving orders, sending email.
There are no constraints on connections between classes of
nodes.
1-mode data are usually singly reported (each person reports on
their friends), but you can use multiple-informant data, which is
more common in child development research (Cairns and
Cairns).
Social Network Data
Basic Data Elements: Modes

Social network data are substantively divided by the number of
modes in the data.
2-mode data represents nodes from two separate classes, where
all relations cross classes. Examples:
People as members of groups
People as authors on papers
Words used often by people
Events in the life history of people
The two modes of the data represent a duality: you can project
the data as people connected to people through joint membership
in a group, or groups to each other through common membership
N-mode data generalizes the constraint on ties between classes to
N groups
Social Network Data

Breiger: 1974 - Duality of Persons and Groups
Argument:
Metaphor: people intersect through their associations,
which defines (in part) their individuality.
The Duality argument is that relations among
groups imply relations among individuals
Social Network Data

Bipartite networks imply a constraint on the mixing, such that ties only cross classes.
Here we see a tie connecting each woman with the party she attended (Davis data)
Social Network Data

Social Network Data
Bipartite networks imply a constraint on the mixing, such that ties only cross classes.
Here we see a tie connecting each woman with the party she attended (Davis data)

By projecting the data, one can look at the shared between people or the common
memberships in groups: this is the person-to-person projection of the 2-mode data.
Social Network Data

Social Network Data
By projecting the data, one can look at the shared between people or the common
memberships in groups: this is the group-to-group projection of the 2-mode data.

Social Network Data
Here I plot faculty overlaps on grad student committees to see how integrated the
department is. The substantive problem was demonstrating to the dean that a new
hire helped integrate the department, but that more work was needed….
Connecting effect of a new hire in an academic department
(not sociology!)
External Readers

Working with two-mode data
A person-to-group adjacency matrix is rectangular, with persons down
rows and groups across columns
1 2 3 4 5
A 0 0 0 0 1
B 1 0 0 0 0
C 1 1 0 0 0
D 0 1 1 1 1
E 0 0 1 0 0
F 0 0 1 1 0
A =
Each column is a group,
each row a person, and
the cell = 1 if the person in
that row belongs to that
group.
You can tell how many
groups two people both
belong to by comparing
the rows: Identify every
place that both rows = 1,
sum them, and you have
the overlap.
Social Network Data

One can get either projection easily with a little matrix multiplication.
First define AT as the transpose of A (simply reverse the rows and
columns). If A is of size P x G, then AT will be of size G x P.
ji
T
ij A
A 
1 2 3 4 5
A 0 0 0 0 1
B 1 0 0 0 0
C 1 1 0 0 0
D 0 1 1 1 1
E 0 0 1 0 0
F 0 0 1 1 0
A =
A B C D E F
1 0 1 1 0 0 0
2 0 0 1 1 0 0
3 0 0 0 1 1 1
4 0 0 0 1 0 1
5 1 0 0 1 0 0
AT =
Working with two-mode data
Social Network Data

G = AT(A)
1 2 3 4 5
A 0 0 0 0 1
B 1 0 0 0 0
C 1 1 0 0 0
D 0 1 1 1 1
E 0 0 1 0 0
F 0 0 1 1 0
A =
A B C D E F
1 0 1 1 0 0 0
2 0 0 1 1 0 0
3 0 0 0 1 1 1
4 0 0 0 1 0 1
5 1 0 0 1 0 0
AT =
(6x5) (5x6)
AT * A = P
(5x6) 6x5) (5x5)
G
1 2 3 4 5
1 2 1 0 0 0
2 1 2 1 1 1
3 0 1 3 2 1
4 0 1 2 2 1
5 0 1 1 1 2
A * AT = P
(6x5)(5x6) (6x6)
P
A B C D E F
A 1 0 0 1 0 0
B 0 1 1 0 0 0
C 0 1 2 1 0 0
D 1 0 1 4 1 2
E 0 0 0 1 1 1
F 0 0 0 2 1 2
Social Network Data
P = A(AT)

G = AT(A)
Social Network Data
P = A(AT)

Theoretically, these two equations define what Breiger means by duality:
“With respect to the membership network,…, persons who are
actors in one picture (the P matrix) are with equal legitimacy viewed as
connections in the dual picture (the G matrix), and conversely for groups.”
(p.87)
The resulting network:
1) Is always symmetric
2) the diagonal tells you how many groups (persons) a person
(group) belongs to (has)
Social Network Data
In practice, most network software (UCINET, PAJEK) will do all of these
operations. It is also simple to do the matrix multiplication in programs
like SAS, SPSS, or R.

Existing Sources of Social Network Data:
There are lots of network data archived. Check INSNA for a listing. The PAJEK data
page includes a number of exemplars for large-scale networks.
2-Mode Data
• One can construct networks from many different data sources if you want to
work with 2-mode data. Any list can be so transformed.
• Director interlocks
• Protest event participation
• Authors on papers
• Words in documents…
1-Mode Data
Local Network data:
• Fairly common, because it is easy to collect from sample surveys.
• GSS, NHSL, Urban Inequality Surveys, etc.
• Pay attention to the question asked
• Key features are (a) number of people named and (b) whether alters are
able to nominate each other.
Social Network Data
Network Data Sources: Existing data sources

1-Mode Data
Partial network data:
• Much less common, because cost goes up significantly once you
start tracing to contacts.
• Snowball data: start with focal nodes and trace to contacts
• CDC style data on sexual contact tracing
• Limited snowball samples:
• Colorado Springs drug users data
• Geneology data
• Small-world network samples
• Limited Boundary data: select data within a limited bound
• Cross-national trade data
• Friendships within a classroom
• Family support ties
Social Network Data

1-Mode Data
Complete network data:
• Significantly less common and never perfect.
• Start by defining a theoretically relevant boundary
• Then identify all relations among nodes within that boundary
• Co-sponsorship patterns among legislators
• Friendships within strongly bounded settings (sororities,
schools)
• Examples:
• Add Health on adolescent friendships
• Hallinan data on within-school friendships
• McFarland’s data on verbal interaction
• Electronic data on citations or coauthorship (see Pajek data
page)
• See INSNA home page for many small-scale networks
Social Network Data

Boundary Specification Problem
Network methods describe positions in relevant social fields, where flows
of particular goods are of interest. As such, boundaries are a
fundamentally theoretical question about what you think matters in
the setting of interest.
See Laumann & Marsden (1983) for a good review of the boundary
specification problem
In general, there are usually relevant social foci that bound the relevant
social field. We expect that social relations will be very clumpy.
Consider the example of friendship ties within and between a high-
school and a Jr. high:
Social Network Data
Network Data Sources: Collecting network data

a) Network data collection can be time consuming. It is better (I think) to
have breadth over depth. Having detailed information on <50% of the
sample will make it very difficult to draw conclusions about the general
network structure.
b) Question format:
• If you ask people to recall names (an open list format), fatigue will
result in under-reporting
• If you ask people to check off names from a full list, you can often get
over-reporting
c) It is common to limit people to a small number if nominations (~5). This
will bias network measures, but is sometimes the best choice to avoid
fatigue.
d) Concrete relational indicators are best (who did you talk to?) over attitudes
that are harder to define (who do you like?)
Social Network Data

Boundary Specification Problem
Social Network Data
While students were
given the option to name
friends in the other
school, they rarely do.
As such, the school likely
serves as a strong
substantive boundary

Local Network data:
• When using a survey, common to use an “ego-network module.”
• First part: “Name Generator” question to elicit a list of names
• Second part: Working through the list of names to get
information about each person named
• Third part: asking about relations among each person named.
Social Network Data
GSS Name Generator:
“From time to time, most people discuss important matters with other people.
Looking back over the last six months -- who are the people with whom you
discussed matters important to you? Just tell me their first names or initials.”
Why this question?
•Only time for one question
•Normative pressure and influence likely travels through strong ties
•Similar to ‘best friend’ or other strong tie generators
•Note there are significant substantive problems with this name generator

Electronic Small World name generator:
Social Network Data

Local Network data:
The second part usually asks a series of questions about each person
GSS Example:
“Is (NAME) Asian, Black, Hispanic, White or something else?”
Social Network Data
ESWP example:
Will generate N x (number of attributes) questions to the survey

Local Network data:
The third part usually asks about relations among the alters. Do this
by looping over all possible combinations. If you are asking about a
symmetric relation, then you can limit your questions to the n(n-1)/2
cells of one triangle of the adjacency matrix:
Social Network Data
1 2 3 4 5
1
2
3
4
5
GSS: Please think about the relations between the people you just mentioned. Some of them may
be total strangers in the sense that they wouldn't recognize each other if they bumped into each
other on the street. Others may be especially close, as close or closer to each other as they are to
you. First, think about NAME 1 and NAME 2. A. Are NAME 1 and NAME 2 total strangers? B.
ARe they especially close? PROBE: As close or closer to eahc other as they are to you?

Local Network data:
The third part usually asks about relations among the alters. Do this
by looping over all possible combinations. If you are asking about a
symmetric relation, then you can limit your questions to the n(n-1)/2
cells of one triangle of the adjacency matrix:
Social Network Data

Snowball Samples:
• Snowball samples work much the same as ego-network modules,
and if time allows I recommend asking at least some of the basic
ego-network questions, even if you plan to sample (some of) the
people your respondent names.
• Start with a name generator, then any demographic or relational
questions.
• Have a sample strategy
• Random Walk designs (Klovdahl)
• Strong tie designs
• All names designs
• Get contact information from the people named
Snowball samples are very effective at providing network context
around focal nodes. New work on “Respondent Driven
Sampling (RDS) makes it possible to get good representation
even with initially biased seed nodes.
Social Network Data
http://www.respondentdrivensampling.org/reports/RDSrefs.htm

Snowball Samples:
Social Network Data

Complete Network data
• Data collection is concerned with all relations within a specified
boundary.
• Requires sampling every actor in the population of interest (all
kids in the class, all nations in the alliance system, etc.)
• The network survey itself can be much shorter, because you are
getting information from each person (so ego does not report on
alters).
• Two general formats:
• Recall surveys (“Name all of your best friends”)
• Check-list formats: Give people a list of names, have them
check off those with whom they have relations.
Social Network Data

Social Network Data
Network Data Sources:
Collecting network data
Complete network surveys require
a process that lets you link answers
to respondents.
•You cannot have anonymous
surveys.
•Recall:
•Need Id numbers & a
roster to link, or hand-
code names to find
matches
•Checklists
•Need a roster for people
to check through

Social Network Data
Network Data Sources:
Collecting network data
Complete network surveys require a process that lets you link answers to respondents.
•Typically you have a number of data tradeoffs:
•Limited number of responses.
•Eases survey construction & coding, lowers density & degree, which affects
nearly every other system-level measure.
•Some evidence that people try to fill all of the slots.
•Name check-off roster (names down a row or on screen, relations as check-
boxes).
•Easy in small settings or CADI, but encourages over-response.
•The “Amy Willis” Problem.
•Open recall list.
•Very difficult cognitively, requires an extra name-matching step in analysis.
•Still have to give slots in pen & paper, can be dynamic on-line.
Think carefully about what you want to learn from your survey items.

Social Network Data
Network Data Sources: Missing Data
Whatever method is used, data will always be incomplete. What are the
implications for analysis?
Example 1. People can name friends out of sample, but no way to match them (Add Health)
Ego
M
M
M
M
Out
Out
If the true network looks like this…
Ego
M
M
M
M
Out
Out
…you cannot distinguish it from this
Out
Out

Example 2:
Node population: 2-step neighborhood of Actor X
Relational population: Any connection among all nodes
1-step
2-step
3-step
F
1.1
1.2
1.3
1.4
1.5
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3.1
3.2
3.3
F 1 2 3 4 5 1 2 3 4 5 6 7 8 1 2 3
Full Full (0) Full (0)
Full Full
Full Full
F
F
(0)
F
(0)
Full (0) Unknown UK
UK
Full (0)
Social Network Data

Example 5
Node population: 2-step neighborhood from 3 focal actors
Relational population: All relations among actors
Full Full (0) Full (0)
Full Full
Full Full
Full
Full
(0)
Full
(0)
Full (0) Unknown UK
UK
Full (0)
Full
Focal
1-Step
2-Step
3-Step
Focal 1-Step 2-Step 3-Step
Social Network Data

Summary:
Data collection design & missing data affect the information at hand to draw
conclusions about the system. Everything we do from now on is built on some
manipulation of the observed adjacency matrix; so we want to understand what are valid
and invalid conclusions due to systematic distortions on the data.
Statistical modeling tools hold promise. We can build models of networks that account
for missing data – we are able to “fix” the structural zeros in or models by treating them
as given. This then lets us infer to the world of all graphs with that same missing data
structure. These models are very new, and not widely available yet….
Social Network Data

Local Network Analysis
Introduction
Local network analysis uses data from a simple ego-network survey. These might include
information on relations among ego’s contacts, but often not. Questions include:
Population Mixing
The extent to which one type of person is tied to another type of
person (race by race, etc.)
Local Network Composition
Peer behavior
Cultural milieu
Opportunities or Resources in the network
Social Support
Local Network Structure
Network Size
Density
Holes & Constraint
Concurrency
Dyadic behavior
Frequency of contact
Interaction content
Specific exchange behaviors
Dyadic Similarity

Introduction
Advantages
•Cost: data are easy to collect and can be sampled
•Methods are relatively simple extensions of common variable-based methods
social scientists are already familiar with
•Provides information on the local network context, which is often the primary
substantive interest.
•Can be used to describe general features of the global network context
•Population mixing, concurrency, exchange frequency, etc.
Disadvantages
•Treats each local network as independent, which is false.
The poor performance of ‘number of partners’ for predicting STD spread is
a clear example.
•Impossible to account for how position in a larger context affects local network
characteristics. “popular with who”
•If “structure matters”, ego-networks are strongly constrained to limit the
information you can get on overall structure

Network Composition
Perhaps the simplest network question is “what types of alters does ego interact with”?
Network composition refers to the distribution of types of people in your network.
Networks tend to be more homogeneous than the population. Using the
GSS, Marsden reports heterogeneity in Age, Education, Race and Gender.
He finds that:
•Age distribution is fairly wide, almost evenly distributed,
though lower than the population at large
•Homogenous by education
•Very homogeneous with respect to race
•Heterogeneous with respect to gender

Claude Fischer’s book To
Dwell Among Friends is a
classic study of urbanism
that makes good use of local
network data.
Age heterogeneity varies by
ego’s age and across urban
settings.
Network Composition

Claude Fischer’s book To
Dwell Among Friends is a
classic study of urbanism
that makes good use of local
network data.
Marital composition
similarly varies across
respondents and settings
Network Composition

Network Composition
Calculating network composition using “GSS style” data.
Generally you have a separate variable for each alter characteristic, and you can
construct items by summing over the relevant variables.
You would, for example, have variables on age of each alter such as:
Age_alt1 age_alt2 age_alt3 age_alt4 age_alt5
15 35 20 12 .
You get the mean age, then, with a statement such as:
meanage=mean(Age_alt1, age_alt2, age_alt3, age_alt4, age_alt5);
Difference from ego:
egodif =mean (abs(EgoAge – age_alt1), abs(EgoAge – age_alt2)…);
Be sure you know how the program you use (SAS, SPSS) deals with missing data.

Calculating local network information from global network data
We often want to construct local-level measures from global level data. This
involves a number of steps & opens more opportunities than GSS-style data:
1) Define the local neighborhood:
•Distance (1-step, 2-steps, what?)
•Direction of tie
•Sent, Received, or both?
2) Pull the relevant alters
3) Match the alters to the variables of interest
Once you decide on a type of tie, you need to get the information of interest in a
form similar to that in the example above.
A number of programs do this for you automatically (SPAN, R, etc.)
Network Composition

An example network:
All senior males from a
small (n~350) public HS.
Network Composition
SPAN will do this for you

Network Composition
Common composition measures:
Level measures:
•Mean of a given attribute (average income of alters)
•Proportion with a particular attribute (proportion who smoke)
•Counts (number of peers who have had sex)
Dispersion measures:
•Heterogeneity index (Racial heterogeneity)
•Index of dissimilarity
•Standard Deviation
•Absolute value of the differences
•Variable range of values
•Composition measures for multiple variables simultaneously
•Average correlation across all alters
•Euclidean / Mahalanobis distance measures
These are then treated like any other measure on your respondent.

Network Mixing
A common interest in network research is identifying how likely persons of
one category are to interact with people of another category.
Examples:
Race mixing: how likely are people of one race to interact with people
of another?
Sexual activity mixing: Are people with many partners likely to
associate with each other?
Neighborhood / location mixing: Are people likely to name friends
from the same neighborhood.
These questions can be answered by cross classifying the category of the
nominator with the category of the nominated in a “mixing matrix”.

Network Mixing
Race mixing in one of the Add Health schools

Network Mixing
White Black Hispan Asian Mix/Other
White 1099 128 53 0 231
Black 97 10218 1032 0 539
Hispanic 54 961 104 1 91
Asian 0 0 0 0 0
Mix/Other 191 560 66 0 106

Network Mixing
Working with mixing matrices:
•Group segregation index (Freeman 1972)
•Associations between rows and columns (valued relations)
•“Assortative mixing”
•Correlations or Q
•Log-linear models – See most recent AJS
Assessing chance levels depends on the data available. If you
have full network data you can look at density between groups,
without you can only focus on the sheer volume of ties (without
information on the size of the “target” groups)

Network Structure
While network structure data are limited, there are a number of features that can be of
interest, assuming you have data on the relations among ego’s contacts.
Basic arguments:
a) “structural amplification:” that some feature of the arrangement of ties
amplifies any peer effect of network composition (see Haynie’s paper)
b) “Network range effects:” that being connected to a diverse set of alters -- who
are not connected to each other – provides profitable returns. Granovetter’s
“Strength of Weak Ties”, Burt’s “Structural Holes”
 Familiar to students of social theory as the Tertius Gaudens argument
from Simmel
In both cases, we use the pattern of ties surrounding ego to characterize the local
structure. We start with volume measures, then move on to more complex pattern
measures.

Network Structure: volume
Network Size
0
5
10
15
20
25
30
0 1 2 3 4 5 6+
1985
2004
From time to time, most people discuss important matters with other people. Looking back
over the last six months—who are the people with whom you discussed matters important to
you? Just tell me their first names or initials. IF LESS THAN 5 NAMES MENTIONED,
PROBE: Anyone else?
X1985: 2.9
X2004: 2.1
Increase in
Social Isolation

Density is the average value of the relation among all pairs of ties.
= T / ((N*N-1)/2)
Density is usually calculated over the alters in the network.
R
2
3
1
4
5
1 2 3 4 5
1
2
3
4
5
1 1
1 1
1
D= 5 / ((5*4)/2)
= 5 / 10
= 0.5
2
3
1
4
5

What does Fischer have to
say about the density of
local nets (by context)?

GSS & Density:
0
5
10
15
20
25
30
35
40
45
<0.25 .25-.49 .50-.74 >.74
1985
2004

In general, dense networks should be more cohesive and we would expect that
“goods” will flow through the network more efficiently
•Social support & peer influence, for example, should be stronger in dense
networks
•Density is a volume measure, however, and can mask significant structural
differences:
These two networks have the same density but very different structures.
Most network analysis programs will calculate ego-network density directly.

Network Structure: Weak Ties & Structural Holes
“The Strength of Weak Ties”
In a classic article, Granovetter (1972) argues that for many purposes (such as
getting a job), the most useful network contacts are through “weak ties.”
This is because weak ties connect you to a more diverse set of alters,
increasing the ‘range’ of your network.
Your strong ties tend to be tied to each other, making them redundant for the
purposes of bringing information.
Essentially this argument works on a spurious relation. The key value of
weak ties is not in the weak affective bond, but in the structural location of the
ties. We can measure this directly, and Ron Burt provides a series of
measures for doing so.

Number of Contacts
Number
of
Non-Redundant
Contacts
Maximum
Efficiency
Minimum
Efficiency
Decreasing Efficiency
Increasing Efficiency

Effective Size
Conceptually the effective size is the number of people ego is connected
to, minus the redundancy in the network, that is, it reduces to the non-
redundant elements of the network.
Effective size = Size - Redundancy
  






j q
jq
iqm
p
1
Where j indexes all of the people that ego i has contact with, and q is every third person
other than i or j.
The quantity (piqmjq) inside the brackets is the level of redundancy between ego and a
particular alter, j.

Effective Size:
  






j q
jq
iqm
p
1
Piq is the proportion of actor i’s relations that are spent with q.
1
2
4 5
3 Adjacency
1 2 3 4 5
1 0 1 1 1 1
2 1 0 0 0 1
3 1 0 0 0 0
4 1 0 0 0 1
5 1 1 0 1 0
P
1 2 3 4 5
1 .00 .25 .25 .25 .25
2 .50 .00 .00 .00 .50
3 1.0 .00 .00 .00 .00
4 .50 .00 .00 .00 .50
5 .33 .33 .00 .33 .00

Effective Size:
  






j q
jq
iqm
p
1
mjq is the marginal strength of contact j’s relation with contact q.
Which is j’s interaction with q divided by j’s strongest interaction
with anyone. For a binary network, the strongest link is always 1
and thus mjq reduces to 0 or 1 (whether j is connected to q or not)
The sum of the product piqmjq measures the portion of i’s relation
with j that is redundant to i’s relation with other primary contacts.

Effective Size:
  






j q
jq
iqm
p
1
1
2
4 5
3
P
1 2 3 4 5
1 .00 .25 .25 .25 .25
2 .50 .00 .00 .00 .50
3 1.0 .00 .00 .00 .00
4 .50 .00 .00 .00 .50
5 .33 .33 .00 .33 .00
Working with 1 as ego, we get the following redundancy levels:
PM1jq
1 2 3 4 5
1 --- --- --- --- ---
2 --- .00 .00 .00 .25
3 --- .00 .00 .00 .00
4 --- .00 .00 .00 .25
5 --- .25 .00 .25 .00
Redundancy = 1
Effective size = 4-1 = 3

Effective Size:
  






j q
jq
iqm
p
1
1
2
4 5
3 When you work it out, in a binary network, redundancy reduces to the
average degree, not counting ties with ego of ego’s alters. Since the
average degree is simply another way to say density, we can calculate
redundancy as:
2t/n
where t is the number of ties (not counting ties to ego) and n is the
number of people in the network (not counting ego).
Meaning that effective size = n - 2t/n
UCINET, STRUCTURE, SPAN and PAJEK all calculate effective size

Effective
Ego Size Size: Efficiency
1 4 3 .75
2 2 1 .50
3 1 1 1.00
4 2 1 .50
5 3 1.67 .55
1
2
4 5
3
Efficiency is simply effective size divided by observed size. Taken from each ego’s
point of view, efficiency in this network would be:

1
2
4 5
3
Constraint
Conceptually, constraint refers to how much room you have to negotiate
or exploit potential structural holes in your network.
“..opportunities are constrained to the extent that (a) another of your
contacts q, in whom you have invested a large portion of your network
time and energy, has (b) invested heavily in a relationship with contact
j.” (p.54)
P
1 2 3 4 5
1 .00 .25 .25 .25 .25
2 .50 .00 .00 .00 .50
3 1.0 .00 .00 .00 .00
4 .50 .00 .00 .00 .50
5 .33 .33 .00 .33 .00
2









 
q
qj
iq
ij
ij p
p
p
C

Constraint
q
i j
pij
piq pqj
Cij = Direct investment (Pij) + Indirect investment (PiqPqj)
2









 
q
qj
iq
ij
ij p
p
p
C

Constraint
2









 
q
qj
iq
ij
ij p
p
p
C
Given the p matrix, you can get indirect constraint (piqpqj) by
simply squaring the matrix:
P
1 2 3 4 5
1 .00 .25 .25 .25 .25
2 .50 .00 .00 .00 .50
3 1.0 .00 .00 .00 .00
4 .50 .00 .00 .00 .50
5 .33 .33 .00 .33 .00
P*P
1 2 3 4 5
1 ... .083 .000 .083 .250
2 .165 ... .125 .290 .125
3 .000 .250 ... .250 .250
4 .165 .290 .125 ... .125
5 .330 .083 .083 .083 ...
1
2
4 5
3

Constraint
2









 
q
qj
iq
ij
ij p
p
p
C
Total constraint between any two people then is:
C = (P + P2)##2
Where P is the normalized adjacency matrix, and ## means to
square the elements of the matrix.

Burt (2004) AJS 110:349-399

Local Network Models: Modeling Issues
Local Network modeling issues
Case independence
In very clustered settings, the alters that each person names will overlap.
This will lead to non-independence among the cases.
•If you have enough cases or over time data, you can use random or
fixed effect models
•If you know the names of alters, you can link them to build in a
direct network autocorrelation effect.
Small network effects
Be aware of the size of your networks. Substantively, having 50% white
networks means something different in a net of size 2 vs a net of size 10.
I often suggest interactions to check for these kinds of effects
Dealing with isolates
•Isolated nodes have no network alters, so none of these measures apply.
Depending on the context, you can either leave them out of the analysis,
or use interaction terms to selectively apply the measures of interest.

•Selection
•That some unobserved factor, z, creates both friendships and the
outcome of interest.
•Endogeneity
•That the causal order of peer relations and outcomes is reversed.
Peers do not cause Y, but Y causes friendship relations

Selection
•What do we know about how friendships form?
•Opportunity / focal factors
- Being members of the same group
- In the same class
- On the same team
- Members of the same church
•Structural Relationship factors
- Reciprocity
- Social Balance
•Behavior Homophily
- Smoking
- Drinking

Selection
How to correct this problem?
•Essentially, this is an omitted variable problem, and the obvious “solution” is
been to identify as many potentially relevant alternative variables as you can find.
•Sensitivity measures (see Ken Frank’s work here)
•Propensity score matching
•Individual-level fixed effect models
•Substantively you only look at change in Y as a function of change in X,
holding constant (because dummied out) any individual level effect.
•This works, but it’s drastic. Any endogenous effect of networks on the
self are essentially removed

Endogeneity
Estimated:
Y = b0 + b1(P) + e
where P = some peer function.
But the actual model may really be:
P = b’
0 + b’
1(f(Y)) + e

Algebraically the relation between y and p should be direct translation of the
coefficients since:
Endogeneity Does it matter?
)
(
:
Then
)
(
:
If
1
1
0 1
1
y
p
p
b
b
y
b
b
b
o





The statistical problem of endogeneity is that when you estimate
b’1, it does not equal 1/b1, because of our assumptions about x, and hence e. There
are other models that make different assumptions, where this direction is irrelevant.
But they are uncommon and hard to work with in the multivariate context.
(see Joel H. Levine, Exceptions are the Rule, for a full discussion of this)

Possible solutions:
• Theory: Given what we know about how friendships form,
is it reasonable to assume a bi-directional cause? That is,
work through the meeting, socializing, etc. process and ask
whether it makes sense that Y is a cause of P.
• Models:
-Time Order. We are on somewhat firmer ground if P
precedes Y in time.
- Simultaneous Equation Models. Model both the
friendship pattern and the outcome of interest
simultaneously. Difficult to identify “instruments” or to
specify orders that do not logically make the model
inestimable.

Haynie asks whether peers matter for delinquent behavior, focusing on:
a) the distinction between selection and influence
b) the effect of friendship structure on peer influence
Two basic theories underlie her work:
a) Hirchi’s Social Control Theory
•Social bonds constrain otherwise criminal behavior
•The theory itself is largely ambivalent toward direction of network
effects
b) Sutherland’s Differential Association
•Behavior is the result of internalized definitions of the situation
•The effect of peers is through communication of the appropriateness
of particular behaviors
Haynie adds to these the idea that the structural context of the network can “boost” the
effect of peers: (a) so transmission is more effective in locally dense networks and (b)
the effect of peers is stronger on central actors.
Local Network Models: Peer influence example

Local Network Models: Peer influence example

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