1. Social Network Analysis
Introduction to
Social Network Analysis
Columbia University
November 2004
James Moody
Ohio State University

2. Introduction
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
These patterns of connection form a social space, that can be seen in multiple contexts:

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

4. High Schools as Networks
Introduction
High Schools as Networks

7. 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.

8. Introduction Why do Networks Matter? Local vision
Consider the following example. Here we have sampled respondents (red dots) reporting on their interaction with romantic partners. A classic local network module would ask about their characteristics and behaviors, then attempt to relate those characteristics to ego’s behavior. All of these sampled nodes have the exact same number of partners.

9. Introduction Why do Networks Matter? Local vision
But these nodes are situated in dramatically different parts of the real underlying global network. Here some of them (lower left) are truly local isolates, but most are embedded in a larger network structure.
These are real data from Add Health, on romantic involvement.

10. Social network analysis is:
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.

11. Introduction
Introduction
Social Network data
Basic data elements
Network data sources
Local (ego) Network Analysis
Network Composition
Network Structure
Local Network Models
Complete Network Analysis
Exploratory Analysis
Network Connections
Network Macro Structure
Stochastic Network Analyses
Social Network Software

12. “Networks as Variables” approaches
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?
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.

13. We represent actors with points and relations with lines.
Social Network 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

14. Social Network data consists of two linked classes of data:
Basic Data Elements
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)

15. 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
Undirected, Valued
Directed, Valued 1 3 4 2
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.

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

17. We can examine networks across multiple levels:
Social Network Data
Basic Data Elements: Levels of analysis
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

18. We can examine networks across multiple levels:
Social Network Data
Basic Data Elements: Levels of analysis
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

19. Social Network Data
Basic Data Structures
Working with pictures.
No standard way to draw a sociogram: each of these are equal:

20. Social Network Data Basic Data Structures
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.
Start w. simply opening CNAT.
Then look at ZOOM, Refresh, etc.
Then re-draw the network based on Degree
Then redraw from a particular ego.

21. a b c d e 1 a b c d e 1 Social Network Data Basic Data Structures a b
From pictures to matrices a b c e d a b c e d
Undirected, binary
Directed, binary a b c d e 1 a b c d e 1

22. a b c d e 1 Arc List Adjacency List a b b a b c c b c d c e d c d e
Social Network Data
Basic Data Structures
From matrices to lists a b c d e 1
Arc List
Adjacency List
a b
b a
b c
c b
c d
c e
d c
d e
e c
e d
a b
b a c
c b d e
d c e
e c d

23. Communication, friendship, giving orders, sending email.
Social Network Data
Basic Data Elements: Modes
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 .
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).

24. People as members of groups People as authors on papers
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

25. Breiger: 1974 - Duality of Persons and Groups
Social Network Data
Basic Data Elements: Modes
Breiger: 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

26. Social Network Data Basic Data Elements: Modes
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)

27. Social Network Data Basic Data Elements: Modes
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)

28. Social Network Data Basic Data Elements: Modes
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.

29. Social Network Data Basic Data Elements: Modes
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.

30. A = Social Network Data Working with two-mode data
Basic Data Elements: Modes
Working with two-mode data
A person-to-group adjacency matrix is rectangular, with persons down rows and groups across columns
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. A B C D E F
A =

31. A = Social Network Data Working with two-mode data
Basic Data Elements: Modes
Working with two-mode data
Compare persons A and F:
Person A is in 1 group, Person F is in two groups, and they are in no groups together. A B C D E F
A = S
A = 1
F = 2
AF = 0
Or persons D and F:
Person D is in 4 groups, Person F is in two groups, and they are in 2 groups together. S
D = 4
F = 4
DF = 2

32. Similarly for Groups: A = Social Network Data
Basic Data Elements: Modes
Working with two-mode data
Similarly for Groups: A B C D E F
A =
1 2 1•2 A B C D E F 
Group 1 has 2 members, group 2 has 2 members and they overlap by 1 members (C).

33. Persons-to-Persons Groups-to-Groups Social Network Data
Basic Data Elements: Modes
Working with two-mode data
In general, you can get the overlap for any pair of groups / persons by summing the multiplied elements of the corresponding rows/columns of the persons-to-groups adjacency matrix. That is:
Persons-to-Persons
Groups-to-Groups

34. AT = A = Social Network Data Working with two-mode data
Basic Data Elements: Modes
Working with two-mode data
One can get these 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. A B C D E F
A =
A B C D E F
AT =

35. P = A(AT) G = AT(A) (5x6) (6x5) A * AT = P AT * A = P (6x5)(5x6) (6x6)
Social Network Data
Basic Data Elements: Modes A B C D E F
A B C D E F
P = A(AT)
G = AT(A)
A =
AT =
(5x6)
(6x5)
A * AT = P
(6x5)(5x6) (6x6) P
A B C D E F A B C D E F
AT * A = P
(5x6) 6x5) (5x5) G

36. The resulting network: 1) Is always symmetric
Social Network Data
Basic Data Elements: Modes
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)
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 or SPSS

37. Social Network Data Existing Sources of Social Network Data:
Network Data Sources: Existing data sources
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
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.

38. Existing Sources of Social Network Data: 1-Mode Data
Network Data Sources: Existing data sources
Existing Sources of Social Network Data:
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

39. Existing Sources of Social Network Data: 1-Mode Data
Network Data Sources: Existing data sources
Existing Sources of 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

40. Boundary Specification Problem
Social Network Data
Network Data Sources: Collecting 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 Marsden (19xx) 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:

41. Social Network Data
Network Data Sources: Collecting network data
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.
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?)

42. Boundary Specification Problem
Social Network Data
Network Data Sources: Collecting network data
Boundary Specification Problem
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

43. When using a survey, common to use an “ego-network module.”
Social Network Data
Network Data Sources: Collecting network data
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.
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

44. Electronic Small World name generator:
Social Network Data
Network Data Sources: Collecting network data
Electronic Small World name generator:

45. The second part usually asks a series of questions about each person
Social Network Data
Network Data Sources: Collecting 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?”
ESWP example:
Will generate N x (number of attributes) questions to the survey

46. Social Network Data Local Network data:
Network Data Sources: Collecting network data
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: 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?

47. Social Network Data Local Network data:
Network Data Sources: Collecting network data
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:

48. Random Walk designs (Klovdahl) Strong tie designs All names designs
Social Network Data
Network Data Sources: Collecting 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. Detailed treatments of snowball sampling estimates are given in Frank ().

49. Social Network Data Snowball Samples:
Network Data Sources: Collecting network data
Snowball Samples:

50. Recall surveys (“Name all of your best friends”)
Social Network Data
Network Data Sources: Collecting 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.

51. 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

52. 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. Ego is a matchable person in the School Un
Out
Out
Out Un Un M
Ego M
Ego M M M M M M
True Network
Observed Network

53. Social Network Data Network Data Sources: Missing Data
Example 2. Ego is not on the school roster M M Un M Un M M M M M M M Un Un Un
True Network
Observed Network

54. Social Network Data Example 3:
Network Data Sources: Missing Data
Example 3:
Node population: 2-step neighborhood of Actor X
Relational population: Any connection among all nodes F
Full
Full (0)
Full (0) 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
Full
Full
Full (0)
1-step F
(0)
Full
Full UK
2-step
3-step F
(0)
Full (0)
Unknown UK

55. Social Network Data Example 4
Network Data Sources: Missing Data
Example 4
Node population: 2-step neighborhood of Actor X
Relational population: Trace, plus All connections among 1-step contacts F
1-step
2-step
3-step
Full
Full (0)
Full (0) 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
Full
Full
Full (0) F
(0)
Full
Unknown UK F
(0)
Full (0)
Unknown UK

56. Social Network Data Example 5.
Network Data Sources: Missing Data
Example 5.
Node population: 2-step neighborhood of Actor X
Relational population: Only tracing contacts F
Full
Full (0)
Full (0) 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
Unknown
Full
Full (0)
1-step F
(0)
Full
Unknown UK
2-step
3-step F
(0)
Full (0)
Unknown UK

57. Social Network Data Example 6
Network Data Sources: Missing Data
Example 6
Node population: 2-step neighborhood from 3 focal actors
Relational population: All relations among actors
Focal
1-Step
2-Step
3-Step
Focal
Full
Full
Full (0)
Full (0)
1-Step
Full
Full
Full
Full (0)
Full
(0)
Full
Full UK
2-Step
3-Step
Full
(0)
Full (0)
Unknown UK

58. Social Network Data Example 7.
Network Data Sources: Missing Data
Example 7.
Node population: 1-step neighborhood from 3 focal actors
Relational population: Only relations from focal nodes
Focal
1-Step
2-Step
3-Step
Focal
Full
Full
Full (0)
Full (0)
1-Step
Full
Unknown
Unknown
Full (0)
Full
(0)
Unknown
Unknown UK
2-Step
3-Step
Full
(0)
Full (0)
Unknown UK

59. 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 Structural
Network Size
Density
Holes & Constraint
Concurrency
Dyadic behavior
Frequency of contact
Interaction content
Specific exchange behaviors

60. Local Network Analysis
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, activity distribution (limited)
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

61. Local Network Analysis
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 (30% differ by less than a year, on average)
Very homogeneous with respect to race (96% are single race)
Heterogeneous with respect to gender

62. Local Network Analysis
Network Composition
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.

63. Local Network Analysis
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

64. Local Network Analysis
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
You get the mean age, then, with a statement such as:
meanage=mean(Age_alt1, age_alt2, age_alt3, age_alt4, age_alt5);
Be sure you know how the program you use (SAS, SPSS) deals with missing data.

65. Local Network Analysis
Network Composition
Calculating local network information from global network data
Define the local neighborhood:
Distance (1-step, 2-steps, what?)
Direction of tie
Sent, Received, or both?
Pull the relevant alters
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.

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

67. Local Network Analysis
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

68. Local Network Analysis
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”.

69. Local Network Analysis
Network Mixing
Race mixing in one of the Add Health schools

70. Local Network Analysis
Network Mixing
White Black Hispan Asian Mix/Other
White
Black
Hispanic
Asian
Mix/Other

71. Local Network Analysis
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
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)

72. Local Network Analysis
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:
“structural amplification:” that some feature of the arrangement of ties amplifies any peer effect of network composition (see Haynie’s paper)
“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.

73. Distribution of total network size, GSS 1985
Local Network Analysis
Network Structure: volume
Network Size
Distribution of total network size, GSS 1985

74. Local Network Analysis
Network Structure: volume
Network Size by:
Age:
Drops with age at an increasing rate. Elderly have few close ties.
Education:
Increases with education. College degree ~ 1.8 times larger
Sex (Female):
No gender differences on network size.
Race:
African Americans networks are smaller (2.25) than White Networks (3.1).

75. Local Network Analysis
Network Structure: volume
What does Fischer have to say about the size of local nets (by context)?

76. D= 5 / ((5*4)/2) = 5 / 10 = 0.5 Local Network Analysis
Network Structure: volume
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. 2 1 2 3 4 5 2 3 1 4 5 1 R 3 4 5
D= 5 / ((5*4)/2)
= 5 / 10
= 0.5

77. Local Network Analysis
Network Structure: volume
What does Fischer have to say about the density of local nets (by context)?

78. Local Network Analysis
Network Structure: volume
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, for example, have the same density but very different structures.
Most network analysis programs will calculate ego-network density directly.

79. Local Network Analysis
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.

80. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Maximum
Efficiency
Decreasing Efficiency
Number of Non-Redundant Contacts
Increasing Efficiency
Minimum
Efficiency
Number of Contacts

81. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
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
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.

82. 3 2 1 5 4 Local Network Analysis Effective Size:
Network Structure: Weak Ties & Structural Holes
Effective Size:
Piq is the proportion of actor i’s relations that are spent with q. 3 2
Adjacency P 1 5 4

83. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Effective Size:
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.

84. Effective Size: 3 2 1 5 4 Redundancy = 1 Effective size = 4-1 = 3
Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Effective Size: 3 2
Working with 1 as ego, we get the following redundancy levels: 1 P
PM1jq 5 4
Redundancy = 1
Effective size = 4-1 = 3

85. Effective Size: 3 2 1 5 4 Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Effective Size: 3 2
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 1 5 4
UCINET, STRUCTURE, SPAN and PAJEK all calculate effective size

86. Ego Size Size: Efficiency 1 4 3 .75 2 2 1 .50 3 1 1 1.00 4 2 1 .50
Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Efficiency is simply effective size divided by observed size. Taken from each ego’s point of view, efficiency in this network would be:
Effective
Ego Size Size: Efficiency 3 2 1 5 4

87. 3 2 1 5 4 Local Network Analysis Constraint
Network Structure: Weak Ties & Structural Holes
Constraint
Conceptually, constraint refers to how much room you have to negotiate or exploit potential structural holes in your network. 3 2
“..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) 1 5 4 P

88. q i j Constraint pij piq pqj
Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Constraint q i j
pij
piq
pqj
Cij = Direct investment (Pij) + Indirect investment (PiqPqj)

89. Local Network Analysis
Network Structure: Weak Ties & Structural Holes 3 2
Constraint 1 5 4
Given the p matrix, you can get indirect constraint (piqpqj) by simply squaring the matrix:
P*P P

90. Total constraint between any two people then is:
Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Constraint
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.

91. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Hierarchy
Conceptually, hierarchy (for Burt) is really the extent to which constraint is concentrated in a single actor. It is calculated as:

92. Hierarchy 3 2 1 5 4 H=.514 Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Hierarchy 3 2 1 C C: 5 4
H=.514

93. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Burt (2004) AJS 110:

94. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Burt (2004) AJS 110:

95. Local Network Analysis
Network Structure: Weak Ties & Structural Holes
Burt (2004) AJS 110:

96. Local Network Analysis
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.

97. Local Network Analysis
Local Network Models: Modeling Issues
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

98. Local Network Analysis
Local Network Models: Modeling Issues
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

99. Local Network Analysis
Local Network Models: Modeling Issues
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

100. where P = some peer function. But the actual model may really be:
Local Network Analysis
Local Network Models: Modeling Issues
Endogeneity
Estimated:
Y = b0 + b1(P) + e
where P = some peer function.
But the actual model may really be:
P = b’0 + b’1(Y) + e

101. Local Network Analysis
Local Network Models: Modeling Issues
Endogeneity
Does it matter?
Algebraically the relation between y and p should be direct translation of the coefficients since:
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)

102. Local Network Analysis
Local Network Models: Modeling Issues
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.

103. Local Network Analysis
Local Network Models: Peer influence example
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.

104. Local Network Analysis
Local Network Models: Peer influence example

105. Local Network Analysis
Local Network Models: Peer influence example