Local Network Analysis
Local Network Analysis
a)
b)
c)
d)
Basic Definitions
Network Composition
Network Structure
Social Capital
a) Bridging Local & Global?
b) Position Generator stuff
If time/interest: some software stuff?
Local Network Analysis
Basic Definitions
Local networks – The network incident to a focal individual.
KA
BN
PP
PC
DC
CM
HF
CF
Local Network Analysis
Basic Definitions
Local networks – The network incident to a focal individual.
Local Network Analysis
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
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
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
Local Network Analysis
General Questions
Questions that you can ask / answer
Mixing
The extent to which one type of person is tied to
another type of person (race by race, etc.)
Aspects of the local context:
Peer delinquency
Cultural milieu
Opportunities
Social Support:
Extent of resources (and risks) present in a type of
network environment.
Structural context (next class)
Local Network Analysis
Mechanics
Calculating local network information.
1) From data, such as the GSS, which has ego-reported information on alter
2) From global network data, such as Add Health, where you have self-reports on
alters behaviors.
Local Network Analysis
Mechanics
Calculating local network information 1: GSS style data.
This is the easiest situation. Here you have a separate variable for each alter
characteristic, and you can construct density 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);
Be sure you know how the program you use (SAS, SPSS) deals with missing data.
Local Network Analysis
Mechanics
Calculating local network information 1: GSS style data.
In addition to the moments of the distribution (mean, std. dev, etc.) we are often
interested in “mixing” and use a “mixing matrix.” This is just a cross tab of ego’s
characteristics by alters' characteristics:
W
White
Black
Hispanic
Other
B
H
O
Local Network Analysis
Mechanics
Calculating local network information 1: GSS style data.
Data from Sexual
Mixing patterns in
China
Local Network Analysis
Mechanics
Calculating local network information 1: GSS style data.
9 – 12 schools (N of schools: 51)
9th
Graders
10th
Graders
11th
Graders
12th
Graders
9th
Graders
80.42
10.23
5.57
3.79
10th
Graders
10.49
69.49
12.35
7.67
11th
Graders
5.68
12.52
67.13
14.67
12th
Graders
3.42
7.76
14.96
73.86
Local Network Analysis
Mechanics
Calculating local network information 2: From a global network.
There are multiple options when you have complete network information.
Type of tie:
Sent, Received, or both?
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.
Local Network Analysis
Mechanics
Calculating local network
information from a global network.
An example network:
All senior males from a
small (n~350) public HS.
Local Network Analysis
Mechanics
Suppose you want to identify ego’s friends, calculate what proportion of ego’s female friends
are older than ego, and how many male friends they have (this example came up in a model of
fertility behavior).
You need to:
•Construct a dataset with
(a) ego's id. This allows you to link each person in the network.
(b) age of each person,
(c) the friendship nominations variables.
•Then you need to:
a) Identify ego's friends
b) Identify their age
c) compare it to ego's age
d) count it if it is greater than ego's.
There is a SAS program described in the exercise that shows you how to do this kind of
work, using the graduate student network data.
Local Network Analysis
Composition - homophily
Highly cited: Google Scholar 6258 as of 1.29.15
Local Network Analysis
Composition - homophily
Key bit here is a duality between settings and relations
Local Network Analysis
Composition - homophily
“Status homophily includes the major socio-demographic dimensions
that stratify society—ascribed characteristics like race, ethnicity, sex, or
age, and acquired characteristics like religion, education, occupation, or
behavior patterns.
Value homophily includes the wide variety of internal states presumed to
shape our orientation toward future behavior.”
Local Network Analysis
Composition - homophily
This is the proportion
different. So 4.7% of
pairs in 1985 were
different race,
compared to 9.8% in
2004.
Smith et al “Social
Distance in the
U.S.A: Sex, Race,
Religion, Age, and
Education:
Note this is *raw*
rates, not adjusted for
chance.
Local Network Analysis
Composition - homophily
Use matched sample
logit model & predict
tie as a function of
social distance.
Smith et al “Social
Distance in the
U.S.A: Sex, Race,
Religion, Age, and
Education”
Local Network Analysis
Composition - homophily
Smith et al “Social Distance in the U.S.A: Sex, Race, Religion, Age, and Education”
Local Network Analysis
Composition - homophily
Local Network Analysis
Composition - homophily
Local Network Analysis
Composition - homophily
Local Network Analysis
Composition - homophily
Local Network Analysis
Composition - homophily
Coefficient for logit model of any cross sex ties, by setting:
highly contextual
Local Network Analysis
Composition - homophily
DiPrete, Thomas A., Andrew Gelman Tyler McCormic Julien Teitler & Tian Zheng. “Segregation in Social Networks based on Acquaintanceship and Trust” American Journal of Sociology 116: 1234-1283
Local Network Analysis
Composition - homophily
DiPrete, Thomas A., Andrew Gelman Tyler McCormic Julien Teitler & Tian Zheng. “Segregation in Social Networks based on Acquaintanceship and Trust” American Journal of Sociology 116: 1234-1283
Local Network Analysis
Composition - homophily
DiPrete, Thomas A., Andrew Gelman Tyler McCormic Julien Teitler & Tian Zheng. “Segregation in Social Networks based on Acquaintanceship and Trust” American Journal of Sociology 116: 1234-1283
Local Network Analysis
Composition - homophily
DiPrete, Thomas A., Andrew Gelman Tyler McCormic Julien Teitler & Tian Zheng. “Segregation in Social Networks based on Acquaintanceship and Trust” American Journal of Sociology 116: 1234-1283
Local Network Analysis
Composition - homophily
Network Scaleup is an old
technique, here I
provide a
calculator for
doing it:
http://www.soc.duke.edu/~jmoody77/Hydra/scaleupcalc.htm
Local Network Analysis
Composition - homophily
Key issue for
homophily is
“choice” versus
AJS Volume 115 Number 2 (September 2009): 405–50
Local Network Analysis
Network structure
Homophily / diversity are functions related to the composition of the network, we
are often interested as well in the structure of the local network.
-Size
-Density
-Tie patterns (reciprocity, transitivity)
- local connectivity (number of components, structural holes)
Local Networks
GSS Networks
Network Size
30
X1985: 2.9
X2004: 2.1
25
20
Increase in
Social Isolation
1985
2004
15
10
5
0
0
1
2
3
4
5
6+
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?
Local Networks
GSS Networks
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).
Local Networks
GSS Networks
Network Density
Recall that density is the average value of the relation among all pairs
of ties. Here, density is only calculated over the alters in the network.
2
1
R
3
1
3
4
5
1 2 3 4 5
2
4
5
D=0.5
1
2
3
4
5
1
1
1
1
1
Local Networks
GSS Networks
45
40
35
30
25
20
15
10
5
0
1985
2004
<.25
.25-.49
.50-.74
Density
>.74
Local Networks
GSS Networks
Network Structure Summary
(Marsden, based on GSS 1985)
Local Networks
GSS Networks
Non-Kin 1985
Non-Kin 2004
Kin 1985
Kin 2004
Local Networks
To Dwell Among Friends
Network Size
One of the best-known
books on the stuff of
local social networks;
framed as an attempt to
test the idea that cities
are socially isolating.
Local Networks
To Dwell Among Friends
Network Composition
One of the best-known
books on the stuff of
local social networks;
framed as an attempt to
test the idea that cities
are socially isolating.
Local Networks
To Dwell Among Friends
Network Composition (non-kin)
One of the best-known
books on the stuff of
local social networks;
framed as an attempt to
test the idea that cities
are socially isolating.
Local Networks
To Dwell Among Friends
One of the best-known
books on the stuff of
local social networks;
framed as an attempt to
test the idea that cities
are socially isolating.
Fischer’s Work.
What does Fischer have to
say about the size of local
nets (by context)?
Fischer’s Work.
What does Fischer have to
say about the density of
local nets (by context)?
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Granovetter argues that, under many circumstances, strong
ties are less useful than weak ties. Why?
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Key element is
the correlation
between tie
strength and
structure:
The argument
rests on the
association, the
mechanism is
diffusion of
information
/opportunity
through nonredundant ties
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Burt. Structural Holes
Similar idea to SWT: Your ties matter because of who
your connects are not connected to.
What is (for Burt) Social Capital?
Relationships with other players
Why does it matter?
“Social capital is as important as
competition is imperfect and investment
capital is abundant.”
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
The basic notion of bridging a structural hole involves connecting people who are
not otherwise connected.
These two networks, for example, the focal nodes (red) have the same number of
ties, but the node on the left spans more structural holes than the node on the right.
As such, we would expect the node to have access to unique information or be able
to play the one side of his network against the other.
The key to power & effectiveness, according to Burt, often depends on having ties
that are non-redundant.
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Intuitively, a tie is redundant if it connects you to somebody you are
already connected to. The most efficient networks will be those where each
tie takes you to an entirely new social world. The limit, of course, is when
every person you are connected to is unconnected to anyone else.
Number of Non-Redundant Contacts
Maximum
Efficiency
Decreasing Efficiency
Increasing Efficiency
Number of Contacts
Minimum
Efficiency
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Intuitively, a tie is redundant if it connects you to somebody you are
already connected to. The most efficient networks will be those where each
tie takes you to an entirely new social world.
Redundant &
constrained
Locally Redundant
Unconstrained
Low
High
Constraint
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Burt’s idea discussion network
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Burt’s idea discussion network
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
The results show a
strong effect of
network constraint
on salary, evaluation
and promotion,
independent of the
job/age
characteristics
related to human
capital explanations.
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
The results show a
strong effect of
network constraint
on salary, evaluation
and promotion,
independent of the
job/age
characteristics
related to human
capital explanations.
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Social Network Mechanisms
Strength of Weak Ties & Structural Holes
Why does this work? 2.5 mechanisms
1) Social Balance & Homophily: A Friend of a Friend is a Friend
• We are most similar to those we are closest to, so strong ties
tend to be redundant & carry no information
2) Information Arbitrage & Control: Simmel’s Tertius Gaudens
•Power comes from the ability to control a resource others need
•Brokers can exploit their holes (up until a limit of legitimacy!)
2.5) Social Skills Training
•Brokers have to navigate multiple worlds, which is hard.
•Doing this for long hones a particular kind of “people skill” that
leads to success. This is Burt’s most recent argument.
Structural Holes & Weak Ties
Calculations
Calculating the measures
Burt discusses 4 related aspects of a network:
1) Effective Size
2) Efficiency
3) Constraint
4) Hierarchy
Structural Holes & Weak Ties
Calculations
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
Structural Holes & Weak Ties
Calculations
Effective Size
Burt’s measures for effective size is:


j 1  q piqm jq 


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.
Structural Holes & Weak Ties
Calculations
Effective Size:


j 1  q piqm jq 


Piq is the proportion of actor i’s relations that are spent with q.
3
2
1
4
5
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
.50
1.0
.50
.33
2
.25
.00
.00
.00
.33
3
.25
.00
.00
.00
.00
4
.25
.00
.00
.00
.33
5
.25
.50
.00
.50
.00
Structural Holes & Weak Ties
Calculations
Effective Size:


j 1  q piqm jq 


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 that is, the adjacency matrix).
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.
Structural Holes & Weak Ties
Calculations
Effective Size:
3
2
Working with 1 as ego, we get the following redundancy levels:
1
4


j 1  q piqm jq 


P
5
1
2
3
4
5
1
.00
.50
1.0
.50
.33
2
.25
.00
.00
.00
.33
3
.25
.00
.00
.00
.00
4
.25
.00
.00
.00
.33
5
.25
.50
.00
.50
.00
1
2
3
4
5
1
-----------
PM1jq
2
3
--- --.00 .00
.00 .00
.00 .00
.25 .00
4
--.00
.00
.00
.25
5
--.25
.00
.25
.00
Sum=1, so
Effective size = 4-1 = 3.
Structural Holes & Weak Ties
Calculations
Effective Size:
3
2
1
4
5


j 1  q piqm jq 


When you work it out, redundancy reduces to the
average degree, not counting ties with ego of ego’s
alters.
Node Degree
2
1
3
0
4
1
5
2
Mean: 4/4 = 1
Structural Holes & Weak Ties
Calculations
Effective Size:
3
2
1
4
5


j 1  q piqm jq 


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
Structural Holes & Weak Ties
Calculations
Efficiency is the effective size divided by the
observed size.
3
2
1
4
5
Node
1
2
3
4
5
Size
4
2
1
2
3
Effective
Size:
3
1
1
1
1.67
Efficiency
.75
.5
1.0
.5
.55
Structural Holes & Weak Ties
Calculations
Constraint
3
2
Conceptually, constraint refers to how much
room you have to negotiate or exploit
potential structural holes in your network.
1
4
5
“..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)
Structural Holes & Weak Ties
Calculations
Constraint
3
2
1
4


Cij   pij   piq pqj 
q


5
P
1
2
3
4
5
1
.00
.50
1.0
.50
.33
2
.25
.00
.00
.00
.33
3
.25
.00
.00
.00
.00
4
.25
.00
.00
.00
.33
5
.25
.50
.00
.50
.00
2
Structural Holes & Weak Ties
Calculations
Constraint
q


Cij   pij   piq pqj 
q


2
piq
i
pqj
pij
Cij = Direct investment (Pij) + Indirect investment
j
Structural Holes & Weak Ties
Calculations
3
2
Constraint
1
5
4


Cij   pij   piq pqj 
q


2
Given the p matrix, you can get indirect constraint (piqpqj) with the
2-step path distance.
P
1
2
3
4
5
1
.00
.50
1.0
.50
.33
2
.25
.00
.00
.00
.33
3
.25
.00
.00
.00
.00
4
.25
.00
.00
.00
.33
5
.25
.50
.00
.50
.00
1
1
2
3
4
5
...
.165
.000
.165
.330
2
.083
...
.250
.290
.083
P*P
3
.000
.125
...
.125
.083
4
.083
.290
.250
...
.083
5
.250
.125
.250
.125
...
Structural Holes & Weak Ties
Calculations
Constraint


Cij   pij   piq pqj 
q


2
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.
Structural Holes & Weak Ties
Calculations
Constraint


Cij   pij   piq pqj 
q


.00
.67
1.0
.67
.66
P+P2
.33 .25
.00 .13
.25 .00
.29 .13
.41 .08
.33
.29
.25
.00
.41
.50
.63
.25
.63
.00
2
.00
.44
1.0
.44
.44
Cij
.11 .06
.00 .02
.06 .00
.08 .02
.17 .01
.11
.08
.06
.00
.17
.25
.39
.06
.39
.00
C
.53
Structural Holes & Weak Ties
Calculations
Hierarchy
Conceptually, hierarchy (for Burt) is really the extent to
which constraint is concentrated in a single actor. It is
calculated as:
 Cij   Cij 
j  C N  ln  C N 
 

H 
N ln( N )
Structural Holes & Weak Ties
Calculations
Hierarchy
3
 Cij   Cij 
j  C N  ln  C N 




H
N ln( N )
2
1
4
5
C:
 Cij 


C N 
2
3
4
5
.11 .06 .11 .25
.83 .46 .83 1.9
H=.514
C
.53
Local Networks
Social Capital
(soc journals, as of 2001)
Local Networks
Social Capital
Relational capital – ability to use social ties to access resources.
Position generator:
Local Networks
Summary
Advantages:
• Can collect as random sample
• Informative for most homophily/segregation measures
• (though comes with reporting bias)
• Structural information for local patterns can be collected
• …and with statistical models can generalize to larger structure
• Position generators & scale-up methods are probably most robust
to particular survey design issues
Disadvantages:
• Ego-network module is complex to implement in a survey
• Fully dependent on ego self-report
• No path-level information