Transcript No Slide Title

Peer Influence
Background:
•long standing research interest in how our relations shape our
attitudes and behaviors.
•One mechanism is that people, largely through conversation, change
each others opinions
•This implies that position in a communication network should be
related to attitudes.
Freidkin & Cook:
•A formal model of influence, based on communication
Cohen:
•An application of a similar peer influence model relating to
adolescent college aspirations
Haynie: Peer influence among adolescents
Topics Covered:
•Basic Peer influence
•Selection and influence
•Dynamic mix of above
•Dyad models
Addendum A new statistic for determining the number of groups in a network.
The basic point still holds: finding groups takes a good deal of judgement.
BUT, some statistics can help.
1) The basic output from PROC CLUSTER
2) A new measure proposed in Molecular Biology: Modularity
Addendum A new statistic for determining the number of groups in a network.
Proc cluster gives you a statistic for the basic “fit” of a cluster solution.
This statistic varies depending on the method used, but is usually something
like an R2. Consider this dendrogram:
Addendum A new statistic for determining the number of groups in a network.
Proc cluster gives you a statistic for the basic “fit” of a cluster solution.
This statistic varies depending on the method used, but is usually something
like an R2. Consider this dendrogram:
The SPRSQ and the RSQ are your fit statistics.
Addendum A new statistic for determining the number of groups in a network.
A sharp change in the statistic is your best indicator.
0.9
RSQ
0.8
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0.1
SPRSQ
0
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Addendum A new statistic for determining the number of groups in a network.
Modularity:
 ls  d s  2 
M     
L
2
L

 
S 1 

Nm
M is the modularity score
S indexes each group (“module”)
ls is the number of lines in group s
L is the total number of lines
ds is the sum of the degrees of the nodes in s
Nm is the number of groups
Addendum A new statistic for determining the number of groups in a network.
Modularity:
Addendum A new statistic for determining the number of groups in a network.
Modularity:
5
Pii=.3
Pii=.2
Pii=.1
10
20
Basic Peer Influence Model
Attitudes are a function of two sources:
a) Individual characteristics
•Gender, Age, Race, Education, Etc. Standard sociology
b) Interpersonal influences
•Actors negotiate opinions with others
Basic Peer Influence Model
Freidkin claims in his Structural Theory of Social Influence that the theory
has four benefits:
•relaxes the simplifying assumption of actors who must either
conform or deviate from a fixed consensus of others (public
choice model)
•Does not necessarily result in consensus, but can have a
stable pattern of disagreement
•Is a multi-level theory:
•micro level: cognitive theory about how people weigh
and combine other’s opinions
•macro level: concerned with how social structural
arrangements enter into and constrain the opinionformation process
•Allows an analysis of the systemic consequences of social
structures
Basic Peer Influence Model
Formal Model
Y
(1)
 XB
(T 1)
Y  αWY
(t )
(1)
 (1  α)Y
(1)
(2)
Y(1) = an N x M matrix of initial opinions on M issues for N
actors
X = an N x K matrix of K exogenous variable that affect Y
B = a K x M matrix of coefficients relating X to Y
a = a weight of the strength of endogenous interpersonal
influences
W = an N x N matrix of interpersonal influences
Basic Peer Influence Model
Formal Model
Y
(1)
 XB
(1)
This is the standard sociology model for explaining anything: the General Linear Model.
It says that a dependent variable (Y) is some function (B) of a set of independent
variables (X). At the individual level, the model says that:
Yi   X ik Bk
k
Usually, one of the X variables is e, the model error term.
Basic Peer Influence Model
(T 1)
Y  αWY
(t )
 (1  α)Y
(1)
(2)
This part of the model taps social influence. It says that each person’s final opinion is
a weighted average of their own initial opinions
(1  α)Y
(1)
And the opinions of those they communicate with (which can include their own current
opinions)
(T 1)
αWY
Basic Peer Influence Model
The key to the peer influence part of the model is W, a matrix of
interpersonal weights. W is a function of the communication structure of the
network, and is usually a transformation of the adjacency matrix. In general:
0  wij  1
w
ij
1
j
Various specifications of the model change the value of wii, the extent to which
one weighs their own current opinion and the relative weight of alters.
Basic Peer Influence Model
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Self weight:
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.33 .33
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Even
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.33 .67
2*self
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degree
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Basic Peer Influence Model
Formal Properties of the model
(T 1)
Y  αWY
(t )
 (1  α)Y
(1)
When interpersonal influence is complete, model reduces to:
Y
(t )
(T 1)
 1WY
 0Y
(1)
(T 1)
 WY
When interpersonal influence is absent, model reduces to:
Y
(t )
(T 1)
 0WY
Y
(1)
Y
(1)
(2)
Basic Peer Influence Model
Formal Properties of the model
If we allow the model to run over t, we can describe the model as:
Y
( )
( )
 αWY
 (1  α)XB
The model is directly related to spatial econometric models:
Y
( )
( )
 αWY
~
 X  e
Where the two coefficients (a and ) are estimated directly (See Doreian,
1982, SMR)
Basic Peer Influence Model
Simple example
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1
2
.33 .33
.33 .33
.25 .25
0
0
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.33
0
.33
0
.25 .25
.50 .50
Y
1
3
5
7
3
2.93
3.33
4.16
5.30
4
2.98
3.38
4.14
5.18
a = .8
4
T: 0
1.00
3.00
5.00
7.00
1
2.60
3.00
4.20
6.20
2
2.81
3.21
4.20
5.56
5
3.00
3.40
4.14
5.13
6
3.01
3.41
4.13
5.11
7
3.01
3.41
4.13
5.10
Basic Peer Influence Model
Simple example
2
1
1
2
3
4
3
1
2
.33 .33
.33 .33
.25 .25
0
0
3
4
.33
0
.33
0
.25 .25
.50 .50
Y
1
3
5
7
a = 1.0
4
T: 0
1.00
3.00
5.00
7.00
1
2
3
4
5
6
7
3.00
3.00
4.00
6.00
3.33
3.33
4.00
5.00
3.56
3.56
3.92
4.50
3.68
3.68
3.88
4.21
3.74
3.74
3.86
4.05
3.78
3.78
3.85
3.95
3.81
3.81
3.84
3.90
Basic Peer Influence Model
Extended example: building intuition
Consider a network with three cohesive groups, and an initially random distribution of
opinions:
(to run this model, use peerinfl1.sas)
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
Basic Peer Influence Model
Extended example: building intuition
Consider a network with three cohesive groups, and an initially random distribution of
opinions:
Now weight in-group ties higher than between group ties
Simulated Peer Influence:
75 actors, 2 initially random opinions, Alpha = .8, 7 iterations, in-group tie: 2
Consider the implications for populations of different structures. For example, we might
have two groups, a large orthodox population and a small heterodox population. We can
imagine the groups mixing in various levels:
Heterodox: 10 people
Orthodox: 100 People
Little Mixing
Moderate Mixing
Heavy Mixing
.95 .001
.001 .02
.95 .008
.008 .02
.95 .05
.05 .02
Light
Heavy
Moderate
Light mixing
Light mixing
Light mixing
Light mixing
Light mixing
Light mixing
Moderate mixing
Moderate mixing
Moderate mixing
Moderate mixing
Moderate mixing
Moderate mixing
High mixing
High mixing
High mixing
High mixing
High mixing
High mixing
In an unbalanced situation (small group vs large group) the
extent of contact can easily overwhelm the small group.
Applications of this idea are evident in:
•Missionary work (Must be certain to send missionaries
out into the world with strong in-group contacts)
•Overcoming deviant culture (I.e. youth gangs vs.
adults)
•Work by Hyojung Kim (U Washington) focuses on the
first of these two processes in social movement models
In recent extensions (Friedkin, 1998), Friedkin generalizes the model so that
alpha varies across people. We can extend the basic model by (1) simply
changing a to a vector (A), which then changes each person’s opinion
directly, and (2) by linking the self weight (wii) to alpha.
(T 1)
Y  AWY
(t )
 (I  A)Y
(1)
Were A is a diagonal matrix of endogenous weights, with 0 < aii < 1. A
further restriction on the model sets wii = 1-aii
This leads to a great deal more flexibility in the theory, and some
interesting insights. Consider the case of group opinion leaders with
unchanging opinions (I.e. many people have high aii, while a few have
low):
Peer Opinion Leaders
Group 1
Leaders
Group 2
Leaders
Group 3
Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Peer Opinion Leaders
Further extensions of the model might:
•Time dependent a: people likely value other’s opinions more
early than later in a decision context
•Interact a with XB: people’s self weights are a function of
their behaviors & attributes
•Make W dependent on structure of the network (weight
transitive ties greater than intransitive ties, for example)
•Time dependent W: The network of contacts does not
remain constant, but is dynamic, meaning that influence
likely moves unevenly through the network
•And others likely abound….
Testing the fit of the general model.
Experimental results
In the Friedkin and Cook paper, they test a version of the general model
experimentally in 50 4 person groups.
Each person was given time to form an initial opinion on a set of
scenarios, and then discuss their opinions with others, based on a given
structure.
Based on the model, they can predict the relation between people’s initial
opinions and the group’s final opinion.
They find that the model does predict well, even controlling for the spread of
initial opinions, the average opinion, and the structure of the network
Testing the fit of the general model
Identifying peer influence in real data
There are two general ways to test for peer influence in an observed
network. The first estimates the parameters (a and ) of the peer
influence model directly, the second transforms the network into a
dyadic model, predicting similarity among actors.
Peer influence model:
For details, see Doriean, 1982, sociological methods and research. Also Roger Gould
(AJS, Paris Commune paper for example)
Y
( )
( )
 αWY
~
 X  e
Peer influence model:
For details, see Doriean, 1982, sociological methods and research.
Also Roger Gould (AJS, Paris Commune paper for example)
The basic model says that people’s opinions are a function of the opinions of
others and their characteristics.
Y
( )
( )
 αWY
~
 X  e
WY = A simple vector which can be added to your model. That
is, multiple Y by a W matrix, and run the regression with WY as a
new variable, and the regression coefficient is an estimate of a.
This is what Doriean calls the QAD estimate of peer influence.
The problem with the above regression is that cases are, by definition,
not independent. In fact, WY is also known as the ‘network
autocorrelation’ coefficient, since a ‘peer influence’ effect is an
autocorrelation effect -- your value is a function of the people you are
connected to. In general, OLS is not the best way to estimate this
equation. That is, QAD = Quick and Dirty, and your results will not be
exact.
In practice, the QAD approach (perhaps combined with a GLS
estimator) results in empirical estimates that are “virtually
indistinguishable” from MLE (Doreian et al, 1984)
The proper way to estimate the peer equation is to use maximum
likelihood estimates, and Doreian gives the formulas for this in his
paper.
The other way is to use non-parametric approaches, such as the
Quadratic Assignment Procedure, to estimate the effects.
An empirical Example: Peer influence in the OSU Graduate Student Network.
Each person was asked to rank their satisfaction with the program, which is the dependent
variable in this analysis.
I constructed two W matrices, one from HELP the other from Best Friend. I treat
relations as symmetric and valued, such that:
 1 if Aijt  1 or A jit  1 


Wijt  2 if Aijt  1 and A jit  1


0
otherwise


W
ij
1
j
Wii  0
I also include Race (white/Non-white, Gender and Cohort Year as exogenous variables in
the model.
(to run the model, see osupeerpi1.sas)
An empirical Example: Peer influence in the OSU Graduate Student Network.
Distribution of Satisfaction with the department.
Parameter Estimates
Variable
Parameter
Estimate
Standardized
Pr > |t| Estimate
Intercept
FEMALE
NONWHITE
y00
y99
y98
y97
PEER_BF
PEER_H
2.60252
-1.07540
-0.22087
0.93176
-0.19375
-0.45912
0.60670
0.23936
0.50668
0.0931
0.0142
0.5975
0.0798
0.7052
0.4637
0.3060
0.0002
0.0277
0
-0.25455
-0.05491
0.21627
-0.04586
-0.08289
0.11919
0.42084
0.23321
Model R2 = .41, compared to .15 without the peer effects
The most common method for estimating peer
effects is to include the mean of ego’s alters in the
network. Under certain specifications of the model,
this is exactly the same as the QAD analysis
sketched above.
Example of mean-peer model: Haynie on Delinquency
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.
In Friedkin’s model, (a) is akin to changing w, such that the effect is greater for
transitive ties, (b) is akin to making a dependent on centrality.
Example of mean-peer model: Haynie on Delinquency
Example of mean-peer model: Haynie on Delinquency
Example of mean-peer model: Haynie on Delinquency
Example of mean-peer model: Haynie on Delinquency
Example of mean-peer model: Haynie on Delinquency
Peer influence through Dyad Models
Another way to get at peer influence is not through the level of Y,
but through the extent to which actors are similar with respect to Y.
Recall the simulated example: peer influence is reflected in how
close points are to each other.
Peer influence through Dyad Models
The model is now expressed at the dyad level as:
Yij  b0  b1 Aij   bk X k  eij
k
Where Y is a matrix of similarities, A is an adjacency matrix,
and Xk is a matrix of similarities on attributes
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The substantive question is how one set of relations (or dyadic attributes)
relates to another.
For example:
• Do marriage ties correlate with business ties in the Medici family
network?
• Are friendship relations correlated with joint membership in a club?
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Assessing the correlation is straight forward, as we simply correlate each
corresponding cell of the two matrices:
Marriage
1 ACCIAIUOL
2
ALBIZZI
3 BARBADORI
4 BISCHERI
5 CASTELLAN
6
GINORI
7 GUADAGNI
8 LAMBERTES
9
MEDICI
10
PAZZI
11
PERUZZI
12
PUCCI
13
RIDOLFI
14 SALVIATI
15
STROZZI
16 TORNABUON
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
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0
0
0
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1
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1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
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0
0
0
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0
1
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0
0
1
1
0
0
0
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0
0
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1
1
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0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
Business
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 1
6 0 0 1
7 0 0 0
8 0 0 0
9 0 0 1
10 0 0 0
11 0 0 1
12 0 0 0
13 0 0 0
14 0 0 0
15 0 0 0
16 0 0 0
0
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
Correlation:
1 0.3718679
0.3718679
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
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1
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0
0
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1
1
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1
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1
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1
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1
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0
0
1
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1
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1
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0
0
0
1
0
0
0
0
0
0
0
0
0
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
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0
0
0
0
0
0
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0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
Dyads:
1 2 0
1 3 0
1 4 0
1 5 0
1 6 0
1 7 0
1 8 0
1 9 1
1 10 0
1 11 0
1 12 0
1 13 0
1 14 0
1 15 0
1 16 0
2 1 0
2 3 0
2 4 0
2 5 0
2 6 1
2 7 1
2 8 0
2 9 1
2 10 0
2 11 0
2 12 0
2 13 0
2 14 0
2 15 0
2 16 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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But is the observed value statistically significant?
Can’t use standard inference, since the assumptions are violated. Instead, we
use a permutation approach.
Essentially, we are asking whether the observed correlation is large (small)
compared to that which we would get if the assignment of variables to nodes
were random, but the interdependencies within variables were maintained.
Do this by randomly sorting the rows and columns of the matrix, then reestimating the correlation.
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When you permute, you have to permute both the rows and the columns
simultaneously to maintain the interdependencies in the data:
ID ORIG
A
B
C
D
E
0
0
0
0
0
1
0
0
0
0
2
1
0
0
0
Sorted
3
2
1
0
0
4
3
2
1
0
A
D
B
C
E
0
0
0
0
0
3
0
2
1
0
1
0
0
0
0
2
0
1
0
0
4
1
3
2
0
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Procedure:
1. Calculate the observed correlation
2. for K iterations do:
a) randomly sort one of the matrices
b) recalculate the correlation
c) store the outcome
3. compare the observed correlation to the distribution of
correlations created by the random permutations.
Complete Network Analysis
Network Connections: QAP
QAP MATRIX CORRELATION
-------------------------------------------------------------------------------Observed matrix:
Structure matrix:
# of Permutations:
Random seed:
PadgBUS
PadgMAR
2500
356
This can be done
simply in UCINET
Univariate statistics
1
Mean
2 Std Dev
3
Sum
4 Variance
5
SSQ
6
MCSSQ
7 Euc Norm
8 Minimum
9 Maximum
10 N of Obs
1
2
PadgBUS PadgMAR
------- ------0.125
0.167
0.331
0.373
30.000 40.000
0.109
0.139
30.000 40.000
26.250 33.333
5.477
6.325
0.000
0.000
1.000
1.000
240.000 240.000
Hubert's gamma: 16.000
Bivariate Statistics
1
Pearson Correlation:
2
Simple Matching:
3
Jaccard Coefficient:
4 Goodman-Kruskal Gamma:
5
Hamming Distance:
1
2
3
4
5
6
7
Value
Signif
Avg
SD P(Large) P(Small)
NPerm
--------- --------- --------- --------- --------- --------- --------0.372
0.000
0.001
0.092
0.000
1.000 2500.000
0.842
0.000
0.750
0.027
0.000
1.000 2500.000
0.296
0.000
0.079
0.046
0.000
1.000 2500.000
0.797
0.000
-0.064
0.382
0.000
1.000 2500.000
38.000
0.000
59.908
5.581
1.000
0.000 2500.000
Complete Network Analysis
Network Connections: QAP
Using the same logic,we can estimate alternative models, such as
regression, logits, probits, etc. Only complication is that you need
to permute all of the independent matrices in the same way each
iteration.
Complete Network Analysis
Peer-influence results on similarity
dyad model, using QAP
Network Connections: QAP
# of permutations:
Diagonal valid?
Random seed:
Dependent variable:
Expected values:
Independent variables:
2000
NO
995
EX_SIM
C:\moody\Classes\soc884\examples\UCINET\mrqap-predicted
EX_SSEX
EX_SRCE
EX_ADJ
Number of valid observations among the X variables = 72
N = 72
Number of permutations performed: 1999
MODEL FIT
R-square Adj R-Sqr Probability
# of Obs
-------- --------- ----------- ----------0.289
0.269
0.059
72
REGRESSION COEFFICIENTS
Un-stdized
Stdized
Proportion Proportion
Independent Coefficient Coefficient Significance
As Large
As Small
----------- ----------- ----------- ------------ ----------- ----------Intercept
0.460139
0.000000
0.034
0.034
0.966
EX_SSEX
-0.073787
-0.170620
0.140
0.860
0.140
EX_SRCE
-0.020472
-0.047338
0.272
0.728
0.272
EX_ADJ
-0.239896
-0.536211
0.012
0.988
0.012
If we break the original peer influence model into it’s components,
the attribute part of the model suggests that any two people with the
same attribute should have the same value for Y.
The Peer influence model says that (a) if you and I are tied to each
other, then we should have similar opinions and (b) that if we are tied
to many of the same people, then we should have similar opinions.
We can test both sides of these (and many other dyadic properties)
directly at the dyad level.
NODE
1
2
3
4
5
6
7
8
9
0
1
1
1
0
0
0
0
0
1
0
1
0
0
0
1
0
0
ADJMAT
1 1 0 0
1 0 0 0
0 0 1 0
0 0 1 0
1 1 0 1
0 0 1 0
1 0 0 0
0 0 1 1
0 0 0 1
0
1
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
1
0
1
0
0
1
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
SAMERCE
0 0 1 0
0 0 1 0
0 1 0 1
1 0 0 1
0 0 0 0
1 1 0 0
1 1 0 1
1 1 0 1
0 0 1 0
0
0
1
1
0
1
0
1
0
0
0
1
1
0
1
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
1
1
0
0
1
SAMESEX
1 1 0 0 1
0 0 1 1 0
0 1 0 0 1
1 0 0 0 1
0 0 0 1 0
0 0 1 0 0
1 1 0 0 0
1 1 0 0 1
0 0 1 1 0
1
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
0
Y
0.32
0.59
0.54
0.50
0.04
0.02
0.41
0.01
-0.17
Distance (Dij=abs(Yi-Yj)
.000 .277 .228 .181 .278
.277 .000 .049 .096 .555
.228 .049 .000 .047 .506
.181 .096 .047 .000 .459
.278 .555 .506 .459 .000
.298 .575 .526 .479 .020
.095 .182 .134 .087 .372
.307 .584 .535 .488 .029
.481 .758 .710 .663 .204
.298
.575
.526
.479
.020
.000
.392
.009
.184
.095
.182
.134
.087
.372
.392
.000
.401
.576
.307
.584
.535
.488
.029
.009
.401
.000
.175
.481
.758
.710
.663
.204
.184
.576
.175
.000
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
SENDER
RCVER
SIM
NOM
SAMERCE
SAMESEX
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
4
5
6
7
8
9
1
3
4
5
6
7
8
0.27694
0.22828
0.18136
0.27766
0.29763
0.09473
0.30671
0.48148
0.27694
0.04866
0.09559
0.55460
0.57457
0.18221
0.58365
1
1
1
0
0
0
0
0
1
1
0
0
0
1
0
1
0
0
1
0
0
0
1
1
0
0
1
0
0
0
0
1
1
0
0
1
1
0
0
0
0
1
1
0
0
The REG Procedure
Model: MODEL1
Dependent Variable: SIM
Analysis of Variance
Source
DF
Sum of
Squares
Model
Error
Corrected Total
4
31
35
0.90657
0.75591
1.66248
Root MSE
Dependent Mean
Coeff Var
0.15615
0.33161
47.08929
Mean
Square
0.22664
0.02438
R-Square
Adj R-Sq
F Value
Pr > F
9.29
<.0001
0.5453
0.4866
Parameter Estimates
Variable
Intercept
NOM
SAMERCE
SAMESEX
NCOMFND
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
1
1
1
1
1
0.51931
-0.17054
0.05387
-0.06535
-0.16134
0.05116
0.05963
0.05916
0.05365
0.03862
10.15
-2.86
0.91
-1.22
-4.18
<.0001
0.0075
0.3696
0.2324
0.0002
Like the basic Peer influence model, cases in a dyad model are
not independent. However, the non-independence now comes
from two sources: the fact that the same person is represented in
(n-1) dyads and that i and j are linked through relations.
One of the best solutions to this problem is QAP: Quadratic
Assignment Procedure. A non-parametric procedure for
significance testing.
QAP runs the model of interest on the real data, then randomly
permutes the rows/cols of the data matrix and estimates the
model again. In so doing, it generates an empirical distribution of
the coefficients.
MULTIPLE REGRESSION QAP W/ MISSING VALUES
-------------------------------------------------------------------------------# of permutations:
Diagonal valid?
Random seed:
Dependent variable:
Expected values:
Independent variables:
2000
NO
533
EX_SIM
c:\moody\Classes\soc884\examples\UCINET\mrqap-predicted
EX_NCOM
EX_ADJ
EX_SRCE
EX_SSEX
Number of valid observations among the X variables = 72
N = 72
Number of permutations performed: 1999
MODEL FIT
R-square Adj R-Sqr Probability
# of Obs
-------- --------- ----------- ----------0.545
0.525
0.029
72
REGRESSION COEFFICIENTS
Un-stdized
Stdized
Proportion Proportion
Independent Coefficient Coefficient Significance
As Large
As Small
----------- ----------- ----------- ------------ ----------- ----------Intercept
0.519314
0.000000
0.012
0.012
0.988
EX_NCOM
-0.161337
-0.541828
0.011
0.989
0.011
EX_ADJ
-0.170539
-0.381186
0.020
0.980
0.020
EX_SRCE
0.053864
0.124551
0.236
0.236
0.764
EX_SSEX
-0.065364
-0.151144
0.180
0.820
0.180
Christakis & Fowler
Christakis & Fowler
Christakis & Fowler
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Christakis & Fowler
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
The network shows significant evidence of weight-homophily
Christakis & Fowler
Effects of peer obesity on ego, by peer type
Used the
friend/relative
tracking data
from a larger
heart-health study
to identify
network contacts,
including friends.
Edge-wise regressions of the form:
EgoCurrent  1 ( AltCurrent )   2 ( Alt previous)  3 ( Ego previous)  Controls
Ego is repeated for all alters; models include random effects on
ego id
Christakis & Fowler
This modeling strategy
pools observations on
edges and estimates a
global effect net of
change in ego/alter as a
control. Here color is a
single ego, number is
wave (only 2 egos and 3
waves represented).
Ego-Current
Effects of peer obesity on ego, by peer type
3
3
3
3
3
2
2
2
1
2
2
2
1
1
1
Peer Effect
3
1
1
Alter Current
EgoCurrent  1 ( AltCurrent )   2 ( Alt previous)  3 ( Ego previous)  Controls
Christakis & Fowler
Effects of peer obesity on ego, by peer type
1
Ego-Current
Alterative specifications
include using changechange models and
allowing for a random
effect of peers. This
allows for greater
variability in peer
effects, and the potential
to model differences.
3
3
3
3
Note none of these
models use information
from non-ties, other
than entering as fixed
wave/place effects.
3
2
2
2
1
2
2
2
1
1
1

3
2
1
1
Alter Current
( Ego previous  EgoCurrent )   e ( Alt previous  Altcurrent )  Controls
 e    e
Dynamics of Political Polarization
When and why do networks & attitudes polarize?
Paradox Problems:
Dynamics of Political Polarization
When and why do networks & attitudes polarize?
Model Setup:
Dynamics of Political Polarization
When and why do networks & attitudes polarize?
Model Setup:
If actors share the same orientation, interaction reinforces the opinion.
If they disagree on an issue, then:
-- if they agree on the majority of others, they compromise
-- if they disagree they push each other further apart
Dynamics of Political Polarization
When and why do networks & attitudes polarize?
Results:
Dynamics of Political Polarization
When and why do networks & attitudes polarize?
Results:
A mixed selection and influence model: Simultaneous balance on friendship and
behavior.
Two linked models:
a) actors seek interpersonal balance among friends
b) actors change their opinions / behaviors as a weighted function of
the people they are tied to, with W weighted by number of transitive
ties