Peer Influence

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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
• Topics Covered
• Basic Peer influence
• Selection and influence
• Dynamic mix of above
• Dyad models

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Friedkin Cook
One piece in a long standing research program.
Other cites include
Friedkin, N. E. 1984. "Structural Cohesion and
Equivalence Explanations of Social Homogeneity."
Sociological Methods and Research 12235-61. .
1998. A Structural Theory of Social Influence.
Cambridge Cambridge. Friedkin, N. E. and E. C.
Johnsen. 1990. "Social Influence and Opinions."
Journal of Mathematical Sociology
15(193-205). . 1997. "Social Positions in
Influence Networks." Social Networks 19209-22.
See also the supplemental reading on the
syllabus. Many particular context examples can
be found as well.
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Friedkin Cook

• Peer influence models assume that individuals
  opinions are formed in a process of interpersonal
  negotiation and adjustment of opinions.
• Can result in either consensus or disagreement
• Looks at interaction among a system of actors
• In this particular paper, look at the opinion
  results in an experimental setup.

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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

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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 others opinions
• macro level concerned with how social structural
  arrangements enter into and constrain the
  opinion-formation process
• Allows an analysis of the systemic consequences
  of social structures

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Basic Peer Influence Model
Formal Model
(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
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Basic Peer Influence Model
Formal Model
(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
Usually, one of the X variables is e, the model
error term.
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Basic Peer Influence Model
(2)
This part of the model taps social influence. It
says that each persons final opinion is a
weighted average of their own initial opinions
And the opinions of those they communicate with
(which can include their own current opinions)
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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
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.
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Basic Peer Influence Model
1
2
Self weight
1 2 3 4 1 .33 .33 .33 0 2 .33 .33
.33 0 3 .25 .25 .25 .25 4 0 0 .50 .50
1 2 3 4 1 1 1 1 0 2 1 1 1 0 3 1 1 1 1 4 0 0 1 1
Even
3
4
2self
1 2 3 4 1 .50 .25 .25 0 2 .25 .50
.25 0 3 .20 .20 .40 .20 4 0 0 .33 .67
1 2 3 4 1 2 1 1 0 2 1 2 1 0 3 1 1 2 1 4 0 0 1 2
degree
1 2 3 4 1 .50 .25 .25 0 2 .25 .50
.25 0 3 .17 .17 .50 .17 4 0 0 .50 .50
1 2 3 4 1 2 1 1 0 2 1 2 1 0 3 1 1 3 1 4 0 0 1 1
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Basic Peer Influence Model
Formal Properties of the model
(2)
When interpersonal influence is complete, model
reduces to
When interpersonal influence is absent, model
reduces to
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Basic Peer Influence Model
Formal Properties of the model
If we allow the model to run over t, we can
describe the model as
The model is directly related to spatial
econometric models
Where the two coefficients (a and b) are
estimated directly (See Doreian, 1982, SMR)
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Basic Peer Influence Model
Simple example
1
2
1 2 3 4 1 .33 .33 .33 0 2 .33 .33
.33 0 3 .25 .25 .25 .25 4 0 0 .50 .50
Y 1 3 5 7
a .8
3
4
T 0 1 2 3 4 5 6 7 1.00
2.60 2.81 2.93 2.98 3.00 3.01 3.01 3.00 3.00
3.21 3.33 3.38 3.40 3.41 3.41 5.00 4.20 4.20
4.16 4.14 4.14 4.13 4.13 7.00 6.20 5.56 5.30
5.18 5.13 5.11 5.10
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Basic Peer Influence Model
Simple example
1
2
1 2 3 4 1 .33 .33 .33 0 2 .33 .33
.33 0 3 .25 .25 .25 .25 4 0 0 .50 .50
Y 1 3 5 7
a 1.0
3
4
T 0 1 2 3 4 5 6 7
1.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81 3.00 3.00
3.33 3.56 3.68 3.74 3.78 3.81 5.00 4.00 4.00 3.92
3.88 3.86 3.85 3.84 7.00 6.00 5.00 4.50 4.21 4.05
3.95 3.90
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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)
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations
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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
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Simulated Peer Influence 75 actors, 2 initially
random opinions, Alpha .8, 7 iterations,
in-group tie 2
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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
Heavy Mixing
Moderate Mixing
.95 .05 .05 .02
.95 .008 .008 .02
.95 .001 .001 .02
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Light
Heavy
Moderate
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Light mixing
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Light mixing
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Light mixing
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Light mixing
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Light mixing
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Light mixing
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Moderate mixing
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Moderate mixing
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Moderate mixing
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Moderate mixing
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Moderate mixing
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Moderate mixing
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High mixing
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High mixing
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High mixing
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High mixing
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High mixing
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High mixing
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• 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

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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 persons opinion directly, and (2)
by linking the self weight (wii) to alpha.
Were A is a diagonal matrix of endogenous
weights, with 0 lt aii lt 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)
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Peer Opinion Leaders
Group 1 Leaders
Group 2 Leaders
Group 3 Leaders
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Peer Opinion Leaders
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Peer Opinion Leaders
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Peer Opinion Leaders
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Peer Opinion Leaders
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Peer Opinion Leaders
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• Further extensions of the model might
• Time dependent a people likely value others
  opinions more early than later in a decision
  context
• Interact a with XB peoples 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.

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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 peoples
initial opinions and the groups 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
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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 b) 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)
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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 peoples opinions are a
function of the opinions of others and their
characteristics.
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.
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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.
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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
I also include Race (white/Non-white, Gender and
Cohort Year as exogenous variables in the model.
(to run the model, see osupeerpi1.sas)
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An empirical Example Peer influence in the OSU
Graduate Student Network.
Distribution of Satisfaction with the department.
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Parameter Estimates
Parameter Standardized Variable
Estimate Pr gt t Estimate Intercept
2.60252 0.0931 0 FEMALE -1.07540
0.0142 -0.25455 NONWHITE -0.22087 0.5975
-0.05491 y00 0.93176 0.0798
0.21627 y99 -0.19375 0.7052
-0.04586 y98 -0.45912 0.4637
-0.08289 y97 0.60670 0.3060
0.11919 PEER_BF 0.23936 0.0002
0.42084 PEER_H 0.50668 0.0277 0.23321
Model R2 .41, compared to .15 without the peer
effects
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The most common method for estimating peer
effects is to include the mean of egos alters in
the network. Under certain specifications of the
model, this is exactly the same as the QAD
analysis sketched above.
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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.
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Peer influence through Dyad Models
The model is now expressed at the dyad level as
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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If we break the original peer influence model
into its 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.
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NODE ADJMAT SAMERCE
SAMESEX 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1
0 0 0 1 0 0 1 1 0 0 1 1 0 2 1 0 1 0 0 0 1
0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1
3 1 1 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0
0 1 0 0 1 1 0 4 1 0 0 0 1 0 0 0 0 0 0 1 0
0 1 1 1 0 1 0 1 0 0 0 1 1 0 5 0 0 1 1 0 1
0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1
6 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
1 0 0 1 0 0 0 1 7 0 1 1 0 0 0 0 0 0 0 0 1
1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 8 0 0 0 0 1
1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0
0 9 0 0 0 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0
0 1 0 0 1 1 0 0 0
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Distance (Dijabs(Yi-Yj) .000 .277 .228 .181 .278
.298 .095 .307 .481 .277 .000 .049 .096 .555 .575
.182 .584 .758 .228 .049 .000 .047 .506 .526 .134
.535 .710 .181 .096 .047 .000 .459 .479 .087 .488
.663 .278 .555 .506 .459 .000 .020 .372 .029
.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
Y 0.32 0.59 0.54 0.50 0.04 0.02 0.41
0.01 -0.17
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Obs SENDER RCVER SIM NOM
SAMERCE SAMESEX 1 1 2
0.27694 1 1 0 2 1
3 0.22828 1 0 1 3
1 4 0.18136 1 0
1 4 1 5 0.27766 0
1 0 5 1 6 0.29763
0 0 0 6 1 7
0.09473 0 0 1 7 1
8 0.30671 0 0 1 8
1 9 0.48148 0 1
0 9 2 1 0.27694 1
1 0 10 2 3 0.04866
1 0 0 11 2 4
0.09559 0 0 0 12 2
5 0.55460 0 1 1 13
2 6 0.57457 0 0
1 14 2 7 0.18221 1
0 0 15 2 8 0.58365
0 0 0
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The REG Procedure
Model MODEL1
Dependent Variable SIM
Analysis of Variance
Sum of Mean Source
DF Squares Square F
Value Pr gt F Model 4
0.90657 0.22664 9.29
lt.0001 Error 31 0.75591
0.02438 Corrected Total 35
1.66248 Root MSE
0.15615 R-Square 0.5453
Dependent Mean 0.33161 Adj R-Sq
0.4866 Coeff Var
47.08929 Parameter
Estimates Parameter
Standard Variable DF Estimate
Error t Value Pr gt t
Intercept 1 0.51931 0.05116
10.15 lt.0001 NOM 1
-0.17054 0.05963 -2.86 0.0075
SAMERCE 1 0.05387 0.05916
0.91 0.3696 SAMESEX 1
-0.06535 0.05365 -1.22 0.2324
NCOMFND 1 -0.16134 0.03862
-4.18 0.0002
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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.
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Cohen The problem of Selection and Influence
Well known that cohesive groups tend to be more
similar (homogeneous) than the population at
large. Why is this so? It may be due either to
influence people change as a function of the
people around them or selection people join
groups based on their behaviors
Cohen re-analyzed data by Coleman on Newlawn a
middle-class white suburban school of about 1000
students, and identified cliques of students in
the school. (His measure of clique is pretty
exclusive only 9 of the males and 40 of the
females in the school fit his definition) He
proposes to answer the selection vs. influence
question by looking at changes in behavior and
changes in group composition over time.
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Design
Ss(S) - Sf(F) change in group homogeneity over
time. Not clear whether it is due to changes
in behavior or changes in members
Ss(F) - Sf(F) change in group homogeneity over
time, for only those people who are in the same
group both times. Removes the selection effect.
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Homophily due to selection is equal to overall
uniformity (U) minus conformity (C). C is the
differences in table 2, and he estimates
selection effects here. (Finding that the
selection effect is less substantial than
conformity effects (p.233))
By comparing newly formed spring cliques, he is
able to conclude that the majority of overall
conformity is due to initial selection.
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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