The Wonders of Specification Possibilities of Stochastic Actor-Oriented Models for Network Dynamics

Intro
The Wonders of Specification Possibilities
of Stochastic Actor-Oriented Models
for Network Dynamics
Tom A.B. Snijders
University of Oxford
Nuffield/OII Seminar on Social Network Analysis, May 19, 2014
Specification Possibilities of SAOMs 1 / 39

Intro
Overview
Sketch of Stochastic Actor-Oriented Model (‘SAOM’),
evaluation–endowment–creation functions;

Intro
Overview
differentiation tie creation termination

Intro
Overview
homophily at distance two

Intro
Overview
homophily at distance two
with examples from Vanina Torlò’s MBA students
and the Glasgow ‘Teenage Friends and Lifestyle Study’.

Intro
Stochastic Actor-Oriented Model
Methodology for analyzing network dynamics:

Intro
⇒ Probability model of network change in continuous time

Intro
⇒ Probability model of network change in continuous time
⇒ Methods for estimation, testing, goodness of ﬁt, etc.
(observations panel data)
.

Intro
Probability Model of SAOM
Since the SAOM is a continuous-time model,
it sufﬁces to model changes of single tie variables.
Changes can be made by actors i in their outgoing ties.
Notation: Xij is the tie variable indicating the tie i → j ,
network X = (Xij) is a random structure, with values x.

Intro
Objective function
Consider the probability of the network changing to state x,
given that currently it is in state x0.
This probability depends on the objective function ui(x0, x) .
The probability that the next network is x,
if actor i makes a change, is given by
exp(ui(x0, x)
x ∈C exp ui(x0, x )
. (1)
C is the set of all networks that could be the next state x.

Intro
Objective function
Consider the probability of the network changing to state x,
given that currently it is in state x0.
This probability depends on the objective function ui(x0, x) .
The probability that the next network is x,
if actor i makes a change, is given by
exp(ui(x0, x)
x ∈C exp ui(x0, x )
. (1)
C is the set of all networks that could be the next state x.
Basic model speciﬁcation: ui(x0, x) does not depend on x0
and is called the evaluation function.
Then tie termination is simply the reverse of tie creation.

Creation versus maintenance of ties
Differentiation tie creation – maintenance
In the more general case for previous state x0 and
new state x, we distinguish between the situations
⇒ tie creation: x has one tie more than x0;
denoted by ∆+(x0, x) = 1 (else ∆+(x0, x) = 0 )
with associated the creation function ci(x);

Differentiation tie creation – maintenance
In the more general case for previous state x0 and
new state x, we distinguish between the situations
⇒ tie creation: x has one tie more than x0;
denoted by ∆+(x0, x) = 1 (else ∆+(x0, x) = 0 )
with associated the creation function ci(x);
⇒ tie termination: x has one tie less than x0;
denoted by ∆−(x0, x) = 1 (else ∆−(x0, x) = 0 )
with associated the endowment function ei(x)
a better name is maintenance function
(cf. gratiﬁcation function in Snijders, Soc. Metho., 2001).

Differentiation tie creation – maintenance (2)
The general deﬁnition of the objective function is
ui(x0
, x) = fi(x) − fi(x0
)
+ ∆+
(x0
, x) ci(x) − ci(x0
)
+ ∆−
(x0
, x) ei(x) − ei(x0
) .
Recall: x0 is old state, x is new state;
∆+(x0, x) = 1 (creation) or 0 (termination);
∆−(x0, x) = 0 (creation) or 1 (termination);
u = objective function
f = evaluation function
c = creation function
e = maintenance (endowment) function.

Differentiation tie creation – maintenance (3)
This means:
tie creation is modeled by
the sum evaluation function + creation function;
tie maintenance is modeled by
the sum evaluation function + maintenance function.

Estimation
The evaluation, creation, and maintenance functions
are deﬁned as linear combinations of ‘effects’
with the weights being the statistical parameters
(as in regression or generalized linear models).
Evaluation function
fi(β, x) =
k
βk sik(x)
where
i = focal actor;
βk = statistical parameter;
x = network;
sik(x) = effect, function of network & other variables.

Short remark on estimation by Method of Moments:
For network data sets with (e.g.) two waves t1, t2:
params. of evaluation fu. estimated from network state t2;
params. of creation fu. estimated from new ties t1 ⇒ t2;
params. of maint. fu. estimated from terminated ties t1 ⇒ t2.
(For effects that can be associated with speciﬁc ties;
unlike, e.g., nbrDist2).

Example 1
Data from Vanina Torlò and Alessandro Lomi.
International MBA program in Italy;
75 students; 3 waves in one year.
1 Friendship
2 Advice:
To whom do you go for help if you missed a class, etc.

Example 1
Data from Vanina Torlò and Alessandro Lomi.
International MBA program in Italy;
75 students; 3 waves in one year.
1 Friendship
2 Advice:
To whom do you go for help if you missed a class, etc.
3 Covariates.
Here the co-evolution of friendship and advice is considered.
These two networks are interdependent dependent variables.

Friendship (1)
Effect create eval maintain (s.e.)
outdegree (density) –2.984∗∗∗ (0.205)
reciprocity 1.088∗∗∗ (0.280)
reciprocity 2.974∗∗∗ (0.274)
trans. triplets 0.473∗∗∗ (0.070)
trans. triplets 0.060 (0.067)
trans. rec. triplets . –0 207∗∗∗ (0.041)
3-cycles –0.071∗ (0.031)
indegree - popularity –0.099∗∗ (0.034)
indegree - popularity 0.109∗∗ (0.035)
outdegree - activity –0.005 (0.008)
gender alter 0.064 (0.093)
gender ego –0.152† (0.083)
same gender 0.219∗ (0.086)

Friendship (2)
Effect create eval maint (s.e.)
same nationality 0.252∗ (0.100)
perfo alter 0.047 (0.075)
perfo alter –0.244∗∗ (0.083)
perfo ego 0.567∗ (0.244)
perfo ego –0.757∗∗ (0.250)
perfo similarity 0.126 (0.569)
perfo similarity 2.278∗∗ (0.726)
advice 2.067∗∗∗ (0.387)
advice 2.389∗∗∗ (0.520)
indegree advice pop. –0.055∗∗∗ (0.013)
outdegree advice act. –0.036∗ (0.017)

Advice (1)
outdegree (density) –4.536∗∗∗ (0.581)
reciprocity 0.581 (0.403)
reciprocity 2.127∗∗∗ (0.502)
transitive triplets 0.535∗∗∗ (0.158)
transitive triplets –0.053 (0.182)
transitive rec. triplets –0.245† (0.126)
3-cycles 0.085 (0.097)
indegree - popularity 0.016 (0.021)
indegree - popularity 0.085∗∗∗ (0.023)
outdegree - activity 0.025 (0.015)
gender alter –0.152 (0.132)
gender ego –0.199† (0.116)
same gender 0.099 (0.120)

Advice (2)
same natio 0.391∗ (0.168)
perfo alter 0.110 (0.072)
perfo ego –0.161∗∗∗ (0.045)
perfo ego x perfo alter 0.091∗∗∗ (0.021)
perfo alter at distance 2 0.574∗ (0.276)
friendship 2.252∗∗∗ (0.385)
friendship 1.883∗∗∗ (0.442)
indegree friendship pop. –0.031∗∗ (0.012)
outdegree friendship act. –0.041∗∗∗ (0.008)
† p < 0.1; ∗ p < 0.05; ∗∗ p < 0.01; ∗∗∗ p < 0.001;
Interactions with time not included in table.

Conclusions: creation maintenance (F)
For Friendship, there are some strong differences:
Reciprocity 3 times stronger for maintenance than
creation (p < 0.0001)

Transitivity only important for creation (p = 0.002)

Indegree popularity (‘Matthew effect’)
negative for creation, positive for maintenance
(p = 0.002)

(p = 0.002)
Performance alter only for maintenance
(negative, p = 0.04)

(p = 0.002)
Performance ego positive for creation,
negative for maintenance (p = 0.01)

(p = 0.002)
Performance ego positive for creation,
negative for maintenance (p = 0.01)
Performance similarity only for maintenance
(but p = 0.08)

Conclusions: creation maintenance (A)
For Advice, there are weaker differences:
Reciprocity only important for maintenance (p = 0.04)

Transitivity only important for creation (but p = 0.07)

Indegree popularity (‘Matthew effect’) only for
maintenance (but p = 0.07)

Indegree popularity (‘Matthew effect’) only for
maintenance (but p = 0.07)
Testing differences between creation and maintenance effects
is difﬁcult because their parameter estimates are negatively
correlated (which increases the s.e. of the difference).

Conclusions: co-evolution
Positive dyad-level effects advice ⇔ friendship,
creation not different from maintenance,
of same order of magnitude as reciprocity maintenance.

Negative actor-level effects friendship ⇔ advice
(cross-network indegree popularity and outdegree activity):
Specialization between friendship / advice,
w.r.t. incoming ties as well as outgoing ties.

Negative actor-level effects friendship ⇔ advice
(cross-network indegree popularity and outdegree activity):
Specialization between friendship / advice,
w.r.t. incoming ties as well as outgoing ties.
Multilevel issue:
association positive at the dyadic level,
negative at the actor level.

General conclusions
about creation maintenance
There is, in this data set, strong evidence
for differences between creation and maintenance
for some of the effects inﬂuencing the network development.
Not for such differences for cross-network effects, by the way.

General conclusions
about creation maintenance
There is, in this data set, strong evidence
for differences between creation and maintenance
for some of the effects inﬂuencing the network development.
Not for such differences for cross-network effects, by the way.
More research, and theoretical elaboration,
is needed for the cumulation of insight into mechanisms.

Homophily and Beyond
Homophily and beyond

Homophily well known
(Lazarsfeld & Merton 1954;
McPherson, Smith-Lovin & Cook 2001):
ties more likely between similar actors.

⇒ I am similar to my friends ;

⇒⇒I am similar to friends of my friends

⇒⇒I am similar to friends of my friends
‘homophily at distance 2’.
.

Various theoretical arguments for
distance-2 homophily, e.g.:

1 social identity : “tell me who your friends are ..."

2 uncertainty reduction :
“if this person gets along with others like me ..."

3 signal unreliability : if ego’s observation of alter’s
attribute is unreliable,
and ego assumes that homophily operates,
then dist.-2 similarity suggests direct similarity;

3 signal unreliability : if ego’s observation of alter’s
attribute is unreliable,
and ego assumes that homophily operates,
then dist.-2 similarity suggests direct similarity;
4 negative diversity, social capital :
alters bridging to different third actors.
.

?
is there a tendency to homophily at distance 2,
while controlling for (regular) homophily ?

?
is there a tendency to homophily at distance 2,
while controlling for (regular) homophily ?
Regular homophily with transitivity
will imply observed distance-2 homophily:
We also have to control for transitivity.
.

Example :
Study of smoking initiation and friendship
Teenage Friends and Lifestyle Study
(following up on P. West, L. Michell, M. Pearson & others;
cf. Steglich, Snijders & Pearson, Sociol. Methodology, 2010).
One school year group from a Scottish secondary school
starting at age 12-13 years, monitored over 3 years;
129 (out of 160) pupils present at all 3 observations;
three waves, at appr. 1 year intervals.
Smoking: values 1–3; drinking: values 1–5;
covariates:
gender, smoking of parents and siblings (binary),
money available (range 0–40 pounds/week).
.

wave 1 girls: circles
boys: squares
node size: pocket money
color: top = drinking
bottom = smoking
(orange = high)

boys: squares
bottom = smoking
(orange = high)

Effects for similarity at distance 2
Direct homophily effects can be represented by
effects sik(x) expressing similarity
between i and i’s personal network,
si,similarity =
j
xij 1 −
| vi − vj |
vmax − vmin

Effects for similarity at distance 2
Direct homophily effects can be represented by
effects sik(x) expressing similarity
between i and i’s personal network,
si,similarity =
j
xij 1 −
| vi − vj |
vmax − vmin
or by an interaction between the attribute of i
and the attributes of those in i’s personal network
(personal network = out-neighbourhood),
si,interaction = vi
j
xij vj .
.

To define distance-two homophily effects , first
define ˘v
(−i)
j as “alters’ v-average”:
average value of vh for those to whom j is tied, excluding i,
˘v
(−i)
j =



h=i xjh vh
xj+
if xj+ − xji > 0
¯v if xj+ − xji = 0.
.

The distance-two homophily effect can be represented by
the similarity between i and
the alter-averages in i’s personal network,
si,simDist2 =
j
xij



1 −
| vi − ˘v
(−i)
j |
vmax − vmin



.
.

The effect of alter’s v- average,
and its interaction with ego-v, are deﬁned as
si,alter average dist. 2 =
j
xij ˘v
(−i)
j
si,ego × alter average dist. 2 = vi
j
xij ˘v
(−i)
j .
The latter interaction may also be regarded as
a kind of distance-two homophily;
it should be controlled for the alter average at distance two.
.

Structural effects
estimate (s.e.)
1 . outdegree (density) −0.92∗∗ (0.29)
2 . reciprocity 2.28∗∗∗ (0.14)
3 . transitive triplets 0.47∗∗∗ (0.06)
4 . 3-cycles −0.17∗ (0.09)
5 . transitive ties 0.75∗∗∗ (0.10)
6 . indegree − popularity (sqrt) 0.08 (0.11)
7 . outdegree − popularity (sqrt) −0.72∗∗∗ (0.12)
8 . outdegree − activity (sqrt) −0.49∗∗∗ (0.07)
.

Attribute effects: sex, money
estimate (s.e.)
9 . sex alter .
10. sex ego .
11. sex ego × sex alter .
12. sex alter at distance 2 .
13. sex ego × sex alter dist. 2 .
14. money alter .
15. money similarity .
.

Attribute effects: sex, money
estimate (s.e.)
9 . sex alter −0.15 (0.16)
10. sex ego 0.05 (0.12)
11. sex ego × sex alter 0.95∗∗∗ (0.29)
12. sex alter at distance 2 −0.27 (0.23)
13. sex ego × sex alter dist. 2 1.20∗∗ (0.46)
14. money alter 0.015∗∗ (0.005)
15. money similarity 1.08∗∗∗ (0.28)
.

Attribute effects: drinking, smoking
estimate (s.e.)
16. drink alter .
17. drink ego .
18. drink ego × drink alter .
19. drink alter at distance 2 .
20. drink ego × drink alter dist. 2 .
21. smo alter .
22. smo ego .
23. smo ego × smo alter .
24. smo alter at distance 2 .
25. smo ego × smo alter dist. 2 .
.

Attribute effects: drinking, smoking
estimate (s.e.)
16. drink alter −0.00 (0.04)
17. drink ego −0.03 (0.04)
18. drink ego × drink alter 0.06∗ (0.03)
19. drink alter at distance 2 0.01 (0.13)
20. drink ego × drink alter dist. 2 0.15∗ (0.07)
21. smo alter −0.08 (0.09)
22. smo ego −0.15∗ (0.07)
23. smo ego × smo alter 0.29∗∗∗ (0.08)
24. smo alter at distance 2 −0.22 (0.26)
25. smo ego × smo alter dist. 2 −0.12 (0.22)
.

Conclusion :
Interaction between attributes of ego
and average attributes of alter’s friends
(i.e., distance-2 homophily)
play a role for sex and drinking
(not for smoking or pocket money).
.

Creation termination of ties
distinguished for Glasgow study
In a model distinguishing creation and maintenance effects,
reciprocity is stronger for creation than maintenance
(2.96 versus 1.62),
but the difference is borderline signiﬁcant (p = 0.08);
also transitivity is stronger for creation than maintenance
(1.28 versus –0.36),
but without signiﬁcance of the difference (p = 0.14).

Other study: Ørebro study
Large-scale study of adolescent development
initiated by Håkan Stattin and Margaret Kerr (Univ. of Ørebro).
Collaboration also with Bill Burk.
All 12-18 year olds in a small town in Sweden.
In a sample study of a cohort of all 13 year olds in given year,
3 yearly waves, 339 individuals:
evidence for distance-two homophily
for sex and delinquent behavior.
.

Distance-2 effects for MBA students
In the example of Vanina Torlò’s MBA students,
there was also evidence for a positive effect
of the performance of the advisors of potential advisors
on the probability of asking advice from the latter
( ˆβk = 0.57, s.e. = 0.28; p. 14)
si,alter average =
j
xij ˘v
(−i)
j .

General conclusions about
homophily at distance 2
1 Homophily at distance 2 is theoretically meaningful,
and there is empirical evidence for it
in some data sets of friendship dynamics.

2 Testing this is only meaningful with control
for direct homophily and transitivity.

2 Testing this is only meaningful with control
for direct homophily and transitivity.
3 Note: ego × alter interactions sometimes are
better interpretable / better ﬁtting than
similarity measures.
In this speciﬁcation, average alter - dist. 2 is an average;
similarity - dist. 2 is a sum.
.

However
In the Glasgow data set,
when creation and maintenance effects are included
for reciprocity and transitivity,
the distance-2 effects lose their signiﬁcance.
For similarity on actor variables in this data set,
estimated creation effects in all cases are
larger than maintenance effects;
but not signiﬁcantly different.

So?
Homophily at distance 2 is theoretically meaningful,
and there is some empirical evidence for it.
Distinguishing between inﬂuences on creation and
termination of ties is meaningful,
These are reﬁnements of usual network models,
and developing theories will need to go
hand in hand with empirical tests.

So?
Such model speciﬁcations are at the boundary of
information extractable from medium sized data sets.

So?
Such model speciﬁcations are at the boundary of
information extractable from medium sized data sets.
wonders but no miracles

The Wonders of Specification Possibilities of Stochastic Actor-Oriented Models for Network Dynamics

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