the wonders of specification possibilities of stochastic actor-oriented models for network dynamics

Intro

The Wonders of Specification Possibilitiesof 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;

differentiation tie creation � termination

homophily at distance two

with examples from Vanina Torlò’s MBA studentsand the Glasgow ‘Teenage Friends and Lifestyle Study’.


Intro

Stochastic Actor-Oriented Model

Methodology for analyzing network dynamics:

⇒

⇒ Methods for estimation, testing, goodness of fit, etc.(observations panel data)

.


Intro

Stochastic Actor-Oriented Model

Methodology for analyzing network dynamics:

⇒ 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 functionConsider 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.


Intro

Objective functionConsider 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.Specification Possibilities of SAOMs 4 / 39

Creation versus maintenance of ties

Differentiation tie creation – maintenance

In the more general case for previous state x0 andnew 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 functionf = evaluation functionc = creation functione = maintenance (endowment) function.



Differentiation tie creation – maintenance (3)

This means:

tie creation is modeled bythe sum evaluation function + creation function;

tie maintenance is modeled bythe sum evaluation function + maintenance function.



EstimationThe evaluation, creation, and maintenance functionsare 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)

wherei = 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.

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 thancreation (p < 0.0001)

Transitivity only important for creation (p = 0.002)Indegree popularity (‘Matthew effect’)negative for creation, positive for maintenance(p = 0.002)Performance alter only for maintenance(negative, p = 0.04)Performance ego positive for creation,negative for maintenance (p = 0.01)Performance similarity only for maintenance(but p = 0.08)




Reciprocity 3 times stronger for maintenance thancreation (p < 0.0001)Transitivity only important for creation (p = 0.002)

Indegree popularity (‘Matthew effect’)negative for creation, positive for maintenance(p = 0.002)Performance alter only for maintenance(negative, p = 0.04)Performance ego positive for creation,negative for maintenance (p = 0.01)Performance similarity only for maintenance(but p = 0.08)




Reciprocity 3 times stronger for maintenance thancreation (p < 0.0001)Transitivity only important for creation (p = 0.002)Indegree popularity (‘Matthew effect’)negative for creation, positive for maintenance(p = 0.002)

Performance alter only for maintenance(negative, p = 0.04)Performance ego positive for creation,negative for maintenance (p = 0.01)Performance similarity only for maintenance(but p = 0.08)




Reciprocity 3 times stronger for maintenance thancreation (p < 0.0001)Transitivity only important for creation (p = 0.002)Indegree popularity (‘Matthew effect’)negative for creation, positive for maintenance(p = 0.002)Performance alter only for maintenance(negative, p = 0.04)

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




Reciprocity 3 times stronger for maintenance thancreation (p < 0.0001)Transitivity only important for creation (p = 0.002)Indegree popularity (‘Matthew effect’)negative for creation, positive for maintenance(p = 0.002)Performance alter only for maintenance(negative, p = 0.04)Performance ego positive for creation,negative for maintenance (p = 0.01)

Performance similarity only for maintenance(but p = 0.08)




Reciprocity 3 times stronger for maintenance thancreation (p < 0.0001)Transitivity only important for creation (p = 0.002)Indegree popularity (‘Matthew effect’)negative for creation, positive for maintenance(p = 0.002)Performance alter only for maintenance(negative, p = 0.04)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 formaintenance (but p = 0.07)

Testing differences between creation and maintenance effectsis difficult because their parameter estimates are negativelycorrelated (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.

Multilevel issue:association positive at the dyadic level,negative at the actor level.



General conclusionsabout creation � maintenance

There is, in this data set, strong evidencefor differences between creation and maintenancefor 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

‘homophily at distance 2’.

.






⇒ I am similar to my friends ;

⇒⇒I am similar to friends of my friends


.






⇒ I am similar to my friends ;⇒⇒I am similar to friends of my friends


.



Various theoretical arguments fordistance-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’sattribute 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 ?

£ Regular homophily with transitivitywill imply observed distance-2 homophily:We also have to control for transitivity.

.



Example :Study of smoking initiation and friendshipTeenage 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 schoolstarting 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).

.Specification Possibilities of SAOMs 22 / 39


wave 1 girls: circlesboys: squares

node size: pocket money

color: top = drinkingbottom = smoking

(orange = high)






(orange = high)



Effects for similarity at distance 2Direct homophily effects can be represented byeffects sik(x) expressing similaritybetween 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 .

.



Effects for similarity at distance 2Direct homophily effects can be represented byeffects sik(x) expressing similaritybetween 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 , firstdefine v̆

(−i)j as “alters’ v-average”:

average value of vh for those to whom j is tied, excluding i,

v̆(−i)j =

∑

h 6=i xjh vh

xj+if xj+ − xji > 0

v̄ if xj+ − xji = 0.

.



The distance-two homophily effect can be represented bythe similarity between i andthe alter-averages in i’s personal network,

si,simDist2 =∑

j

xij

1−| vi − v̆

(−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 v̆(−i)j

si,ego × alter average dist. 2 = vi

∑

j

xij v̆(−i)j .

The latter interaction may also be regarded asa 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, smokingestimate (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, smokingestimate (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 egoand 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 tiesdistinguished 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 developmentinitiated 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 homophilyfor 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 effectof the performance of the advisors of potential advisorson the probability of asking advice from the latter(β̂k = 0.57, s.e. = 0.28; p. 14)

si,alter average =∑

j

xij v̆(−i)j .



General conclusions abouthomophily at distance 2

1 Homophily at distance 2 is theoretically meaningful,and there is empirical evidence for itin some data sets of friendship dynamics.

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

3 Note: ego × alter interactions sometimes arebetter interpretable / better fitting thansimilarity measures.In this specification, average alter - dist. 2 is an average;similarity - dist. 2 is a sum.

.



General conclusions abouthomophily at distance 2

1 Homophily at distance 2 is theoretically meaningful,and there is empirical evidence for itin some data sets of friendship dynamics.

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

3 Note: ego × alter interactions sometimes arebetter interpretable / better fitting thansimilarity 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 includedfor 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 arelarger 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 andtermination of ties is meaningful,and there is some empirical evidence for it.

These are refinements of usual network models,and developing theories will need to gohand in hand with empirical tests.

Such model specifications are at the boundary ofinformation 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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