Intro
The Wonders of Specification Possibilities
of Stochastic Actor-Oriented Models
for Network Dynamics
Tom A.B. Snijders...
Intro
Overview
Sketch of Stochastic Actor-Oriented Model (‘SAOM’),
evaluation–endowment–creation functions;
Specification P...
Intro
Overview
Sketch of Stochastic Actor-Oriented Model (‘SAOM’),
evaluation–endowment–creation functions;
differentiatio...
Intro
Overview
Sketch of Stochastic Actor-Oriented Model (‘SAOM’),
evaluation–endowment–creation functions;
differentiatio...
Intro
Overview
Sketch of Stochastic Actor-Oriented Model (‘SAOM’),
evaluation–endowment–creation functions;
differentiatio...
Intro
Stochastic Actor-Oriented Model
Methodology for analyzing network dynamics:
Specification Possibilities of SAOMs 2 / ...
Intro
Stochastic Actor-Oriented Model
Methodology for analyzing network dynamics:
⇒ Probability model of network change in...
Intro
Stochastic Actor-Oriented Model
Methodology for analyzing network dynamics:
⇒ Probability model of network change in...
Intro
Probability Model of SAOM
Since the SAOM is a continuous-time model,
it suffices to model changes of single tie varia...
Intro
Objective function
Consider the probability of the network changing to state x,
given that currently it is in state ...
Intro
Objective function
Consider the probability of the network changing to state x,
given that currently it is in state ...
Creation versus maintenance of ties
Differentiation tie creation – maintenance
In the more general case for previous state...
Creation versus maintenance of ties
Differentiation tie creation – maintenance
In the more general case for previous state...
Creation versus maintenance of ties
Differentiation tie creation – maintenance (2)
The general definition of the objective ...
Creation versus maintenance of ties
Differentiation tie creation – maintenance (3)
This means:
tie creation is modeled by
...
Creation versus maintenance of ties
Estimation
The evaluation, creation, and maintenance functions
are defined as linear co...
Creation versus maintenance of ties
Short remark on estimation by Method of Moments:
For network data sets with (e.g.) two...
Creation versus maintenance of ties
Example 1
Data from Vanina Torlò and Alessandro Lomi.
International MBA program in Ita...
Creation versus maintenance of ties
Example 1
Data from Vanina Torlò and Alessandro Lomi.
International MBA program in Ita...
Creation versus maintenance of ties
Friendship (1)
Effect create eval maintain (s.e.)
outdegree (density) –2.984∗∗∗ (0.205...
Creation versus maintenance of ties
Friendship (2)
Effect create eval maint (s.e.)
same nationality 0.252∗ (0.100)
perfo a...
Creation versus maintenance of ties
Advice (1)
Effect create eval maint (s.e.)
outdegree (density) –4.536∗∗∗ (0.581)
recip...
Creation versus maintenance of ties
Advice (2)
Effect create eval maint (s.e.)
same natio 0.391∗ (0.168)
perfo alter 0.110...
Creation versus maintenance of ties
Conclusions: creation maintenance (F)
For Friendship, there are some strong difference...
Creation versus maintenance of ties
Conclusions: creation maintenance (F)
For Friendship, there are some strong difference...
Creation versus maintenance of ties
Conclusions: creation maintenance (F)
For Friendship, there are some strong difference...
Creation versus maintenance of ties
Conclusions: creation maintenance (F)
For Friendship, there are some strong difference...
Creation versus maintenance of ties
Conclusions: creation maintenance (F)
For Friendship, there are some strong difference...
Creation versus maintenance of ties
Conclusions: creation maintenance (F)
For Friendship, there are some strong difference...
Creation versus maintenance of ties
Conclusions: creation maintenance (A)
For Advice, there are weaker differences:
Recipr...
Creation versus maintenance of ties
Conclusions: creation maintenance (A)
For Advice, there are weaker differences:
Recipr...
Creation versus maintenance of ties
Conclusions: creation maintenance (A)
For Advice, there are weaker differences:
Recipr...
Creation versus maintenance of ties
Conclusions: creation maintenance (A)
For Advice, there are weaker differences:
Recipr...
Creation versus maintenance of ties
Conclusions: co-evolution
Positive dyad-level effects advice ⇔ friendship,
creation no...
Creation versus maintenance of ties
Conclusions: co-evolution
Positive dyad-level effects advice ⇔ friendship,
creation no...
Creation versus maintenance of ties
Conclusions: co-evolution
Positive dyad-level effects advice ⇔ friendship,
creation no...
Creation versus maintenance of ties
General conclusions
about creation maintenance
There is, in this data set, strong evid...
Creation versus maintenance of ties
General conclusions
about creation maintenance
There is, in this data set, strong evid...
Homophily and Beyond
Homophily and beyond
Specification Possibilities of SAOMs 19 / 39
Homophily and Beyond
Homophily and beyond
Homophily well known
(Lazarsfeld & Merton 1954;
McPherson, Smith-Lovin & Cook 20...
Homophily and Beyond
Homophily and beyond
Homophily well known
(Lazarsfeld & Merton 1954;
McPherson, Smith-Lovin & Cook 20...
Homophily and Beyond
Homophily and beyond
Homophily well known
(Lazarsfeld & Merton 1954;
McPherson, Smith-Lovin & Cook 20...
Homophily and Beyond
Homophily and beyond
Homophily well known
(Lazarsfeld & Merton 1954;
McPherson, Smith-Lovin & Cook 20...
Homophily and Beyond
Various theoretical arguments for
distance-2 homophily, e.g.:
Specification Possibilities of SAOMs 20 ...
Homophily and Beyond
Various theoretical arguments for
distance-2 homophily, e.g.:
1 social identity : “tell me who your f...
Homophily and Beyond
Various theoretical arguments for
distance-2 homophily, e.g.:
1 social identity : “tell me who your f...
Homophily and Beyond
Various theoretical arguments for
distance-2 homophily, e.g.:
1 social identity : “tell me who your f...
Homophily and Beyond
Various theoretical arguments for
distance-2 homophily, e.g.:
1 social identity : “tell me who your f...
Homophily and Beyond
?
is there a tendency to homophily at distance 2,
while controlling for (regular) homophily ?
Specific...
Homophily and Beyond
?
is there a tendency to homophily at distance 2,
while controlling for (regular) homophily ?
Regular...
Homophily and Beyond
Example :
Study of smoking initiation and friendship
Teenage Friends and Lifestyle Study
(following u...
Homophily and Beyond
wave 1 girls: circles
boys: squares
node size: pocket money
color: top = drinking
bottom = smoking
(o...
Homophily and Beyond
wave 2 girls: circles
boys: squares
node size: pocket money
color: top = drinking
bottom = smoking
(o...
Homophily and Beyond
wave 3 girls: circles
boys: squares
node size: pocket money
color: top = drinking
bottom = smoking
(o...
Homophily and Beyond
Effects for similarity at distance 2
Direct homophily effects can be represented by
effects sik(x) ex...
Homophily and Beyond
Effects for similarity at distance 2
Direct homophily effects can be represented by
effects sik(x) ex...
Homophily and Beyond
To define distance-two homophily effects , first
define ˘v
(−i)
j as “alters’ v-average”:
average value ...
Homophily and Beyond
The distance-two homophily effect can be represented by
the similarity between i and
the alter-averag...
Homophily and Beyond
The effect of alter’s v- average,
and its interaction with ego-v, are defined as
si,alter average dist...
Homophily and Beyond
Structural effects
estimate (s.e.)
1 . outdegree (density) −0.92∗∗ (0.29)
2 . reciprocity 2.28∗∗∗ (0....
Homophily and Beyond
Attribute effects: sex, money
estimate (s.e.)
9 . sex alter .
10. sex ego .
11. sex ego × sex alter ....
Homophily and Beyond
Attribute effects: sex, money
estimate (s.e.)
9 . sex alter −0.15 (0.16)
10. sex ego 0.05 (0.12)
11. ...
Homophily and Beyond
Attribute effects: drinking, smoking
estimate (s.e.)
16. drink alter .
17. drink ego .
18. drink ego ...
Homophily and Beyond
Attribute effects: drinking, smoking
estimate (s.e.)
16. drink alter −0.00 (0.04)
17. drink ego −0.03...
Homophily and Beyond
Conclusion :
Interaction between attributes of ego
and average attributes of alter’s friends
(i.e., d...
Homophily and Beyond
Creation termination of ties
distinguished for Glasgow study
In a model distinguishing creation and m...
Homophily and Beyond
Other study: Ørebro study
Large-scale study of adolescent development
initiated by Håkan Stattin and ...
Homophily and Beyond
Distance-2 effects for MBA students
In the example of Vanina Torlò’s MBA students,
there was also evi...
Homophily and Beyond
General conclusions about
homophily at distance 2
1 Homophily at distance 2 is theoretically meaningf...
Homophily and Beyond
General conclusions about
homophily at distance 2
1 Homophily at distance 2 is theoretically meaningf...
Homophily and Beyond
General conclusions about
homophily at distance 2
1 Homophily at distance 2 is theoretically meaningf...
Homophily and Beyond
However
In the Glasgow data set,
when creation and maintenance effects are included
for reciprocity a...
Homophily and Beyond
So?
Homophily at distance 2 is theoretically meaningful,
and there is some empirical evidence for it....
Homophily and Beyond
So?
Homophily at distance 2 is theoretically meaningful,
and there is some empirical evidence for it....
Homophily and Beyond
So?
Homophily at distance 2 is theoretically meaningful,
and there is some empirical evidence for it....
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The Wonders of Specification Possibilities of Stochastic Actor-Oriented Models for Network Dynamics

  1. 1. 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
  2. 2. Intro Overview Sketch of Stochastic Actor-Oriented Model (‘SAOM’), evaluation–endowment–creation functions; Specification Possibilities of SAOMs 1 / 39
  3. 3. Intro Overview Sketch of Stochastic Actor-Oriented Model (‘SAOM’), evaluation–endowment–creation functions; differentiation tie creation termination Specification Possibilities of SAOMs 1 / 39
  4. 4. Intro Overview Sketch of Stochastic Actor-Oriented Model (‘SAOM’), evaluation–endowment–creation functions; differentiation tie creation termination homophily at distance two Specification Possibilities of SAOMs 1 / 39
  5. 5. 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 students and the Glasgow ‘Teenage Friends and Lifestyle Study’. Specification Possibilities of SAOMs 1 / 39
  6. 6. Intro Stochastic Actor-Oriented Model Methodology for analyzing network dynamics: Specification Possibilities of SAOMs 2 / 39
  7. 7. Intro Stochastic Actor-Oriented Model Methodology for analyzing network dynamics: ⇒ Probability model of network change in continuous time Specification Possibilities of SAOMs 2 / 39
  8. 8. 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) . Specification Possibilities of SAOMs 2 / 39
  9. 9. 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. Specification Possibilities of SAOMs 3 / 39
  10. 10. 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. Specification Possibilities of SAOMs 4 / 39
  11. 11. 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. Specification Possibilities of SAOMs 4 / 39
  12. 12. 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); Specification Possibilities of SAOMs 5 / 39
  13. 13. 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); ⇒ 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). Specification Possibilities of SAOMs 5 / 39
  14. 14. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 6 / 39
  15. 15. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 7 / 39
  16. 16. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 8 / 39
  17. 17. Creation versus maintenance of ties 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). Specification Possibilities of SAOMs 9 / 39
  18. 18. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 10 / 39
  19. 19. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 10 / 39
  20. 20. Creation versus maintenance of ties 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) Specification Possibilities of SAOMs 11 / 39
  21. 21. Creation versus maintenance of ties 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) Specification Possibilities of SAOMs 12 / 39
  22. 22. Creation versus maintenance of ties Advice (1) Effect create eval maint (s.e.) 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) Specification Possibilities of SAOMs 13 / 39
  23. 23. Creation versus maintenance of ties Advice (2) Effect create eval maint (s.e.) 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. Specification Possibilities of SAOMs 14 / 39
  24. 24. Creation versus maintenance of ties Conclusions: creation maintenance (F) For Friendship, there are some strong differences: Reciprocity 3 times stronger for maintenance than creation (p < 0.0001) Specification Possibilities of SAOMs 15 / 39
  25. 25. Creation versus maintenance of ties 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) Specification Possibilities of SAOMs 15 / 39
  26. 26. Creation versus maintenance of ties 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) Specification Possibilities of SAOMs 15 / 39
  27. 27. Creation versus maintenance of ties 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) Performance alter only for maintenance (negative, p = 0.04) Specification Possibilities of SAOMs 15 / 39
  28. 28. Creation versus maintenance of ties 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) Performance alter only for maintenance (negative, p = 0.04) Performance ego positive for creation, negative for maintenance (p = 0.01) Specification Possibilities of SAOMs 15 / 39
  29. 29. Creation versus maintenance of ties 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) 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) Specification Possibilities of SAOMs 15 / 39
  30. 30. Creation versus maintenance of ties Conclusions: creation maintenance (A) For Advice, there are weaker differences: Reciprocity only important for maintenance (p = 0.04) Specification Possibilities of SAOMs 16 / 39
  31. 31. Creation versus maintenance of ties 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) Specification Possibilities of SAOMs 16 / 39
  32. 32. Creation versus maintenance of ties 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) Specification Possibilities of SAOMs 16 / 39
  33. 33. Creation versus maintenance of ties 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) Testing differences between creation and maintenance effects is difficult because their parameter estimates are negatively correlated (which increases the s.e. of the difference). Specification Possibilities of SAOMs 16 / 39
  34. 34. Creation versus maintenance of ties Conclusions: co-evolution Positive dyad-level effects advice ⇔ friendship, creation not different from maintenance, of same order of magnitude as reciprocity maintenance. Specification Possibilities of SAOMs 17 / 39
  35. 35. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 17 / 39
  36. 36. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 17 / 39
  37. 37. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 18 / 39
  38. 38. Creation versus maintenance of ties 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. Specification Possibilities of SAOMs 18 / 39
  39. 39. Homophily and Beyond Homophily and beyond Specification Possibilities of SAOMs 19 / 39
  40. 40. Homophily and Beyond Homophily and beyond Homophily well known (Lazarsfeld & Merton 1954; McPherson, Smith-Lovin & Cook 2001): ties more likely between similar actors. Specification Possibilities of SAOMs 19 / 39
  41. 41. 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 ; Specification Possibilities of SAOMs 19 / 39
  42. 42. 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 Specification Possibilities of SAOMs 19 / 39
  43. 43. 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’. . Specification Possibilities of SAOMs 19 / 39
  44. 44. Homophily and Beyond Various theoretical arguments for distance-2 homophily, e.g.: Specification Possibilities of SAOMs 20 / 39
  45. 45. Homophily and Beyond Various theoretical arguments for distance-2 homophily, e.g.: 1 social identity : “tell me who your friends are ..." Specification Possibilities of SAOMs 20 / 39
  46. 46. Homophily and Beyond 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 ..." Specification Possibilities of SAOMs 20 / 39
  47. 47. Homophily and Beyond 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; Specification Possibilities of SAOMs 20 / 39
  48. 48. Homophily and Beyond 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; 4 negative diversity, social capital : alters bridging to different third actors. . Specification Possibilities of SAOMs 20 / 39
  49. 49. Homophily and Beyond ? is there a tendency to homophily at distance 2, while controlling for (regular) homophily ? Specification Possibilities of SAOMs 21 / 39
  50. 50. Homophily and Beyond ? 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. . Specification Possibilities of SAOMs 21 / 39
  51. 51. Homophily and Beyond 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). . Specification Possibilities of SAOMs 22 / 39
  52. 52. Homophily and Beyond wave 1 girls: circles boys: squares node size: pocket money color: top = drinking bottom = smoking (orange = high) Specification Possibilities of SAOMs 23 / 39
  53. 53. Homophily and Beyond wave 2 girls: circles boys: squares node size: pocket money color: top = drinking bottom = smoking (orange = high) Specification Possibilities of SAOMs 24 / 39
  54. 54. Homophily and Beyond wave 3 girls: circles boys: squares node size: pocket money color: top = drinking bottom = smoking (orange = high) Specification Possibilities of SAOMs 25 / 39
  55. 55. Homophily and Beyond 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 Specification Possibilities of SAOMs 26 / 39
  56. 56. Homophily and Beyond 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 . . Specification Possibilities of SAOMs 26 / 39
  57. 57. Homophily and Beyond 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. . Specification Possibilities of SAOMs 27 / 39
  58. 58. Homophily and Beyond 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    . . Specification Possibilities of SAOMs 28 / 39
  59. 59. Homophily and Beyond 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. . Specification Possibilities of SAOMs 29 / 39
  60. 60. Homophily and Beyond 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) . Specification Possibilities of SAOMs 30 / 39
  61. 61. Homophily and Beyond 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 . . Specification Possibilities of SAOMs 31 / 39
  62. 62. Homophily and Beyond 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) . Specification Possibilities of SAOMs 31 / 39
  63. 63. Homophily and Beyond 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 . . Specification Possibilities of SAOMs 32 / 39
  64. 64. Homophily and Beyond 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) . Specification Possibilities of SAOMs 32 / 39
  65. 65. Homophily and Beyond 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). . Specification Possibilities of SAOMs 33 / 39
  66. 66. Homophily and Beyond 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). Specification Possibilities of SAOMs 34 / 39
  67. 67. Homophily and Beyond 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. . Specification Possibilities of SAOMs 35 / 39
  68. 68. Homophily and Beyond 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 . Specification Possibilities of SAOMs 36 / 39
  69. 69. Homophily and Beyond 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. Specification Possibilities of SAOMs 37 / 39
  70. 70. Homophily and Beyond 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. Specification Possibilities of SAOMs 37 / 39
  71. 71. Homophily and Beyond 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. 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. . Specification Possibilities of SAOMs 37 / 39
  72. 72. Homophily and Beyond 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. Specification Possibilities of SAOMs 38 / 39
  73. 73. Homophily and Beyond 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, and there is some empirical evidence for it. These are refinements of usual network models, and developing theories will need to go hand in hand with empirical tests. Specification Possibilities of SAOMs 39 / 39
  74. 74. Homophily and Beyond 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, and there is some empirical evidence for it. These are refinements of usual network models, and developing theories will need to go hand in hand with empirical tests. Such model specifications are at the boundary of information extractable from medium sized data sets. Specification Possibilities of SAOMs 39 / 39
  75. 75. Homophily and Beyond 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, and there is some empirical evidence for it. These are refinements of usual network models, and developing theories will need to go hand in hand with empirical tests. Such model specifications are at the boundary of information extractable from medium sized data sets. wonders but no miracles Specification Possibilities of SAOMs 39 / 39

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