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Statistical Social Network Analysis - Stochastic Actor Oriented Models Johan Koskinen The Social Statistics Discipline Area, School of Social Sciences Mitchell Centre for Network Analysis Workshop: Monday, 29 August 2011

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Stochastic actor-oriented models - data Study of 32 freshman university students, 7 waves in 1 year. (van de Bunt, van Duijn, & Snijders, Computational & Mathematical Organization Theory, 5, (1999), 167 – 192)

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Stochastic actor-oriented models - data

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The mini-step Say that actor i wants to make a change to his network A piece of notation Adjacency matrix before change -010 0-10 10-1 000- x = after change ( i j ) x ( i j ) = -110 0-10 10-1 000- Lets say actor i = 1 changes relation to actor j = 2

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The mini-step Say that actor i wants to make a change to his network A piece of notation Adjacency matrix before change -010 0-10 10-1 000- x = after change ( i j ) x ( i j ) = -110 0-10 10-1 000- Lets say actor i = 1 changes relation to actor j = 2

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The mini-step Say that actor i wants to make a change to his network A piece of notation Adjacency matrix before change -010 0-10 10-1 000- x = after change ( i j ) x ( i j ) = -110 0-10 10-1 000- Lets say actor i = 1 changes relation to actor j = 2

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The mini-step Say that actor i wants to make a change to his network A piece of notation Adjacency matrix before change -010 0-10 10-1 000- x = after change ( i j ) x ( i j ) = -110 0-10 10-1 000- Lets say actor i = 1 changes relation to actor j = 2

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The mini-step A sample y path is a series of toggles Where a toggle set element

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 4

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 1 2 3 4 -110 0-10 10-1 000- x (1 2) =

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 1 2 3 4 -110 0-10 10-1 000- x (1 2) =

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 1 2 3 4 1 2 3 4 -110 0-10 10-1 000- x (1 2) = -000 0-10 10-1 000- x (1 3) =

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 -110 0-10 10-1 000- x (1 2) = -000 0-10 10-1 000- x (1 3) =

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 -110 0-10 10-1 000- x (1 2) = -000 0-10 10-1 000- x (1 3) = -011 0-10 10-1 000- x (1 4) =

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 4 1 3 4 1 2 3 4 1 2 3 4 -110 0-10 10-1 000- x (1 2) = -000 0-10 10-1 000- x (1 3) = -011 0-10 10-1 000- x (1 4) =

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The mini-step In Snijders stochastic actor-oriented model the actors degree of satisfaction with new state is a weighted sum of e.g. The number of outgoing ties The number of reciprocated ties The number of ties to people of type A The number of ties to people like the actor The number of friends that are also friends The number of indirect friends

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The mini-step In Snijders stochastic actor-oriented model the actors degree of satisfaction with new state is a weighted sum of e.g. The number of outgoing ties The number of reciprocated ties The number of ties to people of type A The number of ties to people like the actor The number of friends that are also friends The number of indirect friends

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The mini-step In Snijders stochastic actor-oriented model the actors degree of satisfaction with new state is a weighted sum of e.g. The number of outgoing ties The number of reciprocated ties The number of ties to people of type A The number of ties to people like the actor The number of friends that are also friends The number of indirect friends Assume that we believe that these local structures are important to actors when they change their out-going ties

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 -110 0-10 10-1 000- x (1 2) = -000 0-10 10-1 000- x (1 3) = -011 0-10 10-1 000- x (1 4) = 1 edge 1 reciprocated tie 1 indirect others +1 edge +2 transitive triangles 1 indirect others 1 edge ±0 reciprocated tie ±0 indirect others

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 We model the preference of changing actor: Preference actor i = β 1 × edge + β 2 × reciprocated tie + β 3 × indirect others + ε

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 We model the preference of changing actor: Preference actor i = β 1 × edge + β 2 × reciprocated tie + β 3 × indirect others + ε random component Importance of structure

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 We model the preference of changing actor: Preference actor i = β 1 × edge + β 2 × reciprocated tie + β 3 × indirect others + ε - If parameter positive: preference for creating reciprocated ties - If parameter negative: preference against creating reciprocated ties

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 We model the preference of changing actor: Preference actor i = β 1 × edge + β 2 × reciprocated tie + β 3 × indirect others + ε Random component captures non- systematic, idiosyncratic factors

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The evaluation function -010 0-10 10-1 000- x = 1 2 3 4 1 The evaluation function is the systematic part of the preference (e.g.): f i (y) = β 1 × edge + β 2 × reciprocated tie + β 3 × indirect others

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The mini-step -010 0-10 10-1 000- x = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 -110 0-10 10-1 000- x (1 2) = -000 0-10 10-1 000- x (1 3) = -011 0-10 10-1 000- x (1 4) = The evaluation function gives the probability of change:

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Empirical example The Gerhard van de Bunt data: 32 university freshmen (24 fem, 8 male). 3 observations used (t1, t2, t3) Densities increase from 0.15 at t1 via 0.18 to 0.22 at t3.

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Simple model Rate parameters: about 3 opportunities for change/actor between observations; out-degree parameter negative: on average, cost of friendship ties higher than their benefits; reciprocity effect strong and highly significant (t = 1.79/0.27 = 6.6). estimates.e. Rate t1-t2 3.51(0.54) Rate t2-t3 3.09(0.49) Out-degree 1.10(0.15) Reciprocity 1.79(0.27)

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The evaluation function in the simple model The evaluation function is: This expresses how much actor i likes the network Adding reciprocated tie (i.e., for which x ji = 1) gives i Adding non-reciprocated tie (i.e., for which x ji = 0) gives i.e., this has negative benefits

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Empirical example no 2 Excerpt of 50 girls from the Teenage Friends and Lifestyle Study data set (West and Sweeting 1995) s50-smoke.dat. Smoking: 1 (non), 2 (occasional) and 3 (regular, i.e. more than once per week). s50-drugs.dat. Cannabis use: 1 (non), 2 (tried once), 3 (occasional) and 4 (regular). s50-alcohol.dat. Alcohol: 1 (non), 2 (once or twice a year), 3 (once a month), 4 (once a week) and 5 (more than once a week). s50-sport.dat. Sport: 1 (not regular) and 2 (regular). s50-familyevent.dat. Binary information over whether or not the number of persons has changed with which the pupil shares his home address.

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Empirical example no 2 A script for fitting data: friend.data.w1 <- as.matrix(read.table("s50-network1.dat"))# read data friend.data.w2 <- as.matrix(read.table("s50-network2.dat")) friend.data.w3 <- as.matrix(read.table("s50-network3.dat")) drink <- as.matrix(read.table("s50-alcohol.dat")) smoke <- as.matrix(read.table("s50-smoke.dat")) friendship <- sienaNet( array( c( friend.data.w1, friend.data.w2, friend.data.w3 ), dim = c( 50, 50, 3 ) ) )# create dependent variable smoke1 <- coCovar( smoke[, 1 ] )# create constant covariate alcohol <- varCovar( drink )# create time varying covariate

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Empirical example no 2 A script for fitting data: mydata <- sienaDataCreate( friendship, smoke1, alcohol )# define data myeff <- getEffects( mydata )# create effects structure print01Report( mydata, myeff, modelname = 's50_3_init' )# siena01 for reports myeff <- includeEffects( myeff, transTrip, cycle3 )# add structural effects myeff <- includeEffects( myeff, egoX, altX, egoXaltX, interaction1 = "alcohol" )# add covariate effects myeff <- includeEffects( myeff, simX, interaction1 = "smoke1" ) mymodel <- sienaModelCreate( useStdInits = TRUE, projname = 's50_3' )# define model as data + effects ans <- siena07( mymodel, data = mydata, effects = myeff) # estimate model

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Co-evolution models We model the preference of actor when changing behaviour:

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Co-evolution models We model the preference of actor when changing behaviour:

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Co-evolution models We model the preference of actor when changing behaviour:

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Co-evolution models We model the preference of actor when changing behaviour: Similar evaluation function:

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Wrap-up The package RSiena supports analysis of - un-directed networks - bi-partite networks - multiple networks - … and more forthcoming Updates and news are published on http://www.stats.ox.ac.uk/%7Esnijders/siena/ Next RSiena workshop: 14 Sept in Zurich

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