Tom A.B. Snijders Multilevel Flavours Manchester …snijders/siena/Manchester...The Multiple...

Tom A.B. Snijders Multilevel Flavours Manchester Keynote 1 / 63

The Multiple Flavours of Multilevel Issuesfor Networks

Tom A.B. Snijders

University of OxfordUniversity of Groningen

June 19, 2012

Network analysis,certainly multilevel network analysis,is a cooperative venture ...

Multiple Flavours

As a statistician, for me originallymultilevel analysis was about nested data sets.

Then I learned that for sociologists, there is interestin the theoretical distinction between the levels:e.g., pupils in schools exemplifynot only multiple populations for whichinference sample ⇒ population is important,but also individuals in social contexts, anddifferent sets of actors mutually interacting.

This variety further multiplieswhen you think of network analysis.

Multiple Flavours

Multiple Flavours Whose pictures?

Where are we @?

Multiple Flavours

What is essential for ‘multilevel’ point of view:

Units of different natures;

these have their own type of influence on variables;

random/unexplained variability associated with each ‘level’.

Multiple Flavours

1 levels in one network

2 multiple parallel networks:replication, populations of networks

3 multiple actor setsand multiple types of relation

4 large networks

Multiple Flavours

4 large networks

Multiple Flavours

4 large networks

Multiple Flavours

4 large networks

Levels in One Network

1One Network, Multiple Levels

David Krackhardt, in his 2012 Sunbelt Keynote Lecture,talked about the multiple levels of analysis in a network:

in a network with N actors,level L is represented by those issueswhere the number of cases is of order NL .

Level 1 Actors / Nodes

Level 2 Dyads / Ties

Level 3 Triads

Higher Larger subgroups ∼ hypergraph models

Level 0 Network

Level 3 Triads

Level 0 Network

Level 3 Triads

Level 0 Network

Level 3 Triads

Level 0 Network

Level 3 Triads

Level 0 Network

Level 3 Triads

Level 0 Network

.Tom A.B. Snijders Multilevel Flavours Manchester Keynote 13 / 63

The recognition and distinction of actors and dyadsis fundamental for the graph-theoretic basisof network analysis.

Triads are fundamental for the sociological approachto social networks since Simmel.

Hypergraph models are not used so very much.They are a natural representation, e.g., foractivities occurring in groups andemails with multiple recipients.

Theoretical interest for the distinctionbetween the actor and dyad levelsbecomes even more interesting with multivariate networks.

Levels in One Network Relations impinge on relations

Multilevel issues in multivariate networks

... on the variety of how relations can affect relations ...

(cf. also the algebraic approach;e.g., work by Pattison & Breiger.)

Here the various levels are not nested:

ties, dyads, actors, triads, subgroups, ...,

Different relations can impinge on one anotherin many different ways.

Dyad level

direct association(within tie)entrainment

mixed reciprocity

Dyad level

mixed reciprocity

Dyad level

mixed reciprocity

Dyad level

mixed reciprocity

Dyad level

mixed reciprocity

Actor level

mixed popularity

mixed activity

mixed twopathity

Actor level

mixed popularity

mixed activity

mixed twopathity

Actor level

mixed popularity

mixed activity

mixed twopathity

Actor level

mixed popularity

mixed activity

mixed twopathity

Actor level

mixed popularity

mixed activity

mixed twopathity

Actor level

mixed popularity

mixed activity

mixed twopathity

Triad level

mixedtransitive closure

agreement ⇒red tie

. . . and other tie orientations . . .

Triad level

This type of cross-network dependencies is discussedfor cross-sectional observationsin Wasserman & Pattison (1999),with examples in Lazega & Pattison (1999).

For longitudinal observationsdependencies are multiplied,because we must distinguish betweenthe dependent and the explanatory(antecedent – subsequent) relations.

Levels in One Network Multivariate SAOMs

Stochastic Actor-Oriented Models

Dynamics of multivariate networks can be represented bystochastic actor-oriented models as a direct extensionof such models for single networks.

(Snijders, Lomi & Torlò, Social Networks, in press.)

Multivariate dynamics modeled as continuous-time Markovchain, with state = the multivariate network,where tie variables change one by one.

Levels in One Network Multivariate SAOMs

This can be extended by dependentbehaviour variables and/or two-mode networks.

Note that in Markov process modeling,extending the state space meansrelaxing the Markov assumption:the current state then provides more information.

Levels in One Network Example: Vanina’s MBA study

Example

Research withVanina Torlò, Alessandro Lomi, Christian Steglich.

International MBA program in Italy;75 students; 3 waves.

1 Friendship

2 Advice:To whom do you go for help if you missed a class, etc.

3 Communication:With whom do you talk regularly.

Example

1 Friendship

Example

1 Friendship

Results will be presented comparingunivariate and multivariate models.

The difference shows which parts of the rulesgoverning the dynamics of each of the networksare mediated by the other networks.

Results: Friendship, univariate – multivariateuni multi

Effect par. (s.e.) par. (s.e.)

Out-degree –1.840 (0.233) –4.311 (0.395)Reciprocity 1.604 (0.097) 0.756 (0.184) ⇐Transitive triplets 0.188 (0.017) 0.150 (0.024)3-cycles –0.095 (0.030) –0.065 (0.037)Indegree popularity (p) 0.218 (0.062) 0.292 (0.104)Outdegree popularity (p) –0.383 (0.065) –0.237 (0.085) ←Outdegree activity (p) –0.079 (0.041) 0.156 (0.044) ⇐Sex alter –0.016 (0.070) –0.061 (0.078)Sex ego –0.158 (0.070) –0.102 (0.079)Same sex 0.277 (0.065) 0.144 (0.078) ←Same nationality 0.240 (0.080) 0.208 (0.091)Performance alter –0.015 (0.023) –0.017 (0.030)Performance ego –0.076 (0.024) –0.081 (0.025)Performance similarity 0.764 (0.188) 0.367 (0.224) ←

Results: Friendship (across networks)

Effect par. (s.e.)

Advice ⇒ Friendship 0.842 (0.256)Reciprocal advice ⇒ Friendship 0.222 (0.209)Advice ind. ⇒ Friendship popularity –0.211 (0.071)Advice outd. ⇒ Friendship activity –0.143 (0.077)Communication ⇒ Friendship 1.694 (0.197)Reciprocal Comm. ⇒ Friendship 0.415 (0.225)Comm. ind. ⇒ Friendship popularity 0.112 (0.096)Comm. outd. ⇒ Friendship popularity –0.087 (0.058)Comm. outd. ⇒ Friendship activity –0.231 (0.061)

Results: Advice, univariate – multivariateuni multi

Out-degree –2.267 (0.321) –5.129 (0.661)Reciprocity 1.329 (0.131) 0.314 (0.170) ⇐Transitive triplets 0.320 (0.038) 0.174 (0.045) ⇐3-cycles –0.065 (0.061) –0.103 (0.069)Indegree popularity (p) 0.245 (0.057) 0.293 (0.094)Outdegree popularity (p) –0.346 (0.143) 0.073 (0.225) ←Outdegree activity (p) –0.088 (0.062) 0.150 (0.080) ←Sex alter –0.043 (0.092) –0.002 (0.103)Sex ego –0.269 (0.096) –0.174 (0.102)Same sex 0.168 (0.086) 0.080 (0.102)Same nationality 0.450 (0.124) 0.371 (0.127)Performance alter 0.129 (0.036) 0.164 (0.047)Performance ego –0.107 (0.034) –0.071 (0.037)Performance similarity 0.735 (0.245) –0.003 (0.268) ⇐

Results: Advice (across networks)

Effect par. (s.e.)

Friendship ⇒ Advice 0.670 (0.367)Reciprocal friendship ⇒ Advice –0.113 (0.210)Friendship ind. ⇒ Advice popularity –0.269 (0.130)Friendship outd. ⇒ Advice activity –0.199 (0.077)Communication ⇒ Advice 1.533 (0.339)Reciprocal Comm. ⇒ Advice 0.632 (0.253)Comm. ind. ⇒ Advice popularity 0.100 (0.124)Comm. outd. ⇒ Advice activity –0.189 (0.077)

Communication, univariate – multivariateuni multi

Out-degree –1.116 (0.237) –1.977 (0.433)Reciprocity 1.334 (0.068) 0.941 (0.110) ⇐Transitive triplets 0.127 (0.010) 0.104 (0.011) ←3-cycles –0.035 (0.025) 0.006 (0.023)Indegree popularity (p) 0.216 (0.043) 0.254 (0.063)Outdegree popularity (p) –0.429 (0.077) –0.481 (0.081)Outdegree activity (p) –0.057 (0.027) 0.096 (0.035) ⇐Sex alter –0.016 (0.058) –0.012 (0.057)Sex ego –0.160 (0.054) –0.114 (0.061)Same sex 0.206 (0.054) 0.139 (0.055)Same nationality 0.009 (0.061) –0.083 (0.065)Performance alter –0.007 (0.019) –0.009 (0.023)Performance ego –0.080 (0.019) –0.048 (0.023)Performance similarity 0.580 (0.129) 0.552 (0.161)

Results: Communication (across networks)

Effect par. (s.e.)

Friendship ⇒ Comm. 1.792 (0.216)Reciprocal friendship ⇒ Comm. –0.011 (0.165)Friendship ind. ⇒ Comm. popularity –0.051 (0.055)Friendship outd. ⇒ Comm. activity –0.101 (0.038)Closure friendship ⇒ Comm. –0.127 (0.033)Advice ⇒ Comm. 0.945 (0.256)Reciprocal Advice ⇒ Comm. 0.720 (0.227)Advice ind. ⇒ Comm. popularity 0.027 (0.045)Advice outd. ⇒ Comm. activity 0.051 (0.053)Advice ind. ⇒ Comm. activity –0.127 (0.043)

Results: Communication (across networks)

Effect par. (s.e.)

Friendship ⇒ Comm. 1.792 (0.216)Reciprocal friendship ⇒ Comm. –0.011 (0.165)Friendship ind. ⇒ Comm. popularity –0.051 (0.055)Friendship outd. ⇒ Comm. activity –0.101 (0.038)Closure friendship ⇒ Comm. –0.127 (0.033) ⇐Advice ⇒ Comm. 0.945 (0.256)Reciprocal Advice ⇒ Comm. 0.720 (0.227)Advice ind. ⇒ Comm. popularity 0.027 (0.045)Advice outd. ⇒ Comm. activity 0.051 (0.053)Advice ind. ⇒ Comm. activity –0.127 (0.043)

Dyad level

Dyad level C

All relations positively associated within dyads,direct effects stronger than reciprocal effects.

Actor level, incoming

Actor level:incoming ties(popularity)(red =negative association)

At actor-level for incoming ties:friendship and advice negatively associated,positive association communication ⇒ friendship.

Actor level, outgoing

Actor level:outgoing ties(activity)(red =negative association)

At actor-level for outgoing ties:all relations weakly negatively related;

A → C is mixed effect:incoming advice ties lead to less comm. activity.

Triad level

negative mixed closureFriendsh. ; Comm.

Note that this is an effect additional to thepositive triadic closure of friendshipand the direct dyadic effect of friendship on communication.This will be taken up below.

Conclusions: levels

Dyad level and actor level tell different stories.

Individuals specialize:

Tied dyads tend to have multiplex ties:advice and friendship.

Conclusions: levels

Individuals specialize: learning⇔ sociability

Conclusions: levels

Individuals specialize: advice⇔ friendship

Conclusions: levels

Individuals specialize: advice⇔ friendship

Conclusions: dyad level

Cross-dependencies between networks changethe representation of the internal dynamics:Observed reciprocation and homophily for networksare partly accounted for by multiplex entrainmentand reciprocation & homophily in the other networks;this is the case especially for advice;

homophily in friendship mediated by communication;

performance homophily remains only for communication;

homophily in advice remains only for nationality.

Communicate with those having similar performance,

and they will advise you.

Conclusions: triad level

Most cross-network triadic effects disappearwhen controlling for actor-level dependencies.

This illustrates/generalizes Feld’s (ASR 1982) remarkthat unmodeled degree heterogeneitymay lead to spurious conclusions of transitivity.

The only remaining cross-network triadic effect is thenegative mixed closure Friendship ; Communication,

counteracting the positive dyadic dependency F⇔ C;

they communicate relatively less with friends of friends

who are not also direct friends.

Conclusions: triad level

Most cross-network triadic effects disappearwhen controlling for actor-level dependencies.

This illustrates/generalizes Feld’s (ASR 1982) remarkthat unmodeled degree heterogeneitymay lead to spurious conclusions of transitivity.

The only remaining cross-network triadic effect is thenegative mixed closure Friendship ; Communication,counteracting the positive dyadic dependency F⇔ C;

they communicate relatively less with friends of friends

who are not also direct friends.

Levels in One Network Network Theory

Network Theory Pattern ???There are two main networks (friendship and advice),representing interactions that are connected totwo distinct goals: social well-being and academic success.

The two goals are not contradictory but pursuing themdemands time – a limited resource.

At the dyadic level multiplexity is dominant,leading to positive association between the networks,related to multi-functionality of personal contacts;

at the actor level, specialization is dominant(depending on preferences and comparative advantage),leading to negative association between the networks.

Network Theory Pattern ???There are two main networks (friendship and advice),representing interactions that are connected totwo distinct goals: social well-being and academic success.

The two goals are not contradictory but pursuing themdemands time – a limited resource.

At the dyadic level multiplexity is dominant,leading to positive association between the networks,related to multi-functionality of personal contacts;

at the actor level, specialization is dominant(depending on preferences and comparative advantage),leading to negative association between the networks.

Here only networks are considered;it would be interesting to combine thiswith individual goal achievement results.

Multiple Parallel Networks

2Multiple Parallel Networks

Multiple Parallel Networks Sample from a population of networks

Multilevel network analysis in the sense of analyzingmultiple similar networks, mutually independent,permits research to transcend the level ofnetwork as case studies,and to generalize to a population of networks.

This was proposed by Snijders & Baerveldt(J. Math. Soc. 2003).

Also see Entwisle, Faust, Rindfuss, & Kaneda (AJS, 2007).

Sample from Population of Networks

Suppose we have a sample indexed by j = 1, . . . ,Nfrom a population of networks,where the networks are similar in some sense;stochastic replicates of each other;

they all are regarded as realizations ofprocesses obeying the same model,but having different parameters θ1, . . . , θj, . . . , θN .

Sample from Population of Networks

Suppose we have a sample indexed by j = 1, . . . ,Nfrom a population of networks,where the networks are similar in some sense;stochastic replicates of each other;they all are regarded as realizations ofprocesses obeying the same model,but having different parameters θ1, . . . , θj, . . . , θN .

Multiple Parallel Networks Fixed effects

Meta-Analysis ∼ Fixed Effects Model:

θ1, . . . , θj, . . . , θN are arbitrary values,no assumption about a population is made.

two-stage procedure:

estimate each θj separately, combine the results byFisher’s procedure for combining independent tests:‘is there any evidence for a hypothesized effect?’

Meta-Analysis ∼ Fixed Effects Model:

θ1, . . . , θj, . . . , θN are arbitrary values,no assumption about a population is made.

two-stage procedure:

estimate each θj separately, combine the results byFisher’s procedure for combining independent tests:‘is there any evidence for a hypothesized effect?’

Meta-Analysis ∼ Fixed Effects Model (contd.):

For coordinate k of the parameter, test null hypothesis

H0 : θkj = 0 for all j

against alternative hypothesis

H1 : θkj = 0 for at least one j .

(Two-sided variants also are possible.)

Procedure: see, e.g., Snijders & Bosker Section 3.7.

Mercken, Snijders, Steglich, & de Vries (2009)applied this in a study of smoking initiation:7704 adolescents in 70 schools in 6 countries.

Multiple Parallel Networks Random effects: two-stage procedure

Meta-Analysis ∼ Random Effects Model:θ1, . . . , θj, . . . , θN are drawn randomlyfrom a population P[net] of networks,no further distributional assumptions are made.

Two-stage procedure:

estimate each θj separately,combine the results in a meta-analysis (Cochran 1954),(‘V-known problem in multilevel analysis)which allows testing hypotheses about P[net]

such as, for a coordinate k ,

Htotal0 : all θkj = 0;

Hmean0 : E{θkj} = 0;

Hspread0 : var{θkj} = 0.

Meta-Analysis ∼ Random Effects Model:θ1, . . . , θj, . . . , θN are drawn randomlyfrom a population P[net] of networks,no further distributional assumptions are made.

Two-stage procedure:

estimate each θj separately,combine the results in a meta-analysis (Cochran 1954),(‘V-known problem in multilevel analysis)which allows testing hypotheses about P[net]

such as, for a coordinate k ,

Htotal0 : all θkj = 0;

Hmean0 : E{θkj} = 0;

Hspread0 : var{θkj} = 0.

The input for the meta-analysis consists ofestimates θ̂j and their standard errors s.e.j.

The meta analysis is constructed based on the model

θ̂j = μ + Uj + Ej ,

where μ is the population mean,Uj is the true effect of group j,and Ej is the statistical error of estimation.

Uj and Ej are independent residuals with mean 0,the Uj are i.i.d. with unknown variance,and var(Ej) = s.e.2

j(‘V–known’).

The input for the meta-analysis consists ofestimates θ̂j and their standard errors s.e.j.

The meta analysis is constructed based on the model

θ̂j = μ + Uj + Ej ,

where μ is the population mean,Uj is the true effect of group j,and Ej is the statistical error of estimation.

Uj and Ej are independent residuals with mean 0,the Uj are i.i.d. with unknown variance,and var(Ej) = s.e.2

j(‘V–known’).

Meta-Analysis ∼ Random Effects Model (contd.)

This has been applied in quite many studies: e.g.,

Lubbers (2003):homophily in 57 classrooms with 1466 students (also withintegrated random coefficient p∗ approach);

Baerveldt, van Duijn, Vermeij, van Hemert (2004):ethnic homophily in 20 schools, 1317 students;

Valente, Fujimoto, Chou, and Spruit-Metz (2009):friendship & obesity, 17 classrooms with 617 students;

Mercken, Snijders, Steglich, & de Vries (2009):fr. & smoking, 7704 adolescents, 70 schools, 6 countries;also other studies by Liesbeth Mercken.

Multiple Parallel Networks Random effects: integrated procedure

Meta-Analysis ∼ Integrated Random Effects Model:θ1, . . . , θj, . . . , θN are drawn randomlyfrom a population P[net] of networks,and are assumed to havea common multivariate normal N (μ,Σ) distribution,perhaps conditionally on network-level covariates.

Integrated procedure:

Estimate μ and Σ and consider the‘posterior’ distribution of θj given the data.

Advantage:The analysis of the separate networks draws strength fromthe total sample of networks by regression to the mean.

Useful especially for many small networks.

Meta-Analysis ∼ Integrated Random Effects Model (contd.)

New developmentsin collaboration between Johan Koskinen and T.S.,for the stochastic actor-oriented model for network dynamics.

Recall that this is a model for network dynamics,where the dynamics isan unobserved sequence of ‘micro steps’and the parameters are estimated from network panel data.

This is elaborated following a likelihood-based approach;see Koskinen & Snijders (JSPI 2007),Snijders, Koskinen & Schweinberger (AAS 2010),Schweinberger (PhD thesis 2007, Chapters 4 and 5).

Meta-Analysis ∼ Integrated Random Effects Model (contd.)

New developmentsin collaboration between Johan Koskinen and T.S.,for the stochastic actor-oriented model for network dynamics.

Recall that this is a model for network dynamics,where the dynamics isan unobserved sequence of ‘micro steps’and the parameters are estimated from network panel data.

This is elaborated following a likelihood-based approach;see Koskinen & Snijders (JSPI 2007),Snijders, Koskinen & Schweinberger (AAS 2010),Schweinberger (PhD thesis 2007, Chapters 4 and 5).

Here we discuss a Bayesian approach, where the parametersμ,Σ have a prior distribution. We assume the conjugate prior,

Σ−1 swishartp(Λ−10 , ν0), and conditionally on Σ

μ | Σ s Np(μ0,Σ/κ0) .

Thus, the parameters of the prior are Λ0, ν0, κ0.

The joint p.d.f., for data y1, . . . ,yj, . . . ,yN, is

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

j=1 ϕp(θj | μ,Σ) hierarchical model

×∏N

j=1 pSAOM(yj | θj) network model

Since pSAOM(yj | θj) cannot be calculated directly,we employ data augmentation (Tanner & Wong, 1987):augment the network panel data by the sequence vjof all microsteps connecting the consecutive observations.

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

The joint p.d.f., for data y1, . . . ,yj, . . . ,yN,using data augmentation, is the sum over vj of

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

j=1 pSAOM(vj | θj,yj) . network model

The posterior distribution can be sampledby Markov chain Monte Carlo (MCMC).The unknown random variables are

μ,Σ; θ1, . . . , θN; v1, . . . ,vN

and these are sampled in turn, as follows.

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

μ,Σ; θ1, . . . , θN; v1, . . . ,vN

fInvWish�

Σ | Λ−10 , ν0�

ϕp�

μ | (μ0,Σ/κ0)�

×∏N

μ,Σ; θ1, . . . , θN; v1, . . . ,vN

1 For all j make some Metropolis Hastings stepssampling vj | yj, θj,as in Snijders, Koskinen & Schweinberger (2010).

2 For all j make one or more Metropolis Hastings stepssampling θj | vj, μ,Σ,using a random walk proposal distribution(Schweinberger 2007, Ch. 5.4;Koskinen & Snijders 2007, Sect. 4.4).

3 Sample (μ,Σ) | θ1, . . . , θN,Λ0, ν0, κ0

from the full conditional distribution.

This requires tuning to obtain good mixing – as usual.

3 Sample (μ,Σ) | θ1, . . . , θN,Λ0, ν0, κ0

Multiple Actor Types

3Multiple Actor Sets

andMultiple Relations

Multilevel network analysis with multiple actor typeswas pioneered by Breiger (Soc. Forces, 1974),Hedstrøm, Sandell & Stern (AJS, 2000) andLazega, Jourda, Mounier & Stofer (Social Networks, 2008).

The 2012 Sunbelt Conference saw an avalanche of new work,with contributions by Alessandro Lomi, Peng Wang,and Johan Koskinen.

At this conference we see many further developments.

what can I add???

Longitudinal data of linked networks with multiple actor typesalso can be analyzed using RSiena, thanks tothe flexible design by Krists Boitmanis and Ruth Ripley.

what can I add???

Large Networks

4Large Networks

Large Networks

A collection of many groups ∼ many ‘parallel’ networksis really a case of a large networkwhere between-group ties are ignored.

The applicability of our usual network models to large groups(hundreds of actors and more) seems limited by the factthat we still do not know a lot about howthe large-scale structure of networks differs from thesmall-scale and medium-scale structures;

and large networks must be full of heterogeneitywhere, e.g., ERGM parameters will not be constantin all ‘regions’ of the network– but how to define such ‘regions’?

Large Networks

Models for large networks should incorporatelatent heterogeneity – at which level?

actors / subgroups ?

Work by Johan Koskinen – Michael Schweinberger – ....

Large Networks

the road on and on

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