tom a.b. snijders - oxford statisticssnijders/siena/net_longi_napels2017.pdf · a1. actor-oriented...

Multilevel Longitudinal Network Analysis

Tom A.B. Snijders

University of Groningen

University of Oxford

May, 2017

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Overview

A Introduction: the Stochastic Actor-Oriented Model

and the Siena program;

coevolution.

B Analysis of Multilevel Networks

multiple node sets, multiple networks.

C Multilevel Analysis of Networks

multiple parallel networks.

D Multilevel Network Event Models.

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A1. Actor-Oriented Model for Network Dynamics Network Panel Data

Stochastic Actor-Oriented Model

Kinds of data:

1 Network panel data

= repeated measures of a network on the same node set

(some exogenous node changes are permitted).

2 Network and behavior panel data

= the same, with also behavioral dependent variables measured

⇒coevolution of networks and behavior.

3 Multivariate network / behavior panel data

= the same, multiple networks and/or multiple behaviors

⇒further coevolution.

Networks may also be two-mode networks:

rectangular adjacency matrices.3 / 91


E.g.: Study of smoking initiation and friendship

(following up on earlier work by P. West, M. Pearson & others).

One school year group from a Scottish secondary school

starting at age 12-13 years, was monitored over 3 years,

3 observations, at appr. 1-year intervals,

160 pupils (with some turnover: 129 always present),

with sociometric & behaviour questionnaires.

Smoking: values 1–3;

drinking: values 1–5;

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wave 1 girls: circles

boys: squares

node size: pocket money

color: top = drinking

bottom = smoking

(orange = high)

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boys: squares



bottom = smoking

(orange = high)

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boys: squares



bottom = smoking

(orange = high)

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Questions:

⇒ how to model network dynamics from such data?

⇒ how to model joint dependence between networks

and actor attributes such as drinking and smoking?

The Glasgow cohort data set is a panel,

and it is natural to assume latent change going on

between the observation moments:

continuous time probability model,

discrete time observations.

Panel data sets are common for networks representing

relations between human actors like friendship, advice, esteem,

which can be regarded as states rather than events.

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A1. Actor-Oriented Model for Network Dynamics Continuous-time Markov chains

Continuous-time Markov chains: simplicity

Holland & Leinhardt (1977) framework for network dynamics:

1 continuous-time Markov models for panel data

(changes between observations being unobserved);

this allows expressing feedback: network builds upon itself;

2 decompose change in smallest constituents,

i.e., single tie changes: ministeps.

3 This means that coordination between actors is not modeled; only

feedback, as the actors constitute each others’ changing environment.

Note: continuous-time modeling for non-network panel data

developed by Kalbfleisch & Lawless, Bergstrom, Singer, et al.

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A1. Actor-Oriented Model for Network Dynamics Actor-oriented

Simulations – actor orientation

A simulation approach allows to extend this

to include triadic and other complex dependencies.

Actor-oriented perspective (Snijders, 1996, 2001)

(‘SAOM = Stochastic Actor-oriented Model ’) :

in a directed network,

tie changes are modeled as resulting from

actions by nodes = actors to change their outgoing ties;

An alternative is a tie-oriented perspective (Koskinen & Snijders, 2013)

in the ERGM tradition (‘LERGM ’):

tie changes are modeled as dependent on the current network

without a specific process role for the nodes.

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A1. Actor-Oriented Model for Network Dynamics Principles

Stochastic actor-oriented models: principles

⇒ model for network dynamics;

⇒ probability model is continuous-time stochastic process,

observations are discrete-time;

⇒ unobserved changes are ministeps (one variable at the time)

⇒ estimation theory elaborated for panel data

(i.e., finitely many observation moments, mostly just a few: ≥ 2);

⇒ elaborated also for network & behaviour panel data,

multivariate networks, two-mode networks;

⇒ actor-oriented: in line with social science theories

that focus on choices by nodes = actors

(can be individuals or organizations) ;

⇒ estimation by R package RSiena .

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A1. Actor-Oriented Model for Network Dynamics Notation

Notation

1 Actors i = 1, . . . , n (network nodes).

2 Array X of ties between them : one binary network X ;

Xij = 0 (or 1) if there is no tie (or there is a tie), from i to j .

Matrix X is adjacency matrix of digraph.

Can be extended to multiple networks or discrete ordered values.

Xij is a tie indicator or tie variable.

3 Exogenously determined independent variables:

actor-dependent covariates v , dyadic covariates w .

These can be constant or changing over time.

4 Continuous time parameter t,

observation moments t1, . . . , tM .

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A1. Actor-Oriented Model for Network Dynamics Principles

Model assumptions

1 X (t) is a Markov process.

Strong assumption;

covariates and state space extensions may enhance plausibility.

2 Condition on the first observation X (t1) , do not model it:

no assumption of a stationary marginal distribution.

3 At any time moment, only one tie variable Xij can change.

This precludes swapping partners or coordinated group formation.

Such a change is called a ministep (also ‘micro-step’).

4 Heuristic: Each actor “controls” her outgoing ties

collected in the row vector(Xi1(t), ...,Xin(t)

).

Actors have full information on all variables (can be weakened).

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A1. Actor-Oriented Model for Network Dynamics Rate function

Timing model: rate functions

‘how quick is change?’

At randomly determined moments t,

actors i get opportunity to change a tie variable Xij : ministep.

(Actors are also permitted to leave things unchanged.)

Each actor i has a rate function λi (α, x), with λ+(α, x) =∑

i λi (α, x):

1 Waiting time until next ministep for current state x

∼ Exponential(λ+(α, x)

);

2 P{

Next ministep is for actor i}

=λi (α, x)

λ+(α, x).

Rate functions may be constant between waves (∼ homogeneous Poisson

processes) or depend on actor characteristics or positions.

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Timing model: rate functions

‘how quick is change?’

At randomly determined moments t,

actors i get opportunity to change a tie variable Xij : ministep.

(Actors are also permitted to leave things unchanged.)

Each actor i has a rate function λi (α, x), with λ+(α, x) =∑

i λi (α, x):

1 Waiting time until next ministep for current state x

∼ Exponential(λ+(α, x)

);

2 P{

Next ministep is for actor i}

=λi (α, x)

λ+(α, x).

Rate functions may be constant between waves (∼ homogeneous Poisson

processes) or depend on actor characteristics or positions.14 / 91

A1. Actor-Oriented Model for Network Dynamics Objective function

Choice model: objective functions

‘what is the direction of change?’

The objective function fi (β, xold, xnew) for actor i

models change probabilities to go from xold to xnew

(cf. potential function for xnew).

xold and xnew are two consecutive network states differing by only one tie.

Ministep: When actor i gets an opportunity for change,

s/he has the possibility to change one outgoing tie variable Xij ,

or leave everything unchanged.

By x (±ij) is denoted the network obtained from x

when xij is changed (‘toggled’) into 1− xij .

Formally, x (±ii) is defined to be equal to x .

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A1. Actor-Oriented Model for Network Dynamics Probabilities

Probabilities in ministep

Conditional on actor i being allowed to make a change,

i.e., i taking a ministep,

the probability that Xij changes into 1− Xij is

pij(β, x) =exp

(fi (β, x , x

(±ij)))

n∑h=1

exp(fi (β, x , x

(±ih))) ,

and pii is the probability of not changing anything.

Higher values of the objective function indicate

the preferred direction of changes.

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A1. Actor-Oriented Model for Network Dynamics Algorithm

Simulation algorithm network dynamics

Generate

∆ time

λ

Choose

actor i

λ

Choose

tie change i → j

i , f

Effectuate changes

t, x

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A1. Actor-Oriented Model for Network Dynamics Specification

Model specification :

Objective function fi reflects network effects

(endogenous) and covariate effects (exogenous).

Convenient specification of objective function is a linear combination.

In basic model specifications,

objective function does not depend on the ‘old’ network:

fi (β, xold, xnew = x) =

L∑k=1

βk sik(x) ,

where the weights βk are statistical parameters

indicating strength of ‘effect’ sik(x).

Dependence on actor-dependent covariates (vi )

or dyad-dependent (wij) is left out of the notation.

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Examples of effects (1)

Some possible network effects for actor i , e.g.:

1 out-degree effect, controlling the density / average degree,

si1(x) = xi+ =∑

j xij

2 reciprocity effect, number of reciprocated ties

si2(x) =∑

j xij xji

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Various effects related to network closure:

3 transitive triplets effect,

number of transitive patterns in i ’s ties

(i → j , i → h, h→ j)

si3(x) =∑

j ,h xij xih xhj

i

h

j

transitive triplet

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4 GWESP effect (cf. ERG models)

(geometrically weighted edgewise shared partners)

which gives a more moderate contribution of transitivity

GWESP(i , α) =∑j

xij eα{

1 −(1− e−α

)∑h xihxhj

}.

0 1 2 3 4 5 6

0

2

4

6

s

GW

ES

Pw

eigh

t α =∞α = 1.2α = 0.69α = 0

Figure: Weight of tie i → j for s =∑

h xihxhj two-paths.

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Various objective function effects associated with actor covariate v .

Those to whom ‘ego’ i is tied are called i ’s ‘alters’.

1 covariate-related popularity, ‘alter’

sum of covariate over all of i ’s alters

si1(x) =∑

j xij zj ;

2 covariate-related activity, ‘ego’

i ’s out-degree weighted by covariate

si2(x) = zi xi+;

3 covariate-related interaction, ‘ego × alter’

si3(x) = zi∑

j xij zj ;

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Rate function: basic specification

The rate function λi (x) defines how often actor i gets

opportunities for change / makes microsteps.

The observations are taken at times (‘waves’) t1, t2, . . . , tM (M ≥ 2)

and the time interval [tm, tm+1] is called a period.

A basic feature is to include period-dependent multiplicative parameters

λi (ρ, α, x) = ρm λ0i (α, x)

where λi (ρ, α, x) applies in period [tm, tm+1].

λ0i (α, x) may be constant, or depend on i and x and parameters α;

the free parameters ρm reflect that observations may be taken any time,

without interfering with the network process,

and that we have no prior knowledge about the ‘speed of social time’.

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Rate function: extended specification

A non-constant rate function λi (α, x)

represents that some actors change their ties more quickly than others,

depending on covariates or network position.

Dependence on covariates:

λi (ρ, α, x) = ρm exp(∑h

αh vhi ) .

Dependence on network position, e.g., on an actor variable V :

λi (ρ, α, x) = ρm exp(α1 vi ) .

ρm is a period-dependent base rate.

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A1. Actor-Oriented Model for Network Dynamics Estimation

Estimation

For estimating the parameters, if there are complete continuous-time data

(all ministeps known), we could use maximum likelihood.

For panel data, estimation is less straightforward.

Estimation methods have been developed using

Method of Moments, Generalized Method of Moments,

Bayes, and Maximum Likelihood methods.

Method of Moments is used the most:

statistical efficiency quite good, time efficiency good.

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A1. Actor-Oriented Model for Network Dynamics Method of moments

Estimation: Method of moments

Method of moments (‘estimating equations’) ‘MoM’ :

Choose a suitable statistic Z = (Z1, . . . ,ZK ),

the statistic Z must be sensitive to the parameter θ in the sense that

∂Eθ(Zk)

∂θ> 0 ;

determine value θ of θ for which

observed and expected values of Z are equal:

Eθ {Z} = z .

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Statistics for MoM

Assume that there are M = 2 observation moments,

and rates are constant: λi (x) = ρ.

ρ determines the expected “amount of change”.

A sensitive statistic for ρ is the Hamming distance,

C =

g∑i , j=1i 6=j

| Xij(t2)− Xij(t1) | ,

the “observed total amount of change”.

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For the weights βk in the objective function

fi (β, x) =L∑

k=1

βk sik(x) ,

a higher value of βk means that all actors

‘strive more strongly after’ a high value of sik(x),

so sik(x) will tend to be higher for all i , k .

This leads to the statistic

Sk =n∑

i=1

sik(X (t2)

).

This statistic will be sensitive to βk :

a higher βk will tend to lead to higher values of Sk .

(Recall: the model is conditional on x(t1).)

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A1. Actor-Oriented Model for Network Dynamics Stochastic approximation

How to solve the moment equation?

Moment equation Eθ{Z} = z is difficult to solve, as

Eθ{Z}

cannot be calculated explicitly.

However, the solution can be approximated, e.g., by the

Robbins-Monro (1951) method for stochastic approximation.

Iteration step (cf. Newton-Raphson) :

θN+1 = θN − aN D−1(zN − z) , (1)

where zN is a simulation of Z with parameter θN ,

D is a suitable matrix, and aN → 0 .

This yields (with tuning) a surprisingly stable algorithm.29 / 91

A1. Actor-Oriented Model for Network Dynamics Glasgow data

Example: Glasgow data

The following page presents estimation results for the Glasgow data:

friendship network between 160 pupils, observed at 3 yearly waves.

The model was the result of an extensive goodness of fit exercise,

considering distributions of outdegrees, indegrees, and triad motifs.

Transitive closure is represented by the

geometrically weighted shared partners (‘gwesp’) effect:

i j

k1

k2

k3

•••••••

GWESP 30 / 91


Effect par. (s.e.)

rate (period 1) 11.404 (1.289)

rate (period 2) 9.155 (0.812)

outdegree (density) –3.345∗∗∗ (0.229)

reciprocity: creation 4.355∗∗∗ (0.485)

reciprocity: maintenance 2.660∗∗∗ (0.418)

GWESP: creation 3.530∗∗∗ (0.306)

GWESP: maintenance 0.315 (0.414)

indegree – popularity –0.068∗ (0.028)

outdegree – popularity –0.012 (0.055)

outdegree – activity 0.109∗∗ (0.036)

reciprocated degree – activity –0.263∗∗∗ (0.066)

sex (F) alter –0.130† (0.076)

sex (F) ego 0.056 (0.086)

same sex 0.442∗∗∗ (0.078)

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Some conclusions:

Evidence for reciprocity; transitivity;

reciprocity stronger for creating than for maintaining ties;

transitivity only for creating ties;

gender homophily;

those with many reciprocated ties are less active

in establishing new ties or maintaining existing ties.

Note: definition of reciprocated degree – activity:

sik(x) =∑j

xij xreci+

where

x reci+ =

∑j

xij xji .

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A2. Non-directed networks

A2. Non-directed networks

The actor-driven modeling is less straightforward

for non-directed relations,

because two actors are involved in deciding about a tie.

Various modeling options are possible,

representing different ways of coordination

between the two actors at both sides of the tie.

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A2. Non-directed networks Model options

1 Forcing (dictatorial) model:

one actor takes the initiative and unilaterally imposes

that a tie is created or dissolved.

2 Unilateral initiative with reciprocal confirmation:

one actor takes the initiative and proposes a new tie

or dissolves an existing tie;

if the actor proposes a new tie, the other has to confirm,

otherwise the tie is not created.

(Cf. Jackson & Wolinsky, 1996)

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3 Pairwise conjunctive model:

a pair of actors is chosen and reconsider whether a tie

will exist between them; a new tie is formed if both agree.

4 Pairwise disjunctive (forcing) model:


will exist between them;

a new tie is formed if at least one wishes this.

5 Pairwise compensatory (additive) model:


will exist between them; this is based

on the sum of their utilities for the existence of this tie.

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Option 1 is close to the actor-driven model for directed relations.

In options 3–5, the pair of actors (i , j) is chosen

depending on the product of the rate functions λi λj

(under the constraint that i 6= j ).

The numerical interpretation of the ratio function

differs between options 1–2 compared to 3–5.

In options 2–5, the decision about the tie is taken on the basis of the

objective functions fi , fj of both actors.

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A3. Co-evolution Principles

A3. Co-evolution

In the SAOM for a single network,

the actors change their network neighbourhoods :

these co-evolve as the common changing environment = system state.

This can be extended to a system with multiple variables:

other networks, discrete actor-level variables, two-mode networks.

The basic ideas remain the same:

continuous-time Markov chain, now with larger state space;

heuristic: actors can change outgoing ties and their own variables;

at times of change, only one variable can change;

behavior is discrete, changes –1 / 0 / +1 / (ministeps!).

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Rate functions, objective functions, specified

separately for each dependent variable.

Dependence of rate and objective functions of variable Y

on another dependent variable Z expresses directed dependence:

effect Z ⇒ Y .

Given the longitudinal panel data, this allows

to estimate separately dependence Z ⇒ Y and Y ⇒ Z ;

under the assumption that the model holds...

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Computer simulation algorithm

The co-evolution Markov chain is a succession of ministeps;

variables can be networks or actor-level variables.

Generate

∆ time

λ

Choose

variable h

λ

Choose

actor i

h, λ

Choose

tie change x(h)ij

or behavior change x(h)i

h, i , f

Effectuate changes

t, x

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A3. Co-evolution Networks and behaviour

Networks and Behaviour Studies

Co-evolution of a network and one or more actor variables

representing behavioural tendencies of actors

are Networks and Behaviour Studies that can be used to

study mechanisms of social influence and social selection.

E.g.: network of adolescents,

co-evolution friendship network ⇔ smoking behaviour.

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Actor-oriented models for networks and behavior

Each actor “controls” his outgoing ties

collected in the row vector(Xi1(t), ...,Xin(t)

),

and also her behavior Zi (t) =(Zi1(t), ...,ZiH(t)

),

where it is assumed there are H ≥ 1 behavior variables.

Network change process and behavior change process

run concurrently, with a common state space,

with transition probabilities depending on this joint state.

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Outline of model definition

Microsteps now can be either a change to the network,

or a change to the behavior.

Rate functions are defined separately

for changes in network: λX

for changes in behavior h: λZh .

Objective functions likewise are defined separately

for the network f X and the behaviors f Zh .

X and Z are interdependent because the objective functions

depend on both variables; they are applied as

f Xi(X (t),Z (t)

)and f Zh

i

(X (t),Z (t)

)42 / 91


Behavior ministep

Whenever actor i may make a change in variable h of Z ,

she changes only one behavior, say zih , to the new value v

(recall: changes can be –1, 0, +1).

The new vector is denoted by z(i , h ; v).

Change probabilities are given by

pihv (β, z , x) =exp(f (i , h, v))∑u

exp(f (i , h, u))

where

f (i , h, v) = fZhi (β, z(i , h ; v), x) .

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Specification of behavior model

Objective function for behavior f Zhi also is linear predictor:

f Zhi (β, x , z) =

L∑k=1

βZhk sZh

ik (x , z) ,

Some basic effects:

1 Linear shape ,

sZhi1 (x , z) = zih

2 quadratic shape, ‘effect behavior on itself’,

sZhi2 (x , z) = z2

ih

Quadratic shape effect important for model fit.

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For a negative quadratic shape parameter,

the model for behavior is a unimodal preference model.

zh

fZh

i(β, x, z)

1 2 3 4

For positive quadratic shape parameters ,

the behavior objective function can be bimodal

(‘positive feedback’).

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3 behavior-related similarity,

sum of behavior similarities

between i and his friends

sZhi3 (x , z) =

∑j xij

(1−|zih − zjh |

),

if Zh assumes values between 0 and 1;

may be divided by xi+ =∑

j xij ;

4 average behavior alter — an alternative to similarity:

sZhi4 (x , z) = zih

1xi+

∑j xijzjh

5 popularity-related tendency, (in-degree)

sZhi5 (x , z) = zih x+i

6 activity-related tendency, (out-degree)

sZhi6 (x , z) = zih xi+

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Effects (3) and (4) both express the idea of influence,

but in mathematically different ways.

Theory plus data will have to differentiate between them.

7 dependence on other behaviors (h 6= `) ,

sZhi7 (x , z) = zih zi`

For both the network and the behavior dynamics,

extensions are possible depending on the network position.

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A3. Co-evolution Multivariate networks

Multivariate Networks

Co-evolution of several networks allows studying how these networks

influence each other. The same co-evolution principles apply.

E.g.: networks in organizations,

relevant networks are advice – collaboration – friendship;

other example: bullying in schools,

some relevant networks are friendship – bullying – defending.

A multitude of mixed structural effects are interesting.

E.g.: direct entrainment: advisors become friends;

mixed transitive closure patterns:

advisors of friends become advisors, etc.;

actor-level dependencies:

those who have many friendships give less advice.48 / 91

A3. Co-evolution Two-mode Networks

Two-mode Networks

Networks may also be two-mode networks,

where the first node set is the set of actors

and the second node set can be, e.g., activities, meeting places

representing further contextual aspects; or cognitions.

For co-evolution of friendship, associations, and behavior this opens

the possibility to study whether actors are influenced by their friends

or by their co-members of associations.

Two-mode networks have less structural possibilities

because the second mode is unrelated to the first mode.

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A3. Co-evolution RSiena

The procedures are implemented in the R package

R

S imulation

I nvestigation for

E mpirical

N etwork

A nalysis

which is available from CRAN and (up-2-date) R-Forge

http://www.stats.ox.ac.uk/siena/

Material, papers, can be found on SIENA website.

50 / 91

A4. MAN⇔ AMN Multilevel concepts

A4. Multilevel concepts

Two kinds of combinations multilevel—networks:

MAN = Multilevel Analysis of Networks is

the study of a set of ‘parallel’ networks according to a common model,

like in Hierarchical Linear Modeling etc.;

AMN = Analysis of Multilevel Networks

(Wang, Robins, Pattison, Lazega, 2013):

is the study of networks with nodes of several types, distinguishing

between types of ties according to types of nodes they connect.

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A4. MAN⇔ AMN Multilevel concepts

Emmanuel Lazega and Tom A.B. Snijders (eds).

Multilevel Network Analysis for the Social Sciences.

Cham: Springer, 2016.

Especially p. 36–41.

Special issue of Social Networks ‘Multilevel Social Networks’,

edited by Alessandro Lomi, Garry Robins, and Mark Tranmer,

vol. 44 (January 2016).

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B. Analysis of Multilevel Networks


Multilevel network (Wang, Robins, Pattison, Lazega, 2013):

Network with nodes of several types,

distinguishing between types of ties

according to types of nodes they connect.

Thus, if types of nodes are A, B, C ,

distinguish between A− A, B − B, C − C ties, etc., (within-type)

and between A− B, A− C , etc., ties (between-type).

Some may be networks of interest,

others may be fixed constraints,

still others may be non-existent or non-considered.

This generalizes two-mode networks

and multivariate one mode – two mode combinations.

Emmanuel Lazega and Tom A.B. Snijders (eds)., Multilevel Network

Analysis for the Social Sciences. Cham: Springer, 2016.

Special issue of Social Networks ‘Multilevel Social Networks’, edited by

Alessandro Lomi, Garry Robins, and Mark Tranmer, vol. 44 (January

2016).

53 / 91


Thus, multilevel networks generalize two-mode networks.

Perhaps a better name would be multi-mode networks.

We stick to the nomenclature of Wang et al. (2013).

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B5. Two-sided agency Two-mode networks

B5. Two-mode networks

In the basic RSiena setup,

for two-mode networks only the first mode has agency.

‘Actors choose their activities, activities do not choose the actors.’

This is not always a good representation of the social world;

sometimes the choice is two-sided.

In the implementation of RSiena, two-sided choices are implemented for

symmetric networks:

modelType ≥ 3.

This can be exploited using the RSiena feature of structural zeros,

which are elements in the adjacency matrix / tie variables

that are bound to be 0.

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Two-mode networks represented as one-mode

There are many possibilities for adapting the specification of networks in

RSiena using blocks of structural zeros,

with blocks differentiating actor sets.

A basic option is the embedding of a two-mode network

in a bipartite one-mode network.

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Two-mode network :

two node sets (‘modes’) A and B, ties only A− B;

Bipartite network :

one node set N = A ∪ B with A ∩ B = ∅, ties only A− B.

These are subtly different.

The following page represents the two-mode network X

in two ways as a bipartite network:

directed A⇒ B and non-directed.

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directed

A B

A

B

(0 two-mode A× B

0 0

)

network X

contains the same information as

non-directed

A B

A

B

(0 two-mode A× B

two-mode B × A 0

)

network t(X ) network X

where t(X ) = transpose(X ).

see scripts RscriptSienaTwoModeAsOneMode.R and TwoModeAsSymmetricOneMode Siena.R on

the Siena website

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It is necessary to define dummy variables representing

the two types of node: 1A and 1B , respectively.

It is convenient to have these nodal variables both available.

When the rate effect of 1B is –100 (practically −∞),

only the A nodes will make changes.

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The directed and non-directed embeddings of the

two-mode network contain the same information,

but due to the options available in RSiena

for symmetric = non-directed networks,

they can be used in different ways.

For the usual way of modeling two-mode networks in RSiena,

the agency is in the first node set (‘A’),

represented by ties being directed A⇒ B,

and the nodes in set B ‘just have to accept’.

Representing two-mode networks as one-mode networks

gives additional possibilities.

For the dictatorial model option, the results are identical

to the results for the usual two-mode representation.

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Example of two-sided modeling option

for two-mode networks

For RSiena for non-directed networks,

see Snijders & Pickup (2016) and the manual.

modelType = 3, proposal–confirmation:

1 Ego proposes a tie change, based on a multinomial choice

just like in the directed case.

2 If the proposal is a new tie,

alter decides in a binary choice whether or not to accept.

modelType = 6, tie-based LERGM:

1 A pair of nodes is randomly chosen.

2 For this pair a binary choice is made for the tie,

based on the sum of the objective functions.61 / 91


Thus, the non-directed embedding allows modeling

agency of A nodes combined with B nodes.

However, the possibilities for empirical separation of A and B effects is

limited; e.g., whether dispersion of A degrees is

due to differential activity of A agents

or differential popularity of A nodes for B agents

cannot be distinguished.

This embedding also allows having dependent actor variables

for both modes (trick by James Hollway)

(use dummy variables for both types of actor,

and give these fixed rate effects equal to –100).

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B6. One-mode – two-mode co-evolution

B6. Co-evolution of one-mode and two-mode networks

For the co-evolution of one-mode and two-mode networks:

separate set of slides (Siena onemode twomode s.pdf).

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B7. Multilevel network with two node sets Representation as multivariate network

7. Multilevel network with two node sets

Now consider a multilevel network with two node sets, A and B.

Suppose there are two one-mode networks internal to A and B,

and two two-mode networks X1 from A to B; X2 from B to A.

Specification for RSiena possible by employing

one joint node set A ∪ B and two dependent networks:

A B A B

A

B

(internal A 0

0 internal B

) (0 two-mode A× B

two-mode B × A 0

)

networks A, B network X2 network X1

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B7. Multilevel network with two node sets Representation as multivariate network

Again there have to be dummy variables 1A and 1B

representing the two types of node,

and these can be used to separate the rate functions for A and B.

The separation into two networks is used

to separate the A-A choices from A-B choices,

and similarly the B-B choices from B-A choices.

For example, the ‘upper half’ A and A × B

is just the one-mode – two-mode co-evolution treated earlier;

in this model this is combined with

the co-evolution of B-B and B × A.

Note that this model does not allow double agency:

A actors decide about A-A and A-B ties, B actors about B-B and B-A.

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B7. Multilevel network with two node sets Example

For example:

A a set of organizations, B a set of individuals,

X2 is a fixed membership relation, X1 is not there;

networks A and B could be taken apart

in two distinct networks;

if there are only ties between individuals within organizations, B will be a

network of diagonal blocks

and structural zeros between different organizations;

if there are essential differences between individual ties within

organizations or across organizations,

B can be decomposed in two further distinct networks.

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B8. Nested settings and endogenous actor sets Water resources in 7 villages in Senegal

B8. Nested settings and endogenous actor sets

The following case is a multilevel network with

the following structure:

The node set has a two-level nested ‘group’ structure.

Within-group and between-group relations can be different in nature.

There is an additional network on a subset of nodes;

this subset is endogenous.

There is additional nesting structure within the groups.

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Water resources in 7 villages in Senegal

The research setting is a group of 7 villages in rural Senegal.

These villages have a common water supply resource (deep-well).

This resource is managed by a board elected by the villagers

according to rules of equal representation.

Of the villages, two are Fulani (nomadic cattle breeders)

and five are in large majority Wolof (sedentary agriculture).

The inhabitants live in households (families)

nested in houses (extended families).

A network survey was held in 2010 and 2015 among all inhabitants.

The data analyzed here are of the 406 respondents

for whom data is available for both waves;

of whom 25–28 were elected board members.

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Dependent variables

Many dependent variables can be defined for this rich data set.

We focus here on what is most important:

Sociability within villages

Sociability between villages

Advice between board members

Board membership.

Sociability could also be analyzed as one network,

using village as a covariate;

but the differences may come out more completely

by analyzing them as two dependent networks.

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Within-village and between-village networks

Let us consider all sociability relations together:

advice or discussion or illness help or money or visits.

Relations within villages are of a different nature than between villages,

although same name generator was used.

This could be represented by covariate ‘same village’,

but perhaps more clearly, and with different options,

by separating this as two dependent networks.

Define two networks W = ‘within’ and B = ‘between’,

composed of sociability ties within and between villages,

with structural zero blocks

for the between-village and within-village dyads, respectively.

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The within and between relations can be separated in

two dependent networks, with structural zero blocks.

If there would be two villages, A and B :

A B A B

A

B

(within A 0

0 within B

) (0 between A× B

between B × A 0

)

within-village networks between-village networks

This principle can be extended for more villages.

In this case, we have a 7× 7 block structure.

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Model specification: sociability

For the W and B networks:

1 Outdegree, reciprocity

2 three degree effects

3 transitivity: GWESP

4 actor variable: gender

5 actor variable: wealth,

combined value of harvest, cattle, machines

6 mixed activity effect

7 mixed triadic closure.

Because of the definition of W and B,

mixed dyadic (entrainment) effects are excluded.

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Board membership and relations

The board is a subset of the inhabitants.

There is a separate advice relation between board members.

A limitation of RSiena is in any case,

that the change of board memberships is modeled

as a gradual one-by-one process.

Endogeneity problem:

the actor set for the board advice relations is endogenous!

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Board membership and relations

These can be represent in several ways,

each with its own complications and possibilities.

1 by a binary changing dependent variable

complication: advice relation on entire actor set,

but structurally zero on endogenous subset.

2 by a two-mode network of board seats × inhabitants,

where each board seat has degree ≤ 1.

The node set must then be extended to the union

inhabitants ∪ board seats.

Complication:

change of membership should imply new start of ties.

We choose the first option.74 / 91

B8. Nested settings and endogenous actor sets Model specification

Model specification: board membership

Board membership BM is represented as a binary dependent actor variable

(‘behavior’).

The small number of board members (about 7%)

implies that there is little information to build an explanatory model.

BM can be explained in principle by actor attributes but

because of the equal representation rules they have no direct effects.

For the BM dependent variable:

1 linear shape (modeling the trend in the mean)

2 degree effects of W and B networks.

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Model specification: advice between board members

Advice A between board members is defined on

an endogenous subset (BM) of all nodes.

Because of the definition of the Markov chain process in Siena

where changes occur in ministeps, the simulated state

will contain advice ties also between previous board-members.

These should not be counted as part of the advice relation.

Therefore, all effects for A should be functions of only

the ties between current board members.

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Model specification: advice between board members (2)

This is represented as follows.

1 Basic rate effect is ρ ≡ 1 (fixed).

2 Effect of BM on rate. Effect RateX.

3 Fixed parameters –100 (−∞) for BM-ego and BM-alter effects:

fi (x) = . . . − 100∑j

xij

(I{BMi = 0} + I{BMj = 0}

)+ . . .

New effects egoLThresholdX and altLThresholdX.

4 Instead of transitivity effect,

si (x) =∑j

xij xik xkj I{BMi = BMj = BMk = 1}

Interaction egoRThresholdX × homXtransTrip.

5 etc. for other effects.77 / 91

B8. Nested settings and endogenous actor sets Results

Effect par. (s.e.)

Network Dynamics: Within


reciprocity 1.133∗∗∗ (0.082)

GWESP 1.125∗∗∗ (0.081)

indegree - popularity (sqrt) –0.060 (0.099)

indegree - activity (sqrt) –0.157∗ (0.064)

outdegree - activity (sqrt) –0.411∗∗∗ (0.073)

ethnicity similarity 0.363 (0.249)

Gender.M alter 0.228∗ (0.115)

Gender.M ego –0.301∗∗ (0.102)

same Gender.M 0.442∗∗∗ (0.108)

wealth similarity 0.114 (0.176)

household wealth similarity 0.482∗∗∗ (0.146)

house wealth similarity 0.363∗∗ (0.134)

indegree (sqrt) between popularity 0.202† (0.109)

outdegree (sqrt) between activity 0.117 (0.101)

{i B→ hB→ j} ⇒ {i W→ j} 0.043 (0.747)

board 0.318 (1.018)

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Effect par. (s.e.)

Network Dynamics: Between


reciprocity 2.330∗∗∗ (0.659)

GWESP –0.317 (0.779)

indegree - popularity (sqrt) 0.114 (0.184)

indegree - activity (sqrt) –0.241 (0.371)

outdegree - activity (sqrt) 0.951∗∗∗ (0.125)

ethnicity similarity 0.363 (0.270)

Gender.M alter 0.765∗ (0.340)

Gender.M ego –0.235 (0.558)

same Gender.M 1.100∗ (0.494)

wealth similarity 1.730∗∗ (0.606)

household wealth similarity –0.651 (0.439)

house wealth similarity 0.142 (0.395)

indegree (sqrt) within popularity 0.722∗∗∗ (0.128)

outdegree (sqrt) within activity –0.264 (0.231)

{i W→ hB→ j} ⇒ {i B→ j} 0.573∗∗ (0.200)

{i B→ hW→ j} ⇒ {i B→ j} –0.070 (0.117)

board 1.178 (0.986)

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Effect par. (s.e.)

Network Dynamics: Advice in Board

basic rate parameter board N.A. (N.A.)

effect Boardmember on rate 3.368∗∗∗ (0.776)


reciprocity 1.480† (0.783)

ethnicity alter –0.319 (0.831)

ethnicity ego –0.362 (0.381)

same ethnicity 0.293 (0.298)

Gender.M alter –0.215 (0.536)

Gender.M ego –0.219 (0.292)

same Gender.M 0.126 (0.205)

BM = 0 alter –100. fixed

BM = 0 ego –100. fixed

transitive triplets through Boardmember 0.363 (0.228)

within 1.012† (0.565)

between 1.068 (1.599)

Large standard errors because it is a small network.

If no control for ‘within’, ethnic similarity is significant.

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Effect par. (s.e.)

Behavior Dynamics: Board membership

rate Boardmember period 1 0.410 (0.189)

Boardmember linear shape –3.006∗∗∗ (0.662)

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B8. Nested settings and endogenous actor sets Conclusions of the analysis

Conclusions of the analysis: covariates

Which effects were found to be significant?

1 Rich differences W -B.

2 Few significant results for explaining board membership and advice between

board members: small network.

3 Covariate sex:

homophily W and B, males send more ties W and B, receive fewer ties W ;

4 Covariate wealth homophily:

W only house and household level,

B only individual level.

Perhaps an effect of age; can be checked.

5 Covariate ethnicity: no homophily.

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Conclusions of the analysis: structural

1 reciprocity for all relations.

2 internal transitivity only for W ;

mixed transitivity {i W→ hB→ j} ⇒ {i B→ j} :

‘village friends are introduced to outsiders’,

‘outside friends are not introduced to village friends’

(but note: no distinction creation–maintenance).

3 differential activity negative for W , positive for B.

4 mixed degree effects:

high W indegrees lead to high B outdegrees;

also vice versa, but less strongly so.

5 direct entrainment of W and B ties with A ties always positive,

but (marginally) significant only for W ⇒ A.

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Overall conclusions

⇒ It is important to distinguish between the

within-village and between-village sociability networks.

⇒ Individual wealth and the total wealth of the household or the

extended family play different roles:

analogous to differences within- and between-group regression

in regular multilevel analysis (ecological fallacy issues!).

⇒ With some additional effects, it is possible to analyse

the advice relation on the endogenous subset of nodes,

while remaining in the regular RSiena framework.

⇒ This analysis does not incorporate differences

between the villages (cf. random intercepts/slopes).

⇒ Further analyses should consider effects of age

and mixed degree effects of board advice relation.

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C. Multilevel Analysis of Networks

C. Multilevel Analysis of Networks

For the Multilevel Analysis of Networks and function sienaBayes:

separate set of slides (MultiMetaSAOM s.pdf);

also see Chapter 11.3 of the RSiena manual.

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D. Multilevel Network Event Models

D. Multilevel Network Event Models

The R package Goldfish is a new package for Network Event Models.

It is being developed by Christoph Stadtfeld, James Hollway and Per Block,

and includes options for multilevel network structures.

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References

Some references (time-ordered)

Tom A.B. Snijders (2001). The Statistical Evaluation of Social Network Dynamics.

Sociological Methodology, 31, 361–395.

Johan H. Koskinen and Tom A.B. Snijders (2007).

Bayesian inference for dynamic social network data.

Journal of Statistical Planning and Inference, 13, 3930–3938.

Tom Snijders, Christian Steglich, and Michael Schweinberger (2007),

Modeling the co-evolution of networks and behaviour.

Pp. 41–71 in Longitudinal models in the behavioral and related sciences,

eds. Kees van Montfort, Han Oud and Albert Satorra; Lawrence Erlbaum.

Steglich, C.E.G., Snijders, T.A.B. and Pearson, M. (2010).

Dynamic Networks and Behavior: Separating Selection from Influence.

Sociological Methodology, 40, 329–392.

Tom A.B. Snijders, Johan Koskinen, and Michael Schweinberger (2010).

Maximum Likelihood Estimation for Social Network Dynamics.

Annals of Applied Statistics, 4, 567–588.

Johan H. Koskinen and Tom A.B. Snijders (2013). Longitudinal models.

Pp. 130–140 in Exponential Random Graph Models,

edited by Dean Lusher, Johan Koskinen, and Garry Robins. Cambridge University Press.

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References

Some references (continued)

Tom A.B. Snijders, Alessandro Lomi, and Vanina Torlo (2013). A model for the multiplex

dynamics of two-mode and one-mode networks, with an application to employment

preference, friendship, and advice. Social Networks, 35, 265–276.

Viviana Amati, Felix Schonenberger, Tom A.B. Snijders (2015). Estimation of stochastic

actor-oriented models for the evolution of networks by generalized method of moments.

Journal de la Societe Francaise de Statistique, 156, 140–165.

Tom A.B. Snijders (2016). The Multiple Flavours of Multilevel Issues for Networks, pp.

15-46 in Emmanuel Lazega and Tom A.B. Snijders (eds), Multilevel Network Analysis for

the Social Sciences; Theory, Methods and Applications. Springer.

Tom A.B. Snijders and Mark Pickup (2016). Stochastic actor-oriented models for network

dynamics. In Victor, J. N., Lubell, M., and Montgomery, A. H., editors, Oxford Handbook

of Political Networks. Oxford University Press, still in press.

Tom A.B. Snijders (2017). Stochastic Actor-Oriented Models for Network Dynamics.

Annual Review of Statistics and its Application, 4, 343-363.

See SIENA manual and homepage.

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Extras Convergence criterion

Extra 1. Convergence criterion

In the implementation in RSiena, the algorithm runs for a preset number

of iterations, after which convergence is assessed.

Determine how close we are to Eθ{Z} = z

One way to measure this uses the ‘t-ratios for convergence’,

tconvk =ZNk − zk

s.d.(Z1k , . . . ,ZNk),

where Z = (Zn1, . . . ,ZnK ), and requires, e.g.,

maxk|tconvk | ≤ 0.1 .

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Overall maximum convergence ratio

A better criterion turns out to be the maximum t-ratio for convergence for

any linear combination of the parameters,

tconv.max = maxb

b ′(ZN − z

)√b ′ Cov(Z ) b

.

This is equal to(

use Cauchy-Schwarz, Σ = Cov(Z ))

maxc

{c ′Σ−1/2

(Z − z

)√c ′c

}=(Z − z

)′Σ−1

(Z − z

).

The definition implies that

tconv.max ≥ maxk|tconvk | .

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The current rule for practical use is that

tconv.max ≤ 0.25 and maxk|tconvk | ≤ 0.1 .

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