course on social network analysis' - oxford statisticssnijders/blockmodels.pdf · analysis...

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Course onSocial NetworkAnalysisBlockmodeling

Anuska Ferligoj,Vladimir Batagelj,Andrej Mrvar

University of Ljubljana

Slovenia

Padova, April 10-11, 2003

A. Ferligoj, V. Batagelj, A. Mrvar: Social Network Analysis / Blockmodeling 1'

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Outline1 Introduction to Blockmodeling. . . . . . . . . . . . . . . . . . 1

2 Blockmodeling as a clustering problem. . . . . . . . . . . . . 2

12 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

20 Pre-specified blockmodeling. . . . . . . . . . . . . . . . . . . 20

28 Final Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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Introduction to Blockmodeling

Pajek - shadow 0.00,1.00 Sep- 5-1998World trade - alphabetic order

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Alphabetic order of countries (left) and rearrangement (right)

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Blockmodeling as a clustering problemThe goal ofblockmodelingis to

reduce a large, potentially in-

coherent network to a smaller

comprehensible structure that

can be interpreted more readily.

Blockmodeling, as an empiri-

cal procedure, is based on the

idea that units in a network can

be grouped according to the ex-

tent to which they are equiva-

lent, according to somemean-

ingful definition of equivalence.

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Cluster, clustering, blocks

One of the main procedural goals of blockmodeling is to identify, in a given

networkN = (U, R), R ⊆ U ×U, clusters(classes) of units that share

structural characteristics defined in terms ofR. The units within a cluster

have the same or similar connection patterns to other units. They form a

clusteringC = {C1, C2, . . . , Ck} which is apartition of the setU. Each

partition determines an equivalence relation (and vice versa). Let us denote

by∼ the relation determined by partitionC.

A clusteringC partitions also the relationR into blocks

R(Ci, Cj) = R ∩ Ci × Cj

Each such block consists of units belonging to clustersCi andCj and all

arcs leading from clusterCi to clusterCj . If i = j, a blockR(Ci, Ci) is

called adiagonalblock.

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Equivalences

Regardless of the definition of equivalence used, there are two basic

approaches to the equivalence of units in a given network (Faust, 1988):

• the equivalent units have the same connection pattern to thesameneighbors;

• the equivalent units have the same or similar connection pattern to

(possibly)different neighbors.

The first type of equivalence is formalized by the notion ofstructural

equivalence (Lorrain and White, 1971) and the second by the notion of

regular equivalence (White and Reitz, 1983) with the latter a generalization

of the former.

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Structural Equivalence

Units arestructurally equivalentif they are connected to the rest of the

network in identical ways. From the definition it follows that there are four

possible ideal blocks:

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Regular Equivalence

Intuitively, two units areregularly equivalentif they are equally connected

to equivalent others.

Batagelj, Doreian, and Ferligoj (1992) proved that regular equivalence

produces two types of blocks: null blocks and 1-covered blocks:

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Generalized equivalence / Block Types

Y1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1complete

Y0 1 0 0 0

X 1 1 1 1 10 0 0 0 00 0 0 1 0

row-dominant

Y0 0 1 0 0

X 0 0 1 1 01 1 1 0 00 0 1 0 1

col-dominant

Y0 1 0 0 0

X 1 0 1 1 00 0 1 0 11 1 0 0 0

regular

Y0 1 0 0 0

X 0 1 1 0 01 0 1 0 00 1 0 0 1row-regular

Y0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0col-regular

Y0 0 0 0 0

X 0 0 0 0 00 0 0 0 00 0 0 0 0

null

Y0 0 0 1 0

X 0 0 1 0 01 0 0 0 00 0 0 1 0

row-functional

Y1 0 0 00 1 0 0

X 0 0 1 00 0 0 00 0 0 1

col-functional

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Establishing Blockmodels

The problem of establishing a partition of units in a network in terms of a

selected type of equivalence is a special case ofclustering problemthat can

be formulated as an optimization problem(Φ, P ) as follows:

Determine the clusteringC? ∈ Φ for which

P (C?) = minC∈Φ

P (C)

whereΦ is the set offeasible clusteringsandP is acriterion function.

Since the set of unitsU is finite, the set of feasible clusterings is also

finite. Therefore the setMin(Φ, P ) of all solutions of the problem (optimal

clusterings) is not empty.

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Criterion function

Criterion functions can be constructed

• indirectlyas a function of a compatible (dis)similarity measure between

pairs of units, or

• directly as a function measuring the fit of a clustering to an ideal

one with perfect relations within each cluster and between clusters

according to the considered types of connections (equivalence).

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Indirect Approach

JJJ

�?

?

R

Q

D

hierarchical algorithms,

relocation algorithm, leader algorithm

RELATION

DESCRIPTIONS

OF UNITS

original relation

path matrix

triadsorbits

DISSIMILARITY

MATRIX

STANDARD

CLUSTERING

ALGORITHMS

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Dissimilarities

The dissimilarity measured is compatiblewith a considered equivalence∼if for each pair of units holds

Xi ∼ Xj ⇔ d(Xi, Xj) = 0

Not all dissimilarity measures typically used are compatible with structuralequivalence. For example, thecorrected Euclidean-like dissimilarity

d(Xi, Xj) =

=

√√√√(rii − rjj)2 + (rij − rji)2 +

n∑s=1

s 6=i,j

((ris − rjs)2 + (rsi − rsj)2)

is compatible with structural equivalence.

The indirect clustering approach does not seem suitable for establishing

clusterings in terms of regular equivalence since there is no evident way

how to construct a compatible (dis)similarity measure.

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ExampleThe analyzed network consists of social support exchange relation among

fifteen students of the Social Science Informatics fourth year class

(2002/2003) at the Faculty of Social Sciences, University of Ljubljana.

Interviews were conducted in October 2002.

Support relation among students was identified by the following question:

Introduction: You have done several exams since you are in the

second class now. Students usually borrow studying material from

their colleagues.

Enumerate (list) the names of your colleagues that you have most

often borrowed studying material from. (The number of listed

persons is not limited.)

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Class network

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

class.net

Vertices represent students

in the class; circles – girls,

squares – boys. Opposite

pairs of arcs are replaced by

edges.

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Indirect approachPajek - Ward [0.00,7.81]

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42

Using Corrected Euclidean-like

dissimilarity and Ward clustering

method we obtain the following

dendrogram.

From it we can determine the num-

ber of clusters: ’Natural’ cluster-

ings correspond to clear ’jumps’ in

the dendrogram.

If we select 3 clusters we get the

partitionC.

C = {{b51, b89, b02, b96, b03, b85, g10, g24},{g09, g63, g12}, {g07, g28, g22, g42}}

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Partition in 3 clusters

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

On the picture, ver-

tices in the same

cluster are of the

same color.

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MatrixPajek - shadow [0.00,1.00]

b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b02

b03

g10

g24

C

1

b51

b85

b89

b96

g07

g22

C

2

g28

g42

g09

g12

C

3

g63

The partition can be used

also to reorder rows and

columns of the matrix repre-

senting the network. Clus-

ters are divided using blue

vertical and horizontal lines.

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Direct Approach

The second possibility for solving the blockmodeling problem is to construct

an appropriate criterion function directly and then use a local optimization

algorithm to obtain a ‘good’ clustering solution.

Criterion functionP (C) has to besensitiveto considered equivalence:

P (C) = 0 ⇔ C defines considered equivalence.

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A Criterion Function

One of the possible ways of constructing a criterion function that directly

reflects the considered equivalence is to measure the fit of a clustering to

an ideal one with perfect relations within each cluster and between clusters

according to the considered equivalence.

Given a clusteringC = {C1, C2, . . . , Ck}, letB(Cu, Cv) denote the set of

all ideal blocks corresponding to blockR(Cu, Cv). Then the global error of

clusteringC can be expressed as

P (C) =∑

Cu,Cv∈C

minB∈B(Cu,Cv)

d(R(Cu, Cv), B)

where the termd(R(Cu, Cv), B) measures the difference (error) between

the blockR(Cu, Cv) and the ideal blockB. The functiond has to be

compatible with the selected type of equivalence.

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Local Optimization

For solving the blockmodeling problem we use a local optimization

procedure (a relocation algorithm):

Determine the initial clusteringC;

repeat:if in the neighborhood of the current clusteringCthere exists a clusteringC′ such thatP (C′) < P (C)then move to clusteringC′ .

The neighborhood in this local optimization procedure is determined by the

following two transformations:

• movinga unitXk from clusterCp to clusterCq (transition);

• interchangingunits Xu andXv from different clustersCp andCq

(transposition).

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Pre-specified blockmodelingIn the previous slides the inductive approaches for establishing blockmodels

for a set of social relations defined over a set of units were discussed.

Some form of equivalence is specified and clusterings are sought that are

consistent with a specified equivalence.

Another view of blockmodeling is deductive in the sense of starting with a

blockmodel that is specified in terms of substance prior to an analysis.

In this case given a network, set of types of ideal blocks, and a reducedmodel, a solution (a clustering) can be determined which minimizes thecriterion function.

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Some Blockmodel Types

1. Cohesive subgroupswith intraposition ties and no ties between

positions:

1 0 0

0 1 0

0 0 1

2. A center-peripherymodel with onecore position (center) which is

internally cohesive and connected with all other positions. The other

positions (periphery) are all connected to the core position and are not

connected to each other:

1 1 1

1 0 0

1 0 0

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. . . Some Blockmodel Types

3. A centralizedmodel which, a special case of the center-periphery

model, where all ties are from the core position or toward it:

1 1 1

0 0 0

0 0 0

or

1 0 0

1 0 0

1 0 0

4. A hierarchicalmodel with the positions on a single path:

0 1 0

0 0 1

0 0 0

or

0 0 0

1 0 0

0 1 0

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. . . Some Blockmodel Types

5. A transitivity model similar to hierarchical model with permitted ties

from lower positions to all higher positions:

0 1 1

0 0 1

0 0 0

or

0 0 0

1 0 0

1 1 0

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Prespecified blockmodeling example

We expect that center-periphery model exists in the network: some students

having good studying material, some not.

Prespecified blockmodel: (com/complete, reg/regular, -/null block)

1 2

1 [com reg] -

2 [com reg] -

Using local optimization we get the partition:

C = {{b02, b03, b51, b85, b89, b96, g09},

{g07, g10, g12, g22, g24, g28, g42, g63}}

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2 clusters solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

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ModelPajek - shadow [0.00,1.00]

g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

Image and Error Matrices:

1 2

1 reg -

2 reg -

1 2

1 0 3

2 0 2

Total error = 5

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Benefits from Optimization Approach

• ordinary/inductive blockmodeling: Given a networkN and set oftypes of connectionT , determine the modelM;

• evaluationof the quality of a model,comparingdifferent models,

analyzing the evolutionof a network(Sampson data, Doreian andMrvar 1996): Given a networkN, a modelM, and blockmodelingµ,compute the corresponding criterion function;

• model fitting/deductive blockmodeling: Given a networkN, set oftypesT , and a family of models, determineµ which minimizes thecriterion function (Batagelj, Ferligoj, Doreian, 1998).

• we can fit the network to apartial modeland analyze the residualafterward;

• we can also introduce differentconstraintson the model, for example:unitsX andY are of the same type; or, types of unitsX andY are notconnected; . . .

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Final RemarksThe current, local optimization based, programs for generalized block-

modeling can deal only with networks with at most some hundreds of

units.

What to do with larger networks is an open question. For some specialized

problems also procedures for (very) large networks can be developed

(Doreian, Batagelj, Ferligoj, 1998; Batagelj, Zaversnik, 2002).

Another interesting problem is the development ofblockmodeling of valued

networksor more generalrelational data analysis(Batagelj, Ferligoj, 2000).

The generalized blockmodeling is implemented in Pajek – program for

analysis and visualization of large networks. It is freely available, for

noncommercial use, at:

http://vlado.fmf.uni-lj.si/pub/networks/pajek/

Padova, April 10-11, 2003 ▲ ▲ ❙ ▲ ● ▲ ❙ ▲▲ ☛ ✖

http://vlado.fmf.uni-lj.si/pub/preprint/Joc00.pdf

http://arxiv.org/abs/cs.DS/0202039

http://vlado.fmf.uni-lj.si/pub/preprint/HHBock.pdf

http://vlado.fmf.uni-lj.si/pub/networks/pajek/default.htm

course on social network analysis' - oxford statisticssnijders/blockmodels.pdf · analysis...

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