anuska ferligojˇ - university of...

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%Photo: Vladimir Batagelj, UNI-LJ

Blockmodeling

Anuska FerligojUniversity of Ljubljana

Budapest, 16 September 2009

http://vlado.fmf.uni-lj.si

A. Ferligoj: Cluster Analysis 1'

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Outline1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cluster, clustering, blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

9 Establishing blockmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

11 Indirect approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

19 Direct approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

24 Generalized blockmodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

26 Pre-specified blockmodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

31 Symmetric-acyclic blockmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

34 Blockmodeling of vauled networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

35 Blockmodeling of two-mode network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

39 Blockmodeling of three-way networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

42 Application: Clustering of Slovenian sociologists . . . . . . . . . . . . . . . . . . . . . . . 42

48 Open problems in blockmodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Budapest, September 16, 2009 s s y s l s y ss * 6


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Introduction

The goal of blockmodeling is to reduce alarge, potentially incoherent network to asmaller comprehensible structure that can beinterpreted more readily. Blockmodeling,as an empirical procedure, is based on theidea that units in a network can be groupedaccording to the extent to which they areequivalent, according to some meaningfuldefinition of equivalence (structural (Lorrainand White 1971), regular (White and Re-itz 1983), generalized (Doreian, Batagelj,Ferligoj 2005).



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Cluster, clustering, blocksOne 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 charac-teristics defined in terms of R. The units within a cluster have the same or similarconnection patterns to other units. They form a clustering C = {C1, C2, . . . , Ck}which is a partition of the set U. Each partition determines an equivalence relation(and vice versa). Let us denote by ∼ the relation determined by partition C.

A clustering C partitions also the relation R into blocks

R(Ci, Cj) = R ∩ Ci × Cj

Each such block consists of units belonging to clusters Ci and Cj and all arcs leadingfrom cluster Ci to cluster Cj . If i = j, a block R(Ci, Ci) is called a diagonal block.



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The Everett network

a b c d e f g h i ja 0 1 1 1 0 0 0 0 0 0b 1 0 1 0 1 0 0 0 0 0c 1 1 0 1 0 0 0 0 0 0d 1 0 1 0 1 0 0 0 0 0e 0 1 0 1 0 1 0 0 0 0f 0 0 0 0 1 0 1 0 1 0g 0 0 0 0 0 1 0 1 0 1h 0 0 0 0 0 0 1 0 1 1i 0 0 0 0 0 1 0 1 0 1j 0 0 0 0 0 0 1 1 1 0

a c h j b d g i e fa 0 1 0 0 1 1 0 0 0 0c 1 0 0 0 1 1 0 0 0 0h 0 0 0 1 0 0 1 1 0 0j 0 0 1 0 0 0 1 1 0 0b 1 1 0 0 0 0 0 0 1 0d 1 1 0 0 0 0 0 0 1 0g 0 0 1 1 0 0 0 0 0 1i 0 0 1 1 0 0 0 0 0 1e 0 0 0 0 1 1 0 0 0 1f 0 0 0 0 0 0 1 1 1 0

A B C

A 1 1 0

B 1 0 1

C 0 1 1



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EquivalencesRegardless of the definition of equivalence used, there are two basic approaches to theequivalence of units in a given network (compare Faust, 1988):

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

• the equivalent units have the same or similar connection pattern to (possibly)different neighbors.

The first type of equivalence is formalized by the notion of structural equivalence andthe second by the notion of regular equivalence with the latter a generalization of theformer.



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Structural equivalence

Units are equivalent if they are connected to the rest of the network in identical ways(Lorrain and White, 1971). Such units are said to be structurally equivalent.

In other words, X and Y are structurally equivalent iff:

s1. XRY ⇔ YRX s3. ∀Z ∈ U \ {X, Y} : (XRZ⇔ YRZ)s2. XRX⇔ YRY s4. ∀Z ∈ U \ {X, Y} : (ZRX⇔ ZRY)



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. . . Structural equivalence

The blocks for structural equivalence are null or complete with variations on diagonalin diagonal blocks.



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Regular equivalence

Integral to all attempts to generalize structural equivalence is the idea that units areequivalent if they link in equivalent ways to other units that are also equivalent.

White and Reitz (1983): The equivalence relation ≈ on U is a regular equivalence onnetwork N = (U, R) if and only if for all X, Y, Z ∈ U, X ≈ Y implies both

R1. XRZ⇒ ∃W ∈ U : (YRW ∧W ≈ Z)R2. ZRX⇒ ∃W ∈ U : (WRY ∧W ≈ Z)



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. . . Regular equivalence

Theorem 1.1 (Batagelj, Doreian, Ferligoj, 1992) Let C = {Ci} be a partitioncorresponding to a regular equivalence ≈ on the network N = (U, R). Then eachblock R(Cu, Cv) is either null or it has the property that there is at least one 1 ineach of its rows and in each of its columns. Conversely, if for a given clustering C,each block has this property then the corresponding equivalence relation is a regularequivalence.

The blocks for regular equivalence are null or 1-covered blocks.



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Establishing blockmodelsThe problem of establishing a partition of units in a network in terms of a selectedtype of equivalence is a special case of clustering problem that can be formulated asan optimization problem (Φ, P ) as follows:

Determine the clustering C? ∈ Φ for which

P (C?) = minC∈Φ

P (C)

where Φ is the set of feasible clusterings and P is a criterion function.



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Criterion function

Criterion functions can be constructed

• indirectly as a function of a compatible (dis)similarity measure between pairs ofunits, or

• directly as a function measuring the fit of a clustering to an ideal one with perfectrelations within each cluster and between clusters according to the consideredtypes of connections (equivalence).



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Indirect approach

−→−→

−−−−→−→

R

Q

D

hierarchical algorithms,

relocation algorithm, leader algorithm, etc.

RELATION

DESCRIPTIONSOF UNITS

original relation

path matrix

triadsorbits

DISSIMILARITYMATRIX

STANDARDCLUSTERINGALGORITHMS



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Dissimilarities

The dissimilarity measure d is compatible with a considered equivalence ∼ if for eachpair of units holds

Xi ∼ Xj ⇔ d(Xi, Xj) = 0

Not all dissimilarity measures typically used are compatible with structural equiva-lence. For example, the corrected 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 clusteringsin terms of regular equivalence since there is no evident way how to construct acompatible (dis)similarity measure.



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Example: Support network among informatics students

The analyzed network consists of social support exchange relation among fifteenstudents of the Social Science Informatics fourth year class (2002/2003) at the Facultyof Social Sciences, University of Ljubljana. Interviews were conducted in October2002.

Support relation among students was identified by the following question:

Introduction: You have done several exams since you are in the second classnow. Students usually borrow studying material from their colleagues.

Enumerate (list) the names of your colleagues that you have most oftenborrowed studying material from. (The number of listed persons is notlimited.)



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Class network - graph

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

class.net

Vertices represent studentsin the class; circles – girls,squares – boys. Recipro-cated arcs are represented byedges.



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Class network – matrix

Pajek - shadow [0.00,1.00]

b02

b03

g07

g09

g10

g12

g22

g24

g28

g42

b51

g63

b85

b89

b96

b02

b03

g07

g09

g10

g12

g22

g24

g28

g42

b51

g63

b85

b89

b96



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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-likedissimilarity and Ward clusteringmethod we obtain the followingdendrogram.From it we can determine the num-ber of clusters: ‘Natural’ cluster-ings correspond to clear ‘jumps’ inthe dendrogram.If we select 3 clusters we get thepartition C.

C = {{b51, b89, b02, b96, b03, b85, g10, g24},

{g09, g63, g12}, {g07, g28, g22, g42}}



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Partition into three clusters (Indirect approach)

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

On the picture, ver-tices in the samecluster are of thesame 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 usedalso to reorder rows andcolumns of the matrix repre-senting the network. Clus-ters are divided using bluevertical and horizontal lines.



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Direct approachThe second possibility for solving the blockmodeling problem is to construct anappropriate criterion function directly and then use a local optimization algorithm toobtain a ‘good’ clustering solution.

Criterion function P (C) has to be sensitive to considered equivalence:

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



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Criterion function

One of the possible ways of constructing a criterion function that directly reflectsthe considered equivalence is to measure the fit of a clustering to an ideal one withperfect relations within each cluster and between clusters according to the consideredequivalence.

Given a clustering C = {C1, C2, . . . , Ck}, let B(Cu, Cv) denote the set of all idealblocks corresponding to block R(Cu, Cv). Then the global error of clustering C canbe expressed as

P (C) =∑

Cu,Cv∈C

minB∈B(Cu,Cv)

d(R(Cu, Cv), B)

where the term d(R(Cu, Cv), B) measures the difference (error) between the blockR(Cu, Cv) and the ideal block B. d is constructed on the basis of characterizationsof types of blocks. The function d has to be compatible with the selected type ofequivalence.



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Empirical blocksa b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0f 1 1 1 0 1 0 1g 0 1 1 0 0 0 0

Ideal blocksa b c d e f g

a 0 1 1 0 0 0 0b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0

Number ofinconsistenciesfor each block

A BA 0 1B 1 2

The value of the criterion function is the sum of all incon-sistencies P = 4.



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Local optimization

For solving the blockmodeling problem we use the relocation algorithm:

Determine the initial clustering C;repeat:

if in the neighborhood of the current clustering Cthere exists a clustering C′ such that P (C′) < P (C)then move to clustering C′ .

The neighborhood in this local optimization procedure is determined by the followingtwo transformations:

• moving a unit Xk from cluster Cp to cluster Cq (transition);

• interchanging units Xu and Xv from different clusters Cp and Cq (transposition).



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Partition into three clusters: Direct solution (unique)

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

This is the same par-tition and has thenumber of inconsis-tencies.



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Generalized blockmodeling

1 1 1 1 1 1 0 0

1 1 1 1 0 1 0 1

1 1 1 1 0 0 1 0

1 1 1 1 1 0 0 0

0 0 0 0 0 1 1 1

0 0 0 0 1 0 1 1

0 0 0 0 1 1 0 1

0 0 0 0 1 1 1 0

C1 C2

C1 complete regular

C2 null complete



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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 1

row-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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Pre-specified blockmodelingIn the previous slides the inductive approaches for establishing blockmodels for aset of social relations defined over a set of units were discussed. Some form ofequivalence is specified and clusterings are sought that are consistent with a specifiedequivalence.

Another view of blockmodeling is deductive in the sense of starting with a blockmodelthat is specified in terms of substance prior to an analysis.

In this case given a network, set of types of ideal blocks, and a reduced model, asolution (a clustering) can be determined which minimizes the criterion function.



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Types of pre-specified blockmodels

The pre-specified blockmodeling starts with a blockmodel specified, in terms ofsubstance, prior to an analysis. Given a network, a set of ideal blocks is selected, afamily of reduced models is formulated, and partitions are established by minimizingthe criterion function.

The basic types of models are:

* * *

* 0 0

* 0 0

* 0 0

* * 0

? * *

* 0 0

0 * 0

0 0 *

core - hierarchy clustering

periphery



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Pre-specified blockmodeling example

We expect that core-periphery model exists in the network: some students havinggood 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 = 5center-periphery



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Symmetric-acyclic blockmodelLet us introduce a new type of permitted diagonal block and a new type of blockmodel,respectively labeled the ‘symmetric’ block and the ‘acyclic blockmodel’. The ideabehind these two objects comes from a consideration of the idea of a ranked clustersmodel from Davis and Leinhardt (1972). We examine the form of these modelsand approach them with the perspective of pre-specified blockmodeling (Doreian,Batagelj, Ferligoj 2000, 2005).

We say that a clustering C = {C1, . . . , Ck} over the relation R is a symmetric-acyclicclustering iff

• all subgraphs induced by clusters from C (diagonal blocks) contain onlybidirectional arcs (edges); and

• the remaining graph (without these subgraphs) is acyclic (hierarchical).



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An example of symmetric-acyclic blockmodel

The pre-specified blockmodel is acyclic with symmetric diagonal blocks. An exampleof a clustering into 3 clusters symmetric-acyclic type of models can be specified in thefollowing way:

sym nul nul

nul,one sym nul

nul,one nul,one sym

The criterion function for a symmetric-acyclic blockmodel is simply a count of thenumber of (1) asymmetric ties in diagonal blocks and (2) non-zero ties above thediagonal.



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Table 1: An Ideal symmetric-acyclic network

a . 1 1 1 . . . . . . . . . . . . . . . . . .b 1 . 1 1 1 . . . . . . . . . . . . . . . . .c 1 1 . 1 1 . . . . . . . . . . . . . . . . .d 1 1 1 . 1 . . . . . . . . . . . . . . . . .e . 1 1 1 . . . . . . . . . . . . . . . . . .f 1 1 . 1 1 . 1 1 . . . . . . . . . . . . . .g . 1 1 1 . 1 . 1 . . . . . . . . . . . . . .h 1 1 . 1 1 1 1 . . . . . . . . . . . . . . .i 1 . 1 1 1 . . . . 1 1 1 . . . . . . . . . .j 1 1 1 1 1 . . . 1 . . . . . . . . . . . . .k 1 1 1 . 1 . . . 1 . . 1 . . . . . . . . . .l . 1 1 1 . . . . 1 . 1 . . . . . . . . . . .m . 1 1 1 . . 1 . . . . . . 1 1 1 1 . . . . .n 1 1 . . 1 1 . . . . . . 1 . 1 1 1 . . . . .o 1 . 1 1 . 1 . 1 . . . . 1 1 . 1 1 . . . . .p . 1 1 . 1 . 1 . . . . . 1 1 1 . 1 . . . . .q . 1 . 1 1 . 1 . . . . . 1 1 1 1 . . . . . .r . 1 . . . . . . 1 1 1 . . . . . . . 1 1 . 1s 1 1 . . 1 . . . 1 1 1 1 . . . . . 1 . . 1 1t 1 . 1 1 . . . . 1 . 1 . . . . . . 1 . . 1 1u 1 1 1 . 1 . . . 1 1 . 1 . . . . . . 1 1 . 1v 1 1 . 1 1 . . . . 1 1 1 . . . . . 1 1 1 1 .



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Blockmodeling of vauled networksAnother interesting problem is the development of blockmodeling of valued networks (Batagelj,

Ferligoj 2000, Ziberna 2007).120 A. Ziberna / Social Networks 29 (2007) 105–126

Fig. 4. Optimal partition for valued blockmodeling obtained using null and max-regular blocks with m equal to 5 or using

null and sum-regular blocks with m equal to 10.

6.3. Binary blockmodeling

In this case binary blockmodeling does not produce as good results as the previous approaches.

It was explored on the matrix on Fig. 1 sliced at 1, 2, 3, 5, and 10 (values of ties equal or

grater to this values were recoded into ones). These values were suggested by the barplot on

Fig. 5.

Fig. 5. Histogram of cell values for the matrix on Fig. 1.

Ziberna (2007) proposed several approaches togeneralized blockmodeling of valued networks,where values of the ties are assumed to be mea-sured on at least interval scale.The first approach is a straightforward gen-eralization of the generalized blockmodelingof binary networks (Doreian, P., Batagelj, V.,Ferligoj, A., 2005) to valued blockmodeling.The second approach is homogeneity blockmod-eling. The basic idea of homogeneity blockmod-eling is that the inconsistency of an empiricalblock with its ideal block can be measured bywithin block variability of appropriate values.



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Blockmodeling of two-mode networkIt is also possible to formulate a generalized blockmodeling problem where thenetwork is defined by two sets of units and ties between them.

Therefore, two partitions — row-partition and column-partition have to be determined.

Two-mode generalized blockmodeling can be done.



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Supreme Court voting for twenty-six important decisionsIssue Label Br Gi So St OC Ke Re Sc ThPresidential Election PE - - - - + + + + +

Criminal Law CasesIllegal Search 1 CL1 + + + + + + - - -Illegal Search 2 CL2 + + + + + + - - -Illegal Search 3 CL3 + + + - - - - + +Seat Belts CL4 - - + - - + + + +Stay of Execution CL5 + + + + + + - - -

Federal Authority CasesFederalism FA1 - - - - + + + + +Clean Air Action FA2 + + + + + + + + +Clean Water FA3 - - - - + + + + +Cannabis for Health FA4 0 + + + + + + + +United Foods FA5 - - + + - + + + +NY Times Copyrights FA6 - + + - + + + + +

Civil Rights CasesVoting Rights CR1 + + + + + - - - -Title VI Disabilities CR2 - - - - + + + + +PGA v. Handicapped Player CR3 + + + + + + + - -

Immigration Law CasesImmigration Jurisdiction Im1 + + + + - + - - -Deporting Criminal Aliens Im2 + + + + + - - - -Detaining Criminal Aliens Im3 + + + + - + - - -Citizenship Im4 - - - + - + + + +

Speech and Press CasesLegal Aid for Poor SP1 + + + + - + - - -Privacy SP2 + + + + + + - - -Free Speech SP3 + - - - + + + + +Campaign Finance SP4 + + + + + - - - -Tobacco Ads SP5 - - - - + + + + +

Labor and Property Rights CasesLabor Rights LPR1 - - - - + + + + +Property Rights LPR2 - - - - + + + + +

The Supreme Court Justices andtheir ‘votes’ on a set of 26 “impor-tant decisions” made during the2000-2001 term, Doreian and Fu-jimoto (2002).The Justices (in the order inwhich they joined the SupremeCourt) are: Rehnquist (1972),Stevens (1975), O’Conner (1981),Scalia (1982), Kennedy (1988),Souter (1990), Ginsburg (1993)and Breyer (1994).



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. . . Supreme Court voting / a (4,7) partition

Pajek - shadow

[0.00,1.00]

Cam

paignFi

VotingR

igh

DeportA

lie

IllegalS.1

IllegalS.2

StayE

xecut

PG

AvP

layer

Privacy

Imm

igratio

LegalAidP

o

Crim

inalAl

IllegalS.3

CleanA

irAc

Cannabis

NY

Tim

es

SeatB

elt

UnitedF

ood

Citizenshi

PresE

lecti

Federalism

FreeS

peech

CleanW

ater

TitleIV

Tobacco

LaborIssue

Property

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens

upper – conservative / lower – liberal



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Blockmodeling of three-way networksThe indirect approach to structural equivalence blockmodeling in 3-mode networks.Indirect means – embedding the notion of equivalence in a dissimilarity and deter-mining it using clustering. A 3-mode network N over the basic sets X , Y and Z isdetermined by a ternary relation R ⊆ X×Y ×Z. The notion of structural equivalencedepends on which of the sets X , Y and Z are (considered) the same. There are threebasic cases:

• all three sets are different

• two sets are the same

• all three sets are the same



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Example: Artificial dataset

Randomly generated ideal structure rndTest(c(5,6,4),c(35,35,35)):



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Example: Solutions1 9 b f i k o 3 4 d e j lm u v w q r s 2 5 7 a c g t x z 6 8 h n p y

050

010

0015

0020

00

Dendrogram of agnes(x = dist3m(t, 0, 1), method = "ward")

Agglomerative Coefficient = 1dist3m(t, 0, 1)

Hei

ght

1 3 4 9 f m y 2 7 a d r 8 g j n w x 5 b p t z c e i l o s v 6 h k q u

050

010

0015

0020

00



Hei

ght

1 4 5 6 7 8 a b c q r u 3 9 g j k l o s v x 2 d e f h m p y z i n t w

050

010

0015

0020

00



Hei

ght



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Application: Clustering of Slovenian sociologistsThe described blockmodeling approaches are applied to the collaboration network ofSlovenian sociologists.

The collaboration between sociologists is operationalized by coauthorship of publi-cations. A tie between two researchers is measured by coauthorship of an originalarticle, a chapter of a monograph, or a scientific monograph in the years from 1996 to2007.

The measured coauthorship network measures coauthorship only between Slovenianresearchers inside the field of sociology. Nevertheless we know that they publishtogether also with researchers from the other fields in Slovenia and also with theresearchers outside Slovenia.

The network consists of 95 units and 224 ties. Practically the whole network is onecomponent which means that all except 6 units are connected. These 6 units are inthree dyads and were excluded from further analysis. All analyses presented here arefor the 89 units in the single large component.



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Indirect blockmodeling approach

First the indirect approach for structural equivalence for coauthorship network wasperformed. The Ward dendrogram where the corrected euclidean-like dissimilaritywas used is presented in the dendrogram.



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Pre-specified blockmodeling

In the case of the coauthorship network we can assume a core-periphery structure: oneor several clusters of strongly connected scientists that strongly collaborate amongthemselves. The last cluster consists of scientists that do not collaborate amongthemselves and also do not with the scientists of the other clusters. They publishby themselves or with the researchers from the other scientific disciplines or withresearchers outside of Slovenia.

We performed the core-periphery pre-specified blockmodels (structural equivalence)into two to ten clusters. The highest drop of the criterion function was obtained forthe blockmodels into three, five, and eight clusters. As eight clusters have been shownalso by the dendrogram this clustering is further analyzed.



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Core-periphery structure of the coauthorship network



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If we permute rows and columns in the relational matrix in such a way that first theunits of the first cluster are given, than the units of the second cluster and so on, andif we cut the relational matrix by clusters we obtain the table in the next slide. Herea black square is drawn if the two researchers have at least one joint publication or awhite square if they have not publish any joint publication.

In the table the obtained blockmodel into eight clusters is presented (seven coreclusters and one peripherical one). The seven core clusters are rather small ones andthe peripherical large one.

Only cluster 4 is connected also to the first and second clusters, all other core clustersshow cohesivness inside the cluster and nonconnectivity outside the cluster.

The first two diagonal blocks are complete without any inconsistency compared tothe ideal complete blocks. All white squares in complete blocks and black squares inzero blocks are inconsistencies. The number of all inconsistencies is the value of thecriterion function. For the blockmodel the value of the criterion function is 310.



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Main results

• There are several small core groups of the Slovenian sociologists that publishamong themselves and much less or not at all with the members of the othergroups.

• The core groups overlap with the organizational structure: sociologists of each ofthese core groups are members of a research center.

• The periphery group composed by the sociologists that are mostly not coauthorswith the other Slovenian sociologists is very large (more than 56 %). Most ofthem publish only as a single author. Probably some of them publish with theother Slovenian nonsociologists or with the researchers outside of Slovenia.

• The coauthorship network is quite a sparse one which tells us, that the preferredpublication culture by Slovenian sociologists is to publish as a single author.

• More detailed analysis of the periphery group should be further studied.



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Open problems in blockmodelingThe current, local optimization based, programs for generalized blockmodeling candeal only with networks with at most some hundreds of units. What to do with largernetworks is an open question. For some specialized problems also procedures for(very) large networks can be developed (Doreian, Batagelj, Ferligoj 2000; Zaversnik,Batagelj 2004).


anuska ferligojˇ - university of...

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