Doreian, Batagelj, Ferligoj - Generalized Blockmodeling

8112019 Doreian Batagelj Ferligoj - Generalized Blockmodeling

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Photo Vladimir Batagelj UNI-LJ

Blockmodeling

Anuska FerligojUniversity of Ljubljana

Budapest 16 September 2009



A Ferligoj Cluster Analysis 1

Outline1 Introduction 1

2 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




Introduction

The goal of blockmodeling is to reduce a

large potentially incoherent network to a

smaller comprehensible structure that can be

interpreted more readily Blockmodeling

as an empirical procedure is based on the

idea that units in a network can be grouped

according to the extent to which they are

equivalent according to some meaningful

definition of equivalence (structural (Lorrain

and White 1971) regular (White and Re-

itz 1983) generalized (Doreian Batagelj

Ferligoj 2005)





Cluster clustering blocksOne of the main procedural goals of blockmodeling is to identify in a given network

N = (U R) R sube U times U clusters (classes) of units that share structural charac-

teristics defined in terms of R The units within a cluster have the same or similar

connection patterns to other units They form a clustering C = C 1 C

2 C

k

which is a partition of the set U Each partition determines an equivalence relation

(and vice versa) Let us denote by sim the relation determined by partition C

A clustering C partitions also the relation R into blocks

R(C i C j) = R cap C i times C j

Each such block consists of units belonging to clusters C i and C j and all arcs leading

from cluster C i to cluster C j If i = j a block R(C i C i) is called a diagonal block







Equivalences

Regardless of the definition of equivalence used there are two basic approaches to the

equivalence of units in a given network (compare Faust 1988)

bull the equivalent units have the same connection pattern to the same neighbors

bull 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 and

the second by the notion of regular equivalence with the latter a generalization of the

former





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 hArr YRX s3 forallZ isin U XY (XRZ hArr YRZ)

s2 XRX hArr YRY s4 forallZ isin U XY (ZRX hArr ZRY)






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

in diagonal blocks





Regular equivalence

Integral to all attempts to generalize structural equivalence is the idea that units are

equivalent if they link in equivalent ways to other units that are also equivalent

White and Reitz (1983) The equivalence relation asymp on U is a regular equivalence on

network N = (U R) if and only if for all XYZ isin U X asymp Y implies both

R1 XRZ rArr existW isin U (YRW and W asymp Z)

R2 ZRX rArr existW isinU

(WRY and W asymp Z)







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 of clustering problem that can be formulated as

an optimization problem (Φ P ) as follows

Determine the clustering C isin Φ for which

P (C) = minCisinΦ

P (C)

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











Example Support network among informatics students

The analyzed network consists of social support exchange relation among fifteen

students of the Social Science Informatics fourth year class (20022003) at the Faculty

of Social Sciences University of Ljubljana Interviews were conducted in October

2002Support 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 oftenborrowed studying material from (The number of listed persons is not

limited)





Class network - graph

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet

Vertices represent students

in the class circles ndash girls

squares ndash boys Recipro-

cated arcs are represented by

edges





Class network ndash matrix

Pajek - shadow [000100]

b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42

Using Corrected Euclidean-likedissimilarity and Ward clustering

method we obtain the following

dendrogram

From it we can determine the num-

ber of clusters lsquoNaturalrsquo cluster-

ings correspond to clear lsquojumpsrsquo in

the dendrogram

If we select 3 clusters we get the

partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42





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 same

cluster are of the

same color





MatrixPajek - shadow [000100]

b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3

The partition can be used

also to reorder rows andcolumns of the matrix repre-

senting the network Clus-

ters are divided using blue

vertical and horizontal lines







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 with

perfect relations within each cluster and between clusters according to the considered

equivalence

Given a clustering C = C 1 C 2 C k let B (C u C v) denote the set of all ideal

blocks corresponding to block R(C u C v) Then the global error of clustering C can

be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)

where the term d(R(C u C v) B) measures the difference (error) between the block

R(C u C v) and the ideal block B d is constructed on the basis of characterizations

of types of blocks The function d has to be compatible with the selected type of

equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0

Number of inconsistencies

for each block

A B

A 0 1B 1 2

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









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

C 1 C 2

C 1 complete regular

C 2 null complete





Generalized equivalence block types

Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional





Pre-specified blockmodeling

In 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 specifiedequivalence

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 reduced model asolution (a clustering) can be determined which minimizes the criterion function





Types of pre-specified blockmodels

The pre-specified blockmodeling starts with a blockmodel specified in terms of

substance prior to an analysis Given a network a set of ideal blocks is selected a

family of reduced models is formulated and partitions are established by minimizing

the 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





Pre-specified blockmodeling example

We expect that core-periphery model exists in the network some students having

good studying material some not

Prespecified blockmodel (comcomplete regregular -null block)

1 21 [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





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6

Image and Error Matrices

1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery





Symmetric-acyclic blockmodel

Let us introduce a new type of permitted diagonal block and a new type of blockmodel

respectively labeled the lsquosymmetricrsquo block and the lsquoacyclic blockmodelrsquo The idea

behind these two objects comes from a consideration of the idea of a ranked clusters

model from Davis and Leinhardt (1972) We examine the form of these models

and approach them with the perspective of pre-specified blockmodeling (DoreianBatagelj Ferligoj 2000 2005)

We say that a clustering C = C 1 C k over the relation R is a symmetric-acyclic

clustering iff

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

bull the remaining graph (without these subgraphs) is acyclic (hierarchical)





An example of symmetric-acyclic blockmodel

The pre-specified blockmodel is acyclic with symmetric diagonal blocks An example

of a clustering into 3 clusters symmetric-acyclic type of models can be specified in the

following way

sym nul nul

nulone sym nul

nulone nulone sym

The criterion function for a symmetric-acyclic blockmodel is simply a count of the

number of (1) asymmetric ties in diagonal blocks and (2) non-zero ties above the

diagonal





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





Blockmodeling of vauled networks

Another interesting problem is the development of blockmodeling of valued networks (BatageljFerligoj 2000 Ziberna 2007)

Ziberna (2007) proposed several approaches to

generalized blockmodeling of valued networks

where values of the ties are assumed to be mea-

sured on at least interval scaleThe first approach is a straightforward gen-

eralization of the generalized blockmodeling

of 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 empirical

block with its ideal block can be measured by

within block variability of appropriate values





Blockmodeling of two-mode network

It 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 mdash row-partition and column-partition have to be determined

Two-mode generalized blockmodeling can be done





Supreme Court voting for twenty-six important decisions

Issue Label Br Gi So St OC Ke Re Sc Th

Presidential Election PE - - - - + + + + +Criminal Law Cases

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

Federal Authority Cases

Federalism FA1 - - - - + + + + +Clean Air Action FA2 + + + + + + + + +Clean Water FA3 - - - - + + + + +

Cannabis for Health FA4 0 + + + + + + + +United Foods FA5 - - + + - + + + +NY Times Copyrights FA6 - + + - + + + + +

Civil Rights Cases

Voting Rights CR1 + + + + + - - - -Title VI Disabilities CR2 - - - - + + + + +PGA v Handicapped Player CR3 + + + + + + + - -

Immigration Law Cases

Immigration 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 Cases

Labor Rights LPR1 - - - - + + + + +Property Rights LPR2 - - - - + + + + +

The Supreme Court Justices and

their lsquovotesrsquo on a set of 26 ldquoimpor-

tant decisionsrdquo made during the

2000-2001 term Doreian and Fu-

jimoto (2002)

The Justices (in the order in

which they joined the Supreme

Court) are Rehnquist (1972)

Stevens (1975) OrsquoConner (1981)

Scalia (1982) Kennedy (1988)

Souter (1990) Ginsburg (1993)

and Breyer (1994)





Supreme Court voting a (47) partition

C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens

upper ndash conservative lower ndash liberal





Blockmodeling of three-way networks

The indirect approach to structural equivalence blockmodeling in 3-mode networks

Indirect means ndash 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 is

determined by a ternary relation R sube X timesY timesZ The notion of structural equivalencedepends on which of the sets X Y and Z are (considered) the same There are three

basic cases

bull all three sets are different

bull two sets are the same

bull all three sets are the same





Example Artificial dataset

Randomly generated ideal structure rndTest(c(564)c(353535))





Example Solutions

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

0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

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

H e

i g h t

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

0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t

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

0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t




A F li j Cl A l i 42




Indirect blockmodeling approach

First the indirect approach for structural equivalence for coauthorship network wasperformed The Ward dendrogram where the corrected euclidean-like dissimilarity

was used is presented in the dendrogram


A F li j Cl t A l i 43





In the case of the coauthorship network we can assume a core-periphery structure one

or several clusters of strongly connected scientists that strongly collaborate among

themselves The last cluster consists of scientists that do not collaborate among

themselves and also do not with the scientists of the other clusters They publishby themselves or with the researchers from the other scientific disciplines or with

researchers 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 for

the blockmodels into three five and eight clusters As eight clusters have been shown

also by the dendrogram this clustering is further analyzed






Core-periphery structure of the coauthorship network






If we permute rows and columns in the relational matrix in such a way that first the

units of the first cluster are given than the units of the second cluster and so on and

if we cut the relational matrix by clusters we obtain the table in the next slide Here

a black square is drawn if the two researchers have at least one joint publication or a

white square if they have not publish any joint publication

In the table the obtained blockmodel into eight clusters is presented (seven core

clusters and one peripherical one) The seven core clusters are rather small ones and

the peripherical large one

Only cluster 4 is connected also to the first and second clusters all other core clusters

show 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 in

zero blocks are inconsistencies The number of all inconsistencies is the value of the

criterion function For the blockmodel the value of the criterion function is 310






Main results

bull 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 other

groups

bull The core groups overlap with the organizational structure sociologists of each of

these core groups are members of a research center

bull The periphery group composed by the sociologists that are mostly not coauthors

with the other Slovenian sociologists is very large (more than 56 ) Most of

them publish only as a single author Probably some of them publish with the

other Slovenian nonsociologists or with the researchers outside of Slovenia

bull The coauthorship network is quite a sparse one which tells us that the preferred

publication culture by Slovenian sociologists is to publish as a single author

bull More detailed analysis of the periphery group should be further studied






Open problems in blockmodeling

The current local optimization based programs for generalized blockmodeling 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 2000 Zaversnik

Batagelj 2004)




Outline1 Introduction 1

2 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





Introduction












Ferligoj 2005)









2 C

k













Equivalences





different neighbors



former

















in diagonal blocks





Regular equivalence







(WRY and W asymp Z)












P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




Introduction












Ferligoj 2005)









2 C

k













Equivalences





different neighbors



former

















in diagonal blocks





Regular equivalence







(WRY and W asymp Z)












P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)








2 C

k













Equivalences





different neighbors



former

















in diagonal blocks





Regular equivalence







(WRY and W asymp Z)












P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)






Equivalences





different neighbors



former

















in diagonal blocks





Regular equivalence







(WRY and W asymp Z)












P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)
















in diagonal blocks





Regular equivalence







(WRY and W asymp Z)












P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




Regular equivalence







(WRY and W asymp Z)












P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)











P (C) = minCisinΦ

P (C)




















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)


















limited)






b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96

classnet





edges







b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)






b02

b03

g07

g09

g10g12

g22

g24

g28

g42

b51

g63b85

b89

b96

b 0 2

b 0 3

g 0 7

g 0 9

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

b 5 1

g 6 3

b 8 5

b 8 9

b 9 6





Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




Indirect approach

b51

b89

b02

b96

b03

b85

g10

g24

g09

g63

g12

g07

g28

g22

g42



dendrogram




the dendrogram


partition C

C = b51 b89 b02 b96 b03 b85 g10 g24

g09 g63 g12 g07 g28 g22 g42






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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





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






b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





b02

b03

g10

g24 C1

b51

b85

b89

b96

g07

g22 C2

g28

g42

g09

g12 C3

g63

b 0 2

b 0 3

g 1 0

g 2 4

C 1

b 5 1

b 8 5

b 8 9

b 9 6

g 0 7

g 2 2

C 2

g 2 8

g 4 2

g 0 9

g 1 2

C 3

g 6 3












Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)






Criterion function



equivalence



be expressed as

P (C) =

C uC visinC

minBisinB(C uC v)

d(R(C u C v) B)




equivalence





Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




Empirical blocks

a b c d e f g

a 0 1 1 0 1 0 0b 1 0 1 0 0 0 0

c 1 1 0 0 0 0 0

d 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 blocks

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

b 1 0 1 0 0 0 0c 1 1 0 0 0 0 0

d 1 1 1 0 0 0 0e 1 1 1 0 0 0 0

f 1 1 1 0 0 0 0g 1 1 1 0 0 0 0


for each block

A B

A 0 1B 1 2











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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)









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

C 1 C 2


C 2 null complete






Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





Y 1 1 1 1 1

X 1 1 1 1 11 1 1 1 11 1 1 1 1

complete

Y 0 1 0 0 0

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

row-dominant

Y 0 0 1 0 0

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

col-dominant

Y 0 1 0 0 0

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

regular

Y 0 1 0 0 0

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

row-regular

Y 0 1 0 1 0

X 1 0 1 0 01 1 0 1 10 0 0 0 0

col-regular

Y 0 0 0 0 0

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

null

Y 0 0 0 1 0

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

row-functional

Y

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

0 0 0 00 0 0 1

col-functional






















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





















0 0

0 0

0 0

0

0 0

0 0

0 0


periphery









1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)








1 21 [com reg] -

2 [com reg] -


C = b02 b03 b51 b85 b89 b96 g09

g07 g10 g12 g22 g24 g28 g42 g63





2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




2 Clusters Solution

b02

b03

g07

g09

g10

g12

g22 g24

g28

g42

b51

g63

b85

b89

b96





Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




Model


g07

g10

g12

g22

g24

g28

g42

g63

b85

b02

b03

g09

b51

b89

b96

g 0 7

g 1 0

g 1 2

g 2 2

g 2 4

g 2 8

g 4 2

g 6 3

b 8 5

b 0 2

b 0 3

g 0 9

b 5 1

b 8 9

b 9 6


1 21 reg -

2 reg -

1 21 0 3

2 0 2

Total error = 5

center-periphery












clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)











clustering iff










following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)







following way

sym nul nul

nulone sym nul

nulone nulone sym



diagonal






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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





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





































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)




































Civil Rights Cases











jimoto (2002)







and Breyer (1994)






C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)





C am p ai gnF i

V o t i n gR i gh

D e p or t A l i e

I l l e g al S 1

I l l e g al S 2

S t a y E x e c u t

P GA v P l a y er

P r i v a c y

I mmi gr a t i o

L e g al A i d P o

C r i mi n al A l

I l l e g al S 3

C l e anA i r A c

C ann a b i s

N Y T i m e s

S e a t B el t

U ni t e d F o o d

C i t i z en s h i

P r e s E l e c t i

F e d er al i s m

F r e e S p e e c h

C l e anW a t er

T i t l eI V

T o b a c c o

L a b or I s s u e

P r o p er t y

Rehnquest

Thomas

Scalia

Kennedy

OConner

Breyer

Ginsburg

Souter

Stevens











basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)









basic cases














Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)










Example Solutions


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t


0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0


H e

i g h t























































Main results


groups




















Batagelj 2004)






















































Main results


groups




















Batagelj 2004)









Batagelj 2004)

Doreian, Batagelj, Ferligoj - Generalized Blockmodeling

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Transcript of Doreian, Batagelj, Ferligoj - Generalized Blockmodeling