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Exploratory data analysis and contiguity relations: An outlook Giuseppe Giordano 1 , Gilbert Saporta 2 , Maria Prosperina Vitale 1 1 Dept. of Economics and Statistics, University of Salerno, Italy {ggiordan; mvitale}@unisa.it 2 Conservatoire National des Arts et Métiers, Paris, France [email protected] London – December 13, 2015

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Page 1: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

Exploratory data analysis and contiguity relations: An outlook

Giuseppe Giordano1, Gilbert Saporta2, Maria Prosperina Vitale1

1 Dept. of Economics and Statistics, University of Salerno, Italy{ggiordan; mvitale}@unisa.it

2 Conservatoire National des Arts et Métiers, Paris, [email protected]

London – December 13, 2015

Page 2: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

Outline

Setting: Multidimensional Data Analysis (MDA) and Social Network Analysis (SNA)

Theoretical frameworks: Notion of contiguity, Homophily principle, Social Influence, ...

Aim 1: to present a framework for the treatment of SNA data structures with explorative techniques of MDA

Methods: Smooth factorial analysis-SFA; Factorial Analysis of Local Differences-FALD (PCA, MCA) Benali, H., Escofier, B. (1990

Aim 2: To define ad hoc relational data structures highlighting the effect ofexternal information on networks

Methods: Factorial Contiguity Maps and Auxiliary information in SNA Giordano G., Vitale M.P. (2007) (2011)

Illustrative example on Scientific Collaboration

Exploratory data analysis and contiguity relations: An outlook

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General aims

use of Contiguity Analysis to synthesize and visualize the patterns of social relationships in a metric space

explore the effect of external information on relational data looking for groups of structurally equivalent actors obtained through clustering methods

illustrative example in the framework of scientific collaboration gives a major insight into the proposed strategy of Multidimensional Data Analysis in the framework of Social Network Analysis

Exploratory data analysis and contiguity relations: An outlook

Page 4: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

Background and Aim

Background

- SNA focuses on ties among interacting units (Dyad, Triad, Subgroups)

to describe the pattern of the social relationships in a network

- the techniques of MDA consider statistical observations (at individual

level) to obtain syntheses of variables and units

Aim

to present a framework for the analysis of relational data and

attribute variables through the explorative techniques of

Multidimensional Data Analysis

Exploratory data analysis (EDA) is a detective work, finding and revealing the

clues, i.e. uncovering structures in the data. EDA uses numerical as well as

visual and graphical techniques to accomplish its aims. (Tukey, 1977)

Exploratory data analysis and contiguity relations: An outlook

Page 5: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

Data Structures

Relational data (pairwise links joining two units)

=> SNA

Attribute data (qualitative or quantitative variables)

=> MDA

Two Perspectives (to put together the 2 data structures)

Multidimensional

Data Analysis - MDA

Social Network

Analysis - SNA

Exploratory data analysis and contiguity relations: An outlook

Page 6: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

MDA and SNA: Data structures

Attribute Data Matrix

n1 n2 n3 . ni . . . ng

n1

n2

n3

.

ni

.

.

.

ng

E1 E2 … Ej … Eh

n1

n2

n3

.

ni

.

.

.

ng

Affiliation matrix: 2-mode network

Adjacency matrix: 1-mode network

Relational Data Matrix

MDA SNA

G

(np)(gg)

A(gh)

single set of actors

g = number of actors

Two sets: actors/events

h= number of events E

b1.

.bi..bq

a1… a

j... a

p

f ij

Contingency Table

F ➔

f . j

f i .

b1.

.bi..bq

a1… a

j... a

p

f ij

Contingency Table

F➔

f . j

f i .

Exploratory data analysis and contiguity relations: An outlook

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MDA in SNA Usually different techniques of MDA have been used

to visualise and explore the relationships in the net

structure

- Multidimensional Scaling : representation of similarity or dissimilarity

measures among the actors onto a factorial map (Freeman, 2005)

- Canonical correlation: analysis of associations among actor

characteristics, (i.e. network composition) and the pattern of social

relationships (i.e. network structure) (Wasserman and Faust,

1989)

- Correspondence Analysis and Multiple Factor Analysis: analysis of 2-

mode networks (Roberts, 2000; de Nooy, 2003; Faust, 2005; D’Esposito et

al., 2014; Ragozini et al., 2015)

- Clustering techniques for network data

(Batagelj, Ferligoj, 1982, 2000)

such as…

Exploratory data analysis and contiguity relations: An outlook

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Contiguity analysis is a generalization of linear discriminant analysis in

which the partition of elements is replaced by a more general

graph structure defined a priori of the set of the observations

(disjoint cliques, chain structures, undirected graph).

Contiguity analysis

(Lebart 1969, 2006; Lebart et al., 2000)

Answer

G (n,n) contiguity matrix holding the n vertices in a

contiguity graph (symmetric binary matrix); gii’ = 1 if i is

a neighbour of i’ and gii’ = 0 if not

How to take into account

relational data in MDA?

Exploratory data analysis and contiguity relations: An outlook

Aim 1

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Contiguity analysis in MDA: SFA and FALD

Smooth factorial analysis - SFA

Analysis of the general pattern in the data by

removing local variations

(replacing each point with the centre of gravity

of its neighbors)Factorial Analysis of Local Differences - FALD

Analysis of the local variations

(replacing each point with the differences from

the barycentre of its neighbors)

Contiguity analysis in MDA

- Smooth factorial analysis – SFA

- Factorial Analysis of Local Differences - FALD

(Benali, Escofier, 1990)(Benali et al. 1990)

Exploratory data analysis and contiguity relations: An outlook

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Quantitative Variables: Principal Component Analysis - PCA

Qualitative Variables: Multiple Correspondence Analysis - MCA

X (n,p) attribute matrix

information

of p characteristics on n statistical

units (vertices)

Q (n, k) = full disjunctive

coding matrix (0/1)

G (n,n) contiguity matrix (network structure)

N (n,n) diagonal matrix [N= diag (G’G)] holding the degree

of each vertices

SFA and FALD: matrices definition

Exploratory data analysis and contiguity relations: An outlook

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SFA

FA

LD

GQN 1−

..

'

.ji.1

q

qqGQNQ +− −

Given the triplet Q, G, N for a MCA

multiply the Q matrix by N-1G

Analysis of relational data and auxiliary information

Exploratory data analysis and contiguity relations: An outlook

Page 12: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

SFA

FA

LD

GXN 1−

GXNX 1−−

Analysis of relational data and auxiliary information

Given the triplet X, G, N for a PCA

multiply the X matrix by N-1G

Exploratory data analysis and contiguity relations: An outlook

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Entries in adjacency matrix can be seen as a particular case of contiguity

relation among statistical units defined in G. It produces a fuzzy partitioning of

the units.

Decomposition of the total variance/inertia into two components:

- local variance between the adjacent units

- residual variance

SFA and FALD in SNA

SFA: variability explained by the

presence of a contiguity

structure

FALD: actors with a prominent

role in contiguous groups

to discover patterns in the data

analysis of cohesive sub-group variations

Exploratory data analysis and contiguity relations: An outlook

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Real data set: Scientific collaboration among scientists

… the process generating ties (e.g. co-authorship) in a

collaboration network is somewhat affected by attribute

data on authors (academic position, research specialty,

geographical proximity … ) or on publications (type,

scientific relevance, …).

Expected results:

- Are authors with different academic positions in their institution

(phd. student, assistant professor, full professor) more likely to

collaborate in writing a publication than authors sharing the same

position?

- Are authors who work in the same research specialty (e.g. Social

statistics) more likely to collaborate in writing a publication than authors

from different specialties?

Exploratory data analysis and contiguity relations: An outlook

Illustrative Example

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Co-authorship network in a

scientific community*

GfKl - CLADAG 2010

Co-authorship patterns in Statistics, focusing on academic statisticians in

Italy (792 grouped in 5 subfields, at March 2010):

- target population involved in a discipline which is not yet fully explored in

terms of its scientific collaboration (i.e., co-authorship) behaviour

- no unique bibliographic archive for collecting their publications

- interest to trace co-authorship relations in distinct data sources

Exploratory data analysis and contiguity relations: An outlook

* De Stefano D., Fuccella V., Vitale M.P., Zaccarin S. (2013). The use of different data sources

in the analysis of co-authorship networks and scientific performance, Social Networks, 35, pp.

370-381

Page 16: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

Data source: Italian academic statisticians whit publications in WoS

archive

A = affiliation matrix 481 x 2289481 scientists, 2289 publications and the cells are = 1 if

authors (in rows) authored a paper (in columns)

Relational Data

Data Definition (categorical variables )

For authors

For publications:

X = matrix 481 x 33 columns expanded in dummy coding of the 3 + 5 + 4 categories:

Academic position: Assistant, Associate, full professor

Scientific sub-sector (italian classification): S01-Statistics; S02-

Statistics for experimental and technological research; S03- Economic

statistic; S04-Demography; S05-Social statistics

Geo-localization of university: North-west; North-East; Centre;

South Italy

Z = matrix 2 x 22892 rows expanded in 3 + 3 dummy coding

Number of authors per publication <4 Auth; 4-10 Auth; >10 Auth;

Year of publication (1989-96; 1997-03; 2004-10)

Exploratory data analysis and contiguity relations: An outlook

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Exploratory data analysis and contiguity relations: An outlook

DATA PRE-TREATMENT

Since we are interested in co-authorship among italian scholars we cutout external link reducing the initial dataset to an affiliation matrix of

333 Authors and 526 Papers

Bipartite network

Papers

Authors

Component Size 3 4 5 6 7 8 11 13 14 16 19 23 25 29 552# of components 14 4 4 2 3 4 1 1 2 1 1 1 1 1 1

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Authors x Authors projection (size: 333)

Component Size 2 3 4 5 6 7 8 197 # of components 20 5 5 5 1 2 2 1

Graph density = 0.013

Largest Component density = 0.025

Exploratory data analysis and contiguity relations: An outlook

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Aim: to explore the association structure between

authors characteristics (attribute variables on

actors)

Analysis of Attribute Data in X(MCA):

Exploratory data analysis and contiguity relations: An outlook

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heig

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cluster 1

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Hierarchical clustering on the factor map

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Page 20: Exploratory data analysis and contiguity relations: An outlookcedric.cnam.fr/fichiers/art_3486.pdf · to visualise and explore the relationships in the net structure ... to discover

Looking at association structure on attribute

variables: Smoothed Multiple Correspondence

Analysis - SFA

Exploratory data analysis and contiguity relations: An outlook

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Analysis of N-1GQ

The factorial map highlights the characteristics of

contiguous scholars. For instance the nodes are the

barycentre of their neighbours and lye close to

characteristics which are peculiar of their ego-centered

network (Sector, Role,..)

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Cluster Analysis on the Results of the S-MCA

Exploratory data analysis and contiguity relations: An outlook

0.00

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inertia gain

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Dim 1 (23.87%)

Dim

2 (

21.6

9%

)

299

241

176

78124182231

33

5490329

191

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1162272704262210238

170106199

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111114144177190194204211278292311288

88257

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1852982036917518118720913226016616204426617183237

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4927172476309

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324134214226249

19715923384314164135287265156277290302119772572892011375162225243

155207

117

30

94

180161

10250

157

130196

100

96

3980129 1011361512245332

18168294310

cluster 1

cluster 2

cluster 3

cluster 4

cluster 5

cluster 6

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

0.00

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0.25

0.30

0.35

-1.0

-0.5

0.0

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2.0

Dim 1 (23.87%)

Dim

2 (2

1.69

%)

heig

ht

cluster 1

cluster 2

cluster 3

cluster 4

cluster 5

cluster 6

299

241176

78124182231

33

5490329

191183

1162272704262210238170106199

275

3318521252326111114144177190194204211278292311288

88257

68108

220

29

103

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303

304

138

316

263

105

253109

35

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167127150

244

2226440

1402283053165

232

308148121

163

267327

45

60

28570

307

301

217

11574120186246

113

192

56142216330

200

214314167230

32025555

1121455028312

223139

286

1431213728025

234

30663

147

79218172322

254202

239

291

89

8215215830047276198

417686125206221296317281

26532824827264

2521782842823183134

315364295119240

272937193

122

173

21326115104222

1072459917914973

19

279

1852982036917518118720913226016616204426617183237

229

319

61313236

269

95

51

189

174

333110

46118

23

169143

66

323

92123

259

38154

146205131

12853

8710226

251

193

7791126

184

242

9

58219

3727327419524

3258

48258

160

268

17123528

32

59262

321

256297

81

492717247630913398

32413421422624919715923384314164135287265156277290302119772572892011375162225243

15520711730

94180161

10250

157

130196

10096

3980129 101136151224533218168294310

Hierarchical clustering on the factor map

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FALD: Correspondence Analysis of

Exploratory data analysis and contiguity relations: An outlook

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-0.5

0.0

0.5

1.0

CA factor map

Dim 1 (27.54%)

Dim

2 (

20

.95

%)

12

3

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78

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331332333CentreNorthEastNorthWest

South

AssistantProf

AssociateProf

FullProf

SECS-S/01

SECS-S/02SECS-S/03

SECS-S/04

SECS-S/05

..

'

.ji.1

q

qqGQNQ +− −

Residualinformation afterleaving out the

contiguityrelationships

Attribute far from the origin

explain the residual variance:

They are notimportant in explaining a

network effect

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CONCLUDING REMARKS – AIM 1

SFA and FALD can be used to explore the relationships

between the network structure and attribute variables…

The factorial maps show the author position as a function of

the attributes in Q and the contiguity structure…

How to combine Network Data and external information

in SNA?

Exploratory data analysis and contiguity relations: An outlook

Aim 2

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AIM 2 -Notation and Definition

A (nxp) affiliation matrix (binary); n actors; p events

aij = 1 -< i-th actor is present at the j-th event

(i=1, …, n; j=1, …, p)

X (nxm) n actors; r nominal variables expanded into m

dummy variables

xik = 1 -< i-th actor belongs to the k-th category

(i=1, …, n; k=1, …, m)

Z(qxp) p events; s nominal variables expanded into q

dummy variables

zhj = 1 -< j-th event belong to the h-th category

(h=1, …, q; j=1, …,p)

Exploratory data analysis and contiguity relations: An outlook

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Z E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12

Z11 1 1 1 1 0 0 0 0 0 0 0 0

Z12 0 0 0 0 1 1 1 1 0 0 0 0

Z13 0 0 0 0 0 0 0 0 1 1 1 1

A E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 X X11 X12 X13 X21 X22

1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0

2 1 1 1 1 1 1 1 1 1 0 1 0 2 1 0 0 1 0

3 1 1 1 1 1 0 0 1 0 0 0 0 3 1 0 0 1 0

4 1 1 0 1 1 1 0 0 0 0 0 1 4 1 0 0 1 0

5 0 0 1 1 0 1 1 0 1 0 0 0 5 1 0 0 1 0

6 1 1 1 1 1 1 1 1 0 0 0 0 6 1 0 0 1 0

7 0 1 0 0 1 0 1 1 0 1 0 1 7 0 1 0 1 0

8 0 0 0 1 1 0 1 1 0 0 0 0 8 0 1 0 1 0

9 0 1 0 1 1 0 1 0 0 0 0 0 9 0 1 0 1 0

10 0 0 1 0 1 0 1 1 0 0 0 0 10 0 1 0 0 1

11 0 0 0 0 1 1 1 1 1 0 1 1 11 0 1 0 0 1

12 0 1 0 0 1 1 1 1 0 0 0 0 12 0 1 0 0 1

13 0 1 0 1 0 0 0 0 1 0 0 1 13 0 0 1 0 1

14 0 0 0 0 0 0 0 0 1 1 1 1 14 0 0 1 0 1

15 0 0 0 1 0 1 0 0 1 1 1 1 15 0 0 1 0 1

16 0 0 0 0 0 0 0 0 0 0 1 1 16 0 0 1 0 1

17 0 0 0 0 1 1 0 0 1 1 1 1 17 0 0 1 0 1

18 1 0 0 0 0 0 0 0 1 1 1 1 18 0 0 1 0 1

Data structure

Exploratory data analysis and contiguity relations: An outlook

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GfKl - CLADAG 2010

ZCXBA +

... The Underlying Decomposition Model(Takane, Shibayama, 1991)

Affiliation data

Effect of actor characteristics

Effect of event

characteristics

Interaction Effect

+ XQZ

Exploratory data analysis and contiguity relations: An outlook

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frequency lconditiona astion interpretalstatistica

)]([ˆ

:Xdesignorthogonalofcaseparticular

)(ˆ

1

1

2

AXXXB

AXXXB

XBA(B)

XBA

B

=

=

−=

diag

argminJ

bk,j [0,1]

=> Affiliation effect of the k-th category to the j-th event.

Effect of actor characteristics on relational data

Exploratory data analysis and contiguity relations: An outlook

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GfKl - CLADAG 2010

Effect of events characteristics in determining

the presence of actors at events

AZZZC

CZA)J(

CZA

C

=

−=

−1

2

)]([ˆ

solution particular retain the we

diag

argminC

Chi affiliation effect due to the presence of actors at event

given the h-th event’s category

Exploratory data analysis and contiguity relations: An outlook

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Use of the coefficients B and C in SNA

The B e C coefficients have been used to approximate

the original affiliation matrix A and to derive the

adjacency matrices GX and GZ (actors x actors)

GX and GZ are then analyzed by SNA methods specially to

look for peculiar patterns of ties and homogeneous

groups of actors induced by the effect of the external

information.

GfKl - CLADAG 2010

ZZ

XX

GAC

GAB

ˆˆ

ˆˆ

Exploratory data analysis and contiguity relations: An outlook

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From A to G: Clustering of G

AAG =

Hierarchical clustering of network positions

Exploratory data analysis and contiguity relations: An outlook

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Analysis of X and A

B (m x p)

column maximum value = 1

row marginal maximum theoretical value = r x p

BXAX

ˆˆ =

maximum value of the columns marginal

of B = r x m

(all n actors are present at the j-th event)

maximum value of rows marginal of B = p

(a category fully characterizes all p

events)

Proximity categories and actors

Exploratory data analysis and contiguity relations: An outlook

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whose generic element weights the joint presence at events of actors having

similar characteristics. Maximum theoretical value in GX

If a homophily effect is present, then GX shows a weighting system coherent

with the capability of all actors characteristics to jointly explain the participation

in events.

XXX AAG = ˆˆ

2

1

2 rpr

p

j

==

Analysis of X and A

Clustering / Blockmodeling

Exploratory data analysis and contiguity relations: An outlook

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Analysis of Z and A

C (n x q)

Results …

ZCA ˆˆ =Z

Proximity between Z categories and

actors

Exploratory data analysis and contiguity relations: An outlook

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Analysis of Z and A

To derive from the weighted adjacency GZ:

where the generic element of GZ is the weight of the tie between a pair

of actors related to their presence at the same categories of events

ZZZ AAG = ˆˆZA

Clustering (or Blockmodeling)

Exploratory data analysis and contiguity relations: An outlook

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Authors x Authors projection (size: 333)

Component Size 2 3 4 5 6 7 8 197 # of components 20 5 5 5 1 2 2 1

Graph density = 0.013

Largest Component density = 0.025

Exploratory data analysis and contiguity relations: An outlook

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Authors Attributes in X and Coefficients B

Exploratory data analysis and contiguity relations: An outlook

North-west

North-east

Centre

South

Secs-S01

Secs-S02

Secs-S03

Secs-S04

Secs-S05

Assistant

Associate

Full professor

Analysis of Gx

XXX AAG = ˆˆ

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Structural Equivalence - Cluster AnalysisK Centre NorthEast NorthWest South 1 0.23 0.27 0.23 0.27 2 0.00 1.00 0.00 0.00 3 0.12 0.12 0.00 0.764 0.35 0.18 0.25 0.22 5 1.00 0.00 0.00 0.00 6 0.00 1.00 0.00 0.00

k AssistantProf AssociateProf FullProf1 0.23 0.41 0.36 2 0.00 0.41 0.59 3 0.53 0.24 0.24 4 0.26 0.32 0.42 5 0.00 0.00 1.006 0.00 1.00 0.00

Exploratory data analysis and contiguity relations: An outlook

Analysis of Gx

Importance of Coefficients Bin 6 classes K S/01 S/02 S/03 S/04 S/05

1 0.00 0.00 0.00 1.00 0.002 0.92 0.00 0.08 0.00 0.003 0.00 1.00 0.00 0.00 0.004 0.36 0.00 0.42 0.00 0.225 0.97 0.03 0.00 0.00 0.006 0.00 0.22 0.78 0.00 0.00

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Papers Attributes in Z and coefficients C

k 1989-96 1997-03 2004-10 Auth <4 Auth 4-10 Auth >10

1 0.24 0.09 0.08 0.07 0.27 0.122 0.05 0.03 0.03 0.03 0.06 0.113 0.08 0.07 0.05 0.05 0.06 0.26

Exploratory data analysis and contiguity relations: An outlook

Weighted importanceof coefficients C for the 3 classes

Analysis of Gz

ZZZ AAG = ˆˆ

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Some concluding remarks

Relational and attribute data

- to derive ad hoc relational data structures (affiliation and adjacency matrices)

- to enhance the interpretation of traditional network analysis from a different

point of view:

i) the complementary use of valued graphs defined according to

observed auxiliary information;

ii) the possibility to introduce explicative measures joining external

information and relational data

iii) the interpretation of the results as complex data where groups of

actors are defined and interpreted as "second order"

individuals

Exploratory data analysis and contiguity relations: An outlook

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Some references

Aluja, Banet T., Lebart, L. (1984). Local and Partial Principal Component Analysis and Correspondence Analysis. In: Havranek, T., Sidak, Z., Novak, M. (Eds.), COMPSTAT Proceedings, Phisyca-Verlag, Vienna. pp. 113-118.

Benali, H., Escofier, B. (1990). Analyse factorielle lissèe et analyse factorielle des diffèrences locales. Revue de Statistique Appliquèe. 38, pp. 55-76.

De Stefano D., Fuccella V., Vitale M.P., Zaccarin S. (2013). The use of different data sources in the analysis of co-authorship networks and scientific performance. Social Networks. 35, pp. 370-381.

D’Esposito, M. R., De Stefano, D., Ragozini, G. (2014). On the use of Multiple Correspondence Analysis to visually explore affiliation networks. Social Networks. 38. pp. 28–40.

Ferligoj, A., Kronegger, L. (2009). Clustering of Attribute and/or Relational Data. Metodoloski Zvezki. 6, 135-153.

Giordano G., Vitale M.P. (2007). Factorial Contiguity Maps to Explore Relational Data Patterns. Statistica applicata. 19, pp. 297-306.

Giordano G., Vitale M.P. (2011). On the use of auxiliary information in Social Network Analysis. Advances in Data Analysis and Classification (ADAC). 5, pp. 95-112.

Lazarsfeld, P., Merton, R. K. (1954). Friendship as a Social Process: A Substantive and Methodological Analysis. In: Freedom and Control in Modern Society, Berger, M., Abel, T., H. Page, C. (eds.). Van Nostrand, New York, 18-66.

Maddala, G.S. (1991). A Perspective on the Use of Limited-Dependent and Qualitative Variables Models in Accounting Research. The Accounting Review . 66, pp. 788-807.

McPherson, M., Smith-Lovin, L., Cook., J. (2001). Birds of a Feather: Homophily in Social Networks. Annual Review of Sociology. 27, pp. 415-44.

Opsahl, T., Panzarasa, P. (2009). Clustering in weighted networks. Social Networks. 31, pp. 155-163

Ragozini, G., De Stefano, D., D’Esposito, M. R., (2015). Multiple factor analysis for time-varying two-mode networks. Network Science. 3, pp. 18-36.

Robins, G.L., Pattison, P., Kalish, Y., Lusher,D. (2007). An introduction to exponential random graph (p*) models for social networks. Social Networks. 29, pp. 173-191.

Takane Y. Shibayama T. (1991). Principal component analysis with external information on both subjects and variables, Psychometrika. 56, issue 1, pp. 97-120.

Exploratory data analysis and contiguity relations: An outlook