Algebraic models for social networks:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge u.a.
Cambridge Univ. Press
1993
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Structural analysis in the social sciences
7 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 310 S. graph. Darst. |
| ISBN: | 0521365686 |
Internformat
MARC
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| 100 | 1 | |a Pattison, Philippa |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algebraic models for social networks |c Philippa Pattison |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge u.a. |b Cambridge Univ. Press |c 1993 | |
| 300 | |a XXI, 310 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Structural analysis in the social sciences |v 7 | |
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Datensatz im Suchindex
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Contents
List of figures and tables page xi
Preface xix
1 Algebraic representations for complete social
networks 1
Complete network data 5
Sources of network data 13
The boundary of a network 13
Relational content 14
Network measurement 17
Reliability and validity of network data 18
Structure in social networks 20
Directed graphs 21
Some analyses for social network data 22
Properties of a structural representation 32
An algebra for complete social networks 36
Compound relations and network paths 37
Comparing paths in networks and the Axiom of
Quality 42
The partially ordered semigroup of a network 44
An algorithm for semigroup construction 49
Summary 54
2 Algebraic representations for local social networks 56
Types of local networks 58
Representing local networks 61
An algebra for local social networks 62
Paths in local networks 63
Comparing paths in local networks 64
The local role algebra of a local network 67
An algorithm for constructing a local role algebra 68
The local role algebra of a subset in a local
network 70
vii
viii Contents
Role algebras 73
Relations among role algebras: The nesting relation 75
Presentation of role algebras 78
Local role algebras and role sets 79
Partial networks and partial role algebras 81
The nesting relation for partial role algebras 86
Analysis of local networks 86
Partially ordered semigroups and role algebras:
A summary 88
3 Comparing algebraic representations 90
Isomorphisms of network semigroups 91
Some networks with isomorphic semigroups 93
Comparing networks: Isotone homomorphisms 96
The n relation of an isotone homomorphism 99
Partial orderings among homomorphisms and
n relations 103
Lattices of semigroups and ^ relations 104
The joint homomorphism of two semigroups 110
The common structure semigroup 113
Lattices of semigroups: A summary 114
Local networks with isomorphic local role algebras 116
Comparing local role algebras: The nesting relation 119 i
Other classes of networks with identical algebras 123
Trees 124
Idempotent relations 128
Monogenic semigroups 129
Handbooks of small networks 133
Summary 134
4 Decompositions of network algebras 135
Decompositions of finite semigroups 137
Direct representations 138
Existence of direct representations 141
Subdirect representations 146
Existence of subdirect representations 149
Factorisation 152
Uniqueness of factorisations 155
An algorithm for factorisation 156
Using factorisation to analyse network semigroups 160
The reduction diagram 161
Co ordination of a partially ordered semigroup 162
Relationships between factors 163
Factorisation of finite abstract semigroups 165
Contents ix
A decomposition procedure for role algebras 166
Summary 171
5 An analysis for complete and local networks 172
An analysis for complete networks 173
Relational conditions of semigroup homomorphisms 173
Generalisations of structural equivalence 188
The correspondence definition 190
Searching for minimal derived set associations 199
Analysing entire networks 199
An example: Relational structure in a self analytic
group 201
Local networks 206
Derived local networks 207
A correspondence definition for local role algebras 208
Some applications 212
Local roles in the Breiger Ennis blockmodel 212
A General Social Survey network 216
The snowball network L 220
Local role algebras for two block two generator
models 222
Summary 223
6 Time dependent social networks 224
A language for change 225
Some relational conditions for smooth change 226
An analysis of time dependent blockmodels 228
The development of relational structure 230
A local role analysis of time dependent blockmodels 234
7 Algebras for valued networks 238
The semigroup of a valued network 238
Binary network semigroups from valued networks 243
Local role algebras in valued local networks 247
Using valued network algebras 250
8 Issues in network analysis 251
Describing social context: Positions and roles 251
Positions and roles 252
The structure and content of relations 253
Some models for relational structure 256
Strong and weak ties 256
The balance model 258
The complete clustering model 258
The transitivity model 259
x Contents
Other triad based models 259
The First and Last Letter laws 260
Permutation models for kinship structures 260
Some other models 261
Describing common structure 261
Common relational forms in two self analytic
groups 262
Common relational forms in two community elites 266
Social structure 270
Analysing large networks 271
References 273
Appendix A Some basic mathematical terms 289
Appendix B Proofs of theorems 292
Author index 303
Subject index 307
Figures and tables
Figures
1.1 A directed graph representation of a friendship
network among four members of a work group page 5
1.2 Representations for symmetric network relations 7
1.3 Representations for a valued network relation 8
1.4 A multiple network W 9
1.5 Structural, automorphic and regular equivalence 26
1.6 The compound relation FH 38
1.7 Some compound relations for the network W 40
1.8 Hasse diagram for the partial order of S(W) 47
1.9 Hasse diagram for the partial order of S(N) 51
1.10 The Cayley graph of the semigroup S(N) 52
2.1 Some partial networks 57
2.2 Partial ordering for the local role algebra of the
network L 67
3.1 The lattice Ls of isotone homomorphic images of S(N4) 105
3.2 Hasse diagram for the partial order of S(N4) 107
3.3 The lattice As for the abstract semigroup with
multiplication table of S(N4) 108
3.4 The lattice L,(S(N4)) of ^ relations on S(N4) 111
3.5 Extended automorphic equivalence 118
3.6 The lattice LQ of role algebras nested in Q 120
3.7 The lattice L^(Q) of ^ relations on the role
algebra Q 122
3.8 Some relations in a small work group 125
3.9 Some directed out trees 126
3.10 Some pseudo order relations on four elements 129
3.11 Two transition graphs T and U 131
4.1 The ^ relation lattice LX(T) of the partially ordered
semigroup T 146
4.2 The lattice LT of isotone homomorphic images of T 146
xi
xii Figures and tables
4.3 The ^ relation lattice LK(V) of the partially ordered
semigroup V 151
4.4 The ^ relation lattice Ln{U2) of the semigroup U2 151
4.5 A ;r relation lattice admitting two irredundant subdirect
decompositions 153
4.6 A nondistributive, modular lattice 156
4.7 Reduction diagram for the factorisation of V 162
5.1 Some network mappings 174
5.2 Automorphic, extended automorphic and regular
equivalences in a network 178
5.3 Indegree and outdegree equivalences in a network 182
5.4 Some conditions for semigroup homomorphisms 184
5.5 The central representatives condition 186
5.6 Relations among equivalence conditions 189
5.7 The lattice LX(S(X)) of ^ relations of S(X) 192
5.8 Searching for minimal derived set associations 200
5.9 Analysis of a complete network 200
5.10 Reduction diagram for the Breiger Ennis semigroup
BE1 204
5.11 Analysis of a local network 211
5.12 Reduction diagram for the local role algebra of the
GSS network 217
5.13 Reduction diagram for the local role algebra of the
network L 221
7.1 The decomposition theorem for valued network
semigroups 246
8.1 Reduction diagram for the Ennis semigroup 265
Tables
1.1 The binary matrix of the friendship network in
a small work group 6
1.2 Binary matrix representation of the multiple network W 9
1.3 Types of complete network data 10
1.4 Relational content in a sample of network studies 15
1.5 Some approaches to network analysis 23
1.6 A blockmodel and multiple networks for which it is
a fat fit, a lean fit and an oc blockmodel (a = 0.5) 30
1.7 Some compound relations for the network W in
binary matrix form 40
1.8 The blockmodel N = {L,A} 41
Figures and tables xiii
1.9 Primitive relations and compound relations of lengths
2 and 3 for the blockmodel N 42
1.10 The multiplication table and partial order for the
partially ordered semigroup S(W) 46
1.11 Generating the semigroup of the blockmodel N 50
1.12 Multiplication table for the semigroup S(N) 51
1.13 Edge and word tables and partial order for the
semigroup S(N) 53
2.1 Types of local network 60
2.2 The local network L in binary matrix form 62
2.3 Paths of length 3 or less in the network L having ego
as source 64
2.4 Right multiplication table for the local role algebra of
ego in the network L 66
2.5 Constructing the local role algebra of ego in the
network L 69
2.6 The blockmodel network N 70
2.7 The local role algebra for block 1 in the network N 70
2.8 The local role algebra for the subset {1,2} in the
network N 71
2.9 Distinct submatrices in the local role algebra for
the subset (1, 2} of the network N 72
2.10 Local role algebra for the subset {1, 2, 3, 4} of
the network N 72
2.11 Quasi orders on S(N) corresponding to the role
algebras Q1 and Qii^i 77
2.12 Distinct relations in the semigroup 5(N) of
the network N 78
2.13 Relation plane for ego in the network L 80
2.14 Relation plane for block 1 in the network N 80
2.15 The role set for block 1 in the network N 81
2.16 Truncated relation plane of order 2 for block 1
in the network N 82
2.17 Truncated relation plane of order 2 for ego in
the network L 83
2.18 Partial local role algebra Q] for block 1 in
the network N 84
2.19 Partial local role algebra Q\ for ego in the network L 85
2.20 Partial local role algebra Q\ for ego in the network L 86
3.1 Two comparable networks N, = (A, B) and N2 = {A, B] 92
3.2 The partially ordered semigroups 5(N,) and S(N2)
of the networks N, and N2 93
3.3 The network B, which is an inflation of the network N, 94
xiv Figures and tables
3.4 The network N3, which is the disjoint union of
the networks Nj and N2 of Table 3.1 95
3.5 Two comparable networks N, = [A, B] and N4 = [A, B] 96
3.6 The partially ordered semigroups S^) and S(N4) 97
3.7 The partially ordered semigroups S, T and U 99
3.8 The ;r relation corresponding to the isotone
homomorphism from S onto T 100
3.9 The ;r relation corresponding to the isotone
homomorphism from S(N4) onto S(N,) 100
3.10 Constructing a homomorphic image of the
partially ordered semigroup S 102
3.11 Isotone homomorphic images of S(N4) 106
3.12 Abstract homomorphic images of S(N4) 109
3.13 Finding abstract homomorphic images of 5(N4) 110
3.14 ^ relations corresponding to isotone homomorphisms
ofS(N4) 110
3.15 The joint homomorphic image / and the joint isotone
homomorphic image K of two semigroups V and W 112
3.16 Common structure semigroups for the semigroups
VandW 114
3.17 Lattices of semigroups and ^ relations 115
3.18 Some small local networks with identical role sets 119
3.19 Role algebras nested in the role algebra Q 120
3.20 TT relations in L^(Q) for the role algebra Q 121
3.21 Two element two relation networks with identical
partially ordered semigroups 133
4.1 A partially ordered semigroup T 138
4.2 Two partially ordered semigroups Sj and S2 139
4.3 The direct product Si x S2 of the semigroups Sj and S2 140
4.4 The ^ relation corresponding to the isotone
homomorphism from T onto Sj 142
4.5 Isotone homomorphic images of the partially ordered
semigroup T 144
4.6 ^ relations in Ln(T) 145
4.7 Two partially ordered semigroups Ux and U2 and
their direct product U1xUl 148
4.8 A subsemigroup U of [/, x t/2 that defines a subdirect
product of Ux and U2 149
4.9 A partially ordered semigroup V isomorphic to
the semigroup U 149
4.10 ^ relations in LJV) 150
4.11 The ^ relations Ku generated by the ordering
1 6 on the semigroup V 159
Figures and tables xv
4.12 ^ relations generated by each possible additional
ordering i j on the semigroup V 159
4.13 Atoms in L^(S(N)) and their unique maximal
complements and corresponding factors 161
4.14 Co ordinates for elements of the partially ordered
semigroup V in the subdirect representation
corresponding to [nu n2} 163
4.15 Association indices for factors of the semigroups
T and V 164
4.16 Multiplication table for a semigroup S 165
4.17 ^ relations in AX(S), presented as partitions on S 167
4.18 The 71 relations nst for each possible additional
ordering s ( on Q 169
4.19 Atoms z of the ^ relation lattice of the local role
algebra of the network L and their unique maximal
complements n{z) 170
5.1 The network X on {1, 2, 3} and the derived network
Y on {a,b} 191
5.2 The partially ordered semigroup S(X) and the factors
A and B of S(X) 191
5.3 Distinct relations in S(X) 192
5.4 The partial orderings ^ and ^ associated
with the mapping ft on the node set of X and
the isotone homomorphism j of S(X) 194
5.5 A network R = {A} on five elements 195
5.6 The partially ordered semigroup S(R) of the network
R = {A} and factors of S(R) 195
5.7 Distinct relations generated by the network N 197
5.8 The partial order 0 corresponding to the factor
S(N)//r4 of 5(N) 197
5.9 The partial orders corresponding to the derived sets
{1,2} and {(134), (2)} for the network N 197
5.10 Derived sets associated with the factor S(N)/7r4 of
the semigroup S(N) 198
5.11 Derived networks corresponding to minimal derived
set associations for the factor S(N)/n4 of S(N) 198
5.12 Minimal derived set associations and corresponding
derived networks for other factors of S(N) 199
5.13 The Breiger Ennis blockmodel for a self analytical
group 202
5.14 The semigroup BEX of the Breiger Ennis blockmodel 203
5.15 Factors of BE\ 203
5.16 Other images of BE\ appearing in Figure 5.10 205
xvi Figures and tables
5.17 Minimal derived set associations for some images
of BE\ shown in Figure 5.10 205
5.18 Derived networks for associations with factors of BE1 206
5.19 The partial orders p and T for the local role
algebra of block 1 of the network N 209
5.20 Derived set associations for the factors of the local
role algebra of block 1 of N 210
5.21 Derived local networks corresponding to some minimal
derived set associations for the factors of the local role
algebra of block 1 210
5.22 Local role algebras for blocks in the Breiger Ennis
blockmodel 213
5.23 Factors of the local role algebras of Breiger Ennis
blocks 214
5.24 Minimal subsets for which factors of the Breiger Ennis
local role algebras are nested in the subset partial order 215
5.25 A local network from General Social Survey items 216
5.26 The local role algebra of the General Social Survey
network 217
5.27 Factors for the role algebra of the GSS network 218
5.28 Other role algebras in the reduction diagram of Figure
5.12 219
5.29 Minimal subset associations for role algebras appearing
in the reduction diagram of the GSS network 219
5.30 The local role algebra generated by the snowball
network L 220
5.31 Role algebras identified in Figure 5.13 221
5.32 Some derived set associations for factors of the
network L 222
5.33 Reducible role algebras from two element two relation
local networks 223
6.1 Blockmodels for Newcomb Fraternity, Year 1, Weeks
1 to 15 229
6.2 Incidence of factors of Week 15 semigroup as images
of semigroups for earlier weeks 231
6.3 Minimal partitions associated with identified images of
S15 and corresponding derived networks 232
6.4 Local role algebras for blocks in the Newcomb
blockmodel at Week 15 234
6.5 Factors of the local role algebras for the Week 15
blockmodel 235
6.6 Incidence of Week 15 role algebra factors in earlier
weeks 236
Figures and tables xvii
7.1 A valued network V = [A, B) 241
7.2 Some max min products for the valued relations A and
B of the valued network V 241
7.3 The partially ordered semigroups S(V) and S(B) generated
by the valued network V and the blockmodel B 242
7.4 Components of the valued relations A and B of the
valued network V 244
7.5 Components of the valued relations in S(V) for the
valued network V 244
7.6 Filtering relations for the semigroup S(V) 245
7.7 A valued local network 248
7.8 Distinct relation vectors from the valued local network
of Table 7.7 249
7.9 The local role algebra of node 1 in the valued local
network of Table 7.7 249
8.1 Some models for networks 255
8.2 The Ennis blockmodel 263
8.3 The semigroup BE2 of the Ennis blockmodel 263
8.4 The joint isotone homomorphic image K of BEX and
BE2 and its factors Kl and K2 264
8.5 Derived set associations of Kl and K2 in the
Breiger Ennis and Ennis blockmodels 264
8.6 Images of BE2 appearing in Figure 8.1 265
8.7 The Altneustadt blockmodel 267
8.8 The Towertown blockmodel 267
8.9 The Altneustadt semigroup A 268
8.10 The Towertown semigroup T 268
8.11 The joint isotone homomorphic image L of the
semigroups A and T 268
8.12 The ^ relations na on A and 7Tr on T corresponding
to the joint isotone homomorphic image L 269
8.13 Minimal derived set associations with L in the
Altneustadt and Towertown blockmodels and
corresponding derived networks 269 |
| any_adam_object | 1 |
| author | Pattison, Philippa |
| author_facet | Pattison, Philippa |
| author_role | aut |
| author_sort | Pattison, Philippa |
| author_variant | p p pp |
| building | Verbundindex |
| bvnumber | BV008416925 |
| classification_rvk | MR 2000 MR 6600 |
| ctrlnum | (OCoLC)231581034 (DE-599)BVBBV008416925 |
| discipline | Soziologie |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV008416925 |
| illustrated | Illustrated |
| indexdate | 2024-12-05T15:01:54Z |
| institution | BVB |
| isbn | 0521365686 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005545371 |
| oclc_num | 231581034 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-12 DE-19 DE-BY-UBM DE-384 DE-739 DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-12 DE-19 DE-BY-UBM DE-384 DE-739 DE-11 DE-188 |
| physical | XXI, 310 S. graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Structural analysis in the social sciences |
| series2 | Structural analysis in the social sciences |
| spelling | Pattison, Philippa Verfasser aut Algebraic models for social networks Philippa Pattison 1. publ. Cambridge u.a. Cambridge Univ. Press 1993 XXI, 310 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Structural analysis in the social sciences 7 Netzwerkanalyse Soziologie (DE-588)4205975-6 gnd rswk-swf Soziales Netzwerk (DE-588)4055762-5 gnd rswk-swf Mathematische Logik (DE-588)4037951-6 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Soziales Netzwerk (DE-588)4055762-5 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Netzwerkanalyse Soziologie (DE-588)4205975-6 s Algebra (DE-588)4001156-2 s DE-188 Mathematische Logik (DE-588)4037951-6 s 1\p DE-604 Structural analysis in the social sciences 7 (DE-604)BV002814947 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005545371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Pattison, Philippa Algebraic models for social networks Structural analysis in the social sciences Netzwerkanalyse Soziologie (DE-588)4205975-6 gnd Soziales Netzwerk (DE-588)4055762-5 gnd Mathematische Logik (DE-588)4037951-6 gnd Algebra (DE-588)4001156-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4205975-6 (DE-588)4055762-5 (DE-588)4037951-6 (DE-588)4001156-2 (DE-588)4114528-8 |
| title | Algebraic models for social networks |
| title_auth | Algebraic models for social networks |
| title_exact_search | Algebraic models for social networks |
| title_full | Algebraic models for social networks Philippa Pattison |
| title_fullStr | Algebraic models for social networks Philippa Pattison |
| title_full_unstemmed | Algebraic models for social networks Philippa Pattison |
| title_short | Algebraic models for social networks |
| title_sort | algebraic models for social networks |
| topic | Netzwerkanalyse Soziologie (DE-588)4205975-6 gnd Soziales Netzwerk (DE-588)4055762-5 gnd Mathematische Logik (DE-588)4037951-6 gnd Algebra (DE-588)4001156-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Netzwerkanalyse Soziologie Soziales Netzwerk Mathematische Logik Algebra Mathematisches Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005545371&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002814947 |
| work_keys_str_mv | AT pattisonphilippa algebraicmodelsforsocialnetworks |