Social semigroups: a unified theory of scaling and blockmodelling as applied to social networks
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Fairfax, Va.
George Mason Univ. Press
1991
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 267 S. graph. Darst. |
| ISBN: | 0913969346 |
Internformat
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| 245 | 1 | 0 | |a Social semigroups |b a unified theory of scaling and blockmodelling as applied to social networks |c John Paul Boyd |
| 264 | 1 | |a Fairfax, Va. |b George Mason Univ. Press |c 1991 | |
| 300 | |a XII, 267 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Groupes sociaux - Modèles mathématiques | |
| 650 | 4 | |a Réseaux sociaux - Modèles mathématiques | |
| 650 | 4 | |a Semi-groupes | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Semigroups | |
| 650 | 4 | |a Social groups |x Mathematical models | |
| 650 | 4 | |a Social networks |x Mathematical models | |
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Datensatz im Suchindex
| _version_ | 1817613080021958656 |
|---|---|
| adam_text |
Contents
1 Introduction 1
2 Relations 13
2.1 Sets, Relations, and Functions 14
2.2 Transitive and Reflexive Relations 25
2.3 Lattices 34
2.4 Algebraic Closure and Interior Operations 42
2.5 Galois Connections 47
2.6 Labeled Graphs 54
2.7 Real Composition and Multigraphs 58
3 Semigroups 63
3.1 Background 63
3.2 Special Elements 64
3.3 Special Kinds of Semigroups 69
3.4 Subdirect Products 85
3.5 Structural Similarity 108
3.6 Ideals and Green Relations 127
4 The Sampson Semigroup of Relations 143
4.1 Measurement, Real Matrices, and Correlations 143
4.2 The Sampson Equations and Semigroup 154
4.3 Green Relations and Representations 163
5 Continuous Structural Similarity 175
5.1 Green Relations and Row/Column Spaces 175
5.2 Idempotents and Singular Value Decomposition 181
5.3 Involutions 199
iii
6 Discrete Structural Similarity 209
6.1 Semigroup Clustering 209
6.2 Green Relations and Zareckii Lattices 211
6.3 Semigroup Clustering by Simulated Annealing 223
6.4 Discrete Representations of Monks 230
7 Conclusions 239
7.1 Transitivity and Order 239
7.2 Semigroups and Green Relations 242
7.3 The Sampson Semigroup and its Kernel 245
7.4 Continuous Structure and SVD 246
7.5 Discrete Similarity and Simulated Annealing 248
iv
List of Figures
1.1 The two ternary operations determined by a groupoid are
indicated both by parentheses and by ordered binary trees. . 3
1.2 Groupoid composition. The relations p and a induce a prod¬
uct relation, pa, between individuals a and c in the triad a, b, c.
If the individuals a and c are equated, then the induced prod¬
uct is an attribute, per, of the individual 7
1.3 Associativity in social relations implies that the two paths,
(pa)r and p{ar), are equal 9
2.1 Two ways of picturing relations 18
2.2 A transitive relation and its diagram 27
2.3 Diagram of an idempotent 31
2.4 Diagrams of partial orders, all of which are lattices except for
P6 36
2.5 Diagrams of Heyting algebras. Left: the smallest nonBoolean
algebra; center: the free Heyting algebra with one generator;
right: the free Boolean algebra with one generator 39
2.6 Diagram of an algebraic closure operation 43
2.7 A lattice of closure operations on relations 46
2.8 Row and column spaces, respectively, of the relation p 53
3.1 A sublattice of varieties of semigroups 70
3.2 The cyclic semigroup Cq,^ with stem s, period p, and kernel
{a", as+1,., as+p~1}. The arrow represent multiplication by
the generator, a 72
3.3 Each graph homomorphism p satisfies the indicated condi¬
tion, while failing the one below it 74
3.4 A graph F. All endomorphisms map upper points to upper
and lower to lower. E.g., 6: a,b,c* *b and rf,e,/i »e 79
v
3.5 A lattice diagram of scales with more powerful lower 85 ;
3.6 Diagram for the Second Isomorphism Theorem 92
3.7 Congruence lattice for T 99
3.8 Congruence lattice on R4 107
3.9 Free structural similarity /3s holds from x to x' iff, for all i
relations p, a £ S, the existence of the solid p or tr arrow ;
implies the existence of the dotted arrow of the same kind. . Ill i
3.10 The intersection of the structural similarities Pi and fh is not
a structural similarity. 112 \
3.11 A semigroup of relations showing that no Galois connection is ¦
possible between the lattice of structural equivalences and the ¦
lattice of congruence relations. Portions of the two lattices are
shown in the next figure 114
3.12 Referring to the previous figure, a sublattice of congruence
relations is shown on the left, and the induced sublattice of
structural equivalences, on the right. No Galois connection is
possible because neither join nor meet is preserved 115
3.13 The ordering of the Green relations 131
3.14 The "Egg Box" structure of a typical V Class 134
4.1 Graph of the relation/9.g,4 158
4.2 A group of order two that is not structurally balanced 173
5.1 Plot of the Columns of Ux The dotted line with arrows
indicates the axis of involution, while the other dotted line
indicates the fixed axis 189
5.2 Plot of the columns of Vy The dotted line with arrows indi¬
cates the axis of involution, while the other dotted line indi¬
cates the fixed axis 191
5.3 Plot of the columns of Vz The dotted line with arrows indi¬
cates the axis of involution, while the other dotted line indi¬
cates the fixed axis 193
6.1 Distributive lattice with join irreducible elements, 0, a, b, c,
and d. Each point is labeled by its unique irredundant repre¬
sentation of each element as a join of join irreducible elements. 216
6.2 Two ^ equivalent idempotents, their common column space,
and their union, which is an idempotent that is % equivalent
to e and 77 222
vi
I
I
j
I
I 6.3 The possible partial orders with up to four points, with Z2 as
I their automorphism group, and with no points that are both
I fixed and isolated 227
| 6.4 Idempotent from the moiety model, together with the best
f fitting assignments. The letter prefixes are used to distinguish
'; F or G assignments when they are different. The superscripts
'', indicate alternative assignments 231
6.5 Idempotent and best fitting assignments for the box model. . 238
vii
I
List of Tables
2.1 Examples of lattices on mathematical structures 35
2.2 An algebraic closure operation 43
2.3 The semigroup generated by the operations R, S, and T. . . 44
2.4 Variations on Galois connections 51
2.5 A typical relation, which will illustrate row and column spaces. 53
2.6 Correlations of A with Boolean and real powers of B 61
3.1 Types of relational homomorphisms p : X\ — X2, where
Vs(pi) = P2, and where p = p*p p 79
3.2 Some of the endomorphisms of the graph F 80
3.3 The subsemigroup of the endomorphism monoid End(F) gen¬
erated by {7, r} 80
3.4 A subsemigroup T of the Sampson semigroup 98
3.5 Representation of T as the subdirect product T/(p2 D pi).
This representation is not injective because P2 H pi ^ 1. . 101
3.6 T isomorphic to a subdirect product, which is injective be¬
cause P2 D pz = t 102
3.7 Semigroup with complete congruences whose intersection is
not a complete congruence 118
3.8 Semigroup with a congruence that is additive but not complete. 119
3.9 Semigroup where C and H are not congruences 132
4.1 A silly semigroup caused by a single error and the use of
Boolean multiplication 147
4.2 Correlations of powers of m by m error matrix E with A. . . 149
4.3 Correlations (x 100) among the raw Sampson relations. . . . 156
4.4 Correlations (x 100) among the products of Sampson relations. 159
4.5 The Sampson semigroup 5 161
4.6 The Sampson semigroup represented as a subdirect product. 167
ix
i
i
5.1 Ux, patterns of choice from the SVD of the Sampson kernel. 188
5.2 Vy, patterns of being chosen, the last choice being negative. . 190 j
5.3 Vz, patterns of being chosen, the last choice being positive. . 192 j
5.4 Singular Values dk and Relative Approximation Errors. . . . 192 }
5.5 Projection (xlOOO) on Y along Xx. These are the predicted '
responses for the idempotent E — b2 197
5.6 Projection (xlOOO) on Z along X1. These are the predicted
responses for the idempotent F = b2a 198
5.7 The orthogonal involution Gg, where 9 = .076026 radians. . . 202
5.8 G, the involution for ab 203
5.9 H, the involution for aba 204
5.10 The involution Gy. Angle = 0.14030353; shear = 0.15784515. 205
5.11 The involution Hz. Angle = 0.61741699; shear = 0.38689627. 205
5.12 Correlations (xlOOO) among the fitted kernel matrices and the
averaged data matrices 207
6.1 Observed positive Sampson choices, A in italics, over the best
fitting moiety idempotent, / in bold. The relation / corre¬
sponds to b2a from the Sampson kernel 233
6.2 Observed negative Sampson choices, B in italics, over the
best fitting moiety involution, g in bold. The relation g cor¬
responds to ab from the Sampson kernel 235
x |
| any_adam_object | 1 |
| author | Boyd, John Paul 1939- |
| author_GND | (DE-588)173030203 |
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| author_role | aut |
| author_sort | Boyd, John Paul 1939- |
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| building | Verbundindex |
| bvnumber | BV004575797 |
| callnumber-first | H - Social Science |
| callnumber-label | HM131 |
| callnumber-raw | HM131 |
| callnumber-search | HM131 |
| callnumber-sort | HM 3131 |
| callnumber-subject | HM - Sociology |
| ctrlnum | (OCoLC)22890018 (DE-599)BVBBV004575797 |
| dewey-full | 305 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 305 - Groups of people |
| dewey-raw | 305 |
| dewey-search | 305 |
| dewey-sort | 3305 |
| dewey-tens | 300 - Social sciences |
| discipline | Soziologie |
| format | Book |
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| id | DE-604.BV004575797 |
| illustrated | Illustrated |
| indexdate | 2024-12-05T15:01:51Z |
| institution | BVB |
| isbn | 0913969346 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002814976 |
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| physical | XII, 267 S. graph. Darst. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | George Mason Univ. Press |
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| spelling | Boyd, John Paul 1939- Verfasser (DE-588)173030203 aut Social semigroups a unified theory of scaling and blockmodelling as applied to social networks John Paul Boyd Fairfax, Va. George Mason Univ. Press 1991 XII, 267 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Groupes sociaux - Modèles mathématiques Réseaux sociaux - Modèles mathématiques Semi-groupes Mathematisches Modell Semigroups Social groups Mathematical models Social networks Mathematical models Soziales Netzwerk (DE-588)4055762-5 gnd rswk-swf Gruppe (DE-588)4022378-4 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-188 Gruppe (DE-588)4022378-4 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002814976&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boyd, John Paul 1939- Social semigroups a unified theory of scaling and blockmodelling as applied to social networks Groupes sociaux - Modèles mathématiques Réseaux sociaux - Modèles mathématiques Semi-groupes Mathematisches Modell Semigroups Social groups Mathematical models Social networks Mathematical models Soziales Netzwerk (DE-588)4055762-5 gnd Gruppe (DE-588)4022378-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4055762-5 (DE-588)4022378-4 (DE-588)4114528-8 |
| title | Social semigroups a unified theory of scaling and blockmodelling as applied to social networks |
| title_auth | Social semigroups a unified theory of scaling and blockmodelling as applied to social networks |
| title_exact_search | Social semigroups a unified theory of scaling and blockmodelling as applied to social networks |
| title_full | Social semigroups a unified theory of scaling and blockmodelling as applied to social networks John Paul Boyd |
| title_fullStr | Social semigroups a unified theory of scaling and blockmodelling as applied to social networks John Paul Boyd |
| title_full_unstemmed | Social semigroups a unified theory of scaling and blockmodelling as applied to social networks John Paul Boyd |
| title_short | Social semigroups |
| title_sort | social semigroups a unified theory of scaling and blockmodelling as applied to social networks |
| title_sub | a unified theory of scaling and blockmodelling as applied to social networks |
| topic | Groupes sociaux - Modèles mathématiques Réseaux sociaux - Modèles mathématiques Semi-groupes Mathematisches Modell Semigroups Social groups Mathematical models Social networks Mathematical models Soziales Netzwerk (DE-588)4055762-5 gnd Gruppe (DE-588)4022378-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Groupes sociaux - Modèles mathématiques Réseaux sociaux - Modèles mathématiques Semi-groupes Mathematisches Modell Semigroups Social groups Mathematical models Social networks Mathematical models Soziales Netzwerk Gruppe |
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