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Formal Concept Analysis III Knowledge Discovery Robert J aschke Asmelash Teka Hadgu FG Wissensbasierte Systeme/L3S Research Center Leibniz Universit at Hannover Robert J aschke (FG KBS) Formal Concept Analysis 1 / 33 Agenda

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SLIDE 1

Formal Concept Analysis

III Knowledge Discovery Robert J¨ aschke Asmelash Teka Hadgu

FG Wissensbasierte Systeme/L3S Research Center Leibniz Universit¨ at Hannover

Robert J¨ aschke (FG KBS) Formal Concept Analysis 1 / 33

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SLIDE 2

Agenda

7

Triadic Formal Concept Analysis Motivation Folksonomies Motivation Triadic Formal Concept Concept-Tri-Lattice Visualization of Tri-Lattices Iceberg Tri-Lattices Computing Tri-Concepts Qualitative Evaluation Neighborhoods

Robert J¨ aschke (FG KBS) Formal Concept Analysis 2 / 33

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SLIDE 3

Motivation: Collaborative Tagging Systems

Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 33

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SLIDE 4

Motivation: Collaborative Tagging Systems

Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 33

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SLIDE 5

Motivation: Collaborative Tagging Systems

Robert J¨ aschke (FG KBS) Formal Concept Analysis 3 / 33

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SLIDE 6

Motivation: Collaborative Tagging Systems

manage your web bookmarks and publication references

  • pen for the public since beginning of 2006, → 5 000 active users

developed and operated at L3S Research Center

Robert J¨ aschke (FG KBS) Formal Concept Analysis 4 / 33

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SLIDE 7

Agenda

7

Triadic Formal Concept Analysis Motivation Folksonomies Motivation Triadic Formal Concept Concept-Tri-Lattice Visualization of Tri-Lattices Iceberg Tri-Lattices Computing Tri-Concepts Qualitative Evaluation Neighborhoods

Robert J¨ aschke (FG KBS) Formal Concept Analysis 5 / 33

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SLIDE 8

Folksonomies

data structure of collaborative tagging systems connects users, tags, and resources conceptual structure created by the people

Forschungszentrum L3S

Wissenschaft & Forschung in den Schlüsselbereichen Wissen, Information und Lernen to science l3s center hannover research by jaeschke and 1 other person on 2006-01-27 10:39:07 edit | delete

Robert J¨ aschke (FG KBS) Formal Concept Analysis 6 / 33

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SLIDE 9

Folksonomies: Hypergraph, Tensor

t1 t2 t3 r1 r2 u1 u2 u3 u4

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 33

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SLIDE 10

Folksonomies: Hypergraph, Tensor

t1 t2 t3 r1 r2 u1 u2 u3 u4

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 33

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SLIDE 11

Folksonomies: Hypergraph, Tensor

t1 t2 t3 r1 r2 u1 u2 u3 u4

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 7 / 33

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SLIDE 12

Folksonomies

Definition (Folksonomy)

F :✏ ♣U, T, R, Y q with U, T, R finite sets of users, tags, and resources, resp. Y ❸ U ✂ T ✂ R ternary relation tripartite hypergraph boolean 3-dimensional tensor triadic formal context

t1 t2 t3 r1 r2 u1 u2 u3 u4

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 8 / 33

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SLIDE 13

Motivation

conceptual clustering of folksonomies

find interessting concepts/clusters support browsing, community detection, recommendations get an overview into the structure of a folksonomy

♣ q ❸ ✂ ✂ U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 9 / 33

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SLIDE 14

Motivation

conceptual clustering of folksonomies

find interessting concepts/clusters support browsing, community detection, recommendations get an overview into the structure of a folksonomy

tri-concept ♣A, B, Cq ❸ U ✂ T ✂ R: maximal cuboid in which every user from A has tagged every ressource from C with all tags from B ➞ shared conceptualization U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 9 / 33

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SLIDE 15

Motivation

Triadic Concept Analysis (Lehmann, Wille 1995) Iceberg Concept Lattices / Closed Itemset Mining (Lakhal/Stumme/ Zaki 1999) Association Rules (Agrawal, Srikant 1993) Formal Concept Analysis (Wille 1982) Tri-Concepts Frequent

Robert J¨ aschke (FG KBS) Formal Concept Analysis 10 / 33

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SLIDE 16

Motivation

We regard F ✏ ♣U, T, R, Y q as triadic formal context. In general, the elements of U, T and R are then called objects, attributes and conditions and ♣u, t, rq P Y is read as “object u has the attribute t under condition r”. U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 11 / 33

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SLIDE 17

Triadic Formal Concept

Definition (tri-concept)

triple ♣A, B, Cq with A ❸ U, B ❸ T, C ❸ R and A ✂ B ✂ C ❸ Y , such that none of the three components can be enlarged without violating the condition A ✂ B ✂ C ❸ Y . We call A the extent, B the intent and C the modus of the formal tri-concept. ➞ natural extension of formal concepts U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 12 / 33

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SLIDE 18

Concept-Tri-Lattice

three quasi orders ➚1, ➚2, ➚3: ♣A1, A2, A3q ➚i ♣B1, B2, B3q :ô Ai ❸ Bi, for i ✏ 1, 2, 3. not antisymmetric, i. e. from ♣A1, A2, A3q ➚i ♣B1, B2, B3q and ♣B1, B2, B3q ➚i ♣A1, A2, A3q does not follow ♣A1, A2, A3q ✏ ♣B1, B2, B3q concept tri-lattice B♣Kq of the triadic context K not a real (mathematical) lattice! U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 13 / 33

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SLIDE 19

Visualization of Tri-Lattices

Since it is not really a lattice, we can not draw a lattice diagram Alternative:

every quasi-order is written along the edge of a virtual triangle the tri-concepts are drawn into the triangle

example to the right: smallest non-trivial tri-lattice B3 ✏ B♣t1✉, t1✉, t1✉, ❍q visualization not always possible

satisfied tetrahedron condition violated Thomson condition

1 3 2 O1 O3 ❍ ❍ t1✉ t1✉ ❍ O2 t1✉

Robert J¨ aschke (FG KBS) Formal Concept Analysis 14 / 33

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SLIDE 20

Visualization of Tri-Lattices

Robert J¨ aschke (FG KBS) Formal Concept Analysis 15 / 33

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SLIDE 21

Iceberg Tri-Lattices

Given support constraints τu, τt, τr : tri-concept ♣A, B, Cq frequent :ô ⑤A⑤ ➙ τu, ⑤B⑤ ➙ τt, and ⑤C⑤ ➙ τr ➞iceberg tri-lattice U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 16 / 33

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SLIDE 22

Computing Tri-Concepts

Given

sets U, T, R ternary relation Y ❸ U ✂ T ✂ R support constraints τu, τt, τr

Find ♣A, B, Cq with

A ❸ U, B ❸ T, C ❸ R ⑤A⑤ ➙ τu, ⑤B⑤ ➙ τt, ⑤C⑤ ➙ τr A ✂ B ✂ C ❸ Y such that none of the sets A, B or C can be enlarged without violating the former condition

Robert J¨ aschke (FG KBS) Formal Concept Analysis 17 / 33

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SLIDE 23

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context ✏ t♣ ♣ qq ⑤ ♣ q P ✉ ♣ q ♣ ✂ q

♣ q ♣ q

✏ ♣ ✂ q ♣ q

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 24

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

✏ t♣ ♣ qq ⑤ ♣ q P ✉ ♣ q ♣ ✂ q

♣ q ♣ q

✏ ♣ ✂ q ♣ q

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 25

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ ♣ q ♣ ✂ q

♣ q ♣ q

✏ ♣ ✂ q ♣ q

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 26

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

♣ q ♣ q

✏ ♣ ✂ q ♣ q

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 27

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

♣ q ♣ q

✏ ♣ ✂ q ♣ q

In the example: ♣A, Iq ✏

  • tu2, u3✉, t♣t1, r1q, ♣t1, r2q, ♣t2, r1q✉

✟ U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 28

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

Loop: Find (frequent) concepts ♣B, Cq in ♣T, R, Iq

✏ ♣ ✂ q ♣ q

In the example: ♣T, R, Iq ✏ ♣T, R, t♣t1, r1q, ♣t1, r2q, ♣t2, r1q✉q U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 29

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

Loop: Find (frequent) concepts ♣B, Cq in ♣T, R, Iq

✏ ♣ ✂ q ♣ q

In the example: ♣B, Cq ✏ ♣tt1✉, tr1, r2✉q U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

slide-30
SLIDE 30

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

Loop: Find (frequent) concepts ♣B, Cq in ♣T, R, Iq

If A ✏ ♣B ✂ Cq

˜ Y , then output

♣A, B, Cq

In the example: ♣B ✂ Cq ˜

Y ✏ ♣tt1✉ ✂ tr1, r2✉q ˜ Y

U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 31

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

Loop: Find (frequent) concepts ♣B, Cq in ♣T, R, Iq

If A ✏ ♣B ✂ Cq

˜ Y , then output

♣A, B, Cq

In the example: ♣B ✂ Cq ˜

Y ✏ ♣tt1✉ ✂ tr1, r2✉q ˜ Y

✏ tu2, u3✉ ✏ A U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

slide-32
SLIDE 32

Computing Tri-Concepts

computes the iceberg tri-lattice of a triadic formal context

Algorithm

Let ˜ Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉ Loop: Find (frequent) concepts ♣A, Iq in ♣U, T ✂ R, ˜ Y q

Loop: Find (frequent) concepts ♣B, Cq in ♣T, R, Iq

If A ✏ ♣B ✂ Cq

˜ Y , then output

♣A, B, Cq

In the example: ♣A, B, Cq ✏ ♣tu2, u3✉, tt1✉, tr1, r2✉q U R T

t1 t2 t3 r1 r2 u1 u2 u3 u4 Robert J¨ aschke (FG KBS) Formal Concept Analysis 18 / 33

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SLIDE 33

Computing Tri-Concepts

Require: U, T, R, Y, τu, τt, τr

1: ˜

Y :✏ t♣u, ♣t, rqq ⑤ ♣u, t, rq P Y ✉

2: ♣A, Iq :✏ FirstFrequentConcept♣♣U, T ✂ R, ˜

Y q, τuq

3: repeat 4:

if ⑤I⑤ ➙ τt ☎ τr then

5:

♣B, Cq :✏ FirstFrequentConcept♣♣T, R, Iq, τtq

6:

repeat

7:

if ⑤C⑤ ➙ τr then

8:

if A ✏ ♣B ✂ Cq ˜

Y then

9:

print A,B,C

10:

end if

11:

end if

12:

until not NextFrequentConcept♣♣B, Cq, ♣T, R, Iq, τtq

13:

end if

14: until not NextFrequentConcept♣♣A, Iq, ♣U, T ✂ R, ˜

Y q, τuq

Robert J¨ aschke (FG KBS) Formal Concept Analysis 19 / 33

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SLIDE 34

Computing Tri-Concepts

The FirstFrequentConcept method: Require: ♣G, M, Iq, τ

1: A :✏ ❍I 2: B :✏ AI 3: if ⑤A⑤ ➔ τ then 4:

NextFrequentConcept♣♣A, Bq, ♣G, M, Iq, τq

5: end if 6: return ♣A, Bq

Robert J¨ aschke (FG KBS) Formal Concept Analysis 20 / 33

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SLIDE 35

Computing Tri-Concepts

the NextFrequentConcept method: Require: ♣A, Bq, ♣G, M, Iq, τ

1: i := max(M) 2: while defined♣iq do 3:

A :✏ ♣B ✌ iqI

4:

if ⑤A⑤ ➙ τ then

5:

D :✏ AI

6:

if B ➔i D then

7:

B :✏ D

8:

return true

9:

end if

10:

end if

11:

i :✏ max♣M③B ❳ t1, . . . , i ✁ 1✉q

12: end while 13: return false

Robert J¨ aschke (FG KBS) Formal Concept Analysis 21 / 33

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SLIDE 36

Qualitative Evaluation

BibSonomy Dataset: all publication records until November 23rd, 2006 removed: DBLP, posts with the tag “imported” ⑤U⑤ ✏ 262, ⑤T⑤ ✏ 5 954, ⑤R⑤ ✏ 11 101, ⑤Y ⑤ ✏ 44 944 Result: 13 992 tri-concepts (75 minutes on a 2 GHz PC) with support constraints τu ✏ 3, τt ✏ 2, τr ✏ 2:

21 tri-concepts contain 41 publications, 15 users and 36 tags

Robert J¨ aschke (FG KBS) Formal Concept Analysis 22 / 33

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SLIDE 37

Qualitative Evaluation

ai ant cognition colony dynamics emergent information intelligence lkl−kss networks simulation sociocognitive swarms tags social fca triadic bookmarking semantic wiki web mining semwiki2006 2006 nepomuk myown bibsonomy folksonomy clustering text eswc2006 swikig grahl stumme schmitz markusjunker, nepomuk lkl_kss, yish langec deynard lysander07, xamde hotho jaeschke brotkasting 1 29 28 33 2,3,5,9,15 17,21,31 12 35 22 10 37 4 18 16 23 13 19,30 24,34 7,11,20,32,41 6,14,27,39 8,40 25,26,36,38 edutella elearning p2p watchdog bluedolphin

p u b l i c a t i

  • n

s tags u s e r s

visualisation of the iceberg tri-lattice for τu ✏ 3, τt ✏ 2, τr ✏ 2

Robert J¨ aschke (FG KBS) Formal Concept Analysis 23 / 33

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SLIDE 38

Qualitative Evaluation

ai ant cognition colony dynamics emergent information intelligence lkl−kss networks simulation sociocognitive swarms tags social fca triadic bookmarking semantic wiki web mining semwiki2006 2006 nepomuk myown bibsonomy folksonomy clustering text eswc2006 swikig edutella elearning p2p watchdog

two topical groups: semantic social semantic further di- vided: wiki web folksonomy

Robert J¨ aschke (FG KBS) Formal Concept Analysis 24 / 33

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SLIDE 39

Qualitative Evaluation

Robert J¨ aschke (FG KBS) Formal Concept Analysis 25 / 33

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SLIDE 40

Qualitative Evaluation

1 10 100 1000 10000 100000 100000 200000 300000 400000 500000 600000 time in seconds ⑤Y ⑤ DecJan Feb Mar Apr May Jun triadic Next Closure ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ Trias

  • Robert J¨

aschke (FG KBS) Formal Concept Analysis 26 / 33

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SLIDE 41

1 A Finite-State Model for On-Line Analytical Pro- cessing in Triadic Contexts 2 Annotation and Navigation in Semantic Wikis 3 A Semantic Wiki for Mathematical Knowledge Man- agement 4 BibSonomy: A Social Bookmark and Publication Sharing System 5 Bringing the ”Wiki-Way” to the Semantic Web with Rhizome 6 Building and Using the Semantic Web 7 Conceptual Clustering of Text Clusters 8 Content Aggregation on Knowledge Bases using Graph Clustering 9 Creating and using Semantic Web information with Makna 10 Emergent Semantics in BibSonomy 11 Explaining Text Clustering Results using Semantic Structures 12 Harvesting Wiki Consensus - Using Wikipedia En- tries as Ontology Elements 13 Information Retrieval in Folksonomies: Search and Ranking 14 KAON – Towards a Large Scale Semantic Web 15 Kaukolu: Hub of the Semantic Corporate Intranet 16 Kollaboratives Wissensmanagement 17 Learning with Semantic Wikis 18 Mining Association Rules in Folksonomies 19 On Self-Regulated Swarms, Societal Memory, Speed and Dynamics 20 Ontologies improve text document clustering 21 Proceedings of the First Workshop on Semantic Wikis – From Wiki To Semantics 22

  • Proc. of the European Web Mining Forum 2005

23 Semantic Network Analysis of Ontologies 24 Semantic Resource Management for the Web: An ELearning Application. 25 Semantic Web Mining 26 Semantic Web Mining and the Representation, Anal- ysis, and Evolution of Web Space 27 Semantic Web Mining for Building Information Por- tals (Position Paper) 28 Social Bookmarking Tools (I): A General Review 29 Social Bookmarking Tools (II). A Case Study – Con- notea 30 Social Cognitive Maps, Swarm Collective Perception and Distributed Search on Dynamic Landscapes 31 SweetWiki : Semantic Web Enabled Technologies in Wiki 32 Text Clustering Based on Background Knowledge 33 The ABCDE Format Enabling Semantic Conference Proceedings 34 The Courseware Watchdog: an Ontology-based tool for Finding and Organizing Learning Material 35 Towards a Wiki Interchange Format (WIF) – Open- ing Semantic Wiki Content and Metadata 36 Towards Semantic Web Mining 37 TRIAS - An Algorithm for Mining Iceberg Tri- Lattices 38 Usage Mining for and on the Semantic Web (Book) 39 Usage Mining for and on the Semantic Web (Work- shop) 40 Wege zur Entdeckung von Communities in Folk- sonomies 41 WordNet improves text document clustering back Robert J¨ aschke (FG KBS) Formal Concept Analysis 27 / 33

slide-42
SLIDE 42

Qualitative Evaluation

ai ant cognition colony dynamics emergent information intelligence lkl−kss networks simulation sociocognitive swarms tags social fca triadic bookmarking semantic wiki web mining semwiki2006 2006 nepomuk myown bibsonomy folksonomy clustering text eswc2006 swikig grahl stumme schmitz markusjunker, nepomuk lkl_kss, yish langec deynard lysander07, xamde hotho jaeschke brotkasting 1 29 28 33 2,3,5,9,15 17,21,31 12 35 22 10 37 4 18 16 23 13 19,30 24,34 7,11,20,32,41 6,14,27,39 8,40 25,26,36,38 edutella elearning p2p watchdog bluedolphin

p u b l i c a t i

  • n

s tags u s e r s

Robert J¨ aschke (FG KBS) Formal Concept Analysis 28 / 33

slide-43
SLIDE 43

Qualitative Evaluation

fca triadic hotho schmitz, stumme bluedolphin, grahl jaeschke 18 13 4 37 10 1 16 23 8,40 myown 2006 folksonomy bibsonomy

tags users publications

Robert J¨ aschke (FG KBS) Formal Concept Analysis 28 / 33

slide-44
SLIDE 44

Qualitative Evaluation

ai ant cognition colony dynamics emergent information intelligence lkl−kss networks simulation sociocognitive swarms tags social fca triadic bookmarking semantic wiki web mining semwiki2006 2006 nepomuk myown bibsonomy folksonomy clustering text eswc2006 swikig grahl stumme schmitz markusjunker, nepomuk lkl_kss, yish langec deynard lysander07, xamde hotho jaeschke brotkasting 1 29 28 33 2,3,5,9,15 17,21,31 12 35 22 10 37 4 18 16 23 13 19,30 24,34 7,11,20,32,41 6,14,27,39 8,40 25,26,36,38 edutella elearning p2p watchdog bluedolphin

p u b l i c a t i

  • n

s tags u s e r s

Robert J¨ aschke (FG KBS) Formal Concept Analysis 29 / 33

slide-45
SLIDE 45

Qualitative Evaluation

users tags publications 33 semwiki2006 semantic eswc2006 swikig wiki langec lysander07, xamde deynard 2,3,5,9,15 17,21,31 35 12

publication 33: ”‘The ABCDE Format Enabling Semantic Conference Proceedings”’

Robert J¨ aschke (FG KBS) Formal Concept Analysis 30 / 33

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SLIDE 46

Neighborhoods

The visualization of tri-lattices is . . . at the moment manual work, time-intensive and pretty complicated,

  • r even impossible (cf. tetrahedron condition and Thomson

condition). Thus: easier visualization option desireable

Robert J¨ aschke (FG KBS) Formal Concept Analysis 31 / 33

slide-47
SLIDE 47

Neighborhoods

Idea: We regard tri-concepts as nodes in a graph. We connect two tri-concepts with an edge, when they contain the same tags, users, or resources. More formally: Two tri-concepts ♣A1, A2, A3q and ♣B1, B2, B3q are neighbors, if for an i P t1, 2, 3✉ it holds Ai ✏ Bi. neighbor relation ✒ ❸ ♣B♣Fq ✂ B♣Fqq The neighborhood graph then is ♣B♣Fq, ✒q.

Robert J¨ aschke (FG KBS) Formal Concept Analysis 32 / 33

slide-48
SLIDE 48

Neighborhoods

neighborhood graph for the tri-concept ♣tjaeschke, schmitz, stumme✉, tfca, triadic✉, t1, 37✉q

1, 37 fca, triadic jaeschke, schmitz, stumme 4, 8, 13, 18, 23, 37, 40 2006, myown jaeschke, schmitz, stumme 4, 10, 13, 16, 18, 23 2006, myown hotho, schmitz, stumme 4, 10 2006, bibsonomy, myown hotho, schmitz, stumme bluedolphin, grahl, schmitz, stumme bibsonomy, folksonomy 4, 10 hotho, schmitz, stumme 4, 13, 16, 18 2006, folksonomy, myown hotho, jaeschke, schmitz, stumme 2006, folksonomy, myown 4, 13, 18 4, 13, 18, 23 2006, myown hotho, jaeschke, schmitz, stumme user resource tag

Robert J¨ aschke (FG KBS) Formal Concept Analysis 33 / 33