1. Generating Hierarchical link patterns based on concept lattice for Navigating the Web of Data
Liang Zheng

2. Navigational features have been largely recognized as fundamental for the Web of Data browsing, search and query.
Most existing navigation approaches are based on link-traversal.
However, current approaches for navigating can hardly expose deep relations of the links.
In this paper, a link pattern hierarchy based on concept lattice is proposed which effectively organizes the link space.

3. Preliminaries

4. Let U be the set of all URIs and L the set of all literals.
Definition 1. (Web of Data T )
The Web of Data (over U and L) is the set of triples (s, p, o) in U U  (U  L). We will denote it by T.

5. Let P  U be a finite set of properties.
Definition 2: Property Hierarchy H
A set P together with a partial ordering  p is called a poset, and is denoted by H=( P,  p )
Irreducible property set
A property set X  P, when u, v  X such that neither u  p v nor
v  p u, is called irreducible property set.
Property set

6. A partial ordering on the subset of P
Let X, Y be two irreducible property subsets of P,
X p Y iff  v  Y ,  u  X, such that u p v
{mother} < {parents} 
{mother, influencedBy} < {parents} 
{mother, influencedBy} < {mother} 
{mother, influencedBy} < {parents, influencedBy} 
{mother, influencedBy} < {parents, knows} X

7. Data analysis is performed to help users analyze the deep relations of the links, by taking advantage of a mathematical theory named Formal Concept Analysis (FCA) theory.

8. Definition 3: Formal Context (形式背景)
Let Es  U be a finite set of URIs, indicating the starting points of the navigation.
Definition 3: Formal Context (形式背景)
a triple K=(G,M,I), where G  U denotes a set of entities, M  U a set of properties, and I ⊆ G×M a binary relation between G and M.
The ordered pair ( g, m) ∈ I iff
 es  Es, such that (es, m, g) T or (g, m, es) T
 Using formal concept analysis(FCA), the navigation of linked data
is a generalization of a formal context[].

9. Example: K: G={e1, e2, e3, e4}, M={mother, father, knows, influencedBy}
 Using formal concept analysis(FCA), the navigation of linked data
is a generalization of a formal context[].

10. Definition 4: Link Pattern( a formal concept of the context K)
For X ⊆ G, Y ⊆ M, a pair lp= <X,Y>, such that X ‘= Y and Y’ = X, is called a link pattern (a formal concept of the context K)
In <X, Y>, the set X is called the extent and the set Y the intent of the link pattern lp.
Let LPK be a finite link pattern set of the context K, and Let ≤ be a partial ordering on LPK, lp1 ≤ lp2⇔ (X1,Y1) ≤ (X2,Y2) ⇔ Y1  p Y2 . Obviously:lp1 ≤ lp2  X1 ⊆ X2
Then lp1 is called a sub_linkpattern of lp2,and lp2 is a super_linkpattern of lp1.
For two link pattern lp1 and lp2, if lp1 ≤ lp2 and there is no link pattern lp3 with lp3 lp1, lp3 lp2, lp1 ≤ lp3 ≤ lp2 , the lp1 is called a child of lp2, and lp2 is called a parent of lp1. This relationship is denoted by lp1 ≺ lp2 .

11. Definition 5: Link Pattern Hierarchy (concept lattices)
With respect to the partial order ≺ , the link pattern set LPK forms a lattice called the link pattern hierarchy of the formal context K, denoted by LPHk
The greatest element of LPHk (G, G’)
The least element of LPHk (M’ , M)

12. Problem 1： Link Pattern Hierarchy Construction
Generating formal concepts
Constructing concept lattices

13. Comparing Performance of Algorithms for Generating Concept Lattices []
Batch Algorithm 批生成算法
首先生成形式背景所对应的所有概念,再生成概念之间的连接关系
静态的形式背景
Incremental Algorithm 增量算法
动态形式背景（交易数据库）
Kuznetsov S O, Obiedkov S A. Comparing performance of algorithms for generating concept lattices[J]. Journal of Experimental & Theoretical Artificial Intelligence, 2002, 14(2-3): 被引用次数：415

15. |M| = 100; |g'| = 4.
|g‘ | : the number of attributes per object

16. |M| = 100; |g'| = 25.

17. |M| = 100; |g'| = 50.

19. Bordat算法的基本思想是:对于形式背景K=(G, M,I),若概念节点为(Ak,Bk),找出属性子集Colk=M- Bk, 且Colk在Ak中能保持完全二元组的性质,即Colk为最大的子集,则Bki=Bk  Colk构成了当前节 点的一个子节点的内涵。

22. The worst-case time complexity of Bordat is O(|G||M|2|L|), where |L| is the size of the concept lattice.

23. References
Kuznetsov S O, Obiedkov S A. Comparing performance of algorithms for generating concept lattices[J]. Journal of Experimental & Theoretical Artificial Intelligence, 2002, 14(2-3):
Heath, T., & Bizer, C. (2011). Linked data: Evolving the web into a global data space. Synthesis lectures on the semantic web: theory and technology, 1(1),