Generating Hierarchical link patterns based on
concept lattice for Navigating the Web of Data
Liang Zheng
•
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.
Preliminaries
• 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.
• 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.
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

• 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.
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
•
Example: K: G={e1, e2, e3, e4}, M={mother, father, knows, influencedBy}
mother
e1
e2
e3
e4
father
1
1
knows
influencedBy
1
1
1
1
1
1
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 .
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
•
The least element of LPHk
(G, G’)
(M’ , M)
Problem 1：
Link Pattern Hierarchy Construction
• Generating formal concepts
• Constructing concept lattices
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）
1. 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): 189-216. 被引用次数：415
|M| = 100; |g'| = 4.
|g‘ | : the number of attributes per object
|M| = 100; |g'| = 25.
|M| = 100; |g'| = 50.
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。
{mother}’={e1}
{father}’={e2}
{parents}’={e1, e2}
{knows}’={e1,e2,e3} {influenced}’={e1, e2, e4}
• The worst-case time complexity of Bordat is O(|G||M|2|L|), where |L|
is the size of the concept lattice.
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): 189-216.
•
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), 1-136.