Directed hyper-graphs for RDF documents 

Amadís Antonio Martínez-Morales ^{1, 2} and María-Esther Vidal ²

¹ Departamento de Computación, Facultad Experimental de Ciencias y Tecnología, Universidad de Carabobo. Valencia, Venezuela. Teléfono: +58-241-6004000, Ext. 375200. aamartin@uc.edu.ve.

² Departamento de Computación y Tecnología de la Información, Universidad Simón Bolívar. Valle de Sartenejas, Baruta, Venezuela.
Teléfono: +58-212-9063268. mvidal@ldc.usb.ve  

Abstract 

Resource Description Framework (RDF) is a W3C proposal to express metadata about resources in the Web. The RDF data model has been formalized using several graph-based representations; each one offers different expressive power and support for the tasks of query answering and semantic reasoning. In this paper, we propose a directed hyper-graph formal model to represent and manage RDF documents efficiently. We have developed algorithms that exploit the properties of the proposed representation, and conducted an experimental study to analyze the space and time savings of our solution with respect to the labeled directed graph representation. Our study has been performed over synthetic and real-world RDF documents, and we could observe that our approach reduces the space required to store an RDF document and speeds up the task of query answering, overcoming the labeled directed graph representation. 

Key words: Data model, directed hyper-graphs, Resource Description Framework (RDF).

Hipergrafos dirigidos para documentos RDF 

Resumen

Resource Description Framework (RDF) es una propuesta del W3C para expresar metadatos acerca de recursos en el Web. El modelo de datos de RDF ha sido formalizado utilizando diversas representaciones basadas en grafos, cada una de las cuales ofrece diferente poder expresivo y soporte para las tareas de responder consultas y razonamiento semántico. En este trabajo se propone el desarrollo de un modelo formal, basado en hipergrafos dirigidos, para el almacenamiento y administración eficiente de documentos RDF. A tal fin, se han desarrollado algoritmos que explotan las propiedades de la representación propuesta y se ha realizado un estudio experimental para analizar las mejoras en espacio y tiempo de esta solución con respecto a la representación basada en grafos dirigidos etiquetados. El estudio fue realizado sobre documentos RDF sintetizados y reales, los resultados obtenidos confirman que el enfoque propuesto reduce el espacio requerido para almacenar un documento RDF y acelera la tarea de responder consultas, mejorando los resultados obtenidos por la representación basada en grafos dirigidos etiquetados. 

Palabras clave: Hipergrafos dirigidos, modelo de datos, Resource Description Framework (RDF).

Received: December 10, 2008  Revised: November 23, 2009 

In this paper we propose a directed hyper-graph formal model for RDF to represent, store, and process RDF documents efficiently, overcoming the limitations of the existing representations. Basically, a directed hyper-graph is defined by a set of nodes and a set of hyper-arcs; each hyper-arc connects a set of source nodes to a set of target nodes. Directed hyper-graphs have been successfully used as a modeling tool to represent concepts and structures in many application areas (e.g., formal languages, relational databases, production and manufacturing systems, public transportation systems, and topic maps [8, 9, 10]). 

In an RDF directed hyper-graph, the information is only stored in the nodes, and the hyper-arcs only preserve the role of each node and the concept of direction of RDF graphs. Thus, each resource (subject, property, or value) is stored only once, and the space complexity of an RDF document is reduced if a resource appears several times in the document. Besides, RDF directed hyper-graphs define implicit position-based indexes [11] for an RDF document, which can support efficient evaluation of queries over the document. 

We have developed algorithms that exploit the properties of the proposed approach, and conducted an empirical study to analyze the space and time savings of our solution with respect to the labeled directed graph (LDG) representation. Our study has been performed over a variety of synthetic and real-world RDF documents, and we could observe that our approach scales better than the LDG representation in terms of space and time complexity. These results encourage us to develop algorithms to solve the tasks of query answering and semantic reasoning, and to extend the proposed approach to represent RDF Schema (RDFS) [12, 13] data models. 

The main contributions of this paper are: (1) an efficient representation of RDF documents based on the directed hyper-graph formal model, (2) an analysis of the expressive power of the directed hyper-graph model, (3) a formal space complexity study of the proposed representation to store RDF documents, (4) query answering algorithms that exploit the properties of the directed hyper-graphs, and (5) an empirical study of the impact of our approach on the task of query answering. This paper extends and updates the work reported in [14]. 

The rest of this paper is structured as follows. Section 2 describes the existing approaches to represent the RDF data model, including their limitations. In Section 3, we present our RDF model based on directed hyper-graphs. Section 4 reports our preliminary experimental results. Finally, in Section 5, the concluding remarks and future work are pointed out. 

Related Work 

An RDF document can be represented as a graph, where each node is a resource and each arc represents a property. Formally, an RDF graph is defined as follows [5, 15]: Suppose there is an infinite set U (URI references), an infinite set B = { b_j: j ³ 0 } (blank nodes), and an infinite set L (RDF literals). A triple (s, p, o) Î (U È B) × U × (U È B È L) is called an RDF triple, where s represents a subject, p a predicate, and o an object. 

Definition 1 

An RDF graph T is a set of RDF triples. 

T = {(s, p, o): (s, p, o) Î (U È B) × U × (U È B È L)} 

The universe of T, univ(T), is the set of elements of U È B È L that occur in the triples of T. The vocabulary of T is the set vocab(T) = univ(T) - B. sub(T) (resp. pred(T), obj(T)) is the set of all elements in univ(T) that occur as a subject (resp. predicate, object) in an RDF graph T. The size of T, |T|, is the number of RDF triples in T. An RDF graph is simple if it does not use vocabulary with a predefined semantics in RDF Schema (RDFS). Let Var be a set of variables disjoint from U, B, and L. A triple (v₁, v₂, v₃) Î (U È Var) × (U È Var) × (U È L È Var) is a triple pattern. A graph pattern is a set of triple patterns. Given a graph pattern P, we denote by var(P) the set of variables mentioned in P. An elemental query Q is an expression of the form Q: H ¬ B, where H = (H₁,..., H_n) is a list of variables such that (“i: 1 £ i £ n: H_i Î var(B)) (safety condition), and B is a graph pattern. We denote H by Head(Q) and B by Body(Q). If |B| = 1 then Q is a basic query, if |B| > 1 then Q is a conjunctive query.

Given an RDF graph T, the required space complexity to store the RDF document represented by T is O(|T|). If there are no blank nodes in T, then the required time complexity to answer an elemental query Q against the RDF document represented by T is O(|T|^k), where k ³ 1 is an integer that represents the number of triple patterns in Body(Q). RDF graphs allow several representations: labeled directed graphs [3, 5], undirected hyper-graphs [6], and bipartite graphs [6, 7]. Each one of these representations has its own limitations with respect to the RDF data model, in terms of expressive power, space complexity, and support for the tasks of query answering and semantic reasoning. 

In the labeled directed graph model, given an RDF graph T, the set of nodes W is comprised of elements in sub(T) È obj(T), and the set of arcs E is composed of elements in pred(T) [3, 5]. Thus, each RDF triple (s, p, o) Î T is represented by a labeled arc, s —p ® o. The number of nodes and arcs for directed labeled graphs representing RDF graphs is |W| £ 2 |T| and |E| = |T| [6]. Thus, given an RDF graph T, the required space complexity to store the RDF document represented by T using this model is O(|T|). 

This approach has two main drawbacks [6]. First, a resource may simultaneously appear as a predicate, a subject and/or an object in the same RDF graph. For example, in the RDF graph T₁ = {(Picasso, paints, Guernica), (Guernica, type, Paint), (Zapata, type, Paint), (paints, range, Paint), (paints, domain, Painter)}, the resource paints occurs as a predicate and a subject. This situation can be modeled by allowing multiple occurrences of the same resource in the resulting labeled directed graph, as arcs or nodes labels (Figure 1). However, this compromises one of the more important properties of graph theory: the intersection between the nodes and arcs labels must be empty. Second, a predicate may relate other predicates in an RDF graph. For example, in the RDF graph T₂ = {(Painter, paints, Paint), (Artist, creates, Artifact), (paints, subPropertyOf, creates)}, the predicate subPropertyOf relates the predicates paints and creates. This situation can be modeled by extending the notion of arc by allowing the connection between arcs (Figure 2). However, the resulting structure is not a graph in the strict mathematical sense, because the set of arcs must be a subset of the Cartesian product of the set of nodes. Since these two simple situations violate some graph constraints, it is not possible to use concepts and search algorithms of graph theory to manipulate RDF graphs. Thus, while labeled directed graph model is the most widely used representation, it can not be considered a formal model for RDF [16].

Figure 1. RDF graph allowing multiple occurrences of the same resource.

Figure 2. RDF graph extending the notion of arc.

In the undirected hyper-graph model, given an RDF graph T, each RDF triple t = (s, p, o) Î T is a hyper-edge and each element of t (subject s, predicate p, and object o) is a node (Figure 3) [6]. The number of nodes and hyper-edges for undirected hyper-graphs representing RDF graphs is |W| = |univ(T)| and |E| = |T| [6]. Thus, given an RDF graph T, the required space complexity to store the RDF document represented by T using this model is O(max(|univ(T)|, |T|)). However, this representation loses the concept of direction in RDF graphs, which impacts the task of semantic reasoning. Additionally, it may not be easy to graphically represent large RDF graphs, like the museum example [17].

Figure 3. Undirected hyper-graph for (s, p, o).

In the bipartite graph model, given an RDF graph T, there are two types of nodes in W: statement nodes St (one for each RDF triple (s, p, o) Î T) and value nodes Val (one for each element w Î univ(T)). Arcs in E relate statement and value nodes as follows: each t Î St has three out-coming arcs that point to the corresponding node for the subject, predicate, or object of the RDF triple represented by the statement node t (Figure 4). The number of nodes and arcs of bipartite graphs representing RDF graphs is |St| = |T|, |Val| = |univ(T)|, and |E| = 3 |T| [6, 7]. Thus, given an RDF graph T, the required space complexity to store the RDF document represented by T using this model is O(max(|univ(T)|, |T|)). While bipartite graphs satisfy the requirement of a formal graph representation for RDF, issues such as reification, entailment, and semantic reasoning have not been addressed yet [16].

Figure 4. Bipartite graph for (s, p, o).

Proposed Solution 

In this work we propose a directed hyper-graph formal model for RDF. Basically, a directed hyper-graph is defined by a set of nodes and a set of hyper-arcs, each one of them connecting a set of source nodes to a set of target nodes. Directed hyper-graphs have been successfully used as a modeling tool to represent concepts and structures in many application areas (e.g., formal languages, relational databases, production and manufacturing systems, public transportation systems, and topic maps [8, 9, 10]). An RDF directed hyper-graph is formally defined as follows: 

Definition 2 

Let T be an RDF graph. The RDF directed hyper-graph representing T is a tuple H(T) = (W, E, r) such that: 

• W = { w: w Î univ(T) } is the set of nodes. 

• E = { e_i: 1 £ i £ |T| } is the set of hyper-arcs. 

• r: W × E ® {‘s’, ‘p’, ‘o’} is the role function of nodes w.r.t. hyper-arcs. Let t Î T be an RDF triple, e Î E an hyper-arc, and w Î orig(e) È dest(e) a node. Then, the following must hold: 

• (r(w, e) = ‘s’) Û (w Î orig(e)) Ù (w Î sub({t})) 

• (r(w, e) = ‘p’) Û (w Î orig(e)) Ù (w Î pred({t})) 

• (r(w, e) = ‘o’) Û (w Î dest(e)) Ù (w Î obj({t})) 

Figures 5 and 6 show RDF directed hyper-graphs representing the RDF graphs T₁ and T₂ of Section 2, respectively. To understand the relationship between hyper-arcs and RDF triples consider, for example, the right topmost hyper-arc in Figure 6, which corresponds with the RDF triple e = (Artist, creates, Artifact). In this case, r(Artist, e) = ‘s’, r(creates, e) = ‘p’, and r(Artifact, e) = ‘o’. In our approach, given an RDF graph T, each node corresponds to an element w Î univ(T). Thus, the information is only stored in the nodes, and the hyper-arcs only preserve the role of each node and the concept of direction of RDF graphs. An advantage of this representation is that each resource (subject, property, or value) is stored only once, and the space required to store an RDF document is reduced if a resource appears several times in the document. In this way, the space complexity of our approach may be smaller than the complexity of representations presented in Section 2. In addition, concepts, techniques, and algorithms of hyper-graph theory may be used to manipulate RDF graphs under this representation.

Figure 5. RDF directed hyper-graph for RDF graph T₁.

Figure 6. Directed hyper-graph for T₂.