Introduction

Many social, biological, and information systems can be well described by networks, where nodes represent objects (individuals), and links denote the relations or interactions between nodes. Predicting a possible link in a network is an interesting issue that has recently gained attention, due to the growing interest in social networks. For instance, one may be interested on finding potential friendship between two persons in a social network, or a potential collaboration between two researchers. Thus link prediction [12,20] aims at predicting whether two nodes (i.e. people) should be connected given that we know previous information about their relationships or interests. A common approach is to exploit the network structure, where numerical information about nodes is analyzed [12,20,9]. However, knowledge about the objects represented in the nodes can improve prediction results. For instance consider that the researchers Joe and Mike do not have a publication in common, thus they do not share a link in a collaboration network. Moreover, graph features do not indicate a potential link between them. However, they have published in the same journal and they both teach the same course in their respectively universities. This information can be an indication of a potential collaboration between them. Given this, approaches that are based on the semantics related to the domain of the objects represented by the nodes [21,18] have been proposed. In some of them, an ontology modeling the domain and the object interests were used in the prediction task.

However, there is uncertainty in such predictions. Often, it is not possible to guarantee the relationship between two objects (nodes). This is maybe due to the fact that information about the domain is incomplete. Thus, it would be interesting if link prediction approaches could handle the probability of a link conditioned on the information about the domain. In our example, knowing that the probability of the relationship between Joe and M ike conditioned on the knowledge of them publishing in the same journal and teaching the same course is high implies a link between them in the network; otherwise, a link is not suggested. In graph-based approaches, probabilistic models learned through machine learning algorithms were used for link prediction. Some examples of probabilistic models are Probabilistic Relational Model (PRM) [6], Probabilistic Entity Relationship Model (PERM) [7] and Stochastic Relational Model (SRM) [22]. On approaches based on semantic we claim that ontologies must be used to model the domain. Therefore, to model uncertainty, probabilistic approaches, such as probabilistic ontologies, must be considered.

An ontology can be represented through a description logic [2], which is typically a decidable fragment of first-order logic that tries to reach a practical balance between expressivity and complexity. To encode uncertainty, a probabilistic description logic (PDL) must be contemplated. The literature contains a number of proposals for PDLs [8,10,19,13]. In this paper we adopt a recently proposed PDL, called Credal ALC (crALC) [4,16,5], that extends the popular logic ALC [2]. In crALC one can specify sentences such as P (Professor|Researcher) = 0.4, indicating the probability that an element of the domain is a Professor given that it is a Researcher. These sentences are called probabilistic inclusions. Exact and approximate inference algorithms that deal with probabilistic inclusions have been proposed [4,5], using ideas inherited from the theory of Relational Bayesian Networks (RBN) [11].

In this paper, we propose to use a probabilistic ontology defined with the PDL crALC for semantic link prediction.

The paper is organized as follows. Section 2 reviews basic concepts of PDLs and crALC. Section 3 presents our algorithm for semantic link prediction through the PDL crALC. Experiments are discussed in Section 4, and Section 5 concludes the paper.

Probabilistic Description Logics and crALC

Description logics (DLs) form a family of representation languages that are typically decidable fragments of first order logic (FOL) [2]. Knowledge is expressed in terms of individuals, concepts, and roles. The semantic of a description is given by a domain D (a set) and an interpretation • I (a functor). Individuals represent objects through names from a set N I = {a, b, . . .}. Each concept in the set N C = {C, D, . . .} is interpreted as a subset of a domain D. Each role in the set N R = {r, s, . . .} is interpreted as a binary relation on the domain.

Several probabilistic descriptions logics (PDLs) have appeared in the literature. Heinsohn [8], Jaeger [10] and Sebastiani [19] consider probabilistic inclusion axioms such as P D (Professor) = α, meaning that a randomly selected object is a Professor with probability α. This characterizes a domain-based semantics: prob-abilities are assigned to subsets of the domain D. Sebastiani also allows inclusions such as P (Professor(John)) = α, specifying probabilities over the interpretations themselves. For example, one interprets P (Professor(John)) = 0.001 as assigning 0.001 to be the probability of the set of interpretations where John is a Professor. This characterizes an interpretation-based semantics.

The PDL crALC is a probabilistic extension of the DL ALC that adopts an interpretation-based semantics. It keeps all constructors of ALC, but only allows concept names on the left hand side of inclusions/definitions. Additionally, in crALC one can have probabilistic inclusions such as P (C|D) = α or P (r) = β for concepts C and D, and for role r. If the interpretation of D is the whole domain, then we simply write P (C) = α. The semantics of these inclusions is roughly (a formal definition can be found in [5]) given by:

∀x ∈ D : P (C(x)|D(x)) = α, ∀x ∈ D, y ∈ D : P (r(x, y)) = β.

We assume that every terminology is acyclic; no concept uses itself. This assumption allows one to represent any terminology T through a directed acyclic graph. Such a graph, denoted by G(T ), has each concept name and role name as a node, and if a concept C directly uses concept D, that is if C and D appear respectively in the left and right hand sides of an inclusion/definition, then D is a parent of C in G(T ). Each existential restriction ∃r.C and value restriction ∀r.C is added to the graph G(T ) as nodes, with an edge from r and C to each restriction directly using it. Each restriction node is a deterministic node in that its value is completely determined by its parents. The semantics of crALC is based on probability measures over the space of interpretations, for a fixed domain. Inferences, such as P (A o (a 0 )|A) for an ABox A, can be computed by propositionalization and probabilistic inference (for exact calculations) or by a first order loopy propagation algorithm (for approximate calculations) [5].

Link Prediction by using crALC

In this section we describe how to apply the PDL crALC for semantic link prediction. We borrowed some syntax from the graph-based approach where each node (a person in a social network) is represented by A, B, C, and we are interested in defining whether a link between A and B is suitable given there is no link between these nodes. Interests between the nodes are modeled through a probabilistic ontology represented by the PDL crALC. The prediction link task can be described as:

Given:

• a network defining relationship between objects;

• an ontology represented by crALC describing the domain of the objects;

• the ontology role that defines the semantic of the relationship between objects; • the ontology concept that describes the network objects. Find:

• a revised network defining relationship between objects.

The proposed algorithm for link prediction receives a network of a specific domain. For instance, in a collaboration network the nodes represent researchers and the relationship can have the semantic "has a publication with" or "is advised by". Therefore, the ontology represented by crALC describes the domain of publications between researchers, having concepts like Researcher, Publication, StrongRelatedResearcher and NearCollaborator and roles like hasPublication, hasSameInstitution and sharePublication. This ontology can be learned automatically through a learning algorithm as the ones proposed in [15,17]. Thus, the nodes represent instances of one of the concepts described in the PDL crALC and the semantic of the links is described by one of the roles in the PDL crALC. These concept and role must be informed as inputs to the proposed algorithm. The link prediction algorithm is described in Algorithm 1.

The algorithm starts looking for all pairs of instances of the concept C defined as the concept that provides the semantic for the network nodes. For each pair it checks whether the corresponding nodes exist in the network (this can be improved by exploring graph-based properties). If not the probability of the link is calculated through the probability of the defined role conditioned on evidence. The evidence is provided by the instances of the ontology. As many instances the ontology have the better is the inference performed. The inference is performed through the Relational Bayesian network build from ontology O. If the probability inferred is greater than a threshold then the corresponding link end if 8:

end if 9: end for Algorithm 1: Algorithm for link prediction through crALC.

is added to the network. Alternatively, when the threshold to be considered is not known a priori, a rank of the inferred links based on their probability is done and the top-k, where k would be a parameter, are chosen.

Preliminary Results

Experiments were run over a collaborative network of researchers. Data was gathered from the Lattes curriculum platform3 , the public repository for Brazilian curriculum researchers. In this platform, every researcher has a unique Lattes code that allows one to link to other researchers according to: shared publications, advising tasks, and examination board participations. Given this collaborative network we are interested in predicting further links among researchers in order to either promote further collaborations (suitable co-workers to research tasks would be suggested) or gather information about research groups. Due to form-filling errors there are many missing links among researchers; thus, we are unable to completely state co-working relationships using only the Lattes platform.

To tackle link prediction we firstly have collected information about 1200 researchers and learned a probabilistic ontology [15,17], represented by the PDL crALC, for modeling their research interests. A simplified probabilistic ontology is given by: P (Publication) = 0.3 P (Board) = 0.33 P (sharePublication) = 0.22 P (wasAdvised) = 0.05 P (hasSameInstitution) = 0. In this probabilistic ontology concepts and probabilistic inclusions denote mutual research interests. For instance, a PublicationCollaborator inclusion refers to Researchers who shares a Publication, thus relates two nodes (Researcher) in a collaboration graph. Therefore, the concept Researcher and the role sharePublication are inputs to the algorithm we proposed in Algorithm 1.

To perform inferences and therefore to obtain link predictions, a propositionalization step (a resulting relational Bayesian network) is required.

In addition, a collaboration graph, based on shared publications, was also defined. Statistical information was computed accordingly. Figure 2 depicts collaborations among 303 researchers. Several relationships and clusterings can also be observed.

If we carefully inspect this collaboration graph (Figure 3 shows a subgraph obtained from Figure 2) we could be interested, for instance, in predicting links among researchers from different groups.

Thus, in Figure 3 one could further investigate whether a link between researcher R (red octagon node) and the researcher B (blue polygon node) is suitable. In order to infer this, the probability of a possible link between R and If we took a threshold of 0.60, the link between R and B would not be included.

One could gain more evidence, such as information about nodes that indirectly connect these two groups (Figure 3), denoted by I 1 , I 2 . The inference would be P (PublicationCollaborator(R) |Researcher(R) ∃sharePublication(I 1 ).∃sharePublication(B) ∃sharePublication(I 2 ).∃sharePublication(B)) = 0.65.

Because more information was provided the probability inferred was different. The same threshold now would preserve the link.

Other inferences are possible by considering the suggestion of links between surrounding nodes, i.e. nodes directly linked to the two nodes R and B , denoted by R 1 , . . . , R k , and B 1 , . . . , B n respectively. For each i = 1, ..., k and j = 1, ..., n, calculates P (link(R i , R j )|E) and P (link(B i , B j )|E).

As a rule, if we are interested in discovering whether A and B could be linked, probabilistic inference P (link(A, B)) should be performed.

In a more general framework, graph information could be useful to deal with a large number of link predictions. Note that graph adjacency allow us to address probabilistic inference for promising nodes. In a naive approach, each pair of nodes in the collaboration graph would be evaluated so, multiple probabilistic relational Bayesian inference calls would be required.

On the other hand, if graph-based information is used, such naive scheme could be improved. In our approach, two nodes are probabilistically evaluated if there is a path between them (number of incoming/outgoing edges, number of mutual friends, node distances are also considered). Thus, numerical graphbased information guides the inference process in the relational Bayesian network (linked to the probabilistic ontology). In addition, other candidates sharing any kind of evidence are also evaluated, i.e., interests based features (linked to ontological knowledge) allow us to further explore link prediction.

Alternatively, by completing an overall link predicting task we can devise further functionalities to the resulting collaboration network. The resulting graph can be considered as being a probabilistic network, i.e., probabilities inferred for each link could be denote strenght of the relationship.

Conclusion

We have presented an approach for predicting links that resorts both to graphbased and ontological information. Given a collaborative network, e.g., a social network, we encode interests and graph features through a crALC probabilistic ontology. In order to predict links we resort to probabilistic inference. Preliminary results focused on an academic domain, and we aimed at predicting links among researchers. These preliminary results showed the potential of the idea.

Previous combined approaches for link prediction [3,1] have focused on machine learning algorithms [14]. In such schemes, numerical graph-based features and ontology-based features are computed; then both features are input into a machine learning setting where prediction is performed. Differently from such approaches, in our work we adopt a generic ontology (instead of a hierarchical ontology, expressing only is-a relationships among interests). Therefore, our approach uses more information about the domain to help the prediction.