recall) which corresponds to the “false negative rate” and by its specificity which corresponds to the “false positive
rate” that are two independent measures. Both sensitivity and specificity are generated from a Beta distribution: the
parameters for specificity are “the prior false positive count” and “the prior true negative count” and the parameters
for sensitivity are “the prior true positive count” and “the prior false negative count”. The prior truth probability is
also modeled by a Beta distribution with the parameters “the prior true count” and “the prior false count” for each
distinct (entity, value) pair. The truth value is generated by a Bernoulli distribution with a parameter $\theta$, corresponding
to the prior probability that the value is true. Finally, the truth and the quality of sources are inferred by a Collapsed
Gibbs Sampling.

**Record Fusion** [55] merges knowledge by relying on integrity constraints, quantitative statistics, and provenance information if this latter is available. To find the true value of each table cell, they use one classifier per column (attribute) present in the table (dataset). These softmax classifiers can be modular, for example a logistic regression, a decision tree, a neural network, and so on. Three representation models are explored to create the feature vector, which will then be provided to the classifier. The first representation is the role of a cell at the column level, where three different strategies are proposed: the first acts on the format of the data (e.g., the letters of the alphabet are replaced by the token “A”, numbers by “N” and characters (e.g., “”, “:”, “”, or space) by “S” and they get a vector from a n-gram model), the second strategy is to cluster attribute values, then the last strategy consider a matrix of embeddings which will map all the values of a column (attribute) in a Euclidean space and for each cell, computes the distance between its position in space and the average position of the other values. The second concerns the role of the cell at the row level (tuple), i.e., it captures the relationship of the attribute with other attributes in its row (entity). Two signals are leveraged, the first one includes the counts of pairs of attributes that are seen together, and the second one captures how often a cell occurs among rows within its own entity. Finally, the third concerns the role of the cell in relation to the complete dataset (table), i.e., takes into consideration the number of denial constraint violations, includes the source information only if available since entities can have different provenance and each source can have different levels of trust. Then, the last step consists in training the different classifiers by a stage-wise additive model for a number of *T* iterations: (1) they learn the softmax classifiers with the original dataset, (2) use previous predictions to construct a new dynamic feature, (3) and learn again the classifiers using these new sets of features. For cells where the label is unknown, they assign a majority vote as their weak labels. They obtain good performance on datasets Flight, Stock, Weather, and Address about 94%-98% of precision.

In contrast to many fusion truth inferring approaches, FaitCrowd [92] measures the quality of data sources over
several degrees, with one quality degree for each knowledge topic for a crowdsourcing case. FaitCrowd represents
the expertise of each source for each topic by a Gaussian distribution. Then, it models the true value provided by
a source for a certain question on a given topic as a logistic function depending on the contribution ratio of the
source on the topic, the expertise of the source and a bias. To estimate the parameters, the model uses the Gibbs-EM
inference method that alternates between Gibbs sampling and gradient descent.

**TKGC** [63] takes advantage of prior knowledge from the KG when feeding it and considers that the noise affect-
ing the truth is represented by a probability distribution determined by the data source. To estimate the difference
between the truth value and the value supplied by a source, the authors use a difference function adapted to each
data type, i.e., categorical data, numerical or datetime value, and string. This difference function takes as input the
representation vectors previously learned in a fact scoring setting for KG completion. In fact, the probability of this
latter function follows a Gaussian distribution $N(0, (k_a\sigma_s^2))$ where $k_a$ is a regularization factor and $\sigma_s$ representing
the noise of a data source. Then the truth inference is performed through a semi-supervised algorithm.

**OKELE** [14] models the probability of a fact being true by a latent random variable following a Beta($\beta_0, \beta_1$)
distribution, with $\beta_1$ corresponding to prior true count of the fact and $\beta_0$ corresponding to its prior false count. The
quality of a data source is represented by its error variance $\omega_s$ that follows a scaled inverse chi-squared distribution
Scale-inv-$\chi^2(\nu_s, \tau_s^2)$ representing the number of facts provided by the source $\nu_s$ with variance $\tau_s^2$. The authors argue
that this distribution handles the effect of dataset size in the case of long-tail entities. The truth inference is performed
by leveraging prior knowledge from existing KGs.

**HYBRID** [82] tackles the TI task about knowledge on tail verticals and experiment the fusion by considering
two assumptions: single-truth and multi-truth. Before applying data fusion, they collect “evidences” on entities by
looking for whether a source contains the subject and object of the original triple. To do this, they use three types of
sources: knowledge bases (Freebase and Knowledge Vault), the Web, and query logs. The provenance information
such as the URL where the system found the evidence and the pattern are retained. Once the evidence retrieval is complete, HYBRID leverages the number of truths for each type of data items as a prior probability (for example, a mobile phone has only one year of creation, or we could consider between two and height buttons). Therefore, when a single true value or multiple true values are expected, it applies a single-truth model or a multi-truth model respectively. To assess the quality of data sources, two metrics are used: *precision*, i.e., the probability when a source provides a value, a truth exists and *recall*, i.e., when a truth exists, the source provides a value.

**CATD** [83] addresses the problem of fusion in a context where data sources provide a few claims and where estimating their quality is difficult due to the lack of data. Quality of sources are estimated by a Gaussian distribution whose mean represented the bias of the source, i.e., its intentional behavior to provide false information and variance represents the reliability degree of the source. To cope with the problem of a small amount of data available from a source, they consider a confidence interval of the variance to represent their reliability. Finally, CATD applies an optimization algorithm by initializing the true values with a simple method (e.g., a median of the values) and starts by estimating the quality of the sources, which depends on their claimed values, then estimates the true values.

**KDEm** [140] replaces the concept of true value with the concept of trustworthy opinions on the value of an entity. This model allows several true values for an entity's attribute, and consequently considers a form of knowledge granularity. KDEm leverages the kernel density estimation with a Gaussian kernel and extends it by adding the weights of the sources to estimate the probability distributions of values for each attribute of an entity. To find true values it combines the density estimation with a threshold and detects outliers that are below this threshold.

# 9. Uncertainty Representation

Handling knowledge uncertainty throughout the data integration process also includes its representation in the KG. Uncertainty can be represented on different value scales such as numeric, alphanumeric, textual, or intervals of values. If these different levels of uncertainty are used for reconciliation, they need to be preserved and represented in the KG as metadata in order to retain a history and could possibly be useful for resolving future conflicts [64]. The inclusion of uncertainty in the KG also enables the selection of knowledge in relation to their confidence and contributes to maintain the quality of the graph [145]. Several works deal with querying UKGs. For example, in [50], Hartig presents tSPARQL that extends RDF model and its query language SPARQL to handle uncertainty. In [24], the authors solve the failing RDF query problem, i.e., when a user obtains an empty answer, that can arise when a user queries the graph with a high confidence threshold. To do this, they use tSPARQL [50] and propose answers obtained by Minimal Failing Subquery, i.e., the minimal subquery contained in the failed main query, and Maximal Succeeding Subquery, i.e., the maximal subquery that succeeded under the confidence threshold provided by the user. In [98], the authors propose a reasoner called URDF that solves data uncertainty for SPARQL queries. Another work tackles the task of UKG querying using UKG embeddings [43]. Therefore, it is important to choose the best knowledge representation when building a KG according to few criteria described in Section 9.2. For instance, several sources may provide the same data, hence the model must also be able to include all provenance information (i.e., multiple data sources). For specific applications, some information to assign to triples could be required e.g., provenance information, the uncertainty from extraction algorithms or other, or spatial and temporal information [16, 26, 49, 104]. Some formalisms of the Semantic Web offer possibilities for representing this uncertainty through metadata. Metadata is data about data defined within the RDF model and is important to estimate the validity of the information [26]. We present the uncertainty representation at the ontology level in Section 9.1 and at the data model level in Section 9.2.

## 9.1. *Uncertainty representation at ontology level*

An ontology entails three notions, namely conceptualization, explicit and formal specification, and sharing [45, 46]. The conceptualization is an abstract view of a domain, including the relevant concepts and entities, and relates them together. In this way, ontologies make domain knowledge understandable by the machine and enable reasoning about knowledge by defining rules, constraints, and the domain and range of relations [156]. [1] provides a table that summarizes the usual components that form an ontology (refer to this table for further details). OWL (Ontology
such as the URL where the system found the evidence and the pattern are retained. Once the evidence retrieval is complete, HYBRID leverages the number of truths for each type of data items as a prior probability (for example, a mobile phone has only one year of creation, or we could consider between two and height buttons). Therefore, when a single true value or multiple true values are expected, it applies a single-truth model or a multi-truth model respectively. To assess the quality of data sources, two metrics are used: *precision*, i.e., the probability when a source provides a value, a truth exists and *recall*, i.e., when a truth exists, the source provides a value.

**CATD** [83] addresses the problem of fusion in a context where data sources provide a few claims and where estimating their quality is difficult due to the lack of data. Quality of sources are estimated by a Gaussian distribution whose mean represented the bias of the source, i.e., its intentional behavior to provide false information and variance represents the reliability degree of the source. To cope with the problem of a small amount of data available from a source, they consider a confidence interval of the variance to represent their reliability. Finally, CATD applies an optimization algorithm by initializing the true values with a simple method (e.g., a median of the values) and starts by estimating the quality of the sources, which depends on their claimed values, then estimates the true values.

**KDEm** [140] replaces the concept of true value with the concept of trustworthy opinions on the value of an entity. This model allows several true values for an entity's attribute, and consequently considers a form of knowledge granularity. KDEm leverages the kernel density estimation with a Gaussian kernel and extends it by adding the weights of the sources to estimate the probability distributions of values for each attribute of an entity. To find true values it combines the density estimation with a threshold and detects outliers that are below this threshold.

# 9. Uncertainty Representation

Handling knowledge uncertainty throughout the data integration process also includes its representation in the KG. Uncertainty can be represented on different value scales such as numeric, alphanumeric, textual, or intervals of values. If these different levels of uncertainty are used for reconciliation, they need to be preserved and represented in the KG as metadata in order to retain a history and could possibly be useful for resolving future conflicts [64]. The inclusion of uncertainty in the KG also enables the selection of knowledge in relation to their confidence and contributes to maintain the quality of the graph [145]. Several works deal with querying UKGs. For example, in [50], Hartig presents tSPARQL that extends RDF model and its query language SPARQL to handle uncertainty. In [24], the authors solve the failing RDF query problem, i.e., when a user obtains an empty answer, that can arise when a user queries the graph with a high confidence threshold. To do this, they use tSPARQL [50] and propose answers obtained by Minimal Failing Subquery, i.e., the minimal subquery contained in the failed main query, and Maximal Succeeding Subquery, i.e., the maximal subquery that succeeded under the confidence threshold provided by the user. In [98], the authors propose a reasoner called URDF that solves data uncertainty for SPARQL queries. Another work tackles the task of UKG querying using UKG embeddings [43]. Therefore, it is important to choose the best knowledge representation when building a KG according to few criteria described in Section 9.2. For instance, several sources may provide the same data, hence the model must also be able to include all provenance information (i.e., multiple data sources). For specific applications, some information to assign to triples could be required e.g., provenance information, the uncertainty from extraction algorithms or other, or spatial and temporal information [17, 26, 51, 104, 110]. Some formalisms of the Semantic Web offer possibilities for representing this uncertainty through metadata. Metadata is data about data defined within the RDF model and is important to estimate the validity of the information [26]. We present the uncertainty representation at the ontology level in Section 9.1 and at the data model level in Section 9.2.

## 9.1. *Uncertainty representation at ontology level*

An ontology entails three notions, namely conceptualization, explicit and formal specification, and sharing [45, 46]. The conceptualization is an abstract view of a domain, including the relevant concepts and entities, and relates them together. In this way, ontologies make domain knowledge understandable by the machine and enable reasoning about knowledge by defining rules, constraints, and the domain and range of relations [156]. [1] provides a table that summarizes the usual components that form an ontology (refer to this table for further details). OWL (Ontology
Web Language) enables to describe an ontology of a knowledge domain through individuals, classes or concepts, and properties. It is based on description logic and is part of the W3C's recommendations. Despite its ability to define rich ontologies, OWL cannot natively represent and reason about uncertainty since this latter is based on crisp logic, i.e., a statement is true or false contrary to fuzzy logic for example [25]. Thus, in [23] the authors argue that the lack of ways for handling uncertain information affects the requirements of the Semantic Web. To model and handle uncertainty in an ontology by including their uncertainty theory, most approaches extend the OWL ontology [97]. These uncertainty theories include the probability theory e.g., Bayesian Network (BN), Fuzzy logic, Belief functions e.g., Dempster-Shafer (DS) theory.

**Fuzzy-OWL** [127] extends OWL with fuzzy sets theory for covering vague knowledge.

**OntoBayes** [156] integrates BNs into OWL in order to preserve the advantages of both. Three OWL classes are introduced: *PriorProb*, *CondProb*, and *FullProbDist* of type *ProbValue* (value between 0 and 1) to manage probabilities.

**PR-OWL** [23] aims to provide a probabilistic extension of OWL since the probability theory can represent uncertainty by combining Bayesian probability theory with First Order Logic. In addition to OWL, the ontology includes the statistical regularities that characterize the knowledge domain, the knowledge that is incomplete, inconclusive, ambiguous, unreliable and dissonant, then the uncertainty associated with this knowledge. It has the ability to perform probabilistic reasoning with incomplete or uncertain information conveyed through an ontology but requires RDF Reification (presented in Section 9.2) since a probabilistic model includes more than on individual (N-ary relations).

**BayesOWL** [25] completes OWL for representing and reasoning with uncertainty based on Bayesian Networks. The BayesOWL model includes a set of structural translation rules to convert an OWL ontology into a directed acyclic graph of a Bayesian Network. It provides the encoding of two types of probability, namely priors and pair-wise conditionals through two defined OWL classes *PriorProb* and *CondProb*. Thus, a prior probability for a concept is defined as an instance of class *PriorProb* with two properties named *hasVariable* and *hasProbValue*. Then a conditional probability is represented through an instance of class *CondProb* with the same properties as the above instance and a property *hasCondition*.

**URW3-XG** [78] provides an ontology as a starting point to be refined. A sentence about the world is asserted by an agent. The uncertainty of a sentence has a relation *hasUncertainty* with a *derivationType*, *uncertaintyType*, *UncertaintyModel*, and a *nature*. *UncertaintyType* includes the ambiguity, empirical uncertainty, randomness, vagueness, inconsistency and incompleteness. *UncertaintyModel* includes probability, fuzzy logic, belief functions, rough sets, and other mathematical models for reasoning under uncertainty. And *UncertaintyNature* is whether aleatoric, i.e., ontic or epistemic.

**Poss-OWL 2** [3] provides an extension of OWL 2 to represent incomplete and uncertain knowledge with a probabilistic viewpoint. The ontology has three main classes: *concept*, *role*, and *axiom*. *Concept* is the equivalent of concept constructor of OWL 2 with an added degree that stands for the certainty level of the concept. *Role* represents the properties of objects and data. Then, *Axiom* corresponds to the probabilistic axioms (PossTBoxAxiom and PossABoxAxiom) where each axiom is associated to a real value representing the certainty level of the axiom. The main issue of Poss-OWL 2 is that it only focus on the description of uncertainty at class-level.

**Riali et al.** [119] proposes a probabilistic extension of fuzzy ontologies in order to model vague, imprecise, probabilistic knowledge since fuzzy OWL only models vagueness. Riali et al. also provide a comparison of different approaches for modeling uncertainty in an ontology such as PODM [58], HyProb-Ontology [101], etc.

**mUnc** [29] aims to unify the different uncertainty theories within a single ontology. The ontology includes the following theories: probability, evidence of Dempster-Shafer, and possibility theory. mUnc allows publishing uncertainty theories alongside their features and computation methods. Each uncertainty theory is linked to a set of features and operators. The features correspond to the metrics on which uncertainty theory is based to indicate the degree of truth, credibility, or the likelihood of a sentence.

## 9.2. *Uncertainty representation at data model level*

The basic RDF model cannot natively inject values directly into the edges. On the other hand, there are ways and other graph representations to circumvent this limitation. To detail these graph representations, we consider that we
Web Language) enables to describe an ontology of a knowledge domain through individuals, classes or concepts, and properties. It is based on description logic and is part of the W3C's recommendations. Despite its ability to define rich ontologies, OWL cannot natively represent and reason about uncertainty since this latter is based on crisp logic, i.e., a statement is true or false contrary to fuzzy logic for example [25]. Thus, in [23] the authors argue that the lack of ways for handling uncertain information affects the requirements of the Semantic Web. To model and handle uncertainty in an ontology by including their uncertainty theory, most approaches extend the OWL ontology [97]. These uncertainty theories include the probability theory e.g., Bayesian Network (BN), Fuzzy logic, Belief functions e.g., Dempster-Shafer (DS) theory.

**Fuzzy-OWL** [127] extends OWL with fuzzy sets theory for covering vague knowledge.

**OntoBayes** [156] integrates BNs into OWL in order to preserve the advantages of both. Three OWL classes are introduced: *PriorProb*, *CondProb*, and *FullProbDist* of type *ProbValue* (value between 0 and 1) to manage probabilities.

**PR-OWL** [23] aims to provide a probabilistic extension of OWL since the probability theory can represent uncertainty by combining Bayesian probability theory with First Order Logic. In addition to OWL, the ontology includes the statistical regularities that characterize the knowledge domain, the knowledge that is incomplete, inconclusive, ambiguous, unreliable and dissonant, then the uncertainty associated with this knowledge. It has the ability to perform probabilistic reasoning with incomplete or uncertain information conveyed through an ontology but requires RDF Reification (presented in Section 9.2) since a probabilistic model includes more than on individual (N-ary relations).

**BayesOWL** [25] completes OWL for representing and reasoning with uncertainty based on Bayesian Networks. The BayesOWL model includes a set of structural translation rules to convert an OWL ontology into a directed acyclic graph of a Bayesian Network. It provides the encoding of two types of probability, namely priors and pair-wise conditionals through two defined OWL classes *PriorProb* and *CondProb*. Thus, a prior probability for a concept is defined as an instance of class *PriorProb* with two properties named *hasVariable* and *hasProbValue*. Then a conditional probability is represented through an instance of class *CondProb* with the same properties as the above instance and a property *hasCondition*.

**URW3-XG** [78] provides an ontology as a starting point to be refined. A sentence about the world is asserted by an agent. The uncertainty of a sentence has a relation *hasUncertainty* with a *derivationType*, *uncertaintyType*, *UncertaintyModel*, and a *nature*. *UncertaintyType* includes the ambiguity, empirical uncertainty, randomness, vagueness, inconsistency and incompleteness. *UncertaintyModel* includes probability, fuzzy logic, belief functions, rough sets, and other mathematical models for reasoning under uncertainty. And *UncertaintyNature* is whether aleatoric, i.e., ontic or epistemic.

**Poss-OWL 2** [3] provides an extension of OWL 2 to represent incomplete and uncertain knowledge with a probabilistic viewpoint. The ontology has three main classes: *concept*, *role*, and *axiom*. *Concept* is the equivalent of concept constructor of OWL 2 with an added degree that stands for the certainty level of the concept. *Role* represents the properties of objects and data. Then, *Axiom* corresponds to the probabilistic axioms (PossTBoxAxiom and PossABoxAxiom) where each axiom is associated to a real value representing the certainty level of the axiom. The main issue of Poss-OWL 2 is that it only focus on the description of uncertainty at class-level.

**Riali et al.** [119] proposes a probabilistic extension of fuzzy ontologies in order to model vague, imprecise, probabilistic knowledge since fuzzy OWL only models vagueness. Riali et al. also provide a comparison of different approaches for modeling uncertainty in an ontology such as PODM [58], HyProb-Ontology [101], etc.

**mUnc** [29] aims to unify the different uncertainty theories within a single ontology. The ontology includes the following theories: probability, evidence of Dempster-Shafer, and possibility theory. mUnc allows publishing uncertainty theories alongside their features and computation methods. Each uncertainty theory is linked to a set of features and operators. The features correspond to the metrics on which uncertainty theory is based to indicate the degree of truth, credibility, or the likelihood of a sentence.

## 9.2. *Uncertainty representation at data model level*

The basic RDF model cannot natively inject values directly into the edges. On the other hand, there are ways and other graph representations to circumvent this limitation. To detail these graph representations, we consider that we
want to represent the uncertainty through a confidence score $s \in [0, 1]$, with 0 representing low confidence and 1 representing high confidence. In [2], the authors use 10 criteria to compare five data representation models: RDF, RDF*, Named Graph, Property Graph and their model Multilayer Graph. Among these 10 criteria, we consider that two main criteria are required to represent the confidence score and the provenance of a RDF triple. The first criterion is edge annotation and refers to the ability of the representation model to assign attribute-value pairs to an edge. The second one is edge as nodes, meaning that an edge can be referenced as multiple nodes. Therefore, we review the data representation models w.r.t. these two criteria and their pros and cons. We illustrate the models with the triple <JoeBiden, isPresident, UnitedStates> associated with the confidence score "0.911" in Figure 11.

<figure><img src="image_14.png" /><figcaption>Fig. 11. Illustration of different ways to embed the uncertainty (captured by a confidence score) of a triple in the KG representation: (a) RDF, (b) Singleton Property, (c) Property Graph, (d) Named Graph, (e) N-ary, (f) RDF Reification, (g) RDF-star (RDF*), (h) Multilayer Graph. ⚪ in RDF-star illustration depicts the triple <BarackObama, isPresident, UnitedStates>.</figcaption></figure>

**Singleton Property** [105] uses a new type of property called “singleton property”, which corresponds to a unique property linked to an URI between two entities. This unique property can be used as a node to which additional
want to represent the uncertainty through a confidence score $s \in [0, 1]$, with 0 representing low confidence and 1 representing high confidence. In [2], the authors use 10 criteria to compare five data representation models: RDF, RDF*, Named Graph, Property Graph and their model Multilayer Graph. Among these 10 criteria, we consider that two main criteria are required to represent the confidence score and the provenance of a RDF triple. The first criterion is edge annotation and refers to the ability of the representation model to assign attribute-value pairs to an edge. The second one is edge as nodes, meaning that an edge can be referenced as multiple nodes. Therefore, we review the data representation models w.r.t. these two criteria and their pros and cons. We illustrate the models with the triple <JoeBiden, isPresident, UnitedStates> associated with the confidence score "0.911" in Figure 11.

<figure><img src="image_13.png" /><figcaption>Fig. 11. Illustration of different ways to embed the uncertainty (captured by a confidence score) of a triple in the KG representation: (a) RDF, (b) Singleton Property, (c) Property Graph, (d) Named Graph, (e) N-ary, (f) RDF Reification, (g) RDF-star (RDF*), (h) Multilayer Graph. ⚪ in RDF-star illustration depicts the triple <BarackObama, isPresident, UnitedStates>.</figcaption></figure>

**Singleton Property** [105] uses a new type of property called “singleton property”, which corresponds to a unique property linked to an URI between two entities. This unique property can be used as a node to which additional
relations can be added. For example, the singleton property in Figure 11(b) is “isPresident#1”, then this node is used to add the confidence value “0.911”. Despite the fact that the singleton property is convenient for a compact meta-level representation, this modeling introduces many unique predicates and affects data querying [52, 120].

**Property Graph** puts additional information about triples that are stored as a list of key/value pairs at edges in the graph. For example, in Figure 11(c), the confidence value and the provenance are attached to the relation “:isPresident”.

**Named Graph** [56] extends the RDF triple model and allows to indicate a triple as a subgraph denoted by an IRI. This subgraph with an identifier can be used to add meta-information. In Figure 11(d), the original RDF triple `<JoeBiden, :isPresident, UnitedStates>` is identified by “:statementId#1”, which is used as the subject in the triple `<:statementId#1, :confidenceValue, 0.911>`. This modeling, which corresponds to nested graphs, is well-supported in the SPARQL standard and well-suited for representing provenance data [120].

**N-ary** [93] creates a node to represent a relation concept whose triples linked to this node correspond to the arguments of the relation. For example, in Figure 11(e), an intermediate node “President1” of type “President” characterizes the relation “:isPresident”, then meta-data can annotate the relation. The main drawback of this representation is its cumbersome syntax, which increases the complexity of the KG since the n-ary relation must be divided into several binary relations.

**RDF reification** [53] consists in creating an Internationalized Resource Identifier (IRI) or blank node that plays the role of the subject of all triples, as depicted in Figure 11(f). To represent the former triple it uses three new relations namely *rdf:subject*, *rdf:predicate*, and *rdf:object* then as many relations as it needs to add metadata. This modeling was the first way to make statements about statements [120]. This method is simple, but its syntax is too verbose since each statement must be reified, which considerably increases the size of the KG, makes queries and RDF data exchange more complex [2, 51, 105, 120].

**RDF-star** [52] is an extension of the RDF model proposed by the Semantic Web community. A RDF triple is a tuple $t^* \in (T^* \times E) \times R \times (T^* \times E \times L) \in T^*$ and RDF-star triple is a tuple $t^* \in (T^* \times E) \times R \times (T^* \times E \times L) \in T^*$ where E is the set of entities, R is the set of relations, L the set of literals, and $T^*$ is the set of RDF-star triples. RDF-star can extend an existing RDF model expressively, since the metadata of triples are simply added as objects of them [71]. For example, in Figure 11(g), metadata such as provenance or confidence scores are added directly to the triple `<BarackObama, :isPresident, UnitedStates>` illustrated by ⊑. In addition to the ability to add metadata at the statement level without modifying the remaining data, RDF-star has its own query language called SPARQL-star which reduces compatibility issues [75]. A comparison on Wikidata have shown that RDF-star performs better than reification, n-ary, and named graph representations in terms of the number of triples, loading time, and storage capacity [71]. However, RDF-star cannot consistently represent the same metadata with different values for the same triple [2, 51]. Indeed, in Figure 11(g), there are different start dates that refer to the same triple.

**Multilayer Graph** [2] unifies the various advantages of the other representations we have described so far into a single, simple, and flexible (whether at node or statement level) model by introducing the notion of “layer”. To explain this, we use the notations and definition of a multilayer graph from [2]: given *Obj* a universe of objects that contains strings, numbers, IRIs and so on, a multilayer graph is defined as $G = (O, \gamma)$ where $O \subseteq Obj$ is a set of objects and $\gamma: O \rightarrow O \times O \times O$ is a partial mapping that models directed, labeled and identified edges between objects. The layers in the multilayer graph come from the nested structure of edge ids. The layer of an object $o \in O$, described as layer($o$) is defined as follows: if $o$ is not an edge id, then layer($o$) = 0; otherwise if $\gamma(o) = (n_1, l, n_2)$ then layer($o$) = $\max\{\text{layer}(n_1), \text{layer}(l), \text{layer}(n_2)\} + 1$. Figure 12 depicts the layer representation of (h) from Figure 11(h). This data model enables to unambiguously represent multiple provenances and different confidence scores in a triple.

# 10. Discussion and perspectives

Throughout this survey, we have seen that several approaches exist to represent uncertainty within KGs. This is made possible by the development of ontologies that include multiple theories enabling uncertainty to be manipulated in addition to data models whose flexibility to include metadata about metadata enable additional information to be associated with extracted triples such as confidence scores. However, we can argue that methods for integrating
relations can be added. For example, the singleton property in Figure 11(b) is “isPresident#1”, then this node is used to add the confidence value “0.911”. Despite the fact that the singleton property is convenient for a compact meta-level representation, this modeling introduces many unique predicates and affects data querying [52, 120].

**Property Graph** puts additional information about triples that are stored as a list of key/value pairs at edges in the graph. For example, in Figure 11(c), the confidence value and the provenance are attached to the relation “:isPresident”.

**Named Graph** [56] extends the RDF triple model and allows to indicate a triple as a subgraph denoted by an IRI. This subgraph with an identifier can be used to add meta-information. In Figure 11(d), the original RDF triple <JoeBiden, :isPresident, UnitedStates> is identified by “:statementId#1”, which is used as the subject in the triple <:statementId#1, :confidenceValue, 0.911>. This modeling, which corresponds to nested graphs, is well-supported in the SPARQL standard and well-suited for representing provenance data [120].

**N-ary** [93] creates a node to represent a relation concept whose triples linked to this node correspond to the arguments of the relation. For example, in Figure 11(e), an intermediate node “President1” of type “President” characterizes the relation “:isPresident”, then meta-data can annotate the relation. The main drawback of this representation is its cumbersome syntax, which increases the complexity of the KG since the n-ary relation must be divided into several binary relations.

**RDF reification** [53] consists in creating an Internationalized Resource Identifier (IRI) or blank node that plays the role of the subject of all triples, as depicted in Figure 11(f). To represent the former triple it uses three new relations namely *rdf:subject*, *rdf:predicate*, and *rdf:object* then as many relations as it needs to add metadata. This modeling was the first way to make statements about statements [120]. This method is simple, but its syntax is too verbose since each statement must be reified, which considerably increases the size of the KG, makes queries and RDF data exchange more complex [2, 51, 105, 120].

**RDF-star** [52] is an extension of the RDF model proposed by the Semantic Web community. A RDF triple is a tuple $t^* \in (T^* \times E) \times R \times (T^* \times E \times L) \in T^*$ and RDF-star triple is a tuple $t^* \in (T^* \times E) \times R \times (T^* \times E \times L) \in T^*$ where *E* is the set of entities, *R* is the set of relations, *L* the set of literals, and *T* is the set of RDF-star triples. RDF-star can extend an existing RDF model expressively, since the metadata of triples are simply added as objects of them [71]. For example, in Figure 11(g), metadata such as provenance or confidence scores are added directly to the triple <BarackObama, :isPresident, UnitedStates> illustrated by ⊑. In addition to the ability to add metadata at the statement level without modifying the remaining data, RDF-star has its own query language called SPARQL-star which reduces compatibility issues [75]. A comparison on Wikidata have shown that RDF-star performs better than reification, n-ary, and named graph representations in terms of the number of triples, loading time, and storage capacity [71]. However, RDF-star cannot consistently represent the same metadata with different values for the same triple [2, 51]. Indeed, in Figure 11(g), there are different start dates that refer to the same triple.

**Multilayer Graph** [2] unifies the various advantages of the other representations we have described so far into a single, simple, and flexible (whether at node or statement level) model by introducing the notion of “layer”. To explain this, we use the notations and definition of a multilayer graph from [2]: given *Obj* a universe of objects that contains strings, numbers, IRIs and so on, a multilayer graph is defined as $G = (O, \gamma)$ where $O \subseteq Obj$ is a set of objects and $\gamma: O \rightarrow O \times O \times O$ is a partial mapping that models directed, labeled and identified edges between objects. The layers in the multilayer graph come from the nested structure of edge ids. The layer of an object $o \in O$, described as layer($o$) is defined as follows: if $o$ is not an edge id, then layer($o$) = 0; otherwise if $\gamma(o) = (n_1, l, n_2)$ then layer($o$) = max${layer(n_1)$, layer(l), layer(n_2)} + 1$. Figure 12 depicts the layer representation of (h) from Figure 11(h). This data model enables to unambiguously represent multiple provenances and different confidence scores in a triple.

# 10. Discussion and perspectives

Throughout this survey, we have seen that several approaches exist to represent uncertainty within KGs. This is made possible by the development of ontologies that include multiple theories enabling uncertainty to be manipulated in addition to data models whose flexibility to include metadata about metadata enable additional information to be associated with extracted triples such as confidence scores. However, we can argue that methods for integrating
<figure><img src="image_15.png" /><figcaption>Fig. 12. Illustration of the notion of “layer” where the first layer can be seen as a single triple <BarackObama, isPresident, UnitedStates> while the second layer contains different sets of information about the triple.</figcaption></figure>

knowledge after its extraction, still overlook uncertainty in their modeling, despite the recently developed methods for embedding UKGs to perform link prediction, KG completion, or confidence prediction. Taking into account the provenance information and the different levels of uncertainty operating at different locations in the knowledge integration pipeline, namely in knowledge (i.e., deltas), data sources, and all components of the pipeline (i.e., extraction, alignment, and fusion) would be beneficial to preserve the traceability, strengthens the quality of the KG, and enables graph querying by specifying a confidence level.

For the alignment task, the approaches do not take into account the uncertainty of the knowledge to be aligned and make the assumption that the knowledge are deterministic. On the contrary, many approaches that tackle KG completion tasks take knowledge uncertainty into account in their models. We believe that extending these models to the task of knowledge alignment would be beneficial. For example, using embedding models of uncertain graphs for mapping-based alignment methods where a transformation function between the two embedding spaces of the two KGs to be aligned is learned. Or simply include confidence scores in neighbor aggregation for GNN-based models.

Once the knowledge has been aligned, it needs to be merged. We have seen several methods dealing with different knowledge characteristics such as granularity or numerical values. Knowledge granularity is an essential aspect when building a KG from multiple heterogeneous sources. Indeed, if we leverage several popular data sources such as Wikipedia or Wikidata and one data source specific to the domain the KG is intended to represent, we are likely to face differences in granularity that we need to manage. If we use the simplest fusion approaches, such as majority voting or averaged voting, the graph will not contain the most specific knowledge. We have seen that most fusion methods do not handle this aspect of granularity, and consider that only one true value exists. Only a few methods tackle this aspect by considering a partial order between the values to be fused or a semantic distance for categorical data. We therefore recommend developing this aspect further in the modeling of fusion models, for example by estimating a granularity score for a data source in parallel with its trustworthiness score, depending on the needs of KG builders. One way of solving this problem is to further develop fusion models to capture the correlation between the attributes of the entities and to identify any inconsistencies in one or more of its attributes. Current fusion models incorporate a confidence score that embodies the trustworthiness of data sources to infer truth in knowledge fusion. Nevertheless, the confidence in extraction algorithms and other components is not accounted for. This modeling is not a problem when the same entities can be extracted from multiple sources. However, when we deal with long-tail entities and when few data sources provide knowledge about them (for example, two data sources), if one contradicts the other and their confidence score are close, the fusion model may have difficulty to find the true value while other confidence scores such as in extraction could guide the fusion model. We also advocate for better fusion models, since most proposals are specifically adapted to numerical data (which represents a relative small part of the entire data involved in KGs), only deal with categorical data, or consider both types of data but handle them in the same way. We believe that it would be more advantageous to consider the different data types in different ways, but within a single framework.
<figure><img src="image_14.png" /><figcaption>Fig. 12. Illustration of the notion of “layer” where the first layer can be seen as a single triple <BarackObama, isPresident, UnitedStates> while the second layer contains different sets of information about the triple.</figcaption></figure>

knowledge after its extraction, still overlook uncertainty in their modeling, despite the recently developed methods for embedding UKGs to perform link prediction, KG completion, or confidence prediction. Taking into account the provenance information and the different levels of uncertainty operating at different locations in the knowledge integration pipeline, namely in knowledge (i.e., deltas), data sources, and all components of the pipeline (i.e., extraction, alignment, and fusion) would be beneficial to preserve the traceability, strengthens the quality of the KG, and enables graph querying by specifying a confidence level.

For the alignment task, the approaches do not take into account the uncertainty of the knowledge to be aligned and make the assumption that the knowledge are deterministic. On the contrary, many approaches that tackle KG completion tasks take knowledge uncertainty into account in their models. We believe that extending these models to the task of knowledge alignment would be beneficial. For example, using embedding models of uncertain graphs for mapping-based alignment methods where a transformation function between the two embedding spaces of the two KGs to be aligned is learned. Or simply include confidence scores in neighbor aggregation for GNN-based models.

Once the knowledge has been aligned, it needs to be merged. We have seen several methods dealing with different knowledge characteristics such as granularity or numerical values. Knowledge granularity is an essential aspect when building a KG from multiple heterogeneous sources. Indeed, if we leverage several popular data sources such as Wikipedia or Wikidata and one data source specific to the domain the KG is intended to represent, we are likely to face differences in granularity that we need to manage. If we use the simplest fusion approaches, such as majority voting or averaged voting, the graph will not contain the most specific knowledge. We have seen that most fusion methods do not handle this aspect of granularity, and consider that only one true value exists. Only a few methods tackle this aspect by considering a partial order between the values to be fused or a semantic distance for categorical data. We therefore recommend developing this aspect further in the modeling of fusion models, for example by estimating a granularity score for a data source in parallel with its trustworthiness score, depending on the needs of KG builders. One way of solving this problem is to further develop fusion models to capture the correlation between the attributes of the entities and to identify any inconsistencies in one or more of its attributes. Current fusion models incorporate a confidence score that embodies the trustworthiness of data sources to infer truth in knowledge fusion. Nevertheless, the confidence in extraction algorithms and other components is not accounted for. This modeling is not a problem when the same entities can be extracted from multiple sources. However, when we deal with long-tail entities and when few data sources provide knowledge about them (for example, two data sources), if one contradicts the other and their confidence score are close, the fusion model may have difficulty to find the true value while other confidence scores such as in extraction could guide the fusion model. We also advocate for better fusion models, since most proposals are specifically adapted to numerical data (which represents a relative small part of the entire data involved in KGs), only deal with categorical data, or consider both types of data but handle them in the same way. We believe that it would be more advantageous to consider the different data types in different ways, but within a single framework.
## 11. Conclusion

In our current world, where knowledge may be noisy, contradictory and of different granularity, uncertainty should be taken into account when constructing a KG from multiple and heterogeneous data sources. In fact, since KG construction relies on automatic knowledge extractions, other levels of uncertainty should be accounted for.

In this paper, we proposed a classification of knowledge related uncertainty into two categories: uncertainty leading to contradictions and uncertainty leading to granularity disparities. We then discussed a theoretical pipeline for the refinement of uncertain knowledge to be integrated in KG construction. This pipeline consists of four main tasks: knowledge representation (including uncertainty and provenance in the KG), knowledge alignment, knowledge fusion, and consistency checking. We also discussed challenges and perspectives on the integration of uncertain knowledge into a KG.

In particular, we have pointed out that tasks such as link prediction and KG completion are currently tackled with representational methods (embeddings) that take into account uncertainty. Knowledge alignment is a well-studied topic, with a wide range of models available from rule-based models to deep learning models, for which we provided a brief overview of existing methods. We also revisited knowledge fusion approaches, most of which based on probabilistic models, and estimated both the trustworthiness of data sources and true values. However, knowledge integration remains a challenging topic for future research. While the representation of uncertainty in a KG has received attention over the last few years (both at the ontological level and at the data model), the current knowledge integration approaches addressing both tasks remain limited in their scope (not taking into account all types of uncertainty and of knowledge deltas since they are only concerned with uncertainty).

## References

[1] S.K. Anand and S. Kumar, Uncertainty Analysis in Ontology-Based Knowledge Representation, New Gener. Comput. 40(1) (2022), 339-376.

[2] R. Angles, A. Hogan, O. Lassila, C. Rojas, D. Schwabe, P.A. Szekely and D. Vrgoc, Multilayer graphs: a unified data model for graph databases, in: GRADES-NDA '22: Proceedings of the 5th ACM SIGMOD Joint International Workshop on Graph Data Management Experiences & Systems (GRADES) and Network Data Analytics (NDA), Philadelphia, Pennsylvania, USA, 12 June 2022, ACM, 2022, pp. 11:1-11:6.

[3] S. Bal-Bourai and A. Mokhtari, Poss-OWL 2: Possibilistic Extension of OWL 2 for an Uncertain Geographic Ontology, in: 18th International Conference in Knowledge Based and Intelligent Information and Engineering Systems, KES 2014, Gdynia, Poland, 15-17 September 2014, Procedia Computer Science, Vol. 35, Elsevier, 2014, pp. 407-416.

[4] D. Benslimane, Q.Z. Sheng, M. Barhamgi and H. Prade, The Uncertain Web: Concepts, Challenges, and Current Solutions, ACM Trans. Internet Techn. 16(1) (2016), 1:1-1:6.

[5] V. Beretta, S. Harispe, S. Ranwez and I. Mougenot, How Can Ontologies Give You Clue for Truth-Discovery? An Exploratory Study, in: Proceedings of the 6th International Conference on Web Intelligence, Mining and Semantics, WIMS 2016, Nimes, France, June 13-15, 2016, ACM, 2016, pp. 15:1-15:12.

[6] C. Bizer, J. Lehmann, G. Kobilarov, S. Auer, C. Becker, R. Cyganiak and S. Hellmann, DBpedia - A crystallization point for the Web of Data, J. Web Semant. 7(3) (2009), 154-165.

[7] J. Bleiholder and F. Naumann, Conflict Handling Strategies in an Integrated Information System, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Informatik, 2006. doi:http://dx.doi.org/10.18452/2460.

[8] J. Bleiholder and F. Naumann, Data Fusion, ACM Comput. Surv. 41(1) (2009).

[9] K.D. Bollacker, C. Evans, P.K. Paritosh, T. Sturge and J. Taylor, Freebase: a collaboratively created graph database for structuring human knowledge, in: Proceedings of the ACM SIGMOD International Conference on Management of Data, SIGMOD 2008, Vancouver, BC, Canada, June 10-12, 2008, ACM, 2008, pp. 1247-1250.

[10] A. Bordes, N. Usunier, A. García-Durán, J. Weston and O. Yakhnenko, Translating Embeddings for Modeling Multi-relational Data, in: Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States, 2013, pp. 2787-2795.

[11] K. Boutouhami, J. Zhang, G. Qi and H. Gao, Uncertain Ontology-Aware Knowledge Graph Embeddings, in: Semantic Technology - 9th Joint International Conference, JIST 2019, Hangzhou, China, November 25-27, 2019, Revised Selected Papers, Communications in Computer and Information Science, Vol. 1157, Springer, 2019, pp. 129-136.

[12] F. Brasileiro, J.P.A. Almeida, V.A. de Carvalho and G. Guizzardi, Applying a Multi-Level Modeling Theory to Assess Taxonomic Hierarchies in Wikidata, in: Proceedings of the 25th International Conference on World Wide Web, WWW 2016, Montreal, Canada, April 11-15, 2016, Companion Volume, ACM, 2016, pp. 975-980.
## 11. Conclusion

In our current world, where knowledge may be noisy, contradictory and of different granularity, uncertainty should be taken into account when constructing a KG from multiple and heterogeneous data sources. In fact, since KG construction relies on automatic knowledge extractions, other levels of uncertainty should be accounted for.

In this paper, we proposed a classification of knowledge related uncertainty into two categories: uncertainty leading to contradictions and uncertainty leading to granularity disparities. We then discussed a theoretical pipeline for the refinement of uncertain knowledge to be integrated in KG construction. This pipeline consists of four main tasks: knowledge representation (including uncertainty and provenance in the KG), knowledge alignment, knowledge fusion, and consistency checking. We also discussed challenges and perspectives on the integration of uncertain knowledge into a KG.

In particular, we have pointed out that tasks such as link prediction and KG completion are currently tackled with representational methods (embeddings) that take into account uncertainty. Knowledge alignment is a well-studied topic, with a wide range of models available from rule-based models to deep learning models, for which we provided a brief overview of existing methods. We also revisited knowledge fusion approaches, most of which based on probabilistic models, and estimated both the trustworthiness of data sources and true values. However, knowledge integration remains a challenging topic for future research. While the representation of uncertainty in a KG has received attention over the last few years (both at the ontological level and at the data model), the current knowledge integration approaches addressing both tasks remain limited in their scope (not taking into account all types of uncertainty and of knowledge deltas since they are only concerned with uncertainty).

## References

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[2] R. Angles, A. Hogan, O. Lassila, C. Rojas, D. Schwabe, P.A. Szekely and D. Vrgoc, Multilayer graphs: a unified data model for graph databases, in: GRADES-NDA '22: Proceedings of the 5th ACM SIGMOD Joint International Workshop on Graph Data Management Experiences & Systems (GRADES) and Network Data Analytics (NDA), Philadelphia, Pennsylvania, USA, 12 June 2022, ACM, 2022, pp. 11:1-11:6.

[3] S. Bal-Bourai and A. Mokhtari, Poss-OWL 2: Possibilistic Extension of OWL 2 for an Uncertain Geographic Ontology, in: 18th International Conference in Knowledge Based and Intelligent Information and Engineering Systems, KES 2014, Gdynia, Poland, 15-17 September 2014, Procedia Computer Science, Vol. 35, Elsevier, 2014, pp. 407-416.

[4] D. Benslimane, Q.Z. Sheng, M. Barhamgi and H. Prade, The Uncertain Web: Concepts, Challenges, and Current Solutions, ACM Trans. Internet Techn. 16(1) (2016), 1:1-1:6.

[5] V. Beretta, S. Harispe, S. Ranwez and I. Mougenot, How Can Ontologies Give You Clue for Truth-Discovery? An Exploratory Study, in: Proceedings of the 6th International Conference on Web Intelligence, Mining and Semantics, WIMS 2016, Nimes, France, June 13-15, 2016, ACM, 2016, pp. 15:1-15:12.

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[12] F. Brasileiro, J.P.A. Almeida, V.A. de Carvalho and G. Guizzardi, Applying a Multi-Level Modeling Theory to Assess Taxonomic Hierarchies in Wikidata, in: Proceedings of the 25th International Conference on World Wide Web, WWW 2016, Montreal, Canada, April 11-15, 2016, Companion Volume, ACM, 2016, pp. 975-980.
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[16] A. Carlson, J. Betteridge, B. Kisiel, B. Settles, E.R.H. Jr. and T.M. Mitchell, Toward an Architecture for Never-Ending Language Learning, in: *Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2010, Atlanta, Georgia, USA, July 11-15, 2010*, M. Fox and D. Poole, eds, AAAI Press, 2010, pp. 1306–1313.

[17] J.J. Carroll, C. Bizer, P.J. Hayes and P. Stickler, Named graphs, *J. Web Semant.* 3(4) (2005), 247–267.

[18] M. Chen, Y. Tian, M. Yang and C. Zaniolo, Multilingual Knowledge Graph Embeddings for Cross-lingual Knowledge Alignment, in: *Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19-25, 2017*, C. Sierra, ed., *ijcai.org*, 2017, pp. 1511–1517.

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