The first way to represent conte

2021/05/21

The first way to represent context is to consider it as data no different from other data. Figure2.1 uses nodes like Santiago, but to which Santiago does this node refer? scientific management schema paper publication data entrepreneurship graph knowledge study education smart case

gra schema collaborative ontology Each additional piece of information removes ambiguity regarding which city is being referred to, providing (for example) more options for matching the city with its analogue in external sources. Various other forms of emergent schema not directly based on a quotient graph framework have also been proposed; examples include emergent schemata based on relational tables[Pham et al., 2015], and baseed on formal concept analysis[Gonzlez and Hogan, 2018]. We refer to Hernndez et al. if \((v'p,w') \in E'\) then there exists \(w\) such that \((v,p,w) \in E\) and \((w,w') \in R\). In other cases, the identifiers used may not be human-interpretable by design. Both semantic and validating schemata require a domain expert to explicitly specify definitions and constraints. Letting \(D\) be a semi-ring imposes that, for any values \(a, a_1, a_2, a_3\) in \(A\), the following hold: The requirement that it be idempotent further imposes the following: Finally, the requirement that it be commutative imposes the following: Idempotence induces a partial order: \(a_1 \leq a_2\) if and only if \(a_1 \oplus a_2 = a_2\). Query \(Q\) then asks for flights from Santiago to cities with events; this query will check and return an annotation reflecting the temporal validity of each answer. A quotient graph can merge multiple nodes into one node, keeping the edges of its constituent nodes.

Shapes graphs can be depicted as UML-like class diagrams, where Figure3.3 illustrates an example of a shapes graph based on Figure2.1, defining constraints on four interrelated shapes. Such semantics will be discussed later in Chapter4. The persistence of HTTP IRIs can then be improved by using namespaces defined through PURL services. alphabetical thesaurus While using such a naming scheme helps to avoid naming clashes, the use of IRIs does not necessarily help ground the identity of a resource. If we merge the two knowledge graphs, we will end up with two distinct nodes for the same city, and thus not integrate their data. [2018]. Most practical data models for graphs allow for defining nodes that are datatype values. Figure3.4 simulates but is not bisimilar to the data graph. Shapes can inherit the constraints of parent shapes with inheritance denoted with an \(\triangle\) connector as in the case of City and Venue, whose conforming nodes must also conform to the Place shape. We refer to the survey by ebiri et al. The natural way to define meet here is as the intersection of sets of days, where, for example, applying meet on the event annotation \(\color{blue}\{[150,152]\}\) and the flight annotation \(\color{blue}\{[1,120],[220,365]\}\) for Punta Arenas leads to the empty time interval \(\color{blue}\{\}\), which may thus lead to the city being filtered from the results (depending on the query evaluation semantics). Lastly, we define the notion of a valid graph under a given shapes schema and target based on the existence of a shapes map satisfying certain conditions. The semantics of a shape is defined in terms of the evaluation of that shape over each node of a given data graph. Conversely, EID16 does not conform to Event, as it does not have the start and end properties required by the example shapes graph. A similar notion of schema has been proposed by Angles [2018] for property graphs. Typically for the purposes of validating a graph with respect to a shapes schema, a target is defined that requires certain nodes to satisfy certain shapes. In Wikidata, for instance, Santiago de Chile is identified as wd:Q2887, where such a scheme has the advantage of providing better persistence and of not being biased to a particular human language. However, if we were to define a shapes target \(T\) to ensure that the Event shape targets EID15 and EID16 i.e., to define \(T\) such that \(\{ (\)EID15, Event\(), (\)EID16, Event\() \} \subseteq T\) then the graph would no longer be valid under \(\Sigma\) and \(T\) since EID16 does not satisfy Event. To derive these answers, we require a conjunction of annotations on compatible flight and city edges, using the meet operator to compute the annotation for which both edges hold. \(\sigma(\)EID15, Event\() = 1\), \(\sigma(\)Santa Luca, Venue\() = 1\), \(\sigma(\)Santa Luca, Place\() = 1\), etc., but where \(\sigma(\)EID16, Event\() = 0\) (as it does not have the required values for start and end), etc., then we see that \(G\) is valid under \(\Sigma\) and \(T\). [2012]. A standard way to define a validating schema for graphs is using shapes[Knublauch and Kontokostas, 2017, Prud'hommeaux et al., 2014, Labra Gayo et al., 2018]. Returning to Figure2.1, for example, we may wish to ensure that all events have at least a name, a venue, a start date, and an end date, such that applications using the data e.g., one that sends event notifications to users can ensure that they have the minimal information required. Using this property, we could state the edge chile:Santiagoowl:sameAsgeo:SantiagoDeChile in our RDF graph, thus establishing an identity link between the corresponding nodes in both graphs. RDF utilises XML Schema Datatypes (XSD)[Peterson et al., 2012], amongst others, where a datatype node is given as a pair \((l,d)\) where \(l\) is a lexical string, such as "2020-03-29T20:00:00", and \(d\) is an IRI denoting the datatype, such as xsd:dateTime. When graphs are used to represent diverse, incomplete data at large scale, the OWA is the most appropriate choice for a default semantics. Considering our running example, it would be unreasonable to assume that the tourism organisation has complete knowledge of everything describable in its knowledge graph, and hence adopting the OWA appears more appropriate. Different quotient graphs may provide different guarantees with respect to the structure they preserve. We may further combine contexts, such as to indicate that Arica is a Chilean city (geographic) since 1883 (temporal) per the Treaty of Ancn (provenance). This definition can then instantiate specific domains of context. Other forms of annotation are domain-independent; for example, Annotated RDF[Dividino et al., 2009, Udrea et al., 2010, Zimmermann et al., 2012] allows for representing context modelled as semi-rings: algebraic structures consisting of domain values (e.g., temporal intervals, fuzzy values, etc.) Two shapes languages have recently emerged for RDF graphs: Shape Expressions (ShEx), published as a W3C Community Group Report[Prud'hommeaux et al., 2014]; and SHACL (Shapes Constraint Language), published as a W3C Recommendation[Knublauch and Kontokostas, 2017]. 4.1.4 Ifthen vs. if-and-only-if semantics, 5.3.2 Non-recursive graph neural networks, 6.4.3 Mapping from other knowledge graphs, 10.1.5 Other open cross-domain knowledge graphs, 10.1.6 Domain-specific open knowledge graphs, https://www.wikidata.org/wiki/Property:P112, https://www.wikidata.org/prop/direct/P112, node satisfies the Boolean condition \(\mathrm{cond}\), conjunction of shape \(\phi_1\) and shape \(\phi_2\), between \(min\) and \(max\) outward edges (inclusive) with label \(p\), \(\qualifiedL{name}{\datatypeL{string}}{1}{*}\wedge\qualifiedL{start}{\datatypeL{dateTime}}{1}{1}\wedge\qualifiedL{end}{\datatypeL{dateTime}}{1}{1}\), \(\qquad\wedge\qualifiedL{type}{\top}{1}{*}\wedge\xrightarrow{venue}\), \(\qualifiedL{lat}{\datatypeL{float}}{0}{1}\wedge\qualifiedL{long}{\datatypeL{float}}{0}{1}\), \(\semantics{\datatype{N}}{G}{v}{\sigma}\), \(\semantics{\Psi_{\mathrm{cond}}}{G}{v}{\sigma}\), \(\semantics{\phi_1 \wedge \phi_2}{G}{v}{\sigma}\), \(\min\{\semantics{\phi_1}{G}{v}{\sigma}, \semantics{\phi_2}{G}{v}{\sigma}\}\), \(\semantics{\qualified{p}{\phi}{min}{max}}{G}{v}{\sigma}\), \(1\) iff \(min \leq \lvert \{ (v,p,u)\in E \mid \semantics{\phi}{G}{u}{\sigma}=1 \} \rvert \leq max\). The shapes schema from Figure3.3 can be expressed as: For example, Event is a shape label (an element of \(S\)) that maps to a shape (an element of \(\phi\)). However, for Arica, we find two different non-empty intersections: \(\color{blue}\{[123,125]\}\) for EID16 and \(\color{blue}\{[276,279]\}\) for EID17. A framework often used for defining emergent schema is that of quotient graphs, which partition groups of nodes in the data graph according to some equivalence relation while preserving some structural properties of the graph. While we could use the pattern of turning the edge into a node as illustrated in Figure2.3a to directly represent such context, another option is to use reification, which allows for making statements about statements in a generic manner (or in the case of a graph, for defining edges about edges). Returning to Figure2.1, we may consider that the properties city and venue are sub-properties of a more general property location, such that given an edge SantaLucacitySantiago, for example, we may also infer that SantaLucalocationSantiago must hold as an edge in the graph. The semantics we present here assumes that each node in the graph either satisfies or does not satisfy each shape labelled by the schema. Constraints can then be defined on the targeted nodes, such as to restrict the number or types of values taken on a given property, the shapes that such values must satisfy, etc. A compromise between OWA and CWA is the Local Closed World Assumption (LCWA), where portions of the data graph are assumed to be complete. In contrast, \(n\)-ary relations[Cyganiak et al., 2014] (Figure3.8b) connect the source node of the edge directly to the edge node \(e\) with the label of the edge; the target node of the edge is then connected to \(e\) (via value). If HTTP IRIs are used to identify the graphs entities, when the IRI is looked up (via HTTP), the web-server can return (or redirect to) a description of that entity in formats such as RDF. using IRIs for the city, person, and founder of, distinct from the webpages describing them. [Santiago (IRI)][founded by (IRI)] [Pedro de Valdivia (IRI)]. In the context of the Semantic Web, the RDF data model goes one step further and recommends that global Web identifiers be used for nodes and edge labels. While in these examples context is represented in an ad hoc manner, a number of specifications have been proposed to represent context as data in a more standard way. Aside from classes, we may also wish to define the semantics of edge labels, aka properties. Therefore, from the graph of Figure2.1, we cannot assume that there is no flight between Via del Mar and Arica. The semantics of owl:sameAs defined by the OWL standard then allows us to combine the data for both nodes. Other authors rather call to minimise the use of such nodes in graph data[Heath and Bizer, 2011]. Each dimension is associated with a partial order over its values e.g., 2020-03-22 \(\preceq\) 2020-03 \(\preceq\) 2020 enabling the selection and combination of sub-graphs that are valid within contexts at different granularities. The first is to associate the entity with uniquely-identifying information in the graph, such as its geo-coordinates, its postal code, the year it was founded, etc. We formally define shapes following the conventions ofLabra Gayo et al. Operations such as slice-and-dice (selecting knowledge according to given dimensions), as well as roll-up (aggregating knowledge at a higher level) are supported. Taking again the edge SantiagoflightArica, Figure3.9 illustrates three higher-arity representations of temporal context. When declaring shapes, the data modeller may not know in advance the entire set of properties that some nodes can have (now or in the future). We refer the reader to the respective papers for more details[Serafini and Homola, 2012, Schuetz et al., 2021]. If \(\sigma(v,s) = 1\), then node \(v\) is labelled \(s\) (possibly amongst other labels); otherwise if \(\sigma(v,s) = 0\), then node \(v\) is not labelled \(s\). Systems that do not adopt the CWA are said to adopt the Open World Assumption (OWA). For example, the dates for the event EID15 in Figure2.1 can be seen as representing a form of temporal context, indicating the temporal scope within which edges such as EID15venueSantaLuca are held true. Open shapes may also be preferred in such cases rather than enumerating constraints on all possible properties that may be inferred on a node. Another option might be to create a fresh IRI representing the venue, but semantically this becomes indistinguishable from there being a known venue. We use \(e\) to denote an arbitrary identifier representing the edge itself to which the context can be associated. If we consider a shapes map where (e.g.) Another option is to place constraints on the number of nodes conforming to a particular shape that the conforming node can relate to with a property (thus generating edges between shapes); for example, Eventvenue1..*Venue denotes that conforming nodes for Event must relate to at least one node with the property venue that conforms to the Venue shape. While the dates for buses, flights, etc., can be represented directly in the graph, or using reification, writing a query to manually intersect the corresponding temporal contexts will be difficult. [2019]. We can define such constraints in a validating schema and validate the data graph with respect to the resulting schema, listing constraint violations (if any). The shapes target can be defined in many ways, such as targeting all instances of a class, the domain or range of a property, the result of a query, nodes connected to the target of another shape by a given property, etc. Distinguishing the identifiers for the webpage and the city itself avoids naming clashes; for example, if we use the URL to identify both the webpage and the city, we may end up with an edge in our graph, such as (with readable labels below the edge): https://www.wikidata.org/wiki/Q2887https://www.wikidata.org/wiki/Property:P112https://www.wikidata.org/wiki/Q203534 [Santiago (URL)][founded by (URL)] [Pedro de Valdivia (URL)]. Furthermore, there are many ways in which other similar or bisimilar graphs can be defined, depending on the (bi)simulation relation that preserves the data graphs structure[ebiri et al., 2019]. This further enables RDF graphs to link to related entities described in external RDF graphs over the Web, giving rise to Linked Data[Berners-Lee, 2006, Heath and Bizer, 2011] (discussed in Chapter9). In the context of property graphs, Neo4j[Miller, 2013] also defines a set of internal datatypes on property values that includes numbers, strings, Booleans, spatial points, and temporal values. Conformance dependencies may also be recursive, where the conformance of Santiago to City requires that it conforms to Place, which requires that ViadelMar and Arica conform to Place, and so on. Conversely, we may define the range of properties, indicating the class(es) of entities for nodes to which edges with that property extend; for example, we may define that the range of city is a class City, inferring that AricatypeCity. Often the goal will be to compute the most concise quotient graph that satisfies a given condition; for example, the nodes without outgoing edges in Figure3.5 could be merged while preserving bisimilarity. Henceforth, we refer to a data graph as a collection of data represented as nodes and edges using one of the models discussed in Chapter2. One of the benefits of modelling data as graphs versus, for example, the relational model is the option to forgo or postpone the definition of a schema. Each shape denoted with a box like Place, Event, etc. Labels can be complemented with aliases (e.g., wd:Q2887skos:altLabelSantiagodeChile) or comments (e.g. If no if Bob does not conform to Barber then Bob satisfies the Barber constraint by shaving such a node. If yes if Bob conforms to Barber then Bob violates the constraint by not shaving at least one node conforming to Person and (not Barber). Given a shape and a targeted node, it is possible to check if the node conforms to that shape or not, which may require checking conformance of other nodes; for example, the node EID15 conforms to the Event shape not only based on its local properties, but also based on conformance of SantaLuca to Venue and Santiago to City. To illustrate, consider the following case inspired by the barber paradox[Labra Gayo et al., 2018], involving a shape Barber whose conforming nodes shave at least one node conforming to Person and (not Barber). Another option is to change a relation represented as an edge, such as SantiagoflightArica, into a node, such as seen in Figure2.3a, allowing us to assign additional context to the relation. \((a_1 \oplus a_2) \oplus a_3 = a_1 \oplus (a_2 \oplus a_3)\), \((\bot \oplus a) = (a \oplus \bot) = a\), \((a_1 \otimes a_2) \otimes a_3 = a_1 \otimes (a_2 \otimes a_3)\), \((\top \otimes a) = (a \otimes \top) = a\), \(a_1 \otimes (a_2 \oplus a_3) = (a_1 \otimes a_2) \oplus (a_1 \otimes a_3)\), \((a_1 \oplus a_2) \otimes a_3 = (a_1 \otimes a_3) \oplus (a_2 \otimes a_3)\), \((\bot \otimes a) = (a \otimes \bot) = \bot\), \((a_1 \otimes a_2) = (a_2 \otimes a_1)\). We may subsequently wish to capture some relations between some of these classes. If we first apply inferencing with respect to the class hierarchy of the semantic schema, the Event shape would now target EID15 and EID16. But in some scenarios, we may wish to guarantee that our data graph or specific parts thereof are in some sense complete. Bisimulation (\(\approx\)) is then an equivalence relation on graphs. Imposing these conditions on the annotation domain allow for reasoning and querying to be conducted over the annotation domain in a well-defined manner. There are many ways in which quotient graphs may be defined, depending not only on how nodes are partitioned, but also how the edges are defined. A recently proposed language that can be used as a common basis for both ShEx and SHACL reveals their similarities and differences[Labra Gayo et al., 2019]. Merging the nodes of each partition into one node while preserving edges leads to the quotient graph shown in Figure3.4: the nodes of this quotient graph are the partitions of nodes from the data graph and an edge \(X\)\(y\)\(Z\) is included the quotient graph if and only if there exists \(x \in X\) and \(z \in Z\) such that \(x\)\(y\)\(z\) is in the original data graph. These definitions can then be embedded into a data graph. With respect to geographic context, the graph describes events in Chile. A notable example is that of contextual knowledge repositories[Serafini and Homola, 2012], which allow for assigning individual (sub-)graphs to their own context. Other frameworks have been proposed for modelling and reasoning about context in graphs. We now discuss various representations by which context can be made explicit at different levels. can be described in RDF graphs in an interoperable manner. Bachelet, while Fuzzy RDF[Straccia, 2009] allows for annotating edges with a degree of truth such as Santiagoclimate0.8Semi-Arid, indicating that it is more-or-less true with a degree of \(0.8\) that Santiago has a semi-arid climate. Other forms of context may also be used. A shapes graph is formed from a set of interrelated shapes. With respect to temporal context, Santiago has existed as a city since 1541, flights from Arica to Santiago began in 1956, etc. RDF reification[Cyganiak et al., 2014] (Figure3.8a) defines a new node \(e\) to represent the edge and connects it to the source node (via subject), target node (via object), and edge label (via predicate) of the edge. Hence some graph models permit the use of existential nodes, represented here as a blank circle: chile:EID42chile:venuechile:venuechile:EID43. We can merge two graphs by taking their union. Other forms of reification have been proposed in the literature, including, for example, NdFluents[Gimnez-Garca et al., 2017]. Bachelet, to state that it was valid from 2006 until 2010 and valid from 2014 until 2018, we cannot pair the values, but may rather have to create a node to represent different presidencies (in the other models, we could have used two named graphs or edge ids). Inside each shape box are placed constraints on the number (e.g., [1..*] denotes one-to-many, [1..1] denotes precisely one, etc.) Figure3.5 illustrates a bisimilar version of the quotient graph, splitting the event partition into two nodes reflecting their different outgoing edges. A second option is to use identity links to state that a local entity has the same identity as another coreferent entity found in an external source; an instantiation of this concept can be found in the OWL standard, which defines the owl:sameAs property relating coreferent entities. In order to describe the structure of the graph, we could consider six partitions of nodes: event, name, venue, class, date-time, city.

Hence, for example, in the RDF representation of the Wikidata[Vrandei and Krtzsch, 2014] a knowledge graph proposed to complement Wikipedia, discussed in more detail in Chapter10 while the URL https://www.wikidata.org/wiki/Q2887 refers to a webpage that can be loaded in a browser providing human-readable metadata about Santiago, the IRI http://www.wikidata.org/entity/Q2887 refers to the city itself. Figure3.3 illustrates a shapes graph of this form. Many (arguably all) facts presented in the data graph of Figure2.1 can be considered true with respect to a certain context. Taking the graph \(G\) from Figure2.1 and the shapes schema \(\Sigma\) from Figure3.3, first assume an empty shapes target \(T = \{\}\). Formally, we can say that every quotient graph simulates its input graph (based on the simulation relation of set membership between data nodes and quotient nodes), meaning that for all \(x \in X\) with \(x\) an input node and \(X\) a quotient node, if \(x\)\(y\)\(z\) is an edge in the data graph, then there must exist an edge \(X\)\(y\)\(Z\) in the quotient graph such that \(z \in Z\); for example, the quotient graph of Figure3.4 simulates the data graph of Figure2.1.