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\section{Contextualizing a Relationship}  The dilated form of $x \in R$, denoted $T_x$, provides a knowledge structure that is suited to contextualizing the meaning of an assertion made about the world. For example, consider the asserted triple  \begin{equation}  x = (lanl:marko, foaf:knows, ucla:apepe)  \end{equation}  What is the meaning of \ttt{foaf:knows} in this context? For the human, the meaning is made explicit in the specification document of the FOAF (Friend of a Friend) ontology (\ttt{http://xmlns.com/foaf/spec/}), which states:  \begin{quote}  ``We take a broad view of `knows', but do require some form of reciprocated interaction (i.e.~stalkers need not apply). Since social attitudes and conventions on this topic vary greatly between communities, counties and cultures, it is not appropriate for FOAF to be overly-specific here."  \end{quote}  Unfortunately, the supplementary information that defines the meaning of \ttt{foaf:knows} is not encoded with the URI itself (nor in the greater RDF graph) and it is only through some external medium (the FOAF specification document) that the meaning of \ttt{foaf:knows} is made clear. Thus, such semantics are entirely informal \cite{uschold:sem2001}. However, even if the complexity of the meaning of \ttt{foaf:knows} could be conveyed by an RDF graph, the nuances of ``knowing" are subtle, such that no two people know each other in quite the same way. In fact, only at the most abstract level of analysis is the relationship of ``knowing" the same between any two people. In order for a human or machine to understand the way in which \ttt{lanl:marko} and \ttt{ucla:apepe} know each other, the complexities of this relationship must be stated. In other words, it is important to state ``a constraint on the possible ways the world might be" \cite{rdfsem:hayes2004} even if that constraint is a complex graph of relationships. For example, the meaning of \ttt{foaf:knows} when applied to \ttt{lanl:marko} and \ttt{ucla:apepe} can be more eloquently stated as:  \begin{quote}  ``Marko and Alberto first met at the European Organization for Nuclear Research (CERN) at the Open Archives Initiative Conference (OAI) in 2005. Six months later, Alberto began a summer internship at the Los Alamos National Laboratory (LANL) where he worked under Herbert Van de Sompel on the Digital Library Research and Prototyping Team. Marko, at the time, was also working under Herbert Van de Sompel. Unbeknownst to Herbert, Marko and Alberto analyzed a scholarly data set that Alberto had acquired at the Center for Embedded Networked Sensing (CENS) at the University of California at Los Angeles (UCLA). The results of their analysis ultimately led to the publication of an article \cite{onthe:rodriguez2008} in Leo Egghe's Journal of Informetrics. Marko and Alberto were excited to publish in Leo Egghe's journal after meeting him at the Institute for Pure and Applied Mathematics (IPAM) at UCLA."  \end{quote}  The facts above, when represented as an RDF graph with triples relating such concepts as \texttt{lanl:marko}, \texttt{ucla:apepe}, \texttt{lanl:herbertv}, \texttt{cern:cern}, \texttt{ucla:ipam}, \texttt{elsevier:joi}, \texttt{doi:10.1016/j.joi.2008.04.002}, etc., serve to form the dilated triple $T_x$. In this way, the meaning of the asserted triple (\texttt{lanl:marko}, \texttt{foaf:knows}, \texttt{ucla:apepe}) is presented in the broader context $T_x$. In other words, $T_x$ helps to elucidate the way in which Marko knows Alberto. Figure 2 depicts $T_x$, where the unlabeled resources and relationships represent the URIs from the previous representation. (Note: For the sake of diagram clarity, the supplemented triples are unlabeled in Figure 2. However, please be aware that the unlabeled resources are in fact the URI encoding of the aforementioned natural language example explaining how Marko knows Alberto.)  Even after all the aforementioned facts about Marko and Alberto's ``knowing" relationship are encoded in $T_x$, still more information is required to fully understand what is meant by Marko ``knowing" Alberto. What is the nature of the scholarly data set that they analyzed? Who did what for the analysis? Did they ever socialize outside of work when Alberto was visiting LANL? What was the conversation that they had with Leo Egghe like? Can their \texttt{foaf:knows} relationship ever be fully understood? Only an infinite recursion into their histories, experiences, and subjective worlds could reveal the ``true" meaning of (\texttt{lanl:marko}, \texttt{foaf:knows}, \texttt{ucla:apepe}). Only when $T_\tau = R$, (Note: For the purpose of this part of the argument, $R$ is assumed to be a theoretical graph instance that includes all statements about the world) that is, when their relationship is placed within the broader context of the world as a whole, does the complete picture emerge. However, with respect to those triples that provide the most context, a $|T_\tau| \ll |R|$ suffices to expose the more essential aspects of (\texttt{lanl:marko}, \texttt{foaf:knows}, \texttt{ucla:apepe}). (Note: A fuzzy set is perhaps the best representation of a dilated triple \cite{zadeh:fuzzy1965}. In such cases, a membership function $\mu_{T_\tau}: R \rar [0,1]$ would define the degree to which every triple in $R$ is in $T_\tau$. However, for the sake of simplicity and to present the proposed model within the constructs of the popular named graph formalism, $T_\tau$ is considered a classical set. Moreover, a fuzzy logic representation requires an associated membership valued in $[0,1]$ which then requires further statement reification in order to add such metadata. With classic bivalent logic, $\{0,1\}$ iss captured by the membership or non-membership of the statement in $T_\tau$.) (Note: The choices made in the creation of a dilated triple are determined at the knowledge-level \cite{know:newell1982}. The presentation here does not suppose the means of creation, only the underlying representation and utilization of such a representation.)  Defining a triple in terms of a larger conceptual structure exposes more degrees of freedom when representing the uniqueness of a relationship. For example, suppose the two dilated triples $T_x$ and $T_y$ diagrammed in Figure 3, where   \begin{equation}  y = lanl:marko, foaf:knows, ucla:apepe).  \end{equation}  and  \begin{equation}  y = lanl:marko, foaf:knows, cap:carole).  \end{equation}  Both $T_x$ and $T_y$ share the same predicate \ttt{foaf:knows}. However, what is meant by Marko knowing Alberto is much different than what is meant by Marko knowing his mother Carole (\ttt{cap:carole}). While, broadly speaking, it is true that Marko knows both Alberto and Carole, the context in which Marko knows Alberto is much different than the context in which Marko knows Carole. The supplementary triples that compose $T_y$ may be the RDF expression of:  \begin{quote}  ``Marko was born in Fairfield, California on November 30$^\text{th}$, 1979. Carole is Marko's mother. Marko's family lived in Riverside (California), Peachtree City (Georgia), Panama City (Panama), and Fairfax (Virginia). During his $10^\text{th}$ grade high-school term, Marko moved with his family back to Fairfield, California."  \end{quote}  It is obvious from these two examples that \ttt{foaf:knows} can not sufficiently express the subtleties that exist between two people. People know each other in many different ways. There are family relationships, business relationships, scholarly relationships, and so on. It is true that these subtleties can be exposed when performing a deeper analysis of the graph surrounding a \ttt{foaf:knows} relationship as other paths will emerge that exist between people (e.g.~vacation paths, transaction paths, coauthorship paths, etc.). The purpose of a dilated triple is to contain these corroborating statements within the relationship itself. The purpose of $T_x$ is to identify those aspects of Marko and Alberto's ``knowing" relationship that make it unique (that provide it the most meaning). Similarly, the purpose of $T_y$ is to provide a finer representation of the context in which Marko knows his mother. The supplementary triples of $T_x$ and $T_y$ augment the meaning of \ttt{foaf:knows} and frame each respective triple $x$ and $y$ in a broader context. (Note: Examples of other predicates beyond \ttt{foaf:knows} also exist. For instance, suppose the predicates \texttt{foaf:member} and \texttt{foaf:fundedBy}. In what way is that individual a member of that group and how is that individual funded?)