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\begin{document}
	\begin{frontmatter}
		
		\title{Linked Data Semantic Distance with Global Normalization for evaluating Semantic Similarity in a Taxonomy} 
		
		
		
		
		\author[First]{Anna Formica} 
		\author[Second]{Francesco Taglino} 
		
		
		
		\address[First]{
			National Research Council,
			Istituto di Analisi dei Sistemi ed Informatica ``Antonio Ruberti",
			Via dei Taurini 19, I-00185 Rome, Italy
			(e-mail: anna.formica@iasi.cnr.it)}
		\address[Second]{National Research Council,
			Istituto di Analisi dei Sistemi ed Informatica ``Antonio Ruberti",
			Via dei Taurini 19, I-00185 Rome, Italy
			(e-mail: francesco.taglino@iasi.cnr.it)}
		
		
		
		\begin{abstract}
			In this work the problem of evaluating semantic similarity in a taxonomy by relying on the notion of {\it information content} is investigated. In particular, a measure that takes into account  not only the  {\it generic sense} of a concept but also its  {\it intended sense} in a given context is considered. Such a measure needs a semantic relatedness approach in order to  evaluate the relatedness  between the generic sense and the intended sense of a concept. In this work we show that relying on  the {\it Linked Data Semantic Distance with Global Normalization} leads to  higher Spearman's correlation values  with human judgment with respect to the original proposal of the authors.
		\end{abstract}
		
		\begin{keyword}
			
			Semantic Similarity,  Information Content,  Taxonomy,  Semantic Relatedness, Concept Sense, Linked Data Semantic Distance.
			
		\end{keyword}
		
	\end{frontmatter}
	
	
	
	
	
	
	\maketitle
	
	
	
	
	
	
	
	\section{Introduction} \label{sect:intro}
	Evaluating semantic similarity is a fundamental topic in Computer Science in different application domains, such as health \cite{Abdelrahman2015},   network security \cite{Weller-Fahy2015}, and bibliometrics \cite{Avram12}, just to mention a few examples. In particular, semantic similarity methods 
	based on the information content (IC) approach have been employed in  different research areas, such as Natural Language Processing \cite{Ajumder2016}, Semantic Web \cite{Formica13}, \cite{Meymandpour2016}, Formal Concept Analysis \cite{Formica19}, \cite{Wang2020}, Geographical Information Systems \cite{FP09}, \cite{Formica2020}, \cite{Schwering2008}.
	
	
	The IC approach has been  recognized as ``the state of the art on semantic similarity'' \cite{Adhikari2018}, \cite{Batet2020} and, in general, has shown higher correlation values with human judgment ({\it HJ}) with respect to other proposals which do not originate from it  \cite{Adhikari2018}, \cite{Batet2020}, \cite{Chandrasekaran2020}, \cite{Formica13}. However it has shown some limitations, as for instance,
	in the presence of pairs of concepts sharing the same ``more general'' meaning \cite{Jeong2017}.
	
	
	In \cite{FT2021a} a novel approach has been presented that allows semantic similarity to be computed by taking into account not only the information contents of the concepts but also their \textit{context}, i.e., the meanings of the concepts in the given application domain. 
	Indeed, as shown in \cite{FT2021a}, the context (or {\it perspective}  \cite{Lin}) is fundamental in evaluating  semantic similarity, and
	different contexts can lead to different similarity degrees among the same concepts.
	
	
	It is important to note that also  the approach proposed by Lin is based on the notion of perspective,  but it does not allow to evaluate  similarity by addressing a single perspective at a time,
	and the  information-theoretic definition  of similarity between concepts
	is interpreted as ``a weighted average of their similarities computed from different perspectives''. 
	
	
	In this work, analogously to \cite{FT2021a}, we distinguish  the notion of concept \textit{generic sense}, i.e., the sense of the concept that is not related to any specific context, from the concept {\it intended sense}, i.e., the meaning of the concept in a specific context. In order to compute semantic similarity between concepts, an important activity consists in evaluating the relatedness \cite{Taieb20} between the generic sense and the intended sense of a given concept. 
	
	With this regard, 
	in this paper in  place of the \textit{Information Content-based Measure} (\textit{ICM}) \cite{SchuPonze14} adopted in \cite{FT2021a}, we  rely on the 
	family of methods proposed in \cite{Piao2016},  here referred to as {\it Linked Data Semantic Distance with Global Normalization} ($LDSDGN$). In particular, in the experiment presented in this work semantic relatedness is evaluated according to the $LDSDGN_{\gamma}$ strategy proposed in \cite{Piao2016}, rather than the   {\it combIC} measure presented in \cite{SchuPonze14}. 
	These are both knowledge-based approaches for computing semantic relatedness of concepts in {\it Resource Description Framework} (RDF) knowledge graphs. 
	
	Essentially, with respect to the {\it combIC} proposal that leverages {\it triple weights}, the $LDSDGN_{\gamma}$  measure focuses on {\it triple patterns}, and aims at verifying the existence of specific configurations of paths in the RDF knowledge graph.
	
	
	In the new experiment presented in this paper, besides  {\it combIC} and  \textit{$LDSDGN_{\gamma}$},  the comparison also involves the following knowledge-based relatedness measures: \textit{LDSD}$_{cw}$, which belongs to the \textit{Linked Data Semantic Distance} (\textit{LDSD}) family \cite{Passant2010}, \textit{ASRMP$^a_m$} \cite{ElVaigh2020},   \textit{Wikipedia Link-based Measure} (\textit{WLM}) \cite{Witten2008}, 
	\textit{Exclusivity-based measure} (\textit{ExclM}) \cite{Hulpus2015}, and
	\textit{Linked Open Data Description Overlap} (\textit{LODOverlap}) \cite{Zhou2012}.
	Overall, according to the new experimental results, the $LDSDGN_{\gamma}$ strategy  shows the  best average Spearman's correlation.
	
	
	The paper is organized as follows. In Section \ref{sect:proposedmeasure} the  similarity measure relying on the intended senses of concepts is  presented, and in Section \ref{relatedness} the \textit{LDSDGN}  family of semantic relatedness measures is recalled. In  Section \ref{sect:experimentalresults} the new experiment is illustrated, followed by  the related work  in Section \ref{sect:relatedWork}. The conclusion is given in Section \ref{conclusion}. 
	
	
	
	
	
	\section{The Semantic Similarity Measure} \label{sect:proposedmeasure}
	
	Consider a set of concepts $C$ of an ISA taxonomy (taxonomy for short), and a  function $p$: 
	
	$p$: $C$ $\rightarrow$ [0,1]
	
	\noindent such that, for any $c \in C$, $p(c)$ is the {\it probability} of the concept $c$ computed on the basis of the relative concept {\it frequency}, $freq(c)$, evaluated from large collections of multidisciplinary texts, such as the Brown Corpus of American English. In particular, the probability of a concept $c$ is defined as $p(c) = freq(c)/N$, where $N$ is the total number of concepts in the corpus. Hence, the information content of a concept  $c$, indicated as $IC(c)$, is computed  as:
	
	
	$IC(c)=-log$ $p(c)$
	
	\noindent which means that, intuitively,  as the probability of a concept increases its informativeness decreases. 
	Given two concepts  $c_i,c_j \in C$, the concept semantic similarity proposed by Lin in \cite{Lin}, $sim_L(c_{i},c_{j})$, is defined as follows: \\
	
	
	
	
	$sim_L(c_{i},c_{j}) =  \frac{2\times
		IC(lcs(c_{i},c_{j}))}{IC{(c_{i})}+IC(c_{j})}$\\
	
	
	
	\noindent where {\it lcs} is the {\it least common subsumer} (or the most informative subsumer in \cite{Res95}) of the concepts $c_i,c_j$. 
	
	As extensively discussed in \cite{FT2021a}, the approaches relying on the method mentioned above do not consider the semantic similarity of the meanings of  the concepts according to a given context.
	For this reason,  an enrichment of it has been proposed, by characterizing the meanings of the compared concepts with respect to a given application domain as recalled below. 
	
	
	%Below it is illustrated by using the Lin's formula, but it can be applied to any information content measure, as shown below.
	Suppose we have an application domain, say $D_k$, the semantic similarity of the concepts $c_i,c_j \in C$, indicated as $sim_{D_k}(c_{i},c_{j})$, is defined as follows:\\
	
	
	$sim_{D_k}(c_{i},c_{j})= \\
	sim(c_{i},c_{j})\times(1-\omega_k) +  sim(\mathcal{S}_{D_k}(c_{i}),\mathcal{S}_{D_k}(c_{j})))\times\omega_k$\\
	
	\noindent where $sim$ is an information content based similarity measure (e.g., $sim_L$),  $\omega_k$ is a weight, $0 \leq \omega_k \leq 1$, defined by the domain expert according to $D_k$, and $\mathcal{S}_{D_k}$ is a function, referred to as the {\it intended sense} function. Such a function associates a concept with its  meaning according to $D_k$ as follows:
	
	$\mathcal{S}_{D_k}$: $C$ $\rightarrow$ $C$
	
	\noindent and  $\mathcal{S}_{D_k}(c)$ = 
	$s$  if $s \in C$ is the {\it intended sense} of $c$ in $D_k$, $\mathcal{S}_{D_k}(c)$ = 
	$c$ otherwise.
	
	
	
	
	
	
	\noindent Note that  the weight $\omega_k$, depending on $D_k$, allows a balance between the roles of the generic senses and the intended senses of the concepts, according to the relevance they have in the domain $D_k$.
	
	
	
	
	\noindent For instance,  consider  an application domain, say $D_1$,  where it is important to characterize people on the basis of the time they spend in an edifice per day. Suppose \{{\it employee}, {\it student}, {\it planter}\} are concepts of the taxonomy, and
	let $\mathcal{S}_{D_1}$ be a function associating the intended senses  with  these      concepts  in the domain $D_1$ as follows:
	
	
	
	$\mathcal{S}_{D_1}(employee) = office$  
	
	$\mathcal{S}_{D_1}(student) = college$
	
	$\mathcal{S}_{D_1}(planter) = reward$
	
	\noindent  where \{{\it office}, {\it college}, {\it reward}\} are also concepts of the taxonomy. According to the above measure, on the basis of the intended senses of the concepts, the following holds:
	
	
	$sim(employee,student)>$ $sim(employee,planter)$
	
	
	\noindent because the intended senses of $employee$ and $student$ share a common meaning, such as for instance $building$, whereas this does not hold in the case of the senses of $employee$ and $planter$.
	
	Suppose now we have a second scenario where earnings are more relevant than  workplaces, and consider another application domain, say  $D_2$, for which  the intended sense of $employee$ is   {\it salary}, which is another  concept of the taxonomy. Therefore, the function $\mathcal{S}_{D_2}$ is defined  as follows:
	
	
	$\mathcal{S}_{D_2}(employee) = salary$
	
	\noindent whereas  the  meanings of the concepts $student$  and  $planter$ are the same, i.e.:
	
	
	$\mathcal{S}_{D_2}(student) = college$
	
	$\mathcal{S}_{D_2}(planter) = reward$.
	
	
	\noindent In this second perspective, since   $salary$ and $reward$  are both a form of $payment$, whereas  $salary$ and  $college$, in general, do not share any meaning, as expected the following holds:
	
	
	
	$sim(employee,student)<$ $sim(employee,planter)$
	
	This is an intuitive and very simple example showing how the role of the intended senses of concepts can impact on their similarity scores, subject to the definition of the weight $\omega_k$ according to $D_k$.
	
	
	
	
	
	
	\section{Evaluating the relatedness of concept intended senses} \label{relatedness}
	
	In the previous section we have seen that  the weight $\omega_k$ depends  on the domain $D_k$, and is established by domain experts on the basis of the roles of the 
	generic senses and intended senses of the concepts in  $D_k$. However in our experiment, since finding several experts in different application domains is not feasible, in order to evaluate this proposal we relied on semantic relatedness methods \cite{Taieb20}. In particular we focus on knowledge-based semantic relatedness methods that depend on the availability of a formal knowledge base, such as a knowledge graph, an ontology, or a taxonomy.
	
	In this paper we consider the  methods for evaluating semantic relatedness of resources in RDF\footnote{http://www.w3.org/TR/2014/REC-rdf11-concepts-20140225/} knowledge graphs, where concepts are represented by the nodes  of the graph.
	RDF is a family of specifications designed as a standard model for data interchange on the Web. It is used for the conceptual description or modeling of information of Web  resources, each identified by a Uniform Resource Identifier (URI). RDF is based upon the idea of making statements about resources by means of expressions in the form of triples following  a \textit{subject}$-$\textit{predicate}$-$\textit{object} pattern. The subject denotes the resource that is being described, and the predicate expresses a relation between the subject and the object, which can be a resource or a literal (e.g., a string, a number).
	
	Let $R=\{r_1, r_2, ..., r_n\}$ be a finite set of URIs each representing a resource, and $L=\{l_1, l_2, ..., l_m\}$ a finite set of literals, an RDF triple has the form:
	
	\begin{center}
		%$<$\textit{s}, \textit{p}, \textit{o}$>$
		$\langle s, p, o\rangle$
	\end{center}
	where
	\textit{s} $\in R$ is the subject, \textit{p} $\in R$ is the predicate, and \textit{o} $\in R \cup L$ is the object.
	An RDF knowledge graph is a set of RDF triples,  where subjects and objects are nodes, and predicates are directed arcs (also called links, edges or arrows).  
	
	
	
	
	
	
	\subsection{LDSD with Global Normalization (\textit{LDSDGN})} \label{PiaoBreslin2016}
	In this paper we address the  {\it Linked Data Semantic Distance with Global Normalization} ($LDSDGN$) family of methods \cite{Piao2016}, which is an evolution of the \textit{Linked Data Semantic Distance} ($LDSD$) measures presented by Passant in \cite{Passant2010}.
	Such a family
	%relies on the identification of path patterns satisfying specific
	%conditions in the knowledge graph.  It 
	is based on the following assumptions. 
	
	Given  two resources $r_a$ and $r_b$, with $r_a$ $\neq$ $r_b$, a distance measure $d$ should satisfy the following three axioms:
	
	(i) Equal self-distance, i.e., 
	{\it d}$(r_a,r_a)$ = {\it d}$(r_b,r_b) = 0$.
	
	(ii) Symmetry, i.e., {\it d}$(r_a,r_b)$ = {\it d}$(r_b,r_a)$.
	
	(iii) Minimality, i.e., {\it d}$(r_a,r_a)$ $<$ {\it d}$(r_a,r_b)$. 
	
	\noindent Since the measures introduced by Passant do not satisfy these three axioms,  in order to meet them,  in \cite{Piao2016}  the authors propose the $LDSDGN$ family containing the  $LDSDGN_\alpha$, $LDSDGN_\beta$, and $LDSDGN_\gamma$ distances, that are recalled below.
	
	\vspace{0.2 cm}
	
	\noindent {\bf LDSDGN$_\alpha$}.
	Given a RDF knowledge graph $\mathcal{G}$, let $C_d$ be  a function that computes the number
	of direct links between two resources in the graph as follows. 
	
	Given two resources $r_a$, $r_b$,  and the predicate $p_j$, then $C_d(p_j,r_a,r_b)$ = 1 if in the graph there exists a triple $\langle r_a,p_j,r_b \rangle$, otherwise $C_d(p_j,r_a,r_b)$ = 0. 
	
	Furthermore,   the total number of resources that can be reached from $r_a$ by means of  the predicate $p_j$ is indicated by $C_d(p_j,r_a)$ (analogously, $C_d(p_j,r_b)$).
	Similarly, $C_{io}(p_j,r_a)$ and $C_{ii}(p_j,r_a)$ are the
	total number of resources indirectly linked to $r_a$ via outgoing and incoming links labeled with the predicate $p_j$, respectively.
	
	
	On the basis of the assumption that resources are more related if there is a great number of them linked via a given predicate $p_k$,  the  $C_{io}$ ($C_{ii}$) function defined  by Passant has been generalized by using the function  $C'_{io}$ ($C'_{ii}$) as follows: $C'_{io}(p_k,r_a,r_b)$ ($C'_{ii}(p_k,r_a,r_b)$) computes the total number of 
	resources linked to $r_a$ and $r_b$ via an outgoing (incoming)
	predicate $p_k$. 
	
	Therefore, given the resources $r_a$, $r_b$, the first distance is the $LDSDGN_\alpha(r_a,r_b)$ measure defined in Eq. \ref{eq:LDSDalpha}:
	
	%\begin{footnotesize}
	\begin{equation}
		\label{eq:LDSDalpha}
		LDSD_\alpha(r_a,r_b)= \frac{1}{1+ f_1 + f_2}\\
	\end{equation}
	%\end{footnotesize}
	
	\noindent where:\\
	
	\begin{footnotesize}
		$f_1 = \sum_{p_j \in U}\frac{C_d(p_j,r_a,r_b)}{1+log(C_d(p_j,r_a))}  +
		\sum_{p_j \in V}\frac{C_d(p_j,r_b,r_a)}{1+log(C_d(p_j,r_b))}\\$
		
		$f_2 = \sum_{p_j \in W}\frac{C'_{io}(p_j,r_a,r_b)}{1+log(C_{io}(p_j,r_a))}  +
		\sum_{p_j \in Z}\frac{C'_{ii}(p_j,r_a,r_b)}{1+log(C_{ii}(p_j,r_a))}$\\
	\end{footnotesize}
	
	
	
	\noindent 
	and 
	$U$,$V$,$W$,$Z$ $\subseteq R$ are the sets of the predicates $p_j$ in the graph $\mathcal{G}$ such that $C_{d}(p_j,r_a,r_b) = 1$,
	$C_{d}(p_j,r_b,r_a) = 1$,
	$C'_{io}(p_j,r_a,r_b) > 0$, and
	$C'_{ii}(p_j,r_a,r_b) > 0$, respectively.\\
	
	
	\begin{figure*}
		\centering
		{\includegraphics[width=0.7\textwidth]{LDSDGN.jpg}} 
		\caption{An example of the knowledge graph $\mathcal{G}$}\label{fig:ldsdgn}
	\end{figure*}
	
	
	
	\noindent As an example, consider the resources $r_a$, and $r_b$ in the graph $\mathcal{G}$ of Figure \ref{fig:ldsdgn}. In order to apply the \textit{LDSDGN$_\alpha$} measure to this  pair, the sets $U$, $V$, $W$, and $Z$ in Eq. \ref{eq:LDSDalpha} are defined as follows. 
	
	
	$U = \{p_1\}$ because $C_d(p_j, r_a, r_b) = 1$ only if $j=1$, and $C_{d}(p_1, r_a) = 2$ since $r_b$ and $r_4$ are all the   resources that can be reached from $r_a$ through the predicate $p_1$.
	$W = \{p_2\}$ because $C'_{io}(p_j, r_a, r_b) = 1$ only for $j=2$, and 
	$C_{io}(p_2, r_a) = 2$ since $r_b$ and $r_2$ are indirectly linked to $r_a$ via outgoing links labeled with  $p_2$, thanks to the following pairs of triples:
	
	
	($\langle r_a,p_2,r_1 \rangle$, $\langle r_b,p_2,r_1 \rangle$);\\
	($\langle r_a,p_2,r_1 \rangle$, $\langle r_2,p_2,r_1 \rangle$).  
	
	
	Finally, $V$ and $Z$ are both equal to the empty set.
	%These triples are highlighted with long-dashed thick arrows in the graph.
	
	\vspace{0.2 cm}
	
	
	
	\noindent {\bf LDSDGN$_\beta$}. With respect to  $LDSDGN_\alpha$, in the  following $LDSDGN_\beta$ measure   the last two addenda at the denominator are modified by addressing the averages between  $C_{io}(p_j,r_a)$, $C_{io}(p_j,r_b)$, and  $C_{ii}(p_j,r_a)$, $C_{ii}(p_j,r_b)$ respectively, in order to achieve symmetry. 
	
	Given the  resources $r_a$, and $r_b$, $LDSDGN_\beta(r_a,r_b)$ is defined according to the following Eq. \ref{eq:LDSDbeta}:
	
	%\begin{footnotesize}
	\begin{equation}
		\label{eq:LDSDbeta}
		LDSD_\beta(r_a,r_b) = \frac{1}{1+ f_1 + f_2}
	\end{equation}
	%\end{footnotesize}
	
	
	\noindent where:\\
	
	\begin{footnotesize}
		\noindent \hspace{0.2cm} $f_1 = \sum_{p_j \in U}\frac{C_d(p_j,r_a,r_b)}{1+log(C_d(p_j,r_a))} + 
		\sum_{p_j \in V}\frac{C_d(p_j,r_b,r_a)}{1+log(C_d(p_j,r_b))}$\\
		
		\noindent \hspace{0.2cm}$f_2 = \sum_{p_j \in W}\frac{C'_{io}(p_j,r_a,r_b)}{1+log(\frac{C_{io}(p_j,r_a)+C_{io}(p_j,r_b)}{2})} $
		
		
		$ \hspace{0.9cm} +
		\sum_{p_j \in Z}\frac{C'_{ii}(p_j,r_a,r_b)}{1+log(\frac{C_{ii}(p_j,r_a)+C_{ii}(p_j,r_b)}{2})}$\\
	\end{footnotesize}
	
	\noindent and the sets $U$, $V$, $W$, and $Z$ are the same sets considered by the \textit{LDSDGN$_\alpha$} measure. 
	
	
	
	For example, when applying the \textit{LDSDGN$_\beta$} measure to the resources $r_a$, and $r_b$ in Figure \ref{fig:ldsdgn}, the further function to be computed with respect to \textit{LDSDGN$_\alpha$} is $C_{io}(p_2, r_b)$, which is equal to 3. Indeed, the resources $r_a$, $r_2$, and $r_9$ are the only resources that can be reached from $r_b$ via outgoing links labelled with the predicate $p_2$, thanks to the following pairs of triples:
	
	
	($\langle r_b,p_2,r_1 \rangle$, $\langle r_a,p_2,r_1 \rangle$);\\
	($\langle r_b,p_2,r_1 \rangle$, $\langle r_2,p_2,r_1 \rangle$);\\
	($\langle r_b,p_2,r_8 \rangle$, $\langle r_9,p_2,r_8 \rangle$).
	
	
	
	
	\vspace{0.2 cm}
	\noindent {\bf LDSDGN$_\gamma$}. In \cite{Piao2016} the authors  propose a further measure, namely $LDSDGN_\gamma$, relying on a {\it global normalization} notion that essentially considers the importance of a path between
	two resources according to the number of its occurrences in the whole graph $\mathcal{G}$. In the following, 
	let $C_{dp}(p_j)$ be
	the global occurrences of the link $p_j$ between two resources in $\mathcal{G}$.
	
	
	Furthermore, 
	$C_{io}(p_k,r_j,r_a,r_b)$ = 1 if there exists a resource $r_j$ such that $\langle r_a,p_k,r_j \rangle$ and $\langle r_b,p_k,r_j \rangle$, and
	$C_{ii}(p_k,r_j,r_a,r_b)$ = 1 if there exists a resource $r_j$ such that $\langle r_j,p_k,r_a \rangle$ and $\langle r_j,p_k,r_b \rangle$.
	The normalizations of
	$C_{io}(p_k,r_j,r_a,r_b)$ and  
	$C_{ii}(p_k,r_j,r_a,r_b)$  
	are carried out by using the global occurrences $C_{iop}(p_k,r_j)$ and $C_{iip}(p_k,r_j)$ of $r_j$ as follows.
	$C_{iop}(p_k,r_j)$  returns the global occurrences of  $r_j$ in the undirected paths $[\langle r_n,p_k,r_j \rangle$, $\langle r_s,p_k,r_j \rangle$],  for any  resources $r_n$, $r_s$ in the graph $\mathcal{G}$ and, analogously, 
	$C_{iip}(p_k,r_j)$  computes the global occurrences of  $r_j$ in the undirected paths $[\langle r_j,p_k,r_n \rangle$, $\langle r_j,p_k,r_s \rangle]$,  for any  resources $r_n$, $r_s$ in $\mathcal{G}$.   
	
	According to the above assumptions, given $r_a$ and $r_b$,  $LDSDGN_\gamma(r_a,r_b)$ is  defined in Eq. \ref{eq:LDSDgamma}:
	
	%\begin{footnotesize}
	\begin{equation}
		\label{eq:LDSDgamma}
		LDSD_\gamma(r_a,r_b)= \frac{1}{1+ f_1 + f_2}
	\end{equation}
	%\end{footnotesize}
	
	
	\noindent where:
	
	\begin{footnotesize}
		$f_1 = \sum_{p_j \in U}\frac{C_d(p_j,r_a,r_b)}{1+log(C_{dp}(p_j))} +
		\sum_{p_j \in V}\frac{C_d(p_j,r_b,r_a)}{1+log(C_{dp}(p_j))}$\\
		
		$ f_2 = \sum_{(p_k, r_j) \in W}\frac{C_{io}(p_k,r_j,r_a,r_b)}{1+log(C_{iop}(p_k,r_j))} + 
		\sum_{(p_k, r_j) \in Z}\frac{C_{ii}(p_k,r_j,r_a,r_b)}{1+log(C_{iip}(p_k,r_j))}$\\
		
	\end{footnotesize}
	
	\noindent and 
	$U \subseteq R$ is the set of the predicates $p_j$ in the graph $\mathcal{G}$ such that $C_{d}(p_j,r_a,r_b) = 1$,
	$V \subseteq R$ is the set of the predicates $p_j$ in $\mathcal{G}$ such that $C_{d}(p_j,r_b,r_a) = 1$,
	$W \subseteq R\times R$ is the set of  pairs ($p_k$,$r_j$) such that $C_{io}(p_k, r_j, r_a, r_b) = 1$, and
	$Z \subseteq R\times R$ is the set of  pairs ($p_k$,$r_j$)  such that $C_{ii}(p_k, r_j, r_a, r_b) = 1$.\\
	
	
	When applying the \textit{LDSDGN$_\gamma$} measure to  $r_a$ and $r_b$ in the graph in Figure \ref{fig:ldsdgn}, we have the following.
	
	$U = \{p1\}$, and $V$ is the empty set.
	Then, we have  $C_{dp}(p_1) = 5$ because of the presence of the following triples 
	$\langle r_a,p_1,r_b \rangle$,
	$\langle r_a,p_1,r_4 \rangle$,
	$\langle r_b,p_1,r_6 \rangle$,
	$\langle r_4,p_1,r_5 \rangle$, and
	$\langle r_4,p_1,r_3 \rangle$.
	$W = \{(p_2, r_1)\}$ because $C_{io}(p_k, r_j, r_a, r_b) = 1$ only when $k=2$, and $j=1$, whereas $Z$ is the empty set.
	
	
	
	
	\begin{table*}[h]
		%\footnotesize
		\caption{Average Spearman's correlations in the 28 contexts according to the experiment presented in \cite{FT2021a} based on {\it CombIC}
		}\label{tab:averagecorrelALLSpearman_IC} \centering
		%\begin{scriptsize}
		\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
			\hline
			{\it concept$_1$}, {\it concept$_2$}&$sim_R$ & $sim_{W\&P}$ &$sim_L$&$sim_{J\&C}$& $sim_{P\&S}$ &$sim_A$&$sim_{A\&M}$\\
			% R&W\&P&L&J\&C&P\&S&A&A\&M\\
			\hline  \hline
			car, automobile & 0.81 & 0.80 & 0.84 & 0.85 & 0.85 & 0.84 & 0.86\\
			\hline
			gem, jewel & 0.80 & 0.77 & 0.83 & 0.82 & 0.84 & 0.87 & 0.87\\
			\hline
			journey, voyage & 0.89 & 0.87 & 0.90 & 0.86 & 0.92 & 0.92 & 0.92\\
			\hline
			boy, lad & 0.85 & 0.81 & 0.91 & 0.85 & 0.92 & 0.85 & 0.85\\
			\hline
			coast, shore & 0.94 & 0.85 & 0.93 & 0.90 & 0.92 & 0.93 & 0.91\\
			\hline
			asylum, madhouse & 0.83 & 0.89 & 0.91 & 0.85 & 0.93 & 0.87 & 0.85\\
			\hline
			magician, wizard & 0.92 & 0.86 & 0.89 & 0.90 & 0.91 & 0.87 & 0.87\\
			\hline
			midday, noon & 0.87 & 0.89 & 0.91 & 0.91 & 0.92 & 0.87 & 0.87\\
			\hline
			furnace, stove & 0.65 & 0.38 & 0.63 & 0.50 & 0.67 & 0.66 & 0.66\\
			\hline
			food, fruit & 0.85 & 0.58 & 0.84 & 0.81 & 0.89 & 0.83 & 0.81\\
			\hline
			bird, cock & 0.84 & 0.82 & 0.88 & 0.82 & 0.86 & 0.85 & 0.84\\
			\hline
			bird, crane & 0.83 & 0.83 & 0.88 & 0.85 & 0.87 & 0.86 & 0.85\\
			\hline
			tool, implement & 0.80 & 0.78 & 0.84 & 0.77 & 0.84 & 0.82 & 0.81\\
			\hline
			brother, monk & 0.82 & 0.71 & 0.83 & 0.83 & 0.93 & 0.82 & 0.81\\
			\hline
			crane, implement & 0.69 & 0.67 & 0.73 & 0.74 & 0.79 & 0.73 & 0.75\\
			\hline
			lad, brother & 0.71 & 0.68 & 0.80 & 0.66 & 0.69 & 0.75 & 0.73\\
			\hline
			journey, car & 0.69 & 0.58 & 0.71 & 0.78 & 0.77 & 0.71 & 0.72\\
			\hline
			monk, oracle & 0.77 & 0.59 & 0.65 & 0.67 & 0.70 & 0.76 & 0.75\\
			\hline
			food, rooster & 0.85 & 0.64 & 0.85 & 0.86 & 0.89 & 0.86 & 0.87\\
			\hline
			coast, hill & 0.62 & 0.72 & 0.70 & 0.69 & 0.65 & 0.70 & 0.71\\
			\hline
			forest, graveyard & 0.81 & 0.79 & 0.84 & 0.84 & 0.88 & 0.83 & 0.83\\
			\hline
			monk, slave & 0.70 & 0.59 & 0.62 & 0.62 & 0.67 & 0.63 & 0.63\\
			\hline
			coast, forest & 0.82 & 0.78 & 0.86 & 0.88 & 0.92 & 0.86 & 0.82\\
			\hline
			lad, wizard & 0.64 & 0.59 & 0.64 & 0.54 & 0.72 & 0.59 & 0.59\\
			\hline
			chord, smile & 0.58 & 0.76 & 0.78 & 0.99 & 0.99 & 0.76 & 0.76\\
			\hline
			glass, magician & 0.85 & 0.85 & 0.85 & 0.91 & 0.89 & 0.81 & 0.77\\
			\hline
			noon, string & 0.92 & 0.88 & 0.92 & 0.94 & 0.86 & 0.94 & 0.93\\
			\hline
			rooster, voyage & 0.79 & 0.73 & 0.82 & 0.86 & 0.90 & 0.83 & 0.82\\
			\hline
			{\it Avg Correl.} & 0.79 & 0.74 & 0.81 & 0.80 & 0.84 & 0.81 & 0.80\\
			\hline
		\end{tabular}
		%\end{scriptsize}
	\end{table*}
	
	
	\begin{table*}[h]
		%\footnotesize
		\caption{Average Spearman's correlations in the 28 contexts according to the new experiment based on {\it LDSDGN$_\gamma$}
		}\label{tab:averagecorrelALLSpearman} \centering
		%\begin{scriptsize}
		\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
			\hline
			{\it concept$_1$}, {\it concept$_2$}&$sim_R$ & $sim_{W\&P}$ &$sim_L$&$sim_{J\&C}$& $sim_{P\&S}$ &$sim_A$&$sim_{A\&M}$\\
			% R&W\&P&L&J\&C&P\&S&A&A\&M\\
			\hline  \hline
			Car,Automobile & 0.83 & 0.82 & 0.84 & 0.83 & 0.88 & 0.82 & 0.82\\
			\hline
			Gem,Jewel & 0.83 & 0.82 & 0.84 & 0.83 & 0.88 & 0.82 & 0.82\\
			\hline
			Journey,Voyage & 0.83 & 0.81 & 0.83 & 0.82 & 0.88 & 0.82 & 0.82\\
			\hline
			Boy,Lad & 0.83 & 0.80 & 0.83 & 0.82 & 0.88 & 0.82 & 0.82\\
			\hline
			Coast,Shore & 0.83 & 0.80 & 0.83 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Asylum,Madhouse & 0.83 & 0.81 & 0.83 & 0.82 & 0.88 & 0.82 & 0.82\\
			\hline
			Magician,Wizard & 0.83 & 0.81 & 0.84 & 0.83 & 0.88 & 0.82 & 0.82\\
			\hline
			Midday,Noon & 0.83 & 0.81 & 0.84 & 0.83 & 0.88 & 0.82 & 0.82\\
			\hline
			Furnace,Stove & 0.83 & 0.77 & 0.82 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Food,Fruit & 0.80 & 0.76 & 0.83 & 0.82 & 0.88 & 0.81 & 0.81\\
			\hline
			Bird,Cock & 0.83 & 0.80 & 0.83 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Bird,Crane & 0.83 & 0.79 & 0.83 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Tool,Implement & 0.83 & 0.80 & 0.83 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Brother,Monk & 0.82 & 0.78 & 0.81 & 0.82 & 0.88 & 0.82 & 0.82\\
			\hline
			Crane,Implement & 0.83 & 0.79 & 0.83 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Lad,Brother & 0.82 & 0.79 & 0.83 & 0.82 & 0.87 & 0.82 & 0.81\\
			\hline
			Journey,Car & 0.78 & 0.75 & 0.80 & 0.82 & 0.87 & 0.81 & 0.80\\
			\hline
			Monk,Oracle & 0.82 & 0.77 & 0.81 & 0.82 & 0.87 & 0.82 & 0.81\\
			\hline
			Food,Rooster & 0.79 & 0.79 & 0.81 & 0.82 & 0.87 & 0.81 & 0.80\\
			\hline
			Coast,Hill & 0.83 & 0.79 & 0.83 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Forest,Graveyard & 0.78 & 0.75 & 0.80 & 0.82 & 0.87 & 0.81 & 0.81\\
			\hline
			Monk,Slave & 0.82 & 0.79 & 0.82 & 0.82 & 0.88 & 0.82 & 0.81\\
			\hline
			Coast,Forest & 0.80 & 0.76 & 0.81 & 0.82 & 0.87 & 0.81 & 0.81\\
			\hline
			Lad,Wizard & 0.82 & 0.79 & 0.83 & 0.82 & 0.87 & 0.82 & 0.81\\
			\hline
			Chord,Smile & 0.83 & 0.78 & 0.83 & 0.82 & 0.87 & 0.82 & 0.82\\
			\hline
			Glass,Magician & 0.80 & 0.76 & 0.81 & 0.82 & 0.87 & 0.82 & 0.82\\
			\hline
			Noon,String & 0.79 & 0.75 & 0.80 & 0.82 & 0.87 & 0.81 & 0.81\\
			\hline
			Rooster,Voyage & 0.79 & 0.75 & 0.80 & 0.82 & 0.87 & 0.81 & 0.81\\
			\hline
			{\it Avg Correl.} & 0.82 & 0.78 & 0.82 & 0.82 & 0.87 & 0.82 & 0.81\\
			\hline
		\end{tabular}
		%\end{scriptsize}
	\end{table*}
	
	
	Then we have $C_{iop}(p_2, r_1) = 3$ because of the following three pairs of triples:
	
	
	($\langle r_a,p_2,r_1 \rangle$, $\langle r_b,p_2,r_1 \rangle$);\\
	($\langle r_a,p_2,r_1 \rangle$, $\langle r_2,p_2,r_1 \rangle$);\\
	($\langle r_b,p_2,r_1 \rangle$, $\langle r_2,p_2,r_1 \rangle$).
	
	
	
	
	According to the results presented in \cite{Piao2016},  $LDSDGN_{\gamma}$  performs better than the $LDSDGN_{\alpha}$ and  $LDSDGN_{\beta}$ measures, and this is the measure we have considered in our experiment.
	\pagebreak
	
	\section{Experimental Results} \label{sect:experimentalresults}
	As mentioned in \cite{FT2021a}, the measure addressed in this paper relies on a novel approach for which the experimentation requires, besides the dataset composed of a set of pairs of concepts, further pairs of concepts representing the concept senses.
	Therefore, in order to compare the new experimental results with the ones of the original proposal, the Miller\&Charles ($M\&C$) dataset \cite{MillCha91} has been addressed and, for each pair of concepts of this dataset,  all the pairs of concepts of the same dataset have been considered as possible contexts. 
	
	With regard to the methods, the same  six information content based approaches discussed in \cite{FT2021a}  have been analyzed and, in particular, besides  Resnik ($sim_R$) and Lin ($sim_L$),  also Jiang and Conrath ($sim_{J\&C}$) \cite{JangC}, Pirr\`o and Seco ($sim_{P\&S}$) \cite{Pirro09},  Adhikari et al. ($sim_A$) \cite{Adhikari2018}, and the measure proposed by Adhikari et al. with the information content model computed as Meng ($sim_{A\&M}$) \cite{Adhikari2016}. The Wu and Palmer method ($sim_{W\&P}$) \cite{WuPalmer} has also been addressed, as  representative  of the edge-counting approach \cite{Rada89}. 
	
	
	
	
	Consider the $M\&C$ dataset, and the same  28 pairs of concepts belonging to it  in order to associate each pair of the dataset with 28 possible application domains $D_k$, $k$ = 1...28, in the following referred to as contexts  (therefore we have 28 $\times$ 28 = 784 similarity scores).
	For instance, for the pair of concepts ($boy, lad$), the 28  contexts are:
	
	
	
	$\mathcal{S}_{D_1}(boy) = car$
	
	$\mathcal{S}_{D_1}(lad) = automobile$
	
	$\mathcal{S}_{D_2}(boy) = gem$
	
	$\mathcal{S}_{D_2}(lad) = gewel$
	
	
	
	...
	
	$\mathcal{S}_{D_{28}}(boy) = rooster$
	
	
	$\mathcal{S}_{D_{28}}(lad) = voyage$.
	
	
	As mentioned in Section \ref{sect:proposedmeasure}, in general,  the intended senses of  concepts are supposed to be estimated by  domain experts, together with the related weight $\omega_k$ in the given context $D_k$. 
	In the  experiment presented in \cite{FT2021a}, in order to quantify such a weight, which represents the relevance of a pair of senses with respect to the pair of contrasted concepts, we used the method proposed in \cite{SchuPonze14} whereas, as already mentioned, in this experiment we rely on the \textit{LDSDGN$_\gamma$} measure. In particular,
	given a pair of concepts $c_i$, $c_j$ and a context $D_k$,  we assume that
	$\omega_k$ is defined as follows:
	
	
	
	
	$\omega_k  = (r_1+r_2)/2$
	
	
	
	
	\noindent  where  $r_1$ = $rel(c_{i},\mathcal{S}_{D_k}(c_{i}))$ and $r_2$ = $rel(c_{j},\mathcal{S}_{D_k}(c_{j}))$, and $rel$ is the relatedness degree computed according to the best strategy presented in \cite{Piao2016}.
	
	
	
	
	It is important to recall that in order to compute the 28 tables, one for each pair of the $M\&C$ dataset, each table containing  28 possible contexts for that pair, a disambiguation step has to be performed.
	In fact, it is well-known that in Wikipedia, and consequently in DBpedia, terms are addressed with the possible meanings they have, i.e., a term is associated with multiple senses. For this reason, in the experiment the disambiguation is necessary in order to address  senses in line with the {\it HJ} evaluation in the $M\&C$ experiment.  For instance, $crane$ in Wikipedia has two main senses, that are $bird$ and $machine$, therefore  when paired for instance with $implement$, it is disambiguated by using the sense $machine$. 
	
	
	
	In the experiment, in associating a given pair of concepts with a pair of possible concept senses, in some cases  the weight $\omega_k$, for a given context $D_k$, is null. In addition, there are some particular situations  for which both the concept senses do not have any relevance with the concepts to be compared, i.e., both the values $r_1$, $r_2$  above are null.
	In other words, for some pairs of concepts, there are  contexts (or perspectives) that do not apply to both the compared concepts, i.e., they do not correspond to any specific point of view and, for this reason, in the experimentation these contexts have been ignored.
	This is for instance the case of the pair of concepts ($coast, shore$), when associated with the pairs of senses  ($brother, monk$), or ($boy, lad$).
	Therefore, in order to analyze significant contexts, a threshold for {\it HJ}   has been introduced, in this case equal to 0.5  (in the scale from 0 to 4). 
	
	
	
	
	In Table \ref{tab:averagecorrelALLSpearman_IC},  the average  correlations for all the 28 pairs according to  Spearman are given, where  the relatedness degrees have been computed by leveraging  the semantic relatedness measure presented in \cite{SchuPonze14}. In Table \ref{tab:averagecorrelALLSpearman}
	the same values have been obtained  by relying on the best semantic relatedness strategy proposed in \cite{Piao2016}. 
	
	
	\begin{table*}[h]
		\caption{Averages of the average Spearman's correlations  with different relatedness methods}\label{tab:relComparisonS} \centering
		%\begin{scriptsize}
		\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
			\hline
			& $sim_{R}$ 	& $sim_{W\&P}$ 	& $sim_{L}$	&	$sim_{J\&C}$ & $sim_{P\&S}$	& $sim_A$ 	& $sim_{A\&M}$  & Avg.\\
			\hline
			{\it CombIC} & 0.79 & 0.74 & 0.81 & 0.80 & 0.84 & 0.81 & 0.80
			& {\bf 0.80}\\
			\hline
			$LDSD_{cw}$	& 0.77 & 0.72 & 0.80 & 0.80 & 0.85 & 0.81 & 0.80 & {\bf 0.79}
			\\
			\hline
			\textit{ASRMP$_m^a$} & 0.78 & 0.72 & 0.78 & 0.77 & 0.81 & 0.77 & 0.76 & {\bf 0.77}\\
			\hline
			\textit{WLM} & 0.74 & 0.71 & 0.78 & 0.80 & 0.83 & 0.78 & 0.76 & {\bf 0.77}  \\
			\hline
			\textit{ExclM} & 0.77 & 0.72 & 0.77 & 0.74 & 0.78 & 0.73 & 0.74 & {\bf 0.75}\\
			\hline
			
			\textit{LODOverlap} & 0.78 & 0.73 & 0.80 & 0.79 & 0.83 & 0.80 & 0.79 & {\bf 0.79} \\
			\hline
			
			\textit{$LDSDGN_{\gamma}$} & 0.82 & 0.78 & 0.82 & 0.82 & 0.87 & 0.82 & 0.81 & {\bf 0.82} \\
			\hline
			
			
			
		\end{tabular}
		%\end{scriptsize}
	\end{table*}
	
	
	In Table \ref{tab:relComparisonS} the above average Spearman's correlations are compared to the ones obtained according to the following methods: \textit{LDSD}$_{cw}$  \cite{Passant2010}, \textit{ASRMP$^a_m$} \cite{ElVaigh2020},  \textit{WLM} \cite{Witten2008}, 
	\textit{ExclM} \cite{Hulpus2015}, and 
	\textit{LODOverlap} \cite{Zhou2012}. 
	In the following    all the mentioned approaches are briefly summarized.
	
	The  {\it CombIC} method relies on the IC notion, and in particular focuses on  the computation of the 
	ICs of  the predicates and the objects of the triples of the knowledge graph.  
	
	
	$LDSD_{cw}$ belongs to the $LDSD$ family proposed by Passant which is at the basis of the above illustrated \textit{$LDSDGN_{\gamma}$} measure. As mentioned in  Subsection \ref{PiaoBreslin2016}, the $LDSD$ family does not satisfy the  three axioms of Equal self-distance, Symmetry, and  Minimality, and the \textit{$LDSDGN$} measures originate from it in order to meet these requirements.
	
	
	According to \textit{ASRMP$_m^a$}, the key idea is to evaluate all the directed paths connecting the  resources to be compared, of length equal to $m$. 
	
	The \textit{WLM}  method is inspired by the {\it Normalized Google Distance} measure. It relies on the information gathered by the nodes that are adjacent to the compared resources and, in particular, on those with incoming links to them.
	
	
	\textit{ExclM}  focuses on a given number of undirected paths and, in particular, it is based on a path weighting function that allows the selection of the undirected 
	paths between the compared resources with the greatest weights.
	
	
	Analogously to \textit{WLM}, in the case of \textit{LODOverlap} only the nodes that are adjacent to the compared resources are addressed, although the method also considers those with outgoing links from the compared resources. 
	
	According to the experimental results, \textit{$LDSDGN_{\gamma}$} shows 
	an average increment  of the average Spearman's correlation with  {\it HJ}  of about $0.02$. 
	Hence,   
	the global normalization notion on which  \textit{$LDSDGN_{\gamma}$} is based, that allows a path to be considered on the basis of   the number of its occurrences in the whole knowledge graph, provides the best strategy. 
	
	As a result, the combination of the measure presented in Section \ref{sect:proposedmeasure} with the approach proposed in \cite{Piao2016} provides the best Spearman's correlation values  in order to evaluate semantic similarity in a taxonomy by addressing the concept intended senses in a given context.
	
	The data concerning the new experiment are available at \cite{Taglino22}.
	
	
	
	
	
	
	
	%Such patterns involve further resources, on the basis of the names of the triples’ predicates, by taking into account the combination of both the adjacent  and the non-adjacent nodes to the compared resources. 
	
	
	
	
	
	
	\section{Related Work} \label{sect:relatedWork}
	
	
	One objection to the early IC based measures relies on   the use of large-scale corpora \cite{Adhikari2018}, \cite{Banu2015}, \cite{Batet2020}, \cite{Jeong2017}, \cite{Zhang2018}.   In fact, evaluating the IC on the basis of statistical information taken from textual corpora requires a huge amount of manual effort at level of both design and  maintenance of the corpus. 
	For this reason, in the literature, an evolution of the IC notion has been extensively investigated, referred to as {\it intrinsic information content} (IIC), although there is a lack of a statistically significant difference between the performances of the IIC models and the corpus-based ones \cite{Lastra15}. 
	In particular, the IIC is evaluated independently of textual corpora, and in accordance with the  intrinsic structure of the taxonomy, i.e., on the basis of the number of hyponyms and/or hypernyms of the concepts (see for instance \cite{Adhikari2016}, \cite{Adhikari2018}).
	However, in this paper we do not present a new IC or IIC computing model, and this proposal is independent of it.
	In fact, although the IIC approaches show a high accuracy in the similarity evaluation, they do not involve  concept meaning  and, in particular, the related similarity measures do not address the intended senses of  concepts according to a given application domain.
	
	Note that the semantic   similarity measure proposed in \cite{Jeong2017} originates from the need to overcome the limitations we highlighted in Section \ref{sect:proposedmeasure} in  WordNet. However, the authors propose a solution  by  associating the different kinds of relationships (e.g., ISA and PartOf)  with different weights, which is again a proposal independent of the concept intended senses.
	
	
	The notion of sense has been addressed by Resnik in \cite{Res99}, where  semantic similarity is used to identify and select the appropriate sense of a concept when it appears in a group of related terms. 
	However,  this paper addresses word sense disambiguation in the field of computational linguistics, where  semantic similarity is not the objective of the work but is used in order to associate a noun with the right sense in a given context. On the contrary, we use the concept intended senses to improve the computation of semantic similarity.
	
	
	
	
	
	
	\section{Conclusion and Future Work} \label{conclusion}
	
	In this paper the \textit{$LDSDGN_{\gamma}$} measure \cite{Piao2016}  has been adopted in order to evaluate  the relatedness of the generic sense of a concept with its intended sense. This approach allowed us to improve the Spearman's correlation with {\it HJ} with respect to the proposal presented  in \cite{FT2021a}, for evaluating  semantic similarity of concepts organized according to a taxonomy. 
	
	
	
	
	For future work we would like to investigate   the use of the  $SemSim$ semantic similarity method \cite{Formica13}, in order  support  the disambiguation step which is a fundamental aspect at the basis of  both semantic similarity and the more general notion  of semantic relatedness.
	
	
	
	
	
	
	
	
	
	
	\bigskip
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