## 2 Bibliometric Analysis
### Publication Networks  
Every bibliometric network that contains academic publications as nodes is considered to be a _publication network_. For publications, the affiliation network is described by a binary matrix \(A_{pc}\) of size \(m\times n\), with \(m\) denoting the number of publications of a given dataset, and \(n\), denoting the amount of disjoint citations that are listed in the publications. Every entry \(a_{ij}\) yields whether the publication \(i\) has cited the citation \(j\) or not. A co-citation of two publications is given if both publications are cited by a third publication [1] (see Figure 3). The two publications are then considered a knowledge base [13]. In a co-citation network, the similarity matrix \(S_{cc}\) is calculated by \(S_{cc}=A_{pc}^{T}A_{pc}\) with size \(n\times n\)[1].  
_Bibliographic coupling_ is another similarity measure for academic publications. If one or more citations of two publications are similar, the two publications are considered bibliographically coupled [1] (see Figure 3). Bibliographically coupled publications form a research front [13]. To compute a bibliographic coupling network, the same bipartite graph and, thus, matrix \(A_{pc}\) as for co-citation networks is used; however, the similarity matrix \(S_{bc}\) is given by \(S_{bc}=A_{pc}A_{pc}^{T}\) with size \(m\times m\)[1].  
In addition to those two similarity measures, which are based on the citations of publications, _lexical measures_ can be used to define the similarity of publications. Lexical measures compare the textual content of publications, such as their title, keywords, abstract, and full text. If two publications contain the same word, they are connected by the latter. In information retrieval theory, term occurrences in documents are described by a document-feature-matrix (DFM) \(A_{tf}\) whose entry \(a_{ij}\) yields a _term frequency_, i.e., how often publication \(i\) contains word \(j\).  
As a DFM has the same properties as an affiliation matrix of a publication-word network, \(A_{tf}\) can be used to describe this affiliation network. The similarity matrix of the term frequency network is then given by \(S_{tf}=A_{tf}A_{tf}^{T}\)[1]. In order to optimize the results, special characters and stop words in publication texts may be removed, and the remaining words may be compressed before \(A_{tf}\) is created [13].  
Figure 3: Co-citation and bibliographic coupling [13].  
Lexical similarity measures pose a problem that needs to be addressed: simple counting of words (which is done when creating a DFM) favors those words that often occur within many publications, which may be counterproductive, as those words may not reflect an interesting similarity between publications. Intuitively, more meaningful are words that often occur in only a few publications. To account for this problem, the _term frequency-inverse document frequency_ (TF-IDF) measure can be used [19]. Let \(m\) be the number of publications and \(n\) the number of disjoint words, which occur in any of the \(m\) publications, \(W\) a matrix of size \(m\times n\) with entries \(w_{ij}\) and \(IDF\) a vector of length \(n\). The inverse document frequency (IDF) is defined as [19]:  
\[IDF_{j}=log(\frac{n}{a_{ij}}) \tag{1}\]  
and the entries of the TF-IDF matrix \(W\) are defined as  
\[w_{ij}=a_{ij}*IDF_{j} \tag{2}\]  
Given the IDF, an improved lexical similarity measure can be used to create a TF-IDF network by calculating its similarity \(S_{tfidf}=WW^{T}\).