Table 3 Analytical metrics for structural network and co-occurrence analysis

Degree centrality

\({{DC}}_{i}=\mathop{\sum }\limits_{j\in U(i)}{I}_{i,j}(j\ne i)\)

Identification of those pivot node \(i\) based on the “star effect”, with the assumption of an indicator function as \({I}_{i,j}\left(\cdot \right)\) (\({I}_{i,j}\) = 1 only when there a direct edge linking node \({i}\) to node \(j\), else \({I}_{i,j}\) = 0) and the neighbor node sets of \(i\) as \(U(i)\)

Betweeness centrality

\({{BC}}_{i}=\mathop{\sum }\limits_{s\ne i,i\ne j,s < j}\frac{{\sum }_{s\to i\to j}{I}_{s,j}(s\ne j)}{{\sum }_{s\to j}{I}_{s,j}(s\ne j)}\)

Identification of those pivot node \(i\) based on the “bridge effect”, with the assumption of an indicator function as \({I}_{s,j}\left(\cdot \right)\) same to the above. Note that \(s\to i\to j\) means all the paths from \(j\) to s through \(i\)

Co-occurrence strength

\({C}_{i,j}=\frac{{F}_{i}\cdot {F}_{j}}{{F}_{i,j}}\)

Measuring the linking strength for the relationship between any pair of knowledge nodes, with the assumption of the paper occurrence of \(i\) as \({F}_{i}\), and the paper co-occurrence of \({i}\) and \(j\) as \({F}_{i,j}\).

Maximum co-citation Interval

\({MCI}=\mathop{\max }\limits_{{t}_{i,{co}}\in \left[T,T+\Delta T\right]}{t}_{i,{co}}\)

measuring the longest time interval of co-cited papers in the timepoint range of \(T\) to \(T+\Delta T\)

Network diameter

\(D=\mathop{\max }\limits_{p\in [1,P]}{d}_{p}\)

Calculation of the maximum value of all distances.

Average path length

\({AAL}=\frac{1}{P}\mathop{\sum }\limits_{p=1}^{P}{d}_{p}\)

Calculation of the mean value of all distances (e.g. a distance denoted as \({d}_{p}\)).

Density

\({D}_{{density}}=\frac{2m}{N\cdot \left(N-1\right)}\)

Measuring the ratio of the actual total number of edges to the number of edges in its fully connected graph, supposing that \(m\) is the total number of edge and \(N\) is the total number of nodes.

Average clustering coefficient

\({ACC}=\frac{1}{I}\mathop{\sum }\limits_{i=1}^{I}\frac{2{e}_{i}}{{{DC}}_{i}\cdot \left({{DC}}_{i}-1\right)}\)

Describing the clustering degree between vertices in a graph whose size is \(I\), which is obtained based on the degree to which adjacent nodes of a node are connected to each other. Note that \({e}_{i}\) is the number of the connections between all neighbors of node \(i\).

Modularity

\(Q=\frac{1}{2m}\mathop{\sum }\limits_{i,j}\left({A}_{i,j}-\frac{{{DC}}_{i}\cdot {{DC}}_{j}}{2m}\right)\cdot {\delta }_{i,j}\)

measuring the tightness of nodes within each cluster, which is regarded as the most commonly used evaluation indicator for evaluating the effectiveness of community detection-based clustering, with the assumption of an indicator function as \({\delta }_{i,j}\) (\({\delta }_{i,j}\) = 1 only when node \({i}\) and node \(j\) belong to the same cluster, else\(\,{\delta }_{i,j}\) = 0), the total number of edge \(m,\) and an adjacent matrix \({A}_{i,j}\) for all nodes.