Table 1 Network topology and topic centrality metrics [42,43,45]

Modularity

Measures non-trivial grouping structure within a network, based on the observed number of edges within a subset of nodes, to the number of edges expected from random assignment

\(\mathop \sum \limits_{k = 1}^{k} [f_{kk} \left( G \right) - f_{kk}^{*} ]^{2}\)

where \(f_{kk}^{*}\) is the expected value of \(f_{kk}\) under some model of random edge assignment

Transitivity

Measures the extent to which nodes in a network cluster together, based on the ratio of the number of triangles and the number of connected triples

\(\frac{{3\tau_{\Delta } \left( G \right)}}{{\tau_{3} \left( G \right)}}\)

where \(3\tau_{\Delta } \left( G \right)\) is the number of triangles in the graph, and \(\tau_{3} \left( G \right)\) is the number of connected triples

Density

Measures the ratio of the number of edges in a graph to the maximum number of possible edges

\(\frac{{\left| {E_{H} } \right|}}{{\left| { V_{H} } \right| \left( { \left| {V_{H} } \right| - 1 } \right)/ 2}}\)

where |E| is the number of edges and |V| is the number of nodes in the graph

Average Path Length

Measures the mean for the shortest paths between all nodes in a network

\(_{{ \frac{1}{{n \cdot \left( {n - 1} \right)}} \cdot \mathop \sum \limits_{i \ne j} d\left( { v_{i} ,v_{j} } \right)}}\)

where d(v_i, v_j) is the shortest path between nodes v_i and v₂, and n is the number of nodes in the graph

Diameter

Measures the largest distance between any pair of nodes in a network

\(max_{u, v} d\left( {u, v} \right)\)

where d(u, v) is the distance between nodes u and v

Betweenness

The fraction of shortest paths that pass through a node

\(\mathop \sum \limits_{s \ne t \ne v \in V} \frac{{\sigma {(}s, t {|} v)}}{{\sigma \left( {s, t} \right)}}\)

where \(\sigma {(}s, t {|} v)\) is the number of shortest paths between s and t that pass through v, and \(\sigma \left( {s, t} \right) = \mathop \sum \limits_{v} \sigma {(}s, t {|}v)\)

Degree

The number of edges connected to a node

\(g\left( v \right) = {\text{deg}}\left( v \right)\)

PageRank

A measure of node importance based on the likelihood of reaching a given node when randomly following links within a network

\(\alpha \mathop \sum \limits_{j} \alpha_{ij} \frac{{x_{j} }}{L\left( j \right)} + \beta\)

where \(L\left( j \right) = \mathop \sum \limits_{i} a_{ij}\) is the number of neighbors of node j, and \(\alpha\) is a damping factor