Table 1 Notation

\({\mathbf {G}}\)

Undirected unweighted input graph

\(\hat{{\mathbf {G}}}\)

A connected sub-graph of \({\mathbf {G}}\)

\({\mathbf {V}}\)

Set of vertices of \({\mathbf {G}}\) (\(|{\mathbf {V}}|=n\))

\({\mathbf {V}}_{\hat{{\mathbf {G}}}}\)

Set of vertices of \({\mathbf {G}}\) inducing \(\hat{{\mathbf {G}}}\) (Set of vertices of \(\hat{{\mathbf {G}}}\))

\(\overline{{\mathbf {V}}_{\hat{{\mathbf {G}}}}}\)

Set of vertices in \({\mathbf {V}} \setminus {\mathbf {V}}_{\hat{{\mathbf {G}}}}\)

\({\mathbf {V}}_{HSN}\)

Set of vertices of HSN

\({\mathbf {E}}\)

Set of edges of \({\mathbf {G}}\) (\(|{\mathbf {E}}|=m\))

\(e_{s,t}\)

Edge connecting vertices s and t

\(d_{{\mathbf {G}}}(s,t)\)

Distance between vertices s and t in \({\mathbf {G}}\)

\({\hat{d}}_{{\mathbf {G}}}(s,t)\)

Normalized distance between vertices s and t in \({\mathbf {G}}\)

\(\sigma _{s,t}\)

Number of shortest paths between vertices s and t

\(\sigma _{s,t}(v)\)

Number of shortest paths between vertices s and t which cross vertex v

\({\hat{\sigma }}_{s,t}\)

Normalized number of shortest paths between vertices s and t

\({\mathbf {P}}_{s}(v)\)

Set of direct predecessors of vertex v on shortest paths from vertex s

\({\mathbf {P}}_{s}({\mathbf {V}})\)

Set of direct predecessors of vertices in \({\mathbf {V}}\) on shortest paths from vertex s

BC(v)

Betweenness centrality of vertex v

\(\delta _{s,t}(v)\)

Pair-dependency of pair of vertices (s, t) on the intermediary vertex v

\(\delta _{s,\cdot }(v)\)

Dependency score of vertex s on vertex v due to all destination vertices

\(\delta _{s,{\mathbf {V}}_{\hat{{\mathbf {G}}}}}(v)\)

dependency score of vertex s on vertex v due to all destination vertices in \({\mathbf {V}}_{\hat{{\mathbf {G}}}}\)

\({\mathbf {C}}\)

Set of clusters of \({\mathbf {G}}\)

\({\mathbf {C}}_i\)

A generic cluster in \({\mathbf {C}}\)

\({\mathbf {C}}(v)\)

The cluster vertex v belongs to

\({\mathbf {C}}^{*}\)

Set of extended clusters in \({\mathbf {G}}\)

\({\mathbf {C}}^{*}_{i}\)

A generic extended cluster in \({\mathbf {C}}^{*}\)

\({\mathbf {K}}\)

Set of all the equivalence classes

\({\mathbf {K}}_i\)

An equivalence class

\({\mathbf {K}}_{{\mathbf {C}}_i}\)

Set of equivalence classes of cluster \({\mathbf {C}}_i\)

\({\mathbf {P}}\)

Set of all the pivots

\(k_{i}\)

Pivot node of the equivalence class \({\mathbf {K}}_{i}\)

\(\mathbf {EN}\)

Set of all the external nodes

\(\mathbf {EN}_{{\mathbf {C}}_{i}}\)

Set of external nodes of cluster \({\mathbf {C}}_{i}\)

\(\mathbf {BN}\)

Set of all the border nodes

\(\mathbf {BN}_{{\mathbf {C}}_{i}}\)

Set of border nodes of cluster \({\mathbf {C}}_{i}\)

\(\mathbf {BN}_{{\mathbf {C}}_{i}}(s,t)\)

Set of border nodes of cluster \({\mathbf {C}}_{i}\) on shortest paths from \(s\in {\mathbf {V}}_{{\mathbf {C}}_{i}}\) to \(t\in \overline{{\mathbf {V}}_{{\mathbf {C}}_{i}}}\)

\(b_i\)

A generic border node in \(\mathbf {BN}\)

\(\delta ^\gamma _{s,\cdot }(v)\)

Global dependency score of s on v due to all \(t\in \overline{{\mathbf {V}}_{{\mathbf {C}}(s)}}\) (same as \(\delta ^\gamma _{s,\overline{{\mathbf {V}}_{{\mathbf {C}}(s)}}}(v)\))

\(\delta ^\gamma _{s,{\mathbf {V}}_{{\mathbf {C}}(v)}}(v)\)

Global dependency score of s on v due to all \(t\in (\overline{{\mathbf {V}}_{{\mathbf {C}}(s)}} \cap {\mathbf {V}}_{{\mathbf {C}}(v)})\)

\(\delta ^\gamma _{s,{\mathbf {V}}_{{\mathbf {C}}(v)}}( {\mathbf {V}}_{{\mathbf {C}}(v)})\)

Global dependency score of s on vertices in \({\mathbf {V}}_{{\mathbf {C}}(v)}\) due to all \(t\in (\overline{{\mathbf {V}}_{{\mathbf {C}}(s)}} \cap {\mathbf {V}}_{{\mathbf {C}}(v)})\)

\(\delta ^\gamma (v)\)

Sum of all the global dependency scores (global BC) on v

\(\delta ^\gamma ({\mathbf {V}})\)

Sum of all the global dependency scores (global BC) on vertices in \({\mathbf {V}}\)

\(\delta ^\lambda _{s,\cdot }(v)\)

Local dependency score of s on v due to all \(t\in {\mathbf {V}}_{{\mathbf {C}}(s)} = {\mathbf {V}}_{{\mathbf {C}}(v)}\)

\(\delta ^\lambda _{s,\cdot }({\mathbf {V}})\)

Local dependency score of s on vertices in \({\mathbf {V}}\) due to all \(t\in {\mathbf {V}}_{{\mathbf {C}}(s)} = {\mathbf {V}}_{{\mathbf {C}}(v)}\)

\(\delta ^\lambda (v)\)

Sum of all the local dependency scores (local BC) on v

\(\delta ^\lambda ({\mathbf {V}})\)

Sum of all the local dependency scores (local BC) on vertices in \({\mathbf {V}}\)

\(\delta ^\epsilon _{s,\cdot }(v)\)

Dependency score of s on v, as external node, due to all \(t\in {\mathbf {V}}_{{\mathbf {C}}(s)}\)

\(\delta ^\epsilon _{s,\cdot }(\mathbf {EN})\)

Dependency score of s on external nodes \(\mathbf {EN}\) due to all \(t\in {\mathbf {V}}_{{\mathbf {C}}(s)}\)

\(\delta ^\epsilon (v)\)

Sum of all the dependency scores on v as external node

\(\delta ^\epsilon (\mathbf {EN}_{{\mathbf {C}}(s)})\)

Sum of all the dependency scores on external nodes of cluster \({\mathbf {C}}(s)\)