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Table 1 Notation

From: Fast cluster-based computation of exact betweenness centrality in large graphs

Notation Description
\({\mathbf {G}}\) 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 (st) 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 \({\mathbf {C}}(s)\)