# Table 1 Notation

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 $${\mathbf {C}}(s)$$ 