▁This ▁measures ▁how ▁many ▁edges ▁are ▁present ▁between ▁nodes ▁of ▁degree ▁at ▁least ▁k , ▁normal ized ▁by ▁how ▁many ▁edges ▁there ▁could ▁be ▁between ▁these ▁nodes ▁in ▁a ▁complete ▁graph . ▁When ▁this ▁value ▁is ▁close ▁to ▁ 1 ▁for ▁values ▁of ▁k ▁close ▁to ▁, ▁it ▁is ▁interpreted ▁that ▁high ▁degree ▁nodes ▁of ▁the ▁network ▁are ▁well ▁connected . ▁The ▁associated ▁sub graph ▁of ▁nodes ▁with ▁degree ▁at ▁least ▁k ▁is ▁also ▁called ▁the ▁" Rich ▁Club " ▁graph . ▁ ▁Normal ized ▁for ▁topology ▁random ization ▁ ▁A ▁criticism ▁of ▁the ▁above ▁metric ▁is ▁that ▁it ▁does ▁not ▁necessarily ▁imply ▁the ▁existence ▁of ▁the ▁rich - club ▁effect , ▁as ▁it ▁is ▁monot on ically ▁increasing ▁even ▁for ▁random ▁networks . ▁In ▁certain ▁degree ▁distributions , ▁it ▁is ▁not ▁possible ▁to ▁avoid ▁connecting ▁high ▁degree ▁hub s . ▁To ▁account ▁for ▁this , ▁it ▁is ▁necessary ▁to ▁compare ▁the ▁above ▁metric ▁to ▁the ▁same ▁metric ▁on ▁a ▁degree ▁distribution ▁pres erving ▁random ized ▁version ▁of ▁the ▁network . ▁This ▁updated ▁metric ▁is ▁defined ▁as : ▁ ▁where ▁ ▁is ▁the ▁rich - club ▁metric ▁on ▁a ▁maxim ally ▁random ized ▁network ▁with ▁the ▁same ▁degree ▁distribution ▁ ▁of ▁the ▁network ▁under ▁study . ▁This ▁new ▁ratio ▁disc ount s ▁un av oid able ▁struct ural ▁correl ations ▁that ▁are ▁a ▁result ▁of ▁the ▁degree ▁distribution , ▁giving ▁a ▁better ▁indicator ▁of ▁the ▁significance ▁of ▁the ▁rich - club ▁effect . ▁ ▁For ▁this ▁metric , ▁if ▁for ▁certain ▁values ▁of