es ▁based ▁on ▁real - world ▁data ▁sets ▁claim ing ▁despite ▁the ▁fact ▁that ▁most ▁of ▁the ▁observed ▁networks ▁have ▁fat - ta iled ▁degree ▁distributions , ▁they ▁devi ate ▁from ▁being ▁scale - free . ▁ ▁Ex cess ▁degree ▁distribution ▁ ▁Ex cess ▁degree ▁distribution ▁is ▁the ▁probability ▁distribution , ▁for ▁a ▁node ▁reached ▁by ▁following ▁an ▁edge , ▁of ▁the ▁number ▁of ▁other ▁edges ▁attached ▁to ▁that ▁node . ▁ ▁Suppose ▁a ▁network ▁has ▁a ▁degree ▁distribution ▁, ▁by ▁selecting ▁one ▁node ▁( random ly ▁or ▁not ) ▁and ▁going ▁to ▁one ▁of ▁its ▁neighb ors ▁( ass uming ▁to ▁have ▁one ▁neighbor ▁at ▁least ), ▁then ▁the ▁probability ▁of ▁that ▁node ▁to ▁have ▁ ▁neighb ors ▁is ▁not ▁given ▁by ▁. ▁The ▁reason ▁is ▁that , ▁whenever ▁some ▁node ▁is ▁selected ▁in ▁a ▁heter ogeneous ▁network , ▁it ▁is ▁more ▁probable ▁to ▁reach ▁the ▁h obs ▁by ▁following ▁one ▁of ▁the ▁existing ▁neighb ors ▁of ▁that ▁node . ▁The ▁true ▁probability ▁of ▁such ▁nodes ▁to ▁have ▁degree ▁ ▁is ▁ ▁which ▁is ▁called ▁the ▁excess ▁degree ▁of ▁that ▁node . ▁In ▁the ▁configuration ▁model , ▁which ▁correl ations ▁between ▁the ▁nodes ▁have ▁been ▁ignored ▁and ▁every ▁node ▁is ▁assumed ▁to ▁be ▁connected ▁to ▁any ▁other ▁nodes ▁in ▁the ▁network ▁with ▁the ▁same ▁probability , ▁the ▁excess ▁degree ▁distribution ▁can ▁be ▁found ▁as : ▁ ▁where ▁ ▁is ▁the ▁mean - deg ree ▁( aver age ▁degree ) ▁of ▁the ▁model . ▁It ▁follows ▁to ▁that ▁fact ▁that ▁the ▁average ▁degree