DEGREE DISTRIBUTION OF A RANDOM GRAPH -pr0-pn <k> <k>=pN-1) p= (N-) For large N and small k,we can use the following approximations: w 1=W-10W-1-1W-1-2)(W-1-k+10W-1-k1_W-I) k(N-1-)川 m0-pr-=w-1-n0-=(w1l=-<k>0-N点）产-< (1-p)N-1t=e< P-pa-pr-p心-"气长e=e Network Science:Random Graphs DEGREE DISTRIBUTION OF A RANDOM GRAPH Network Science: Random Graphs P(k) = N -1 k æ è ç ö ø ÷ p k (1- p) (N-1)-k < k >= p(N -1) p = < k > (N -1) For large N and small k, we can use the following approximations: N -1 k æ è ç ö ø ÷ = (N -1)! k!(N -1- k)! = (N -1)(N -1-1)(N -1- 2)...(N -1- k +1)(N -1- k)! k!(N -1- k)! = (N -1) k k! ln[(1- p) (N -1)-k ] = (N -1- k)ln(1- < k > N -1 ) = -(N -1- k) < k > N -1 = - < k > (1- k N -1 ) @ - < k > (1- p) (N-1)-k = e -<k> P(k) = N -1 k æ è ç ö ø ÷p k (1- p) (N-1)-k = (N -1) k k! p k e -<k> = (N -1) k k! < k > N -1 æ è ç ö ø ÷ k e -<k> = e -<k> < k > k k! ln(1+ x) = (-1) n+1 n n=1 ¥ å x n = x - x 2 2 + x 3 3 - ... for x £1