10 0 10 2 10 −4 10 −3 10 −2 10 −1 10 0 z y q=0.25 q=0.2 q=0.1 q=10 −2 q=10 −3 q=10 −4 10 0 10 2 0 0.2 0.4 0.6 0.8 1 1.2 x 10 −4 z y 10 0 10 2 0 0.5 1 1.5 2 2.5 x 10 −9 q=10 −10 q=10 −15 q=10 −20 q=10 −15 q=10 −20 FIG. 5: Plot of y $^{=}$− q log $^{q}$− g ( z ). From the plot we can see that Lemma is true. More over when z is small g ( z ) > − q log q is also correct. and More over e ( $^{x}$−zq ) 2 2 zq ( $^{q}$− 1) x > 0 (19) ∫ z 1 [log x z ( q $^{2}$z2 − 3 q $^{2}$z + 3 qz − x $^{2}$) − 2 zq $^{2}$+ 2 zq ] dx = ( 1 9 − q $^{2}$) z 3 + Θ ( z $^{2}$log z ) > 0 (20) when q < 1 $_{3}$, where, Θ ( z $^{2}$log z ) denotes the same order of z $^{2}$log z . Thus, g ′$_{z}$ ( z ) > 0 which implies that g ( z ) is a monotonic increasing function and − q log q > − z ∑ x = 1 x z log x z C x $_{z}$q$^{x}$(1 − q ) $^{z}$−$^{x}$. (21) . For case 2, according to Lemma and Levy stable distribution property (the distance between the next nearest neighbor and the origin is also obey P ( r ) ∝ r − α when α > 1). So for large enough friends number we have: E ( ε$_{α}$ ) < L ∑ r = 1 4 r r − α 4 rZ ( α ) log r − α 4 rZ ( α ) (22) = 1 Z ( α ) L ∑ r = 1 r − $^{α}${ $^{(}$− α − 1) log r − log[4 Z ( α $^{)]}$} . (23) lim $^{L}$→ $^{+}$∞ E ( ε$_{α}$ ) = ( a − 1)(2 log 2 + log Z ( a )) + a + 1 2( a − 1) 2 (24) More over we can get: 15