where c is constants, for L → $^{+}$∞ , P ( r ) ∝ r − 1 is the optimal distribution for all P ( r ) ∝ r − α distributions. When w = fL log( L $_{)}$, L is the edge length of the lattice, f denotes the number of friends when α = 1. 1. Symbol and Expression Descriptions Therefore, for a given w = cL , where c is a constant, we have Because, and R$_{α}$ = ∫ L 1 x $^{1}$− $^{α}$dx + O (1) L 1 x − $^{α}$dx + O (1) = 1 $^{2}$− $_{α}$( L $^{2}$− α − 1) + O (1) 1 $^{1}$− $_{α}$( L $^{1}$− α 1) + O (1) ≈ 1 − α 2 α L . (7) lim $^{L}$→∞ f$_{α}$ = lim $^{L}$→∞ w R$_{α}$ = c (2 − α ) 1 α . (8) lim $^{L}$→∞ max q α i , j ≤ lim $^{L}$→∞ 1 ∫ L 1 x − $^{α}$dx + O (1) = lim $^{L}$→∞ 1 1 $^{1}$− $_{α}$( L $^{1}$− α − 1) + O (1) = 0 (9) q$_{F}$ $^{α}$, i ≤ max q α i , $_{j}$. (10) 13 2. Case 1: α < 1 P ( r ) ∝ r − $^{α}$, the distribution of distance between friendships. R$_{α}$ , the expectation of the distance which holds P ( r ) ∝ r − $^{α}$. f$_{α}$ = w R$_{α}$ is the expectation of number of friends. q α i , j denotes the probability of the connection between node i and j for a given α . F α $^{=}${ θ α $_{1}$, θ α $_{2}$, · · · , θ α f$_{α}$ } , denotes the set of friends of node 1, where f$_{α}$ = w $_{R$_{α}$}$. q$_{F}$ $^{α}$, i = 1 f$_{α}$ ∑ f$_{α}$ v = 1 q α θ $_{v}$, $_{i}$, denotes the probability that node i is one of friends of F $^{α}$. f 2 α x x x x f 2 x ∑ α x = 1 f 2 α log f 2 $_{α}$C x f 2 α q x F $^{α}$, $^{$_{i}$(1}$− q$_{F}$ $^{α}$, $_{i}$) f α − x denotes the expectation of entropy of node i when the chosen probability of node i is q$_{F}$ $^{α}$, i and the time of choosing is f 2 $_{α}$. ε$_{α}$ = ∑ n i = 1 ∑ f 2 α x = 1 x f 2 α log x f 2 $_{α}$C x f 2 α q x F $^{α}$, $_{i}$(1 − q$_{F}$ $^{α}$, $_{i}$) f 2 α − $^{x}$, denotes the expectation of entropy for a given F $^{α}$. E ( ε$_{α}$ ), denotes the expectation ε$_{α}$