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where c is constants, for L → $^{+}$∞ , P ( r ) ∝ r − 1 is the optimal distribution for all P ( r ) ∝ r − α distributions. 

1. Symbol and Expression Descriptions 

P ( r ) ∝ r − $^{α}$, the distribution of distance between friendships. 

R$_{α}$ , the expectation of the distance which holds P ( r ) ∝ r − $^{α}$. w 

f$_{α}$ = R$_{α}$ is the expectation of number of friends. 

When w = fL log( L $_{)}$, L is the edge length of the lattice, f denotes the number of friends when α = 1. 

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}$}$. 1 f$_{α}$ α 

q$_{F}$ $^{α}$, i = f$_{α}$ ∑ v = 1 q θ $_{v}$, $_{i}$, denotes the probability that node i is one of friends of F $^{α}$. f 2 α x 2 log x 2 $_{α}$C x 2 q x q$_{F}$ $^{α}$, $_{i}$) f 2 α − x denotes the expectation of entropy of node i 

∑ x = 1 f α f f α F $^{α}$, $^{$_{i}$(1}$− 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$_{α}$ 

2. Case 1: α < 1 

<img src='content_image/344979.jpg'>

Therefore, for a given w = cL , where c is a constant, we have 

Because, 

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and 

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