content_image/350750.jpg content_image/350751.jpg content_image/350752.jpg content_image/350749.jpg content_image/350756.jpg content_image/350754.jpg content_image/350755.jpg content_image/350757.jpg
Thus, for any F $^{α}$, 

It implies that 

Thus 

which is a monotonic increasing function with α < 1. 

3. Case 2: α > 1 

Lemma : if q ∈ (0 , 1 $_{3}$), for any large enough z we have 

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$^{where}$− ∑ z x = 1 x z log x $_{z}$C x $_{z}$q $^{$^{x}$(1}$− q ) $^{z}$−x denotes the expectation of entropy of a node with the probability q to be chose and the total choosing time is z (as shown in Fig. 5). Proof : 

$^{According to Law of Large Numbers, lim$_{z}$}$→∞ − ∑ z x = 1 x z log x $_{z}$C x $_{z}$q$^{x}$(1 − q ) $^{z}$−x $^{=}$− q log q . Thus, we just need to prove 

is a monotonic increasing function. 

For large enough z , normal distribution is a well approximation to binomial distribution then we have z 

where σ 2 = zq (1 − q ) , µ = zq . 

<img src='content_image/350756.jpg'>

Obviously, 

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