of friendships without any constraints of energy, we expect P ( r ) ∝ r to be the optimal entropy information since each node has the same probability in the information sequence. In practice, individuals naturally have a limited energy w . Our numerical results shown in Fig. 2. a indicate that when α ≈ 1, the information entropy ε is near its maximum value for a very broad range of w . For the range w ∈ (5 × 10$^{4}$, 10$^{6}$) and f ∈ (50 , 1000), we find the optimal α to be α = 1 ± 0 . 05. When the size of the lattice is L and P ( r ) ∝ r − $^{1}$, the mean distance between friends is L log $_{L}$. Therefore, we can find the average number of friends f to be f = w log L L (5) which gives one to one correspondence between f and w at the optimal state. When L = 10000 and w ∈ (5 × 10$^{4}$, 10$^{6}$) the average number of friends is f ∈ (50 , 1000) which indeed corresponds to reality [30]. In particular, when considering the average number of friends we contact in one year, f = 300 [30], the optimal value of α is α $^{=}$− 0 . $^{99}$± 0 . 03 (as shown in Fig. 2). Our results suggest that P ( r ) ∝ r − 1 is the optimal distribution for collecting information be- tween all power law distributions. Is P ( r ) ∝ r − 1 the optimal distribution when considering all kinds of distributions? We will demonstrate, based on the following evolutionary model (EM), that among all kinds of distributions, P ( r ) ∝ r − 1 is still the optimal one. In the EM, we also con- struct a network on a lattice of size $^{L}$× L . A node u$_{i}$ is one of the neighbors of node u when there is a direct link from u to u$_{i}$ . Each node u has friends at distances r ( u , u$_{i}$ ) subject to ∑ $^{u$_{i}$}$∈U r ( u , u$_{i}$ ) ≤ w , where U is the set of all neighbors of node u . In the initial stage of the EM, P ( r ) is set to be a uniform distribution. Then we employ the extremal optimization method [31], to maximize the entropy through the evolution of network architecture. At each step, a node is chosen randomly. For a chosen node u , we make two operations, deleting and adding neighbors according to the marginal improvement of entropy. Suppose u has k neighbors. For the deleting execution, we first calculate the marginal entropies of each neighbor of no de u , { △ E$_{u}$$_{1}$ r ( u , u$_{1}$ $_{)}$, △ E$_{u}$$_{2}$ r ( u , u$_{2}$ $_{)}$, · · · , △ E$_{u}$$_{k}$ r ( u , u$_{k}$ ) } , where △ E$_{u}$$_{i}$ means the change in the entropy of node u when we delete node u$_{i}$ from the neighborhood of node u with other parameters being unchanged. Then we randomly select a comparatively small | △ E$_{u}$$_{i}$ r ( u , u$_{i}$ ) | with probability Pr ( u$_{i}$ ) proportional to ( $^{rank}$| △ E$_{u}$$_{i}$ r ( u , u$_{i}$ ) | ) − $^{1}$− log ( k ) [31] and delete u$_{i}$ from u ’s neighborhood. For the adding link execution, suppose v$_{1}$ , v$_{2}$ , · · · , v$_{h}$ are all the candidates which are currently next nearest neighbors of node u . We first calculate the marginal entropies of each of the candidate, { △ E$_{v}$$_{1}$ r ( u , v$_{1}$ $_{)}$, △ E$_{v}$$_{2}$ r ( u , v$_{2}$ $_{)}$, · · · , △ E$_{v}$$_{h}$ r ( u , v$_{h}$ ) } , then we also employ the extremal optimization method to choose a node whose marginal entropy is comparatively large among all candidates’ marginal 5