ε α w −0.5 0 0.5 1 1.5 2 500 1000 1500 2000 3 4 5 6 7 0 0.5 1 1.5 5.5 6 6.5 7 α ε f = 50 f = 40 f = 30 0 1 2 3 4 x 10 4 6 6.2 6.4 6.6 6.8 7 7.2 Time ε 0 2 4 x 10 4 10 −2 10 −3 10 −4 10 −5 Time Diff 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 r P ( x > r ) a b c d FIG. 3: The results of evolutionary model when L = 100 and f = 50. a . The simulation results of OM on a toroidal lattice with the preset power law distribution P ( r ) ∝ r − $^{α}$. b . the dependence of the information entropy ε or α for f around 40 in the OM. We can see that when f = 50, the optimal exponent is 0.95 and it is very close to − 1. c . The changes of entropy in the EM with the evolution time. The entropy is fixed and the system archives a steady state. The fixed entropy is 7. 15 which is very close to the entropy 7.18 in the network of L = 100 which we preset the distribution is P ( r ) ∝ r − $^{1}$. The inset denotes the di ff erence of the time-entropy curve which implies that the di ff erence decays exponentially. From it we can see that for a su ffi cient long time evolution, the entropy converges to a fixed va lue and the system achieves a steady state. d . The cumulative distribution of the distance in EM is shown in log-linear plot in the steady state. We can see that this distribution is very close to P ( r ) ∝ r − 1 (dashed line). term evolution from the initial uniform distribution (each node modify the neighborhood more than 40000 times), the system achieves its stationary state (Fig. 3e). The maximum entropy is 7.15 and the corresponding PDF of the distance between the friends scales as P ( r ) ∝ r − 1 (Fig. 3d and supplementary IIIB), which are very close to the results obtained by OM. So we conclude that P ( r ) ∝ r − 1 is the optimal PDF of distances of friendships for collecting maximal information. It implies that, the spatial structure of the real social networks is the most optimal structure which leads to the maximum diversity of the friends’ location and can help individuals to collect infor- mation e ffi ciently. We note that, it can be proved analytically, under the assumption that the energy 7