Supplementary Information IV. EXPLANATION FOR SPATIAL SCALING OF LIVEJOURNAL In the empirical study of the LiveJournal data set [1], for each distance r , Q ( r ) is the fraction of friendships among all pairs u , v of LiveJournal users with r ( u , v ) = r . Q ( r ) = F ( r ) S ( r ) ∝ r − $^{1}$. Here, F ( r ) denotes the total number of friendships with distance r and S ( r ) is the total number of pairs of nodes that have distance r . The LiveJournal social network has a fractal dimension of about 0 . 8 (they define the fractal dimension of a network as the expone nt d of the best-fit function rank$_{u}$ ( v ) = c · r ( u , v ) $^{d}$, where rank$_{u}$ ( v ) is the number of people who live closer to u than v and c is a constant). We know that for any d -dimensional lattice, the number of nodes that have the same distance r to a given node is proportional to r $^{d}$− $^{1}$. In fractal networks, d should be the fractal dimension. Thus the probability density function P ( r ) of the geographic distance r between friends is about P ( r ) ∝ r $^{d}$− 1 · Q ( r ) = r 0 . $^{8}$−1 · r − 1 = r − 1 . 2 , which is close to r − $^{1}$. We assume that the information obtained from the social network is actually related with the influence of friendships. Indeed, in our social life, our fri ends always talk something about their friends. Thus, we assume that friends and next nearest friends are most important and is enough to consider them in our model. However, Christakis and Fowler have found recently that the influence is mainly within three degrees of separation and ca ll this finding the “Three Degrees of Influence Rule” [2]. It is computationally di ffi cult to take into account more than two degrees of separation of friends to study a system of 10 4 × 10$^{4}$. We have therefor performed the numerical experiments of the OM in 3000 × 3000 size lattice with w ≃ 10 4 ( f = 300) and found that the simulated results were similar when we took into account friends and next nearest friends, and three degrees of separation (as shown in Fig. 4). A. Why We Only Consider Friends and Next Nearest Friends? V. ABOUT THE OPTIMIZATION MODEL (OM) 11