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## B. Algorithm of OM 

When the lattice size is 10000 × 10000, it is hard to record all nodes’ links information. Thus, we first represent each node an index running from 1 to 10 $^{8}$. This way is easy to obtain a function r ( u , v ) to calculate the lattice distance between any pair of nodes u and v , where u , v are now the running index. 

In the OM model all nodes are equivalent. Without losing generality, we can set any node as u = 1. To construct the spatial network on the lattice, each time we first randomly generate a distance r according to the distribution P ( r ) ∝ r − $^{α}$, r ∈ { 1 , 2 , · · · $^{L}$} . Then from the set of nodes which have distance r from node 1, a node is chosen randomly as a friend of node 1 and a directed link is constructed. Repeating the execution until the energy achieves the limit constraint. After the executions we can get all the friends of node 1. Employing the same approach, we can also get all the next nearest friends of node 1. 

## C. Analysis on OM 

In this section we will prove that if energy hold 

<img src='content_image/330442.jpg'>

