Figure 1. Illustration of the undirected Kleinberg’s spatial network constrained on a 2-dimensional periodic square lattice.

Each node has four short-range links (colored blue for node i) and one long-range link (colored black). The probability density function (PDF) of a node to have a long-range link at Manhattan distance r scales as . For the target node i (colored red), when r = 2, there are eight candidate nodes (colored green), from which we can choose an uncoupled node j to make a connection. For another target node u, we can choose to connect it with node v when r = 3.