Random Graph

In the random graph model, nodes are randomly distributed over a plane of size scale x scale. The number of nodes in each network is uniformly drawn from the interval on [min, max]. We divide the plane into fixed size regions according to the parameter $\tx_{scope}$. The interconnection probability $\tx_p$ decides if a pair of networks interconnect; we choose $\tx_p$ based on the size of the two networks:

$$\tx_p = \alpha e^{\beta \frac{\sqrt{n_1 n_2}}{max}}$$

where $n_1$ and $n_2$ are the number of nodes in the two networks, $\alpha$ and $\beta$ determine the scale and shape parameters of the probability distribution, respectively. So, two large networks are more likely to interconnect than two smaller networks.

If two networks interconnect, we randomly select a number of regions to interconnect according to the interconnection density $\tx_{ds}$. If there are multiple nodes from each network in the same region, we select the closest pair of nodes; if a region is selected, but one of the network does not have any node in that region, we choose another region until we meet the peering density criterion, or we have considered all regions. We allow co-location if nodes from different networks are within a geometric distance (the $vicinity$ parameter) of each other. A server placed at a co-location can send traffic to all these networks with no additional cost.