Practical Problem 17: Network Growth

In this practice problem you will implement network growth models. In these models a system is usally initialized with a few nodes, say 2 or 3 and at every simulation step new nodes come in and attach with their incoming “link stubs” to the nodes that already exist in the network according to a set of rules that define the model. In some models the growth does not go on indefinitely, typically, when a certain number of nodes is reached, nodes are removed randomly from the network every time a new node enters.

Preparations

• Prepare netlogo with the usual setup and go function templates.

• Create a small network with 3 nodes, that are all connected. The nodes should have circle shape and color black. Make sure you paint all the patches white in your setup function, otherwise you won’t see the nodes.

• Prepare a function called spring that gets called at every call of the go function and manages the layout, like in the previous example. Make sure you provide sliders for the layout parameters, so you can adapt the layout.

Model 1: Random targets

In this models new nodes come in with a single link and attach to a randomly chose neighbor. You can generate new nodes with crt 1 [...] and use the function create-link-with to connect with existing nodes.

Now try to scale the size of the nodes according to their degree, that means the number of connections a node as.

Once your network has reached a size of 500 nodes, start removing a random node for every node that comes in.

Now generalize the model such that an incoming node either comes in with 1 or with 2 links, each with a 50% chance.

Model 2: Random neighbors

This model is identical to the previous one with one important difference:

• An incoming node picks a random target

• It connects with probability $Q$ to the target and with probability $1-Q$ to a random link neighbor of the target.

When you are done with it, make sure you see what the difference in outcome is to Model 1.

Model 3: Preferential Attachment

This model is very famous and also controversial. It is based on the “rich get richer” effect. This is how it works: Incoming nodes attach to existing nodes with a probability that is proportional to the degree of the network node. Like: Friends with a lot of friends are more attractive to be befriended.

How can this be done?

First you need to add what is known as a netlogo extension. The extension we need here is called Rnd. Read the manual to learn how to use extensions.

When the extension is loaded you can use the function rnd:weighted-one-of to pick nodes according to their degree.

Model 4: Preferential attachment to clustering coefficient

This is a variant of Model 3, only that incoming nodes connect preferentially to nodes with a high clustering coefficient.

For this, another netlogo extension will come in handy: the Network extension. In it you will find that for nodes in a network one can compute the clustering coefficient. This is a measure that computes how well connected your neighbors are. More precisely it computes the links among the neighbors of a node devided by the maximum number of links possible among the neighbors.

Modify your model so that new links attach to existing nodes proportional to the clustering coefficient.