Degree landscapes

Instructions

This Java applet
(
Java 1.4) was written by Jacob Bock Axelsen.

The simulation starts automatically by rewiring a scale-free network to degree hierarchy.

The buttons: The Start button toggles rewiring/non-rewiring of the network. Press the Initialize button to make a new network from the given parameters. The Degree/Random button toggles whether the rewiring is according to degree or quenched randomly assigned ranks. The Layout/Landscape button toggles whether the view is direct graph layout or the corresponding degree landscape (see right on this page). The error rate textfield sets the rate with which rewirings disobey the rewiring constraint (1.0 = randomly rewiring). The exponent  textfield sets the power-law exponent for generating the scale-free network. The size textfield sets the number of nodes in the network. The Shake button shakes the network a bit in case the layout is stuck in a local minimum.

To the lower right is seen the Zipf plot of the network. Notice that this only changes when the network is initialized (resampled).

The layout is computed using a homemade version of the Kamada-Kawai algorithm for optimizing graph layout while considering the geodesic lengths in the graph. The landscape is computed using bicubic interpolation and shading negative values of the x-component of the gradient.


Degree landscapes in scale-free networks

J.B. Axelsen, S Bernhardsson, M. Rosvall, K. Sneppen and A Trusina

Physical Review E 74:36119 (2006)


(A mathematical model for creating networks of widely different topology with a conserved degree-distribution)
 

The model deals with the organization of hubs in networks with a broad distribution of degrees in the following way: suppose a network is dynamically rearranged according to local properties of each node. The two simplest possibilities are either considering the degree/connectivity or some other feature of the node. This other feature is most interesting if it is completely unrelated to degree. The model explores what happens if nodes of similar degree (hot color code) are brought closer together. This is called degree hierarchy. The model also explores what happens if nodes of similar rank (jet color code) is brought together. We assign ranks completely randomly only when the network is initialized.
 
The interesting findings are that 1) the network separates the hubs from each other maximally when rewiring according to random ranks (jet color code) and 2) it turns into a small world as well when introducing small errors (~0.02) in the rewiring scheme. These feature are captured by the degree landscape mode (smooth mountain island landscape).
 
Walk-thru tutorial:
1) Degree hierarchy. The model starts in this configuration.
 
 
 
- this is a narrow 'mountain' where all commands passes properly through the hierarchy as determined by degree.
 
2) Random rank hierarchy. Click the Degree/Random rank - button. After some time watching the rewirings you will end up with something like this:
 
 
 
- this is a whole mountain range with distinctively separated peaks. All distances are great so this is not a small world.
 
3) Random network. Enter 1.0 into the error textfield and press the Shake 'n Read button. The network will now perform random rewirings indefinitely so after some time you should click the Start/Stop button.
 
 
 
- this is a broad rugged single mountain. Notice that the network is a small world automatically and that the random ranks are completely mixed. The high degree nodes have so many links that they will connect to each other quite often by random chance. Notice that the network is quite close to a perfect degree hierarchy.
 
3) Random rank with an error. Click the Start/Stop button to make the network rewire. Click the Degree/Random rank - button to make the network use the random rankings (jet color code). Then enter an error of order .02-.05 into the error textfield and click the Shake 'n Read button. After some time you should click on the Start/Stop button.
 
 
 
- notice how the mountain peaks i.e. the hubs are separated from each other while the network is still a small world? At the same time the random ranks are still somewhat close... this is the case in biology. Click here for more insights