Degree landscapes in scale-free networks
K. Sneppen and
Review E 74:36119 (2006)
(A mathematical model for
creating networks of widely different topology with a conserved
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.
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
1) Degree hierarchy. The model starts in this
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
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.
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
3) Random rank with
Click the Start/Stop button to make the
network rewire. Click the Degree/Random
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.
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