Small-World models
An important class of models, called small-world models is concerned about
explaining the simultaneous existence of small average path lengths and the
high clustering in the real-world networks. The first model in this class was
proposed in 1998 by Duncan Watts and Steven Strogatz, and hence many times
their model is called 'the' small-world model.
* It is relatively easy to construct models of networks with high transitivity
or clustering: e.g. triangular lattice and non-locally coupled ring. However,
such networks are "large-world", meaning that the average distance between the
vertices grows much faster than what is observed in real networks.
* On the other hand, completely random networks like the ER graph have very
small average path lengths that vary as log(n). However, these models have
zero clustering in the limit n->infinity.
* The WS model views these two types as two extremes of a more general model.
The model is constructed as follows:
- Start with a network with high clustering, like the non-locally coupled
ring
- With a small probability p, rewire every link in the network. To rewire,
a link, we simply delete it from the network and put it between any two
randomly chosen vertices.
- When p = 0, we retain the original lattice, whereas when p = 1, we get
ER graph. However, for intermediate values of p, the model exhibits both
high clustering and the small average path length!
- The rewired links act like shortcuts in the network, and hence reduce
node to node distance quickly. For this reason, even for very small values
of p, we get small average path lengths.
- This also explains the fact that in most real-world networks, the
average path lengths are small. Since one needs to rewire only a tiny
fraction of links to reduce the average node to node distance, almost
surely every network has small node to node distance!
* Unfortunately, WS model is difficult to handle mathematically, and hence it
is better to work with a slightly different variant of the model. In this
variant, one adds the random shortcuts (each pair is connected with
probability p) but doesn't remove any existing link. This means that for p = 1,
we don't get ER graph. However, since we are usually interested only in the
small values of p, this model is essentially equivalent to the original WS
model when p is small. Moreover, since no links are ever removed, it is far
easier to treat mathematically.
This means that in the limit