The small-world model of Watts  and Strogatz

In order to model the real-world networks, we need to find a way of generating graphs which have both the clustering and small-world properties. As we know, random graphs show the small-world effect, possessing average vertex-to-vertex distances which increase only logarithmically with the total number N of vertices, but they do not show clustering—the property that two neighbors of a vertex will often also be neighbors of one another.

The Watts-Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering. It was proposed by Duncan J. Watts and Steven Strogatz in their joint 1998 Nature paper. The model also became known as the (Watts) beta model after Watts used B  to formulate it in his popular science book Six Degrees.

Though  Erdős–Rényi (ER) graphs, offer a simple and powerful model with many applications. However the ER graphs do not have two important properties observed in many real-world networks:

1.  They do not generate local clustering and triadic closures. Instead because they have a constant, random, and independent probability of two nodes being connected, ER graphs have a low clustering coefficient.

2.  They do not account for the formation of hubs. Formally, the degree distribution of ER graphs converges to a Poisson distribution, rather than a power law observed in many real-world, scale-free networks.

The Watts and Strogatz model was designed as the simplest possible model that addresses the first of the two limitations. It accounts for clustering while retaining the short average path lengths of the ER model. It does so by interpolating between an ER graph and a regular ring lattice. Consequently, the model is able to at least partially explain the "small-world" phenomena in a variety of networks, such as the power grid, neural network of C. elegans, and a network of movie actors.

Algorithm:

Given the desired number of nodes N, the mean degree K (assumed to be an even integer), and a special parameter, satisfying and 

the model constructs an undirected graph with N nodes and NK/2 edges in the following way:

1. Construct a regular ring lattice, a graph with N nodes each connected to K neighbors, K/2 on each side. 
2. For every node  take every edge (ni,nj) with i<j, and rewire it with probability.Rewiring is done by replacing (ni,nj) with (ni,nk)  where K is chosen with uniform probability from all possible values that avoid loops (k!=i) and link duplication (there is no edge (ni,nk') with K'=K at this point in the algorithm).

Average path length:

For a ring lattice the average path length is l(0)= N/2K >> 1 and scales linearly with the system size. In the limiting case of B -->1 the graph converges to a classical random graph with l(1)=ln N/ln K. However, in the intermediate region 0<B<1 the average path length falls very rapidly with increasing B, quickly approaching its limiting value.

Clustering coefficient:

For the ring lattice the clustering coefficient is C(0) = 3/4 which is independent of the system size. In the limiting case of B -->1 the clustering coefficient attains the value for classical random graphs, C(1) = K/N and is thus inversely proportional to the system size. In the intermediate region the clustering coefficient remains quite close to its value for the regular lattice, and only falls at relatively high B. This results in a region where the average path length falls rapidly, but the clustering coefficient does not, explaining the "small-world" phenomenon.

Degree distribution:

The degree distribution in the case of the ring lattice is just a Dirac delta function centered at K. In the limiting case of B-->1 it is Poisson distribution, as with classical graphs. The degree distribution for 0<B<1 can be written as,

where f(k,K) = min(k-K/2,K/2) and the shape of the degree distribution is similar to that of a random graph and has a pronounced peak at k = K and decays exponentially for large |k-K|. The topology of the network is relatively homogeneous, and all nodes have more or less the same degree.

Limitations:

The major limitation of the model is that it produces an unrealistic degree distribution. In contrast, real networks are often scale-free networks inhomogeneous in degree, having hubs and a scale-free degree distribution. Such networks are better described in that respect by the preferential attachment family of models, such as the Barabási–Albert (BA) model.