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.[1] The model also became known as the (Watts) beta model after Watts used to formulate it in his popular science book Six Degrees.
Contents |
The formal study of random graphs dates back to the work of Paul Erdős and Alfréd Rényi.[2] The graphs they considered, now known as the classical or 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:
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.
Given the desired number of nodes , the mean degree (assumed to be an even integer), and a special parameter , satisfying and , the model constructs an undirected graph with nodes and edges in the following way:
The underlying lattice structure of the model produces a locally clustered network, and the random links dramatically reduce the average path lengths. The algorithm introduces about non-lattice edges. Varying makes it possible to interpolate between a regular lattice () and a random graph () approaching the Erdős–Rényi random graph with and .
The three properties of interest are the average path length, the clustering coefficient, and the degree distribution.
For a ring lattice the average path length is and scales linearly with the system size. In the limiting case of the graph converges to a classical random graph with . However, in the intermediate region the average path length falls very rapidly with increasing , quickly approaching its limiting value.
For the ring lattice the clustering coefficient is which is independent of the system size. In the limiting case of the clustering coefficient attains the value for classical random graphs, 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 . This results in a region where the average path length falls rapidly, but the clustering coefficient does not, explaining the "small-world" phenomenon.
The degree distribution in the case of the ring lattice is just a Dirac delta function centered at . In the limiting case of it is Poisson distribution, as with classical graphs. The degree distribution for can be written as,[3]
where is the number of edges that the node has or its degree. Here , and . The shape of the degree distribution is similar to that of a random graph and has a pronounced peak at and decays exponentially for large . The topology of the network is relatively homogeneous, and all nodes have more or less the same degree.
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. (On the other hand, the Barabási–Albert model fails to produce the high levels of clustering seen in real networks, a shortcoming not shared by the Watts and Strogatz model. Thus, neither the Watts and Strogatz model nor the Barabási–Albert model should be viewed as fully realistic.)
The Watts and Strogatz model also implies a fixed number of nodes and thus cannot be used to model network growth.