However the
ER graphs do not have two important properties observed in many real-world networks:
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.
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 a randomized structure close to ER graphs 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, networks of movie actors, or fat-metabolism communication in
budding yeast.[4]
Algorithm
Watts–Strogatz graph
Given the desired number of nodes , the mean
degree (assumed to be an even integer), and a parameter , all satisfying and , the model constructs an
undirected graph with nodes and edges in the following way:
Construct a regular ring lattice, a graph with nodes each connected to neighbors, on each side. That is, if the nodes are labeled , there is an edge if and only if
For every node take every edge connecting to its rightmost neighbors, that is every edge such that , and rewire it with probability . Rewiring is done by replacing with where is chosen uniformly at random from all possible nodes while avoiding self-loops () and link duplication (there is no edge with at this point in the algorithm).
For a ring lattice, the average path length[1] is and scales linearly with the system size. In the
limiting case of , the graph approaches a random graph with , while not actually converging to it. In the intermediate region , the average path length falls very rapidly with increasing , quickly approaching its limiting value.
Clustering coefficient
For the ring lattice the
clustering coefficient[5], and so tends to as grows, independently of the system size.[6] In the limiting case of the clustering coefficient is of the same order as the clustering coefficient 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.
If we use the Barrat and Weigt[6] measure for clustering defined as the fraction between the average number of edges between the neighbors of a node and the average number of possible edges between these neighbors, or, alternatively,
then we get
Degree distribution
The degree distribution in the case of the ring lattice is just a
Dirac delta function centered at . The degree distribution for a large number of nodes and can be written as,[6]
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, meaning that all nodes are of similar 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. (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.