Introduction to the Modeling and Analysis of Complex Systems

350 CHAPTER 16. DYNAMICAL NETWORKS I: MODELINGbased on a random edge rewiring procedure to gradually modify network topology from acompletely regular, local, clustered topology to a completely random, global, unclusteredone. In the middle of this random rewiring process, they found networks that had the“small-world” property yet still maintained locally clustered topologies. They named thesenetworks “small-world” networks. Note that having the “small-world” property is not asufficient condition for a network to be called “small-world” in Watts-Strogatz sense. Thenetwork should also have high local clustering.Watts and Strogatz didn’t propose their random edge rewiring procedure as a dynamicalprocess over time, but we can still simulate it as a dynamics on networks model. Hereare their original model assumptions:1. The initial network topology is a ring-shaped network made of n nodes. Each nodeis connected to k nearest neighbors (i.e., k/2 nearest neighbors clockwise and k/2nearest neighbors counterclockwise) by undirected edges.2. Each edge is subject to random rewiring with probability p. If selected for randomrewiring, one of the two ends of the edge is reconnected to a node that is randomlychosen from the whole network. If this rewiring causes duplicate edges, the rewiringis canceled.Watts and Strogatz’s original model did the random edge rewiring (step 2 above) by sequentiallyvisiting each node clockwise. But here, we can make a minor revision to themodel so that the edge rewiring represents a more realistic social event, such as a randomencounter at the airport of two individuals who live far apart from each other, etc.Here are the new, revised model assumptions:1. The initial network topology is the same as described above.2. In each edge rewiring event, a node is randomly selected from the whole network.The node drops one of its edges randomly, and then creates a new edge to a newnode that is randomly chosen from the whole network (excluding those to which thenode is already connected).This model captures essentially the same procedure as Watts and Strogatz’s, but therewiring probability p is now represented implicitly by the length of the simulation. If thesimulation continues indefinitely, then the network will eventually become completely random.Such a dynamical process can be interpreted as a model of social evolution inwhich an initially locally clustered society gradually accumulates more and more globalconnections that are caused by rare, random long-range encounters among people.