Line graph operation and small worlds

Complex networks, such as small world networks, are the focus of recent interest because of their potential as models for the interaction networks of complex systems. Most of the well-known models of small world networks are stochastic. The randomness makes it more difficult to gain a visual understanding of how networks are shaped, and how different vertices relate to each other. In this paper, we present and study a method for constructing deterministic small worlds using the line graph operator. This operator introduces cliques at every vertex of the original graph, which may imply larger clustering coefficients. On the other hand, this operator can increase the diameter at most by one and assure the small world property.

► We study line graph operation as a possible tool for constructing small world networks.
► We consider the relationships among the sparseness, diameter, and clustering coefficient of line graphs.
► We show that line graphs of many interconnecting networks are small worlds.