Graph Theory and Small-World Networks

Abstract

Dynamical and adaptive networks are the backbone of many complex systems. Examples range from ecological prey-predator networks to the gene expression and protein providing the grounding of all living creatures The brain is probably the most complex of all adaptive dynamical systems and is at the basis of our own identity, in the form of a highly sophisticated neural network. On a social level we interact through social and technical networks like the Internet. Networks are ubiquitous through the domain of all living creatures. A good understanding of network theory is therefore of basic importance for complex system theory. In this chapter we will discuss the most important notions of graph theory, like clustering and degree distributions, together with basic network realizations. Central concepts like percolation, the robust-ness of networks with regard to failure and attacks, and the "rich-get-richer" phenomenon in evolving social networks will be treated.