Random graphs models for complex networks, and the brain

In this chapter, we discuss complex networks as a prime example where the ideas from complexity theory can be successfully applied. Complex networks show emergent behavior in their connectivity, and they have intricate feedback mechanisms leading to non-linearities, particularly in settings where the network structure is highly heterogeneous. We draw motivation from real-world networks about the properties of such networks. We formulate random graph models for real-world networks and investigate the properties of these models, such as their degree structure, their connectivity and their small-world properties, as well as the behavior of stochastic processes on them. We focus on some models that have received the most attention in the literature, namely, the Erdos-Renyi random graph, inhomogeneous random graphs, the configuration model and preferential attachment models. We also discuss some of their extensions that have the potential to yield more realistic models for real-world networks. We close this chapter by speculating on applications of random graphs to the brain, which is arguably the most complex network that exists.