Motivation

In this part, we are concerned with certain global properties of a network or graph, such as the existence of a giant connected component. The main new mathematical tool that we discuss is percolation theory. Percolation theory started some 50 years ago as a mathematical framework to study the behavior of porous media. It has been used to address questions such as the following.

• If a stone gets wet, does the water penetrate the stone?

• If a material consists of two components, one of which is a perfect insulator, what is the probability that the resistance is finite?

• If we drill for oil, what is the probability that a large number of oil chambers is connected to the one we drilled into?

• What is the probability that a forest fire spreads across an entire forest?

• What is the probability that a virus spreads globally?

• What is the probability that most of a network is connected?

It turns out that certain key events, such as the existence of a giant connected component in a network, emerge rather suddenly as a network parameter is changed. Such phenomena are called phase transitions.

In the context of graphs or networks, percolation is related to connectivity and coverage. These are the other two topics in this part.