There is a close relationship between random graphs and percolation. In fact, percolation and random graphs have been viewed as “the same phenomenon expressed in different languages” (Albert and Barabási, ). Early ideas on percolation (although not under that name) in molecular chemistry can be found in the articles by Flory () and Stockmayer ().

Percolation can be defined more generally than as a process on ${\mathbb{Z}}^d$, $d\ge 2$. In this chapter, we motivate the main ideas and theory of percolation on more general graphs by application to polymer gelation and amorphous computing.

In this chapter, we discuss various issues that arise when networks increase in size. What does it mean for a network to increase in size and how would we visualize that process? Can a sequence of networks, increasing in size, converge to a limit, and what would such a limit look like? We discuss the transformation of an adjacency matrix to a pixel picture and what it means for a sequence of pixel pictures to increase in size. If a limit exists, the resulting function is called a limit graphon, but it is not itself a network. Estimation of a graphon is also discussed and methods described include an approximation by SBM and a network histogram.

As we have seen, networks, such as the Internet and World Wide Web, social networks (e.g., Facebook and LinkedIn), biological networks (e.g., gene regulatory networks, PPI networks, networks of the brain), transportation networks, and ecological networks are becoming larger and larger in today’s interconnected world. Some of these networks are truly huge and difficult, if not impossible, to analyze completely and efficiently. In this chapter, we discuss some of the issues involving comparing networks for similarity or differences, including choice of similarity measures, exchangeable random structures of networks, and property testing in networks.

In this chapter, we introduce a number of parametric statistical models that have been used to model network data. The social network literature has named them $p_1$, $p_2$, and $p^{*}$ models, the last of which has also been referred to as an ERGM (exponential random graph model).

When a network is too large to study completely, we sample from that network just as we would sample from any large population. The structure of network data, however, is more complicated than that of standard statistical data. The main question is, how can one sample from a network that has nodes and edges? Should we sample the nodes? Or should we sample the edges? The answers to these questions depend upon the complexity of the network. In this chapter, we examine various methods of sampling a network.

Random graphs were introduced by the Hungarian mathematicians Erdős and Rényi (, ), who imposed a probabilistic framework on classical combinatorial graph theory. At the same time, Edgar N. Gilbert () also studied the theoretical properties of random graphs.