A valuable feature of networks is their ability to quantify structural relationships. Much like a chemist who, having purified their sample, can bring to bear a wide range of tools for understanding their sample by transforming data into a network, a network scientist can leverage a range of tools that provide precise measurements of their data’s structure. This includes node-level measures like degree and centrality, meso-scale measures like community formation, and macro-scale measures like density and modularity. This chapter introduces these measures and gives a brief guide to their use.

Words, like biological species, are born and then, someday, they die. The half-life of a word is roughly 2,000 years, meaning that in that interval about half of all words are replaced with an unrelated (noncognate) word. Where do the new words come from? There are numerous dimensions along which new words could vary from old words, so it may not be easy to see how to enter this problem. However, extending our small worlds metaphor and the observation of clusters in language, we tell a simple story that mirrors biological theories about the origin of species. Language has urban centers with well-populated and well-connected meanings (like *food* and *red*). It also has rural fringes, where words live more isolated lives as hermits with limited connections to other words (like *twang* and *ohm*). Are new words more likely to be born in urban centers or in the rural fringes?

Why did the United States establish an early American Empire in the Pacific (1856-1898)? This chapter summarizes the argument of this book, explaining why patterns of imperialism demonstrate the influence of commodity prices and entrepreneurs for distinctive patterns of American imperialism. It then addresses 1898. Scholars often suggest that 1898 was the moment when the United States became an empire. This chapter argues that this view is misplaced. Instead, 1898 marks a shift in the US approach to empire, when the US Navy replaced the small entrepreneur as the key figure in US expansion. It then addresses the lessons learned from this book, with an emphasis on the politics of race in the contemporary Pacific and struggles for recognition in the region.

In this chapter, we shall study techniques for analyzing social networks. An important question is how to identify “communities,” that is, subsets of the nodes (people or other entities that form the network) with unusually strong connections. Some of the techniques used to identify communities are similar to the clustering algorithms we discussed in Chapter 7. However, communities almost never partition the set of nodes in a network. Rather, communities usually overlap. For example, you may belong to several communities of friends or classmates. The people from one community tend to know each other, but people from two different communities rarely know each other. You would not want to be assigned to only one of the communities, nor would it make sense to cluster all the people from all your communities into one cluster. Also in this chapter we explore efficient algorithms for discovering other properties of graphs. We look at “simrank,” a way to discover similarities among nodes of a graph. We then explore triangle counting as a way to measure the connectedness of a community. In addition, we give efficient algorithms for exact and approximate measurement of the neighborhood sizes of nodes in a graph, and we look at efficient algorithms for computing the transitive closure.