Degree is the simplest of the node-level measures, but its simplicity often hides its power. Here we will apply degree to the problem of mental structure. Specifically, what is the structure of the relationships between information in the mind? George Kingsley Zipf observed that word frequencies in natural language tend to a follow a scale-free distribution: The most frequent words are few, while the less frequent words are many with a specific linear relationship on a log-log plot. It has also been suggested that this power-law distribution applies to the relationships between words as well as to their meanings. Some words share meanings with many other words while others share few. This is a hypothesis based on the structural distribution of shared meanings, or polysemy (words with multiple meanings). This chapter will explain the theory underlying Zipf’s law of meaning and power laws. It will also show how we can combine these ideas with the most basic node-level network measure: degree.

We present examples to show that people and economic activities are unevenly distributed across space. The variations in density result in strong agglomeration in some important centres. We briefly analyse urban development and illustrate ‘spikiness’ at different spatial scales (global regions, countries, provinces, and counties). We also show that the distribution across space is not random but often displays a remarkably stable and uniform pattern across time and for various levels of geographical aggregation. These observations suggest that similar spatial economic forces are relevant for explaining agglomeration and the regularities of distribution and interaction across space.

We present examples to show that people and economic activities are unevenly distributed across space. The variations in density result in strong agglomeration in some important centres. We briefly analyse urban development and illustrate ‘spikiness’ at different spatial scales (global regions, countries, provinces, and counties). We also show that the distribution across space is not random but often displays a remarkably stable and uniform pattern across time and for various levels of geographical aggregation. These observations suggest that similar spatial economic forces are relevant for explaining agglomeration and the regularities of distribution and interaction across space.

Power laws (and the special case of Zipf’s law) allow us to characterize cities on a map with a single number, namely the slope of a rank-size curve. These rank-size curves show the rank of a city as a function of its size: the biggest city gets rank 1, the second largest city rank 2, and so on. The slope tells us whether cities are relatively similar in size (small slope) or unevenly sized (steep slope). But what explains the existence of different sized cities? This chapter introduces some urban theories that explain the existence of different sized cities; a graphical representation of the Henderson model and a graphical representation of a state-of-the-art model of differentiated cities as developed by Davis and Dingle.

Power laws (and the special case of Zipf’s law) allow us to characterize cities on a map with a single number, namely the slope of a rank-size curve. These rank-size curves show the rank of a city as a function of its size: the biggest city gets rank 1, the second largest city rank 2, and so on. The slope tells us whether cities are relatively similar in size (small slope) or unevenly sized (steep slope). But what explains the existence of different sized cities? This chapter introduces some urban theories that explain the existence of different sized cities; a graphical representation of the Henderson model and a graphical representation of a state-of-the-art model of differentiated cities as developed by Davis and Dingle.