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This chapter discusses the theory of geopolitics, beginning with an overview of some elementary facts from nonlinear dynamics. In particular, it describes boundless growth, examples of which are linear and exponential growth. It then considers the kinds of dynamics illustrated by the asymptotic and logistic models, in which the system is always attracted to a unique stable equilibrium, along with metastable dynamics, boom–bust dynamics, and sustained oscillations. It also examines three fundamental types of dynamics characterizing purely endogenous systems: zero-order dynamics, first-order dynamics, and second-order dynamics. The chapter goes on to analyze Randall Collins' theory of geopolitics, which postulates three main mechanisms explaining the territorial dynamics of states: geopolitical resources, logistical loads, and the marchland position. Finally, it reviews the simulation model of conflict legitimacy dynamics developed by Hanneman et al.

This chapter describes a theory that specifies the variables responsible for the increase or decline of asabiya—the metaethnic frontier theory. According to the metaethnic frontier theory, metaethnic frontiers are zones or regions that are created when metaethnic fault lines coincide with imperial boundaries. One of the main assumptions of the metaethnic frontier theory is that an ethnic group characterized by high asabiya advances up the social scale from subethnie to ethnie and finally to metaethnie. A common occurrence in metaethnic frontiers is ethnogenesis. The chapter first provides an overview of the role of frontiers as incubators of group solidarity before discussing the factors that cause solidarity to increase. It then introduces a pseudospatial model of the dynamic relationship between empires and frontiers and concludes by performing a spatially explicit simulation of a world-system of polities whose territorial dynamics are driven by geopolitical forces and asabiya.

This chapter provides an introduction to the use of quantitative theories in historical sociology. It first explains why we need a mathematical theory in history before making a case for historical dynamics as a research program, with emphasis on the important role of mathematical models. Using the case of the territorial dynamics of agrarian polities, it proposes a research program for theory building in historical dynamics that involves selecting which of the questions need to be addressed as well as the hierarchical modeling approach and the mathematical framework to be employed. In addition to defining the problem to be addressed, other steps include identifying the primary data set, identifying a set of hypotheses, translating all hypotheses in the list into mathematical models, identifying secondary data, and solving the models using appropriate technology.

This concluding chapter summarizes the main developments discussed in the book, beginning with the proposed research program for investigating historical dynamics and how the approach can be used to study the territorial dynamics of agrarian polities. It then considers asabiya and its impact on metaethnic frontiers; the results of empirical tests on ethnic assimilation; the prediction of demographic-structural theory that population dynamics causes state breakdown; and the theory of geopolitics, applied to test of the effect of a region's positional advantage on the maximum polity size originating from it. The chapter also proposes a combined modeling/empirical approach to investigating history and how the focus of the research program may be expanded beyond the dynamics of agrarian polities (for example, to cases involving nomadic pastoralists, thalassocratic polities such as classical Athens, or hegemonic cycles). Finally, it suggests the term “cliodynamics” (by analogy with cliometrics) as an alternative to “historical dynamics”.