**Introduction. Why Do We Need. Mathematical Models of Historical Processes (by Peter Turchin, Andrey Korotayev, and Leonid Grinin)** |

*Peter Turchin*. Scientific Prediction in Historical Sociology: Ibn Khaldun meets Al Saud |

*J"urgen Kl"uver*. Logical and Explanative Characteristics of Evolutionary Theories |

*Andrey Korotayev*. The World System Urbanization Dynamics: A quantitative analysis |

*Leonid Grinin, Andrey Korotayev*. Political Development of the World System: A formal quantitative analysis |

*Andrey Korotayev, Leonid Grinin*. Urbanization and Political Development of the World System: A comparative quantitative analysis |

*Victor C. de Munck*. Experiencing History Small: An analysis of political, economic and social change in a Sri Lankan village |

*Charles S. Spencer*. Modeling (and Measuring) Expansionism and Resistance: State formation in Ancient Oaxaca, Mexico |

*Artemy Malkov*. The Silk Roads: A mathematical model |

**List of Contributors** |

Many historical processes are dynamic (a *dynamic* process
is one that changes with time). Populations increase and
decline, economies expand and contract, while states grow and
collapse. How can we study mechanisms that bring about temporal
change and explain the observed trajectories? A very common
approach, which has proved its worth in innumerable applications
(particularly, but not exclusively, in the natural sciences),
consists of taking a holistic phenomenon and mentally splitting
it up into separate parts that are assumed to interact with each
other. This is the dynamical systems approach, because the whole
phenomenon is represented as a system consisting of several
interacting elements (or subsystems).

In the dynamical system's approach, we must describe
mathematically how different subsystems interact with each
other. This mathematical description is the model of the system,
and we can use a variety of methods to study the dynamics
predicted by the model, as well as attempt to test the model by
comparing its predictions with the observed dynamics.

Generally speaking, models are simplified descriptions of
reality that strip away all of its complexity except for a few
features thought to be critical to the understanding of the
phenomenon under study. Mathematical models are such
descriptions translated into a very precise language which,
unlike natural human languages, does not allow for any double
(or triple) meanings. The great strength of mathematics is that,
once we have framed a problem in mathematical language, we can
deduce precisely what are the consequences of the assumptions we
made -- no more, no less. Mathematics, thus, is an indispensable
tool in true science; a branch of science can lay a claim to
theoretical maturity only after it has developed a body of
mathematical theory, which typically consists of an interrelated
set of specific, narrowly-focused models.

The conceptual representation of any holistic phenomenon as
interacting subsystems is always to some degree artificial. This
artificiality, by itself, cannot be an argument against any
particular model of the system. All models simplify the reality.
The value of any model should be judged only against
alternatives, taking into account how well each model predicts
data, how parsimonious the model is, and how much violence its
assumptions do to reality. It is important to remember that
there are many examples of very useful models in natural
sciences whose assumptions are known to be wrong. In fact, all
models are by definition wrong, and this should not be held
against them.

Mathematical models are particularly important in the study of
dynamics, because dynamic phenomena are typically characterized
by nonlinear feedbacks, often acting with various time lags.
Informal verbal models are adequate for generating predictions
in cases where assumed mechanisms act in a linear and additive
fashion (as in trend extrapolation), but they can be very
misleading when we deal with a system characterized by
nonlinearities and lags. In general, nonlinear dynamical systems
have a much wider spectrum of behaviors than could be imagined
by informal reasoning. Thus, a formal mathematical apparatus is
indispensable when we wish to rigorously connect the set of
assumptions about the system to predictions about its dynamic
behavior.

Modeling of any particular empirical system is as much art as
science. Models can be used for a variety of purposes: a compact
description of the system structure, an investigation into the
logical coherence of the proposed explanation, and derivation of
specific predictions from theory that can be tested with data.
Depending on the purpose, we can develop different models for
the same empirical system.

There are several heuristic rules that aid development of useful
models. One rule is: do not attempt to encompass in your model
more than two hierarchical levels. A model that violates this
rule is the one that attempts to model the dynamics of both
interacting subsystems within the system *and* interactions
of subsubsystems within each subsystem. For example, using an
individual-based simulation to model interstate dynamics
violates this rule (unless, perhaps, we model extremely simple
societies). From the practical point of view, even powerful
computers take a long time to simulate systems with millions of
agents. More importantly, from the conceptual point of view it
is very difficult to interpret the results of such a multilevel
simulation. Practice shows that questions involving multilevel
systems should be approached by separating the issues relevant
to each level, or rather pair of levels (the lower level
provides mechanisms, one level up is where we observe patterns).

The second general rule is to strive for parsimony. Probably the
best definition of parsimony was given by Einstein, who said
that a model should be as simple as possible, but no simpler
than that. It is very tempting to include in the model
everything we know about the studied system. Experience shows,
again and again, that such an approach is self-defeating.

Model construction, thus, always requires making simplifying
assumptions. Surprisingly, however, the resultant models are
often quite robust with respect to these initial assumptions.
That is, "first-cut" models can be investigated mathematically
as to the consequences of relaxing the initial assumptions for
theoretical predictions. Repeated applications of this process
can extend theory and simultaneously increase confidence in the
answers that it provides. The end result is an interlocked set
of models, together with data used to estimate model parameters
and test model predictions. Once a critical mass of models and
data has been accumulated, the scientific discipline can be
thought of as having matured (however, it does not mean that all
questions have been answered).

The hard part of theory building is choosing the mechanisms that
will be modeled, making assumptions about how different
subsystems interact, choosing functional forms, and estimating
parameters. Once all that work is done, obtaining model
predictions is conceptually straightforward, although technical,
laborious, and time consuming. For simpler models, we may have
analytical solutions available. However, once the model reaches
even a medium level of complexity we typically must use a second
method: solving it numerically on the computer. A third approach
is to use agent-based simulations. These ways of obtaining model
predictions should not be considered as strict alternatives. On
the contrary, a mature theory employs all three approaches
synergistically.

* * *

One of the main causes for the expansion of the application of
formal mathematical methods to the study of history and society
is the deep changes that have taken place during recent decades
in the field of information production, collection, and
processing (as well as in the field of information technologies,
in general). These changes affect more and more fields of
academic research. The enhanced possibilities for the
development of databases, the increasing speed of their
processing, in conjunction with the growing availability of many
forms of digital information, the diffusion of personal
computers and more and more sophisticated software provide all
the grounds needed to forecast not only the expansion for the
formalization of new methods of information processing and
presentation, but also the expanding application of formal
mathematical methods in such fields that seem to have nothing to
do with mathematics. We may note some serious changes in the
attitudes of the "humanitarians" toward formal mathematical
methods. The application of formal mathematical methods in the
humanities is not just a fashion, or the way to make one's
research faster and more comfortable. It becomes evident that
such methods create necessary conditions for intellectual
breakthroughs, for the establishment of new paradigms, for the
discovery of new research directions. To a considerable extent
this is accounted for by the very character of many historical
processes.

The times of "Pure History" when historians were only interested
in the deeds of kings and heroes passed long ago. A more and
more important role is played by new directions in historical
research that study long-term dynamic processes and quantitative
changes. This kind of history can hardly develop without the
application of mathematical methods.

This almanac continues a series of edited volumes dedicated to
various aspects of the application of mathematical methods to
the study of history and society (Grinin, de Munck, and
Korotayev 2006; Гринин, Коротаев, Малков 2006; Малков, Гринин,
Коротаев 2006; Коротаев, Малков, Гринин 2006). This edited
volume considers historical dynamics and development of complex
societies. Its constituent articles treat historical processes
at very different levels of scale. Some articles study global
dynamics during the last millennia covering the formation and
development of the World System. Other articles focus on the
dynamics of single societies, or even communities. In general,
this issue of the almanac constitutes an integrated study of
a number of important historical processes through the application
of various mathematical methods. In particular, these articles
trace the trajectories of political development from the early
states to mature statehood. This almanac also traces
trajectories of urban development, and important demographic,
technological, and sociostructural changes.

The almanac demonstrates that the application of mathematical
methods not only facilitates the processing of historical
information, but can also give to a historian a deeper
understanding of historical processes.

**Leonid Grinin** is a senior research fellow of the Volgograd
Center for Social Research, a vice-editor of the journals *History and Modernity* and *Philosophy and Society*, and
a co-editor of the *Social Evolution & History*. Current
research interests include the long-term trends in the evolution
of technologies and their influences on sociocultural evolution,
periodization of history, and long-term development of the
political systems. He is author of over 100 scholarly
publications, including such books as "Philosophy, Sociology,
and Theory of History", "Productive Forces and Historical
Process", "Formations and Civilizations", "The State and
Historical Process". His journal articles include "The Early
State and Its Analogues" (*Social Evolution & History* 1:
131--176), "Democracy and Early State" (*Social Evolution
& History* 3[2]: 93--149), "Early State and Democracy" (in
*The Early State, Its Alternatives and Analogues*, pp.419--463. Volgograd: Uchitel), and "The Early State and Its
Analogues: A Comparative Analysis" (in *The Early State,
Its Alternatives and Analogues*, pp.88--136. Volgograd:
Uchitel). Email: LGrinin@mail.ru.

**J"urgen Kl"uver** is Professor Emeritus of Information
Technologies and Educational Processes, Essen, Germany. Current
research interests include the analysis of social and cognitive
complex systems by computer based mathematical models, in
particular the evolutionary unfolding of sociocultural
complexity by ontogenetic learning processes. J"urgen Kl"uver is
the author of a lot of books and articles. Among these are *An Essay Concerning Sociocultural Evolution. Theoretical
Principles and Mathematical Models* (Dordrecht: Kluwer Academic
Publishers, 2002), "The Logical Structure of Evolutionary
Theories" (in *Computational and Mathematical Organization
Theory* 9 [2003]), *Computerimulationen und soziale
Einzelfallstudien* (Bochum-Herdecke: w3l, 2006), and *On
Communication. An Interdisciplinary and Mathematical Approach*
(together with Christina Kl"uver, Dordrecht: Springer, 2007).
Email: Juergen.Kluever@Uni-Due.de.

**Andrey Korotayev** is Director and Professor of the
"Anthropology of the East" Center, Russian State University
for the Humanities, Moscow, as well as Senior Research Fellow of
the Institute for Oriental Studies and the Institute for African
Studies of the Russian Academy of Sciences. He is author of over
250 scholarly publications, including *Ancient Yemen*
(Oxford: Oxford University Press, 1995), *Pre-Islamic Yemen*
(Wiesbaden: Harrassowitz Verlag, 1996), *Social Evolution*
(Moscow: Nauka, 2003), *World Religions and Social Evolution
of the Old World Oikumene Civilizations: a Cross-Cultural
Perspective* (Lewiston, NY: Mellen, 2004), *Introduction to
Social Macrodynamics: Compact Macromodels of the World System
Growth.* (Moscow: URSS, 2006, with Artemy Malkov and Daria
Khaltourina), *Introduction to Social Macrodynamics: Secular
Cycles and Millennial Trends* (Moscow: URSS, 2006, with Artemy
Malkov and Daria Khaltourina), *Introduction to Social
Macrodynamics: Secular Cycles and Millennial Trends in Africa*
(Moscow: URSS, 2006, with Daria Khaltourina). Email:
AKorotayev@mail.ru.

**Artemy Malkov** is Research Fellow of the Keldysh Institute
for Applied Mathematics, Russian Academy of Sciences. His
research concentrates on the modeling of social and historical
processes, spatial historical dynamics, genetic algorithms,
cellular automata. He has authored and co-authored over 45
scholarly publications, including such monographs as *Introduction to Social Macrodynamics: Compact Macromodels of the
World System Growth.* (Moscow: URSS, 2006), *Introduction to
Social Macrodynamics: Secular Cycles and Millennial Trends*
(Moscow: URSS, 2006), as well as such articles as "History and
Mathematical Modeling" (2000), "Mathematical Modeling of
Geopolitical Processes" (2002), "Mathematical Analysis of
Social Structure Stability" (2004) that have been published in
the leading Russian academic journals. Email: AS@Malkov.org.

**Victor de Munck** is an Associate Professor in the
Anthropology Department of the State University of New York -- New Paltz. His specialty is cognitive anthropology; he has
published one monograph (*Culture, Self and Meaning*) on
this subject and 15 articles on describing the dynamics between
cognitive processes and culture, as well as a number of articles
in cross-cultural research, including "Sexual Equality and
Romantic Love: A Reanalysis of Rosenblatt's Study on the
Function of Romantic Love" (*Cross-Cultural Research* 33
[1999]: 265--277, with Andrey Korotayev), ""Galton's Asset"
and "Flower's Problem": Cultural Networks and Cultural Units
in Cross-Cultural Research (or, the Male Genital Mutilations and
Polygyny in Cross-Cultural Perspective)" (*American
Anthropologist* 105 [2003]: 353--358, with Andrey Korotayev) and
"Valuing Thinness or Fatness in Women: Reevaluating the Effect
of Resource Scarcity" (*Evolution and Human Behavior* 26
[2005]: 257--270, with Carol R.Ember, Melvin Ember, and Andrey
Korotayev). Professor de Munck has conducted three years of
fieldwork in Sri Lanka which has so far yielded one ethnography
(*Seasonal Cycles*. Delhi: Asian Educational Services, 1993)
and over forty articles. Most recently Dr. de Munck received
grants from the National Science Foundation and the Fulbright
Foundation grant. These have been used to conduct field work on
romantic love, marriage choices and sexual practices in Russia,
Lithuania and the U.S.This research has thus far yielded one
edited volume and a number of articles on cultural models of
romantic love. Email: DeMunckV@NewPaltz.edu.

**Charles S.Spencer** is Curator of Mexican and Central
American Archaeology in the Division of Anthropology, American
Museum of Natural History, New York, NY, USA. His research
focuses on the cultural evolution of complex societies in
prehistory. He has conducted archaeological fieldwork in Mexico
and Venezuela. He is the author of numerous articles and
monographs, including it The Cuicatl'an Ca nada and Monte Alb'an:
A Study of Primary State Formation (New York, NY: Academic
Press, 1982), "On the Tempo and Mode of State Formation:
Neoevolutionism Reconsidered" *Journal of Anthropological
Archaeology* 9 [1990]: 1--30), "War and Early State Formation
in Oaxaca, Mexico" in *Proceedings of the National Academy
of Sciences* (100 [2003]: 11185--11187) and, with his frequent
collaborator Elsa M.Redmond, "Multilevel Selection and
Political Evolution in the Valley of Oaxaca, 500--100 B.C." in
*Journal of Anthropological Archaeology* (20 [2001]:
195--229), "The Chronology of Conquest: Implications of New
Radiocarbon Analyses from the Ca nada de Cuicatl'an, Oaxaca" in
*Latin American Antiquity* (12 [2001]: 182--202),
"Militarism, Resistance, and Early State Development in Oaxaca,
Mexico in *Social Evolution & History* (2 [2003]: 25--70),
"A Late Monte Alb'an I Phase (300--100 B.C.) Palace in the
Valley of Oaxaca" in *Latin American Antiquity* (15 [2004]:
441--455), "Primary State Formation in Mesoamerica" in *Annual Review of Anthropology* (33 [2004]: 173--199),
"Institutional Development in Late Formative Oaxaca: The View
from San Mart'in Tilcajete" in *New Perspectives on
Formative Mesoamerican Cultures* (Oxford, UK: Archaeopress,
2005), and "Resistance Strategies and Early State Formation in
Oaxaca, Mexico" in *Intermediate Elites in Precolumbian
States and Empires* (Tucson: University of Arizona Press, 2006).
Email: CSpencer@AMNH.org.