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Mathematics LibreTexts

2.5: A Historical Perspective


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As the final section in this chapter, I would like to present some historical perspective of how people have been developing modeling methodologies over time, especially those for complex systems (Fig. 2.5). Humans have been creating descriptive models (diagrams, pictures, physical models, texts, etc.) and some conceptual rule-based models since ancient times. More quantitative modeling approaches arose as more advanced mathematical tools became available. In the descriptive modeling family, descriptive statistics is among such quantitative modeling approaches. In the rule-based modeling family, dynamical equations (e.g., differential equations, difference equations) began to be used to quantitatively formulate theories that had remained at conceptual levels before.

During the second half of the 20th century, computational tools became available to researchers, which opened up a whole new area of computational modeling approaches 


for complex systems modeling. The first of this kind was cellular automata, a massive number of identical finite-state machines that are arranged in a regular grid structure and update their states dynamically according to their own and their neighbors’ states. Cellular automata were developed by John von Neumann and Stanisław Ulam in the 1940s, initially as a theoretical medium to implement self-reproducing machines [11], but later they became a very popular modeling framework for simulating various interesting emergent behaviors and also for more serious scientific modeling of spatio-temporal dynamics [18]. Cellular automata are a special case of dynamical networks whose topologies are limited to regular grids and whose nodes are usually assumed to be homogeneous and identical.

Dynamical networks formed the next wave of complex systems modeling in the 1970s and 1980s. Their inspiration came from artificial neural network research by Warren McCulloch and Walter Pitts[19] as well as byJohn Hopfield[20,21], and also from theoretical gene regulatory network research by Stuart Kauffman [22]. In this modeling framework, the topologies of systems are no longer constrained to regular grids, and the components and their connections can be heterogeneous with different rules and weights. Therefore, dynamical networks include cellular automata as a special case within them. Dynamical networks have recently merged with another thread of research on topological analysis that originated in graph theory, statistical physics, social sciences, and computational science, to form a new interdisciplinary field of network science [23, 24, 25]. 

Finally, further generalization was achieved by removing the requirement of explicit network topologies from the models, which is now called agent-based modeling(ABM). In ABM, the only requirement is that the system is made of multiple discrete “agents” that interact with each other (and possibly with the environment), whether they are structured into a network or not. Therefore ABM includes network models and cellular automata as its special cases. The use of ABM became gradually popular during the 1980s, 1990s, and 2000s. One of the primary driving forces for it was the application of complex systems modeling to ecological, social, economic, and political processes, in fields like game theory and microeconomics. The surge of genetic algorithms and other population-based search/optimization algorithms in computer science also took place at about the same time, which also had synergistic effects on the rise of ABM. 

I must be clear that the historical overview presented above is my own personal view, and it hasn’t been rigorously evaluated or validated by any science historians (therefore this may not be a valid model!). But I hope that this perspective is useful in putting various modeling frameworks into a unified, chronological picture. The following chapters of this textbook roughly follow the historical path of the models illustrated in this perspective.

Exercise 2.6

 Do a quick online literature search to find a few scientific articles that develop or use mathematical/computational models. Read the articles to learn more about their models, and map them to the appropriate locations in Fig. 2.5.