复杂性思考：探索信息技术的深度

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"复杂性思考 Think Complexity" 是一本由Allen B. Downey编著的书籍，出版于O'Reilly Media，探讨了复杂性这一主题。该书可能涉及计算机科学、算法、网络理论以及系统动态等多个与复杂性相关的领域。本书旨在帮助读者理解和处理复杂性问题，作者Allen B. Downey通过深入浅出的方式介绍了复杂性科学的基本概念。书中可能涵盖了复杂系统的特性，如涌现性、非线性关系、不可预测性和自组织等。这些特性在自然界和社会系统中广泛存在，比如生物生态系统、经济网络、社会交互和计算机程序。在内容部分，可以预期会讨论到各种复杂的计算问题，包括NP完全问题和近似算法。此外，可能还会讲解如何使用数据结构和算法来分析和解决复杂问题，例如图论中的最短路径算法、并行计算和分布式系统的设计原则。书中的章节可能涵盖以下几个方面： 1. 复杂性理论的基础：介绍复杂性的基本概念，如混沌理论、分形几何和复杂网络。 2. 模拟和建模：讨论如何构建模型来理解和预测复杂系统的行为，可能包括计算机模拟和统计建模。 3. 计算复杂性：阐述计算复杂性理论，包括P类和NP类问题，以及NP完全问题的性质。 4. 算法分析：深入研究算法的时间和空间复杂性，学习如何评估和优化算法效率。 5. 生物和自然界的复杂性：探讨生物学中的复杂性，如基因网络、生态系统的稳定性以及生命的起源。 6. 社会和经济复杂性：分析社会网络、市场动态和集体行为的复杂性。此外，该书可能还包含实际案例研究，以帮助读者将理论应用于实际问题。通过这些案例，读者可以学习如何在面对现实世界中的复杂挑战时应用所学知识。本书适合对复杂性科学感兴趣的读者，无论是计算机科学家、工程师、数学家还是社会科学家，都能从中受益。由于采用的是Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License，读者可以在遵守许可协议的情况下自由分享和使用书中的内容。 “复杂性思考 Think Complexity”是一本深度探讨复杂性科学的著作，旨在提升读者理解和处理复杂问题的能力，通过理论与实践的结合，帮助他们在各自的领域应对复杂挑战。

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discrete models of physical systems. In particular, it focuses on complex systems,

which are systems with many interacting components.

Complex systems include networks and graphs, cellular automata, agent-based models

and swarms, fractals and self-organizing systems, chaotic systems, and cybernetic sys-

tems. These terms might not mean much to you at this point. We will get to them soon,

but you can get a preview at http://en.wikipedia.org/wiki/Complex_systems.

A New Kind of Science

In 2002, Stephen Wolfram published A New Kind of Science, where he presents his and

others’ work on cellular automata and describes a scientific approach to the study of

computational systems. We’ll get back to Wolfram in Chapter 6, but I want to borrow

his title for something a little broader.

I think complexity is a “new kind of science” not because it applies the tools of science

to a new subject, but because it uses different tools, allows different kinds of work, and

ultimately changes what we mean by “science.”

To demonstrate the difference, I’ll start with an example of classical science: suppose

someone asked you why planetary orbits are elliptical. You might invoke Newton’s law

of universal gravitation and use it to write a differential equation that describes plane-

tary motion. Then you could solve the differential equation and show that the solution

is an ellipse. Voilà!

Most people find this kind of explanation satisfying. It includes a mathematical deri-

vation—so it has some of the rigor of a proof—and it explains a specific observation,

elliptical orbits, by appealing to a general principle, gravitation.

Let me contrast that with a different kind of explanation. Suppose you move to a city

like Detroit that is racially segregated, and you want to know why it’s like that. If you

do some research, you might find a paper by Thomas Schelling called “Dynamic Models

of Segregation,” which proposes a simple model of racial segregation (a copy is available

from http://statistics.berkeley.edu/~aldous/157/Papers/Schelling_Seg_Models.pdf).

Here is a summary of the paper (from Chapter 10):

The Schelling model of the city is an array of cells where each cell represents a house.

The houses are occupied by two kinds of “agents,” labeled red and blue, in roughly equal

numbers. About 10% of the houses are empty.

At any point in time, an agent might be happy or unhappy, depending on the other agents

in the neighborhood. In one version of the model, agents are happy if they have at least

two neighbors like themselves, and unhappy if they have one or zero.

The simulation proceeds by choosing an agent at random and checking to see whether

it is happy. If so, nothing happens; if not, the agent chooses one of the unoccupied cells

at random and moves.

2 | Chapter 1: Complexity Science

If you start with a simulated city that is entirely unsegregated and run the model for a

short time, clusters of similar agents appear. As time passes, the clusters grow and

coalesce until there are a small number of large clusters and most agents live in homo-

geneous neighborhoods.

The degree of segregation in the model is surprising, and it suggests an explanation of

segregation in real cities. Maybe Detroit is segregated because people prefer not to be

greatly outnumbered and will move if the composition of their neighborhoods makes

them unhappy.

Is this explanation satisfying in the same way as the explanation of planetary motion?

Most people would say not, but why?

Most obviously, the Schelling model is highly abstract, which is to say it is not realistic.

It is tempting to say that people are more complex than planets, but when you think

about it, planets are just as complex as people (especially the ones that have people).

Both systems are complex, and both models are based on simplifications; for example,

in the model of planetary motion, we include forces between the planet and its sun and

ignore interactions between planets.

The important difference is that, for planetary motion, we can defend the model by

showing that the forces we ignore are smaller than the ones we include. And we can

extend the model to include other interactions and show that the effect is small. For

Schelling’s model, it is harder to justify the simplifications.

To make matters worse, Schelling’s model doesn’t appeal to any physical laws, and it

uses only simple computation, not mathematical derivation. Models like Schelling’s

don’t look like classical science, and many people find them less compelling, at least

at first. But as I will try to demonstrate, these models do useful work, including pre-

diction, explanation, and design. One of the goals of this book is to explain how.

Paradigm Shift?

When I describe this book to people, I am often asked if this new kind of science is a

paradigm shift. I don’t think so, and here’s why.

Thomas Kuhn introduced the term “paradigm shift” in The Structure of Scientific Rev-

olutions in 1962. It refers to a process in the history of science where the basic assump-

tions of a field change, or where one theory is replaced by another. He presents as

examples the Copernican revolution, the displacement of phlogiston by the oxygen

model of combustion, and the emergence of relativity.

The development of complexity science is not the replacement of an older model, but

(in my opinion) a gradual shift in the criteria by which models are judged and in the

kinds of models that are considered acceptable.

Paradigm Shift? | 3

For example, classical models tend to be law-based, expressed in the form of equations,

and solved by mathematical derivation. Models that fall under the umbrella of com-

plexity are often rule-based, expressed as computations, and simulated rather than

analyzed.

Not everyone finds these models satisfactory. For example, in Sync, Steven Strogatz

writes about his model of spontaneous synchronization in some species of fireflies. He

presents a simulation that demonstrates the phenomenon, but then writes:

I repeated the simulation dozens of times, for other random initial conditions and for

other numbers of oscillators. Sync every time. … The challenge now was to prove it. Only

an ironclad proof would demonstrate, in a way that no computer ever could, that sync

was inevitable; and the best kind of proof would clarify why it was inevitable.

Strogatz is a mathematician, so his enthusiasm for proofs is understandable, but his

proof doesn’t address what is, to me, the most interesting part of the phenomenon. In

order to prove that “sync was inevitable,” Strogatz makes several simplifying assump-

tions, in particular that each firefly can see all the others.

In my opinion, it is more interesting to explain how an entire valley of fireflies can

synchronize despite the fact that they cannot all see each other. How this kind of global

behavior emerges from local interactions is the subject of Chapter 10. Explanations of

these phenomena often use agent-based models, which explore (in ways that would be

difficult or impossible with mathematical analysis) the conditions that allow or prevent

synchronization.

I am a computer scientist, so my enthusiasm for computational models is probably no

surprise. I don’t mean to say that Strogatz is wrong, but rather that people disagree

about what questions to ask and what tools to use to answer them. These decisions are

based on value judgments, so there is no reason to expect agreement.

Nevertheless, there is rough consensus among scientists about which models are con-

sidered good science and which others are fringe science, pseudoscience, or not science

at all.

I claim—and this is a central thesis of this book—that the criteria upon which this

consensus is based change over time, and that the emergence of complexity science

reflects a gradual shift in these criteria.

The Axes of Scientific Models

I have described classical models as based on physical laws, expressed in the form of

equations, and solved by mathematical analysis; conversely, models of complexity sys-

tems are often based on simple rules and implemented as computations.

4 | Chapter 1: Complexity Science

We can think of this trend as a shift over time along two axes:

Equation-based simulation-based

Analysis computation

The new kind of science is different in several other ways. I present them here so you

know what’s coming, but some of them might not make sense until you have seen the

examples later in the book.

Continuous discrete

Classical models tend to be based on continuous mathematics like calculus; models

of complex systems are often based on discrete mathematics, including graphs and

cellular automata.

Linear non-linear

Classical models are often linear or use linear approximations to non-linear sys-

tems; complexity science is more friendly to non-linear models. One example is

chaos theory.

Deterministic stochastic

Classical models

are usually deterministic, which may reflect underlying philo-

sophical determinism, discussed in Chapter 6; complex models often feature

randomness.

Abstract

detailed

In classical

models, planets are point masses, planes are frictionless, and cows are

spherical (see http://en.wikipedia.org/wiki/Spherical_cow). Simplifications like

these are often necessary for analysis, but computational models can be more

realistic.

One, two

many

In celestial mechanics, the two-body problem can be solved analytically; the three-

body problem cannot. Where classical models are often limited to small numbers

of interacting elements, complexity science works with larger complexes (which is

where the name comes from).

Homogeneous composite

In classical models, the elements tend to be interchangeable; complex models more

often include heterogeneity.

These are generalizations, so we should not take them too seriously. And I don’t mean

to deprecate classical science. A more complicated model is not necessarily better; in

fact, it is usually worse.

Also, I don’t mean to say that these changes are abrupt or complete. Rather, there is a

gradual migration in the frontier of what is considered acceptable, respectable work.

Some tools that used to be regarded with suspicion are now common, and some models

that were widely accepted are now regarded with scrutiny.

1. Chaos is not covered in this book, but you can read about it at http://en.wikipedia.org/wiki/Chaos_theory.

The Axes of Scientific Models | 5

For example, when Appel and Haken proved the four-color theorem in 1976, they used

a computer to enumerate 1,936 special cases that were, in some sense, lemmas of their

proof. At the time, many mathematicians did not consider the theorem truly proved.

Now, computer-assisted proofs are common and generally (but not universally)

accepted.

Conversely, a substantial body of economic analysis is based on a model of human

behavior called “economic man,” or, with tongue in cheek, Homo economicus. Research

based on this model was highly regarded for several decades, especially if it involved

mathematical virtuosity. More recently, this model is treated with more skepticism,

and models that include imperfect information and bounded rationality are hot topics.

A New Kind of Model

Complex models are often appropriate for different purposes and interpretations.

Predictive explanatory

Schelling’s model of segregation might shed light on a complex social phenom-

enon, but it is not useful for prediction. On the other hand, a simple model of

celestial mechanics can predict solar eclipses down to the second, years in the

future.

Realism instrumentalism

Classical models

lend themselves to a realist interpretation; for example, most

people accept that electrons are real things that exist. Instrumentalism is the view

that models can be useful even if the entities they postulate don’t exist. George Box

wrote what might be the motto of instrumentalism: “All models are wrong, but

some are useful.”

Reductionism

holism

Reductionism is the view that the behavior of a system can be explained by un-

derstanding its components. For example, the periodic table of the elements is a

triumph of reductionism, because it explains the chemical behavior of elements

with a simple model of the electrons in an atom. Holism is the view that some

phenomena that appear at the system level do not exist at the level of components,

and cannot be explained in component-level terms.

We get back to explanatory models in Chapter 5, instrumentalism in Chapter 7, and

holism in Chapter 9.

6 | Chapter 1: Complexity Science

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