In 2007, Bruce Willis made a comeback after 12 years with Die Hard 4.0. The original Die Hard premiered in 1988, making the series span over 20 years, and solidifying Bruce Willis as a true veteran. Despite nearly two decades passing, his character’s guts, sarcastic humor, and action prowess remain unchanged. John McClane, a New York cop played by Willis, always finds himself at the wrong place at the wrong time, battling terrorists and suffering greatly, often during holidays like Christmas Eve. Die Hard 4.0 takes place on the 4th of July, American Independence Day. However, as the series progresses, the exaggerations and implausibilities grow. For instance, McClane takes down a helicopter with a car and jumps from a plane without a parachute. This time, the terrorist threat targets the entire nation, not just a specific building, airport, or city. Thomas Gabriel, a former government agent aiming to seize control of America, starts killing all hackers who could thwart his plans while attacking the nation’s networks. Transportation, communication, finance, and electricity networks fall into the terrorists’ hands, plunging America into panic.
How could such an attack be possible without foreign armies or aliens invading? While fiction, the possibility arises from the interconnectivity of all systems. Today, systems like transportation, communication, finance, and energy are interconnected into a single network, which is precisely the problem.
We often hear the term “complex systems.” It’s fair to call our world a complex system due to its intricacies. Conversely, what is a simple system? A simple system is one “whose movements can be precisely described by clear laws.” In other words, a system where one cause determines one effect. Mechanical components moving according to design or substances composed of molecules are simple systems. Understanding the components of a simple system allows complete understanding of its principles. The issue is that such systems are rare. Most of the world comprises complex systems. The intricately connected subway, global air networks, and the internet are prime examples. Additionally, countries, cities, economic systems, and organizations composed of diverse individuals are all complex systems, making our daily lives part of these systems.
To explain complex systems, scholars have introduced numerous new terms. However, let’s start with the relatively familiar word “network.” The term itself provides insights into today’s complex world and serves as a keyword for understanding natural and social phenomena, as revealed by recent complexity scientists. A network presupposes interaction. Adding even one new component can create countless new links. For instance, opening a new airport creates new routes with all existing airports. If there were no interactions among components, no matter how many there are, there would be no complexity. Metcalfe’s Law applies to networks, stating that the value of a network increases “exponentially” with its size. For example, the internet’s usefulness is zero with only one user but becomes four when there are two users since both can send and receive information. With one million connections, its utility is the square of one million. However, complexity also increases proportionally. A complex system is a group of numerous components that constantly interact, resulting in new phenomena and orders differing from when viewed independently.
The butterfly effect described by chaos theory can be understood through network structures. Every action has a cause and effect. When causes and effects are clear, precise predictions are possible. However, in a network’s complexity, pinpointing which cause leads to which effect becomes challenging. A small change can expand through the network, creating significant effects. Consider a video uploaded to a website becoming a global sensation. This is the butterfly effect. The term originated from meteorologist Edward Lorenz’s 1972 lecture title, “Predictability: Can the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”
We’ve long said the world is complex. Human society has always been a complex system. But why has complexity become a hot topic today? Has the world become more complex? Yes, and it has become “suddenly” complex, primarily due to the explosion of networks. Stephen Hawking, author of A Brief History of Time and physicist, explains the emergence of complexity as “an increasing number of independent variables beginning to interact in interdependent and unpredictable ways.”
Imagine looking at a globe lit only in cities connected by highways or flight routes. In early history, only scattered spots would light up. As more highways and routes are built, we eventually witness the entire globe lighting up simultaneously. Connections between previously isolated cities create a point where the entire globe suddenly becomes interconnected. This phenomenon, termed “network explosion” by Eric Beinhocker in The Origin of Wealth, reflects changes that seem gradual but hit a critical point where they suddenly “explode” into rapid transformation. This critical point, or tipping point as Malcolm Gladwell describes it, is when liquid turns to gas or other rapid transitions occur.
In practical terms, consider dropping grains of sand on a table, forming a pile that grows until grains start sliding down the slopes. At a critical angle, a single grain can cause the pile to collapse. This state, where a minor cause triggers drastic changes, is a critical state. Because components are tightly interconnected, minor disruptions can cause chain reactions or domino effects, distinct from dominoes because they represent artificially created critical states. Such critical states, leading to “explosions,” are ubiquitous in nature and human life. Think of avalanches or explosive anger. Contained anger doesn’t disappear; it erupts once it exceeds a threshold. Having a higher threshold for such critical points leads to a better life, evidenced by regretful thoughts like “I should have held back.” Mark Buchanan in Ubiquity highlights that critical states apply to various phenomena: earthquakes, wars, wildfires, hurricanes, mass extinctions, traffic jams, stock market crashes, and even academic citations.
Globalization and the internet today were created similarly. What shape does today’s network take? Research shows natural and social networks resemble today’s air networks. While every city is connected to at least one other, not all have the same number of links. Networks are not entirely random but lack clear patterns, concentrating routes at a few hub airports, like Chicago O’Hare or New York JFK, with numerous smaller airports. The internet, with massive portal sites acting as hubs and countless small sites, mirrors this. Human-created social networks and natural networks, like proteins or cells, show similar structures. Visualizing brain cell connections contrasts with imagining global connectivity, but both represent network structures.
KAIST physics professor Hawoong Jeong highlights that “half of the entire internet network is related to 1% of key hubs,” emphasizing that these hubs naturally emerged despite the World Wide Web’s decentralized design. Thus, terrorists in Die Hard 4.0 don’t need to attack all sites. Targeting key hubs can cripple the entire network. Someone knowledgeable about the internet structure could attack just 1% of crucial nodes, paralyzing half the internet, and targeting 4% could fragment it entirely.
Scientists call such networks scale-free networks, meaning they lack a typical size, making the concept of averages meaningless. Scale-free also implies similar phenomena can be observed regardless of network size. The Pareto principle, or 80-20 rule, becomes a natural occurrence, explaining that 80% of results come from 20% of causes. For instance, 20% of employees generate 80% of a company’s work, or 20% of products yield 80% of sales. Similarly, 20% of customers create 80% of revenue, and in economic terms, 20% of the population holds 80% of wealth. This reflects a power-law distribution where small items are numerous, but large items are rare, forming a steep curve when graphed.
The power-law distribution, while complex, can be visualized as curves differing significantly. Using logarithmic functions, this curve can appear linear, illustrating that the scale and frequency of phenomena are inversely proportional. For instance, weak earthquakes are frequent, but strong ones are rare. Initially discovered in earthquake research, scientists found that larger earthquakes occur at specific fractions of smaller ones. This distribution applies to various phenomena, from broken frozen potatoes to wealth distribution.
Mark Buchanan’s potato example demonstrates the power-law: large pieces are few, small ones many, and the smaller the pieces, the more numerous they are. This power-law enables us to understand the Pareto principle as a natural phenomenon, applicable across internet, biology, and social phenomena. While these distributions coexist, their relationship remains uncertain, potentially stemming from differences in observation scope. Power-law may apply to parts rather than wholes, like observing only the moving crust in earthquakes. Stock price changes also follow this, with tail distributions adhering to power laws.
Power-law thus relates to changes, with critical points causing abrupt transformations. Systems can move predictably until sudden complexity arises, creating new patterns. Even in apparent chaos, regularities exist.
Scale-free networks imply similar observations regardless of scale, suggesting a fractal structure. Mark Buchanan’s potato analogy implies consistent patterns at all sizes. Scale-free networks lack a typical scale, presenting similar views at any level. Thus, understanding complex systems requires recognizing these regularities, impacting fields from natural sciences to social sciences and arts. Integrating mathematical and statistical techniques from diverse disciplines enhances our understanding of the complex world, offering new insights. This interconnectedness appears in brain cells, memory, human relationships, and more.
Knowledge comes by taking things apart: analysis. But wisdom comes by putting things together. – John Morrison
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