In the digital age, the Internet is a virtual world of information. Information organizes itself in nonlinear network patterns, like the Internet. Networking theory is the simplest tool for understanding nonlinear systems behaviors.
Basic Properties of Nonlinear Networks
Nonlinear networks contain both order and randomness
Nonlinear networks are highly connected and therefore able to transfer and process information efficiently. The efficiency of high connectivity comes from implementing both order (to achieve the highest level of clustering) and randomness (to achieve the fewest degrees of separation) among their components. All nonlinear networks have emerging and flexible structures. Randomness allows networks to experience instability, giving them the ability to function far from equilibrium, at the edge of chaos. Upon reaching a critical point, networks spontaneously give rise to a new order of greater complexity (Graph). This process constantly occurs in the real world. For instance, if we heat a container of water, the water molecules at the bottom of the container start moving upward randomly. The more heat they receive, the more chaotic their upward movement becomes. Upon reaching a critical point, a new order suddenly emerges, in which each water molecule joins one of many three-dimensional hexagons. Molecules at the bottom travel vertically to the top and then move down the sides of the hexagons. They move in such an orderly fashion that it seems as if they could all be aware of each other (Diagram).
This is just one example of the self-organizing capability inherent in nonlinear networks that allows for the spontaneous emergence of order out of chaos, which has been formulated in chaos theory. Chaos and instability make networks flexible, which is a useful quality as the pace of change continues to increase. By understanding the behavior of networks, we can change our pessimistic and linear perception of randomness, instability, and the chaotic nature of change into an optimistic, nonlinear perception.
The whole is greater than the sum of the parts in a nonlinear network
New order emerges when all elements within a given system synchronize. Their cooperation creates a synergic relationship among them, elevating the quality and functionality of the network to the whole which is greater than sum of the parts. We have all experienced this holistic, cooperative effect when among large groups of people at concerts, games, political demonstrations, or religious events. To help explain this concept further, I will use excerpts from Sync: The Emerging Science of Spontaneous Order (2003) by Steven Strogatz.
At the heart of the universe is a steady, insistent beat: the sound of cycles in sync. It pervades nature at every scale from the nucleus to the cosmos. Every night along the tidal rivers of Malaysia, thousands of fireflies congregate in the mangroves and flash in unison, without any leader or cue from the environment. Trillions of electrons march in lockstep in a superconductor, enabling electricity to flow through it with zero resistance. In the solar system, gravitational synchrony can eject huge boulders out of the asteroid belt and toward Earth; the cataclysmic impact of one such meteor is thought to have killed the dinosaurs. Even our bodies are symphonies of rhythm, kept alive by the relentless, coordinated firing of thousands of pacemaker cells in our hearts. And that raises a profound mystery: scientists have long been baffled by the existence of spontaneous order in the universe. The law of thermodynamics seems to dictate the opposite, that nature should inexorably degenerate toward a state of greater disorder, greater entropy. Yet all around us we see magnificent structure-galaxies, cells, ecosystems, human beings, that have somehow managed to assemble themselves. This enigma bedevils all of science today. The tendency to synchronize is one of the most pervasive drives in the universe, extending from atoms to animals, from people to planets. All the examples are variations on the same mathematical theme: self-organization, the spontaneous emergence of order out of chaos.
The best modern example of the whole being greater than the sum of its parts is the function of light in the form of a laser. The intense, coherent, needle-thin beam of a laser is a result of trillions of atoms emitting light waves in sync. The atoms themselves are no different from those in an ordinary light bulb; the trick is in the way they cooperate. Instead of cacophonous light of different colors and phases, laser light is one color and one phase, a chorus singing the same note. The resulting laser is composed of the atoms that have the same color and phase, which when combined makes them into a beam that is more than what the individual atoms are capable of.
The integrative, synchronized, and emergent behavior of the components results in the holistic, cooperative, and qualitative characteristic of nonlinear networks. The synergistic character of nonlinear systems is also what makes them so rich. Every major unsolved problem in science, from consciousness to cancer to the collective craziness of the economy, is nonlinear. As is evident in a laser, the synchronized whole has a quality and functionality which none of the individual components had.
There is no proportionality between cause and effect
Again, excerpting Sync: The Emerging Science of Spontaneous Order by Steven Strogatz:
“Linear equations describe simple, idealized situations where causes are proportional to effects, and forces are proportional to responses. Linear equations are tractable because they are modular: they can be broken into pieces. Each piece can be analyzed separately and solved, and finally all the separate answers can be recombined – literally added back together – to give the right answer to the problem. In a linear system, the whole is exactly equal to the sum of the parts. But linearity is often an approximation of a more complicated reality. Most systems behave linearly only when they are close to equilibrium, and only when we don’t push them too hard.”
Since nonlinear systems perform far from equilibrium, cause and effect are not proportional. In a nonlinear system, small changes may have dramatic effects because they may be amplified by self-regulating feedback loops. In highly connected nonlinear networks, sometimes small changes can have major implications, while at other times even major changes can be absorbed with remarkably little disruption. When we observe the real world from a nonlinear perspective, we see examples all around us in which cause and effect are not proportional. In the following paragraphs I will demonstrate this property on the cosmic and personal levels.
The behavior of water illustrates this principle of network theory. Water has one exceptional property when compared to any other liquid: when it freezes, its volume increases and its density decreases. This ability to gain volume while transitioning between liquid and solid makes ice lighter than water. Scientists believe that because of this property, during the early ice ages, ice on the top of the oceans protected the water underneath from volatile, unlivable conditions on the surface while being thin enough to allow some sunlight through, thus enabling the early phases of microbiological development to survive. (Pollard & Kasting, 2005) Without this exceptional property of water, we would not exist to know about it; human life was made possible by one seemingly insignificant property of water.
A car accident can also illustrate the lack of proportionality between cause and effect. For example, if I received a phone call just as I was heading out the door and it was a wrong number, it would delay me by just a few seconds. This delay would have me reaching intersections at a slightly different time than I would have otherwise. Someone might run a red light and hit me. Linear logic freezes the event at the moment of the accident and makes its judgment about the dominant cause based on information available in that particular moment: the other driver might have been drunk or sleepy. But in reality, there are many factors that led to the crash in that intersection. For example, if I had not answered that wrong number I would have passed through that intersection a few seconds earlier, avoiding the accident. The phone call was such an insignificant cause, but it had a major effect.
Nonlinear networks have a power law distribution
In nonlinear networks, not all composed elements participate equally. For instance, the sun is composed of 71 percent hydrogen and 27 percent helium. The remaining 2 percent is comprised of a variety of elements. Elements in the final 2 percent are not equally distributed either. Oxygen is the most abundant (42.9 percent), followed by carbon (17.7 percent) and iron (9.7 percent). Various metals make up the tiny remainder.
This uneven and scale-free type of distribution is called a power law distribution. (See graph above). It is the most surprising characteristic of nonlinear networks. All nonlinear networks follow the power law distribution principle. In non-linear network such as the Internet, those sites with the most connections have the lowest degree of participation, and the sites that exhibit the highest degree of participation have the fewest links or connections. Take Google, Yahoo, or other large websites. There are only a few of them participating at that level, so they have the most connections with other websites. Because those few large sites have so many connections, they have more power within the network and their effectiveness to reach other websites within the network is that much stronger. But those sites that have the highest degree of participation, such as the millions of small personal websites, have the fewest connections. In nonlinear networks, the nodes with higher connectivity in the network have more power and more control.
Power law distribution applies to all nonlinear networks, including those of wealth, social power, blood circulation, the nervous system, and the natural world. Consider the Mississippi River: thousands of creeks contribute small amounts of water, but only a few large rivers contribute most of the water that forms the Mississippi River. The law applies even to written texts, where a few words such as “the” and “and” participate the most, and all the other words, which are the majority, each participate rarely and randomly.
Nonlinear networks display self-similarity and universality
Nonlinear systems perform at the edge of chaos, and fractal mathematics is the geometry of chaos, so I will offer you a brief introduction to that subject. Capra observed, “In nonlinear systems, small changes may have dramatic effects because they may be amplified repeatedly by self-reinforcing feedback. Such nonlinear feedback processes are the basis of the instabilities and the sudden emergence of new forms of order that are so characteristic of self-organization. Mathematically, a feedback loop corresponds to a special kind of nonlinear process known as iteration, in which a function operates repeatedly on itself.” The Web of Life (1997)
If we take a nonlinear function and assume initial values for constants and variables for the first round, then start applying the iteration process by repeatedly taking the resulting value and feeding it back into the system, we may see that the patterns are diverging, converging, or constant. But then we reach a zone where the results provide chaotic values without any traceable pattern. With the help of a computer, we can divide that chaotic zone into thousands of points and plot the obtained values. The resulting plotted values reveal self-similar patterns in which any magnified part looks just like the larger section from which it was taken. As we observe this process, we will notice that even though we see chaos on the surface, at deeper levels of observation the elements are creating an emerging order of self-similar patterns. If examined at different scales, they all display the same degree of order. In other words, they look the same at all scales.
The most surprising quality of fractals is that they are created by the repetition of basic operational rules. That signifies a deep simplicity at the core of the creation of this complex universe. Self-similarity is a powerful property of nonlinear systems.
The observation that very different systems can exhibit fundamental similarities is generally referred to as universality, and its apparent validity presents one of the deepest and most powerful mysteries of modern physics. It is mysterious because there is no obvious reason why systems as different as superconductors, ferromagnets, freezing liquids, and underground oil reservoirs should have anything in common at all. And it is powerful precisely because they do have something in common, which tells us that at least some of the properties of extremely complicated systems can be understood without knowing anything about their detailed structure or governing rules.