As a follow-up on my previous piece on scale-free networks and Web3, I want to dive deeper into complexity and chaotic systems and zoom in on the role of chaotic attractors in this essay. I attempt to link the discussion of chaotic systems back to Web3 through framing the Web as a dynamical system. I am certainly no expert on these topics and the goal of my writing is to organize my thoughts into one repository that is vulnerable to criticisms and receptive to feedback. All that is to say, please be as disagreeable and critical as you can and pass along any resource of even tangential value.
Britannica defines a complex system by the existence of emergent behavior, meaning behavior of a system that is inexplicable by knowledge of each constituent in isolation. Chaos, or more precisely deterministic chaos, is not synonymous with complexity but an epiphenomenon that characterizes a complex system with high sensitivity to initial conditions. In the now-famous Butterfly Effect story, Edward Lorenz created a toy model to describe the atmospheric convection if the inputs were simplistically reduced to three variables, say temperature, wind speed, and humidity. What Lorenz discovered was that in a system with a set of three equations with three variables that change over time (i.e., dynamical), the existence of nonlinearity could cause the system to behave chaotically. In other words, the trajectory of a chaotic system would be highly sensitive to the starting point – a tiny perturbation (i.e., a measurement precision difference) in the starting point would lead to widely different future trajectories of the system. A future dictated by deterministic laws may be indistinguishable from indeterministic systems.
What is lesser-known about the story, however, is that the “butterfly” in the name, whose proverbial flap of wings in Japan could lead to a tornado in Brazil, did not come from a natural phenomenon, but the shape of the attractor in the Lorenz system. In any dynamical system, an attractor describes the set of states towards which the system evolves, regardless of the initial starting point. In the Lorenz system, which we now know has chaotic solutions, the attractor can take on the shape of a butterfly. While all trajectories are different depending on the initial starting point, they all accumulate on the same butterfly-shaped object irrespective of where they have started. In other words, even though a small perturbation may tip the system into a widely different trajectory, the new trajectory would still be attracted to follow a similar, yet not identical orbit. The attractor of a chaotic system is thought of to contain an infinite number of aperiodic orbits that do not cross one another at any point, leading the system to be globally stable (i.e., organized around the attractor) but locally unstable (i.e. any small perturbation could tip over the trajectory).
The Lorenz attractor was a fascinating discovery and turned out to be one example of a strange attractor. A strange attractor lies at the heart of a chaotic system and sheds light on its underlying order. The easiest way to examine a strange attractor is in the phase space of the system, which represents all possible states of a dynamical system and gives a global view of its long-term behavior. Each parameter of the system is represented by an axis and characterizes the attractor. The Lorenz system, for instance, has a three-dimensional phase space (i.e., three parameters), and may have attractors of varying degree of complexity. A phase space of the initial “Butterfly Effect” system Lorenz was analyzing would show that trajectories of all possible states are infinitely different but all accumulate on the same butterfly-shaped attractor in a seemingly finite bound. This ability to generate an infinite set of orbits within a finite space is attributable to the fractal nature of any strange attractor. Most importantly, the discovery of the concept of a strange attractor accelerated the notion that there may just exist some order underlying a chaotic system. While a chaotic system never settles into an “equilibrium” and never repeats itself, giving rise to the unpredictability of such a system and the feeling of chaos, there is an overall pattern that becomes more tangible when we map out its phase space.
So how does this relate to our discussion of Web3? I want to expand our definition of the “Web” or the “Internet” more broadly to this connected, digital space, as that is after all what we think about when we discuss the Web. I argue that the Web is a complex and chaotic system, similar to many other networks in real life. The Web evolves over time as a result of an uncountable number of complex feedback loops between codes and user actions. It is hard for us to picture anything with more than three dimensions, let alone a possibly infinite number of dimensions due to a possibly infinite number of parameters. Therefore, it is practically impossible for us to map out said attractor of a real network like the Web or its phase space, but the thought experiment gives rise to an interesting proposition — that perhaps on this infinite number of dimensions, networks such as the Web evolve along trajectories that do ultimately all accumulate on a certain attractor with an unknown number of unknown parameters. Indeterministic as they may seem, there may exist some order under complex networks in real life, and the prevalence of power law is a breadcrumb pointing to this universality underlying the unpredictable chaos.