Extralogical Complexity Part 1: The Nonlinearity of Nature

 

For an environment to evolve into something like Nature, Nature can’t be what people think. 

 

Vertebrate cognition’s need to make comprehensible and motivating models of reality leads to a host of fallacies and illusions about Nature, the World, and human beings. Linear change—constant, predictable change characterized by bell curves and straight lines--is too weak to account for the self-organizing, adaptive powers of Nature. Nor can change reliably accommodate neat, satisfying explanations, where initiating events always lead to proportional outcomes. This treatise illustrates the prevalence of nonlinear change in Nature, which gives rise to the characteristic phenomena of Complex systems theory presented in the upcoming part two.  

 

You’ve heard “the balance of Nature,” “balance in the Force,” and the like. This is almost exactly what Nature isn’t. Some say it’s chaos or disorder. While common, disorder and mathematical chaos don’t characterize Nature, either. Between balance and disorder, there is dynamism. To host life, an environment must persist “out-of-equilibrium” yet maintain recurrent structure. 

 

ER defines a living thing as a system that maintains its approximate identity by persisting in a periodic-like process, especially within a less ordered environment. This includes constituents of other large systems/networks like industries and economies. For a system to qualify as biological life, it must, among other things, organize and consume energy while persisting out of thermal equilibrium with its surroundings. Life forms are, in other words, oases of periodic order. 

 

Within the life-hosting environment, energies and forces must flow, and flows require thermal, electric, and mechanical gradients--or imbalances. Too little disorder also doesn’t give the hosting environment enough shocks to force it to adapt, to evolve. But it shouldn’t be disordered, either; otherwise, oases of periodic-like order don’t have sufficient time to evolve. You need recurrences, emergences of species and their specimen, businesses and industries. You need recurrent patterns—seasons and geochemical, migration, business, and market cycles, which are often entrained or synced. Since these patterns include periods of suboptimal conditions, they give life-hosting environments antifragility--robustness and adaptability--but they remain predictable enough to avoid readily dismantling the environment (antifragility from Nassim Taleb).   

 

Above all, the environment must readily give rise to self-amplification or self-reinforcing feedback: Occurrences, phenomena, and patterns that reinforce and reproduce themselves, which can bootstrap themselves into existence and out of crises. And this can’t happen in places characterized by linearity and Gaussian distributions. Only in places dominated by nonlinear change, environments characterized by power laws and “fat-tail” distributions, can you have life as humanity knows it.

 

 

The “Physics” of Complexity

 

Complex networks have high connectivity. People in society are separated by six; the animals in the global biosphere by ten; nervous systems have an average of 360 direct synaptic connections; and internet sub-networks (Google, Verizon, Harvard, etc.) are connected by less than four. If a system of variables has too much independence, they dissolve into separate systems; too much dependence, they ossify into a single variable. It’s in the in- between where most of Nature lies. When this happens, you get Complexity: emergence, synergy, and the self-organizing system. As a general rule, Complex networks function best when Complexity is high.

 

A neuron with 360 direct connections, for example, grants a newly connected neuron 360 secondary connections and 129,600 tertiary ones. In a network like a nervous system, these secondary connections easily lead to the dominance of subsequent interactions, overawing the influences of initiating events. Owing to the growing disproportionality between direct and indirect connections, Complexity can be seen as the resultant dominance of secondary interactions/events. From this dominance comes an emergent, self-organizing system, which will be introduced in part two. Connectivity is multiplicative, not additive and can, therefore, easily give rise to nonlinear cascades. This is not to say that all systems’ potential connections and interactions will happen or be “strong” enough to matter, nor that additive/linear effects are rare. But given the potential, a small percentage can produce extraordinary nonlinearity.

 

All factors anywhere near equal, increasing the strength and number of initial events, the size of the system, and/or connectivity will tend to disproportionately increase secondary interactions and, therefore, Complexity. Decreasing initial interactions, network size, etc. results in disproportional decreases in Complexity. However, excessive coupling, or strength of connections, increases the propagation of effects, but you’d rigidify interactions and get tremendous redundancy in signaling and effects. You’d have higher propagation, analogous to how sound travels faster in water, and more total Complexity and synergy; but they’d have too much redundancy to be “useful.” Analogizing the network to music, you’d get loud rhythmic thrums rather than a symphony (see article on law of synergy). Thus, USEFUL Complexity can be seen as the dominance of DIFFERENTIATED secondary interactions/effects.

 

The next segment illustrates these effects and their consequences in action.             

 

 

Preferential Attachment

 

Nonlinearity frequently manifests when a small disparity in ability leads to a much greater disparity in success. A slightly better product leads to more return customers; more return customers lead to more word-of-mouth; more word-of-mouth leads to a social trend; more profits lead to new investments and products, stores, and, perhaps, other businesses and/or public companies. And “success begets success” as readily in Nature as it does in business. If a specimen is slightly better fit than another and survives, it can potentially have far more decedents, which applies to everything from animals, plants, bacteria, viruses, to single strands of self-replicating genetic material. 

 

The formal term for success begetting success is preferential attachment, and it accounts for “hubs” within a network, constituents with an unusual number of connections/interactions. A more attractive flower will initially attract pollinators by smell or sight. Pollinators tend to return to the same sites. Conspecifics (members of the same species) and heterospecific pollinators (different species) often follow, decreasing parity in success and disproportionally increasing heterospecific interactions. The socially successful not only tend to be better at making friends, but also have more chances to them due to meeting more people—then people want to befriend them so they can make other friends. A slightly more fit predator species has disproportionality larger population growth and, therefore, disproportional expansion, increasing heterospecific encounters and, thus, interactions. This type of propagation occurs in “normal” periods as well as in crises.      

 

Social giants are society’s hubs. Throughout the country, people are connected by an average of six degrees of separation, with a single degree being the friend of friend. In Earth’s biosphere, species are connected by ten degrees. Resulting from their disproportional propagation, connections are unevenly scaled, leading to disparity in connections and numbers. Disparity ensures that at least a few constituents are strong enough to survive most crises, increasing an ecosystem’s antifragility. At the same time, excessive hub-orientation can lead to nonlinear runaway extinctions. As confirmed by Complexity experiments, cataclysmic events, like asteroid collisions or hyperactive volcanos, aren’t required. Nor are the economic equivalents required for economic crashes. 

 

Belief propagation in society can work this way if each person convinces more than one person on average. Since they’re composed of entangled packages in an interconnected network, propagation in a personal pool of beliefs is similar. Personal and societal belief propagation may be comparable with information cascades, as well (see Ubiquity by Marc Buchanan). Everyone has at least some tendency to conform--some more, some less, some different. A conformity threshold is the number of people that need to conform to a given behavior (e.g., a riot, the purchase of a new product) before a given person succumbs. A person with a threshold of one conforms after the first person conforms; a person with five succumbs after five conform. This has resulted in entirely unexpected behavior, such as riots. Of course, cascades aren’t completely reliable because the absence of a single threshold at the wrong time can stop the trend dead in its tracks. Information cascades, unsurprisingly, tend to be accelerative: begin slowly, increase at an ever-increasing rate, slow down, then end quickly. 

 

And mimicry isn’t the only cause of unexpected interactions between a network’s parts. Noise, or random stimulus, is more influential than believed, as well (“random” is defined more by how its perceived, not necessarily whether it “is”). In 2021, Daniel Kahneman, et al released Noise, an account of experiments and real World data that shows how profoundly and unpredictably people are influenced by background stimulus. When people are given numbers randomly, for example, it always effects their estimates of the number of African countries or nations in the UN. Criminal sentences and insurance payouts are notoriously erratic, even from the same issuers. Studies show expert real estate appraisals are always quantitively influenced by prior estimates, including when it conflicts with their qualitative assessment. Noise can, thus, cause one trend or effect to influence your choice in another area. Trends in computer esthetics, for instance, might not affect your choice of computer, but it could affect the car you buy. This amplifies and suppresses trends, adding to unpredictably.        

 

The perpetual change of numerous related variables and subnetworks acted upon multiplicatively leads to positive and negative amplification and reinforcement, including self. Small “causes,” like the flap of a butterfly’s wings, can multiply into large effects, and preventing small causes prevents large effects. This is the self-amplification that life and Nature require.

 

The laws of causality and energy conservation still apply, of course, just as the flap of the butterfly wings would be better described as a trigger. Complex networks susceptibility to runaway effects allows “small” planned or unplanned events to have unexpectedly “larger” consequences. Physical laws are satisfied only after accounting for intermediary events, ones unlikely to be ascertained for accounting. Multiplying effects merge with other effects; mutual dependencies develop; and feedback loops stabilize--resulting in organized self-perpetuating phenomena. 

 

From this comes sensitivity to perturbations and initial conditions, and, therefore, intractability and minimal predictability. Because initial conditions are so sensitive, so isn’t the precision and quantity of information necessary to predict large systems/networks, which put to shame any hypothetical human database. But since networks obey certain laws—economic, biological, ecological, etc.--and maintain their approximate identities for extended periods, they do have guess-ability. The dynamics of Complex networks contradict the intuitive view of causation: Assumed proportionality in cause and effect is part of the causation bias, the assumption of satisfactory explanations.

 

The result: 

Scaling relationships, mathematical relationships such as power laws, formulas connecting the frequency and magnitude of events like earthquakes, extinctions, and economic crashes; power law distributions like Pareto’s law of the distribution of wealth; exponentially scaled connections between constituents of large systems/networks such as connectivity and interactions between species and businesses. Power laws emerge in computational experiments exactly as they do in the real World.  

 

Most importantly, nonlinear change and secondary interactions are ultimately responsible for the self-organization, synergy, and emergence of Complex networks. This is the Complex system. This will be introduced in part two.