Sep 22, 2014 | By

How can There be Order in Randomness?

Thomas Pynchon, in his sprawling novel Gravity’s Rainbow (1973), exhibited a fascination for the peculiar mathematics of the Poisson Distribution, a pattern exhibited in certain random sequences (including the location of German rocket strikes in London during WWII). Sometimes referred to as a “law of rare events”, the Poisson distribution has proven to apply to such disparate phenomena as: the volume of Internet traffic; deaths per year in a given age group; DNA mutations resulting from radiation; goals scored in sports with two competing teams; etc.

How is it that mathematical patterns exist for random events? This is the puzzle explored in the mathematics of complexity, a series of related fields that play an increasingly important role in the study of complex systems including physics, biology, economics and sociology. For an excellent, non-technical resource, I highly recommend Melanie Mitchell’s 2009 book – Complexity: A Guided Tour. More recently, Ms. Mitchell posted an article on “Complexity” in The Templeton Foundations Big Questions Online series where you can see both the article and the exchange of comments.

Screen Shot 2014-09-22 at 10.34.17 AMPatterning in peacock feathers. Random variation?

Quite remarkably, complex systems often exhibit self-organizing behavior, arising from the independent and uncoordinated actions of the entities that comprise that system. For example, an army ant is an individual organism with fixed and quite simple behavior patterns. Yet in a colony of millions of individual army ants, intelligent and sophisticated adaptive behaviors appear spontaneously at the collective level, without benefit of either top-down control or intelligence at the individual level. There are no generals in this army, yet the behavior appears coordinated. The behavior is referred to as “emergent.”

How does intelligent behavior appear in a system with no apparent command? The answer seems to be in the way the system explores for information about the environment and then adapts based on that information. For example, food gathering by an army ant colony relies on the foraging behavior of individual ants. Ants on a scouting mission will begin searching in random patterns, but if one finds food it will return straight to the colony, leaving a pheromone trail. In addition to random searching, ants will also sometimes switch to following pheromone trails – tending to reinforce successful pathways to food. Through a balance of random and following behavior by a large number of individual ants, a complex map of food sources and pheromone trails develops. This map is not in the mind of any of the ants, but in the collective behavior of all the ants in the colony, as if the colony itself had a mind.

Screen Shot 2014-09-22 at 10.34.00 AMA murmuration of starlings showing emergent patterns.

Similar complex behaviors are demonstrated in the human immune system as it responds to infection, and in cellular metabolism as the soup of organic chemicals responds to changing conditions inside and outside the living cell. In all these cases, the processes that we observe appear to be finely tuned, reflecting what must be a long and incredibly complex evolutionary history. The Darwinian explanation for these emergent phenomena is that incremental changes or mutations at the component level are tested against the environmental conditions (the fitness landscape), and the optimal changes are selected for by reproduction. In short, the behaviors arise as a result of the evolutionary benefits to the colony, or the body, reinforced through a selection process that finely tunes the system over many, many generations.

Another feature of many complex systems, including the three examples in the last paragraph, is that they can be viewed as a network of connected parts. Army ants operate as a network of foragers – the immune system as a network of cells – the metabolic system as a network of chemicals. As the individual parts bump into each other, they communicate, creating a network of nodes and channels. There are two characteristics of networks that commonly develop in nature. First, they are what are known as “small world networks”, which demonstrate a lot of interconnectedness among nodes. This means there is a small average “degree of separation” between any two parts. In these cases, transmission of information from one node to all nodes will be relatively efficient, compared with a network with higher average degree of separation. Second, they exhibit a pattern referred to as “scale-free”. Scale-free networks demonstrate a consistent pattern of connectedness – very few nodes are highly connected and very few are minimally connected. The ratio of connections to nodes in a scale-free network can be drawn as a smooth curve that looks the same on all parts of the graph. The curve is described mathematically in what is known as a power –law distribution (e.g. P = ( 1 / kx ), where P is the fraction of nodes having k connections and x is the scaling factor). Power law distributions seem to be common in both information networks and biological systems.

Screen Shot 2014-09-22 at 10.33.42 AMMultiple spirals in a sunflower blossom. Efficient and beautiful.

Melanie Mitchell presents a striking example of a power law distribution and how it develops in discussing the question of metabolic scaling. Put simply, metabolic scaling is the observed phenomenon that larger animals have slower metabolisms. Naturally, we would expect this result and intuitively recognize that as body mass increases, surface area increases, but not as fast. Specifically, surface area increases as the square of height and body mass as the cube – a power law ratio of 2/3 would result from a simple geometrical relationship between the two. Yet the expected power law exponent of (2/3) for metabolic scaling proves to be incorrect – the measured result is (3/4).   Larger animals are able to sustain a higher metabolism than one would expect. Why is this the case? After considerable sleuthing, researchers realized that the fractal branching pattern of capillaries in the lungs produces a higher oxygen uptake than mere surface area. Fractal structures (a related branch of mathematical complexity) are able to maximize how much surface area can be packed into a three dimensional volume.

It seems quite remarkable to find these odd pieces of mathematical complexity theory including fractals, power law distributions, scale-free, small-world networks, and self-organizing behaviors throughout the different branches of science. These are beautiful examples of what Eugene Wigner in 1960 called “the unreasonable effectiveness of mathematics in the natural sciences”. Physical reality seems to conform to mathematical principles or, to express it differently, mathematics seems to constrain the forms which physical reality displays. In this sense, then, mathematics is a metaphysical necessity of the physical universe. (see: Do the Puzzles of Mathematics Prove God)

While the mathematics of complexity has been helpful in identifying and describing the mathematical forms that emergent systems exhibit in many fields, it does not explain why the world demonstrates such an inexorable and universal directionality towards diversity and increasing complexity. Just as physicists cannot explain why the universe follows the arrow of time towards higher entropy and decreasing order (see The Puzzle of Entropy), neither have complexity theorists offered an explanation for why the universe displays an inexorable (and seemingly contradictory) path towards increasing diversity and complexity in ordered patterns.

We are not going to solve that problem any time soon. Among other things, it raises important questions about how we think of causation.  (see Causation)   Indeed, the enormity of these questions suggests that we will need to develop an entirely new conception of the fundamental nature of reality before we will be able to explain or understand the phenomena we are observing.

2 Responses to “How can There be Order in Randomness?”

  1. Marius Dejess says:

    I like to read any one here telling me what his concept of complexity is, and also order, and also randomness.

    You see, I cannot accept that there can exist in the same place at the same time and in the same aspect of consideration, an event that is composed of random components in random agitation among themselves and components in an orderly interaction among themselves: and all these components are unconscious and not controlled in whatsoever sense we can understand control; so that all components are into all manners of vicissitude of shall I say, movement, in all directions without ever any component repeating its movement in the same direction for the same time and with the same strength and speed.

    So, please, someone here, anyone at all, tell me what is your concept of randomness, order, and complexity.

    I seem to see that some people are of the idea that complexity and randomness are convertible, which I see to be nonsensical.

    • George Gantz says:

      Marius – You ask a very big question. Randomness, order and complexity are the dance of the universe. Randomness is an exploration, a seeking process, as the agitation and jostling of individual components leads to a sorting out and the formation of an orderly structure. This process is referred to as “complexity theory” because the simple things (components) form incredibly complex structures. This is the way I described it in “The Empirical Standard of Knowing”:

      “Local structure and order emerges by exporting entropy to the larger environment. The entire universe as a whole continues to run down, towards some icy and inevitable death state, but as it does, local pockets of increasing organization and structure emerge.

      An explanation for how this counter-entropic process operates to bring order, structure and growth can be found in the theory of non-linear dynamic systems pioneered by Ilya Prigione and others. This theory posits that in open dynamic systems where energy is in flux, stable structures tend to emerge in the otherwise chaotic flow. These structures succeed by dissipating the energy within the chaotic flow efficiently. For example, when we open the drain at the bottom of a sink the water molecules rush for the drain, bouncing and jostling in a disorganized and turbulent chaos – yet we can then see the swirl of a whirlpool – a stable and orderly structure – emerge. What we are watching is the transformation that the dynamic nonlinear system of flowing water is going through as it seeks out a state that maximizes the efficient local dissipation of energy. The result is a stable, persistent structure.

      Remarkably, all of the structures we see and study in the universe — from cosmological and astronomical, to geological and meteorological, to chemical and biological, indeed including all of life, ecology, even economics — are the result of dynamic systems that exhibit this behavior.”

      To my mind, while the behavior of an individual component can be described as “random”, in the sense that the component does not plan or calculate its interactions with other components (unless it is a conscious human), however it is still “purposeful” – as the component is under some pressure or force to “get somewhere”. This may seem contradictory, but as I explore in the essay, much in this grand and glorious universe is paradoxical, and that is one of its mysterious beauties.

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