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Glimpses at Mathematics and Physics of Social Complexity

Vladimir Dimitrov

University of Western Sydney
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Philosophy is written in this grand book - I mean universe - which stands continuously open
to our gaze, but which cannot be understood unless one first learns to comprehend the language
in which it is written. It is written in the language of mathematics, and its characters are triangles,
circles and other geometric figures, without which it is humanly impossible to understand
a single word of it; without these, one is wandering about in a dark labyrinth.
Galileo (1623)

I.   Dynamics: Forces and Interactions
II.  Phase Space, Dimension, Trajectory
III. Strange Attractors
IV. Bifurcation Diagram
V.  Self-Organized Criticality
VI. Butterfly Effect



We know that the atoms are made of protons, neutrons, and electrons - the "building blocks" of matter. Protons and neutrons are made of quarks, discovered in 1964 by Murray Gell-Mann - one of the founders of the Santa-Fe Institute of Complexity 20 years . There are SIX types of quarks, which can never be found separately, but only inside composite particles called hadrons.

What holds the matter together? FORCES. All forces are due to the underlying interactions of the particles. Five interaction types are all that are needed to explain all observed physical phenomena.

Four types of interactions underlie four types of forces: gravitational, electromagnetic, strong, and weak.

Gravity is perhaps the most familiar force to us; its effects are tiny in particle processes. On the largest scales, gravity is the most important force.

Electromagnetic forces are also familiar; they are responsible for binding the electrons to the nucleus to form electrically-neutral atoms. Atoms combine to form molecules or crystals because of electromagnetic effects due to their charged substructure. Most everyday forces (such as the support of the floor or friction) are due to the electromagnetic forces in matter that resist displacement of atoms or electrons from their equilibrium positions in the material. The carrier particle of the electromagnetic force is the photon.

The strong force holds quarks together to form hadrons; its carrier particles are whimsically called gluons because they so successfully "glue" the quarks together. The binding of protons and neutrons to form nuclei is the result of the residual strong interaction due to their strongly interacting quark and gluon constituents.

The weak forces are the only processes in which a quark (or some other elementary particles) can CHANGE to another type of quark (or particle); they are responsible for the fact that all the more massive quarks decay to produce lighter quarks. In a nucleus where there is sufficient energy, the weak forces make a neutron become a proton and give off an electron and an anti-electron neutrino. It is because of the weak forces that the stable matter around us contains only electrons and the lightest two types of quarks.

The fifth fundamental interaction is responsible for the masses of the fundamental particles (such as quarks, gluons, photons, electrons, etc.).

Possible Social Interpretations

The forces which sustain our universe reflect the dynamics of human complexity. Different scales of this complexity manifest in different ways the forces underlying physical exitence of matter. Human process of thinking and feeling are inseparable from the rhythm of our  brains and hearts, from the pulsation of every single cell belonging to human body. The molecules and atoms of cells are products of interactions between the elementary fuilding blocks of matter.

Society is a product of interactions between people and these interactions are shaped by our life together; therefore society and social interactions are in self-referential relationship. Interactions manifest subtle and yet enormously powerful social forces rooted in the depth of human nature - desires, motives, endeavors, attachments.

According to R. Rummel , "social interactions are the acts, actions, or practices of two or more people mutually oriented towards each other's selves, that is, any behaviour that tries to affect or take account of each other's subjective experiences or intentions." For "Symbolic Interactional Theory and Nonlinear Dynamics", see: .

Strong interactive forces are rooted in our instincts as "social animals" and directed towards sex, reproduction, active search for food and other means and conditions for physical survival as individual species.

Weak forces relate to changes occurring with of our state-of-being: physical, mental, emotional, psychological (spiritual).

Electromagnetic forces manifest in mutual exchange of mental, emotional or/and spiritual "energies" in any process of communication between each others and within ourselves.

Gravity is often expressed in mutual sympathy and desire to share feelings, experience and knowledge, to be in friendly collegial or other type of mutually respectful relationships, etc. Often gravity can express underlying strong forces.

The fifth interactional force might be responsible for an individual's integrity, identity and ability to stay connected with ("grounded in") his or her inner self and deep spiritual essence.

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Let us consider a very simple physical system, a single particle with no properties beside its position in space and the rate of change of its position.

Even such a simple object requires SIX numbers to describe its state at any given time; three numbers -how high above or below us, how far to the east or west of us, and how far from the north or south of us - to describe its spatial location, and three additional numbers to describe how rapidly this spatial location is changing. It is usual to give the last three numbers as moments rather than velocities, where the "momentum" of a particle in a given direction is the mass of the particle times the velocity in that direction. At each instant of time, the state of this single particle is completely described by six numbers. As time advances, the state of the system changes, and this change is completely described by the change of these six numbers. Each of the six numbers can be thought of as a dimension, just as height, length and breadth are the usual three dimensions of ordinary space. The other three numbers, the moments, describe the "position" of the particle in "momentum space".

The STATE SPACE of the particle is thus six-dimensional: the three dimensions of ordinary space, and the additional three dimensions of momentum space. As the particle changes its state from instant to instant, it "moves" in the six-dimensional state space. A "point" in this state space thus represents the complete description of the particle at any instant of time. A trajectory in this six-dimensional space represents the history of the particle's states throughout time. This state space is the particle's PHASE SPACE.

Now let us consider a more complicated physical system: a system consisting of N particles just like the single particle we previously considered. ("N" is just some arbitrary whole number; any physical system can be described as consisting of N particles). The state space - the phase space - of this more complicated system has 6N dimensions, and as before, the state of the entire system at any instant of time is a point in the 6N-dimensional space. The state of this physical system is completely described by giving its location in this 6N-dimensional space. As before, a trajectory in this 6N-dimensional space is a history of the system's states over time.

The phase space includes all possible states of a dynamic system - all its history, all its present and future trajectories.

Possible Social Interpretation

According to R. Rummel quated previously. "(social) space can be considered a domain of human potentialities, dispositions, determinables, and powers, such as an arena within which these intersecting latents form patterns and clusters. We can thus perceive a psychological space, a sociocultural space, a space of meanings or of language, a behavioral space, etc.
(see: )

For application of the concepts of human experiential and mental space in studying social complexity, see the papers "Intrapersonal Autopoiesis" and "Swarm-like Dynamics and their Use in Organizations and Management".

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Strange Attractor is a basic notion in the mathematical theory of chaos, which studies nonlinear dynamics characterised by extreme sensitivity to tiny changes in the initial conditions. The term ëstrange attractorí was firstly used by D. Ruelle and F. Takens (1971) when describing dynamic behaviour of their mathematical model of fluid turbulence.

In 1975 the American mathematicians Li and Yorke firstly used "chaos" as a mathematical term in their widely cited paper "Period Three Implies Chaos" (Amer. Math. Monthly 82, 985) to characterize the emergence of an uncountably infinite number of asymptotically aperiodic ("chaotic") orbits in system dynamics. The theorem says that in any one-dimensional system, if a regular cycle of period three ever occurs, it will also display regular cycles of every other length, as well as chaotic cycles. This was not a new theorem - Li and Yorkeís result was already proven (11 years earlier) in a paper written by the Russian mathematician Sarkovski.

In the theory of dynamical systems, an attractor is defined as a preferred position for the system, such that if the system is started from another state it will evolve until it arrives at the attractor, and will stay there in the absence of other factors. An attractor can be a point (e.g., the centre of a bowl containing a ball), a regular path (e.g., a planetary orbit), a complex series of states (e.g, the metabolism of a cell) or an infinite sequence of chaotic trajectories located in a bounded region of phase space (called strange attractor).

While on the strange attractors, initially close trajectories of system dynamics can separate (diverge) and initially distant trajectories can come close (converge).

Mathematically, the strange attractor is defined as an attracting set with zero measure (that is, a set capable to be enclosed in intervals with arbitrary small total length) in an embedding phase space, and has fractal structure (that is, a structure, which displays self-similarity on all scales of its manifestation). The trajectories, that is, the traces of the energies and forces whirling within the strange attractor, appear to skip around randomly.

Here is the place to recall the formulation of a fractal as given by the mathematician Benoit Mandelbrot, the "farther" of fractals and fractal geometry - "The Geometry of Nature", as he titled his first book on fractals (1983): "A rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole"

Social Interpretation

We use the concept of strange attractor to describe emergence of meanings in the mental space of an individual or a group (organization).

The phase space where meaning emerges is the 'multi-dimensional' mental space of an individual or a 'swarm' of individuals - a non-material ("transcendental" in Kantian term) space energized by continuously generated thoughts and feelings.

Meaning has fractal structure - once a certain dynamical sign makes sense to somebody, s/he can 'zoom' deeper and deeper into the meaning of this sign. Although each level ('scale') of meaning-exploration may differ from any other level, there is similarity between the levels, as they all relate to the dynamics of one and the same sign interpreted by one and the same individual.

The strange attractors of meanings can exhort human actions. Although, the actions may appear randomly skipping around, they relate to the attractors of meanings, which propel them. If there is no attractors of meanings behind one's actions, the actions are simply meaningless; they are running at physical level only. The lack of intelligent support, be it mental, emotional or spiritual, is incompatible with one's growth as a holistic individuality.

More about the Strange Attractors of Meaning, see "Strange Attractors of Meaning".

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The bifurcation diagram was not created by Mitchell Feigenbaum but he found a way to understand it than no one had though of. The bifurcation diagram represents an idealized version of how a system can become chaotic. The diagram splits into two after a certain point (a bifurcation) and thus into four at a later point. The bifurcations come faster and faster until the system becomes chaotic. Feigenbaum discoveres that the bifurcations were occurring at a ratio that approached an irrational number that is approximately 4.669 in the bifurcation diagram. This was found to be true by experiment in real life example. 4.669 is a universal constant in much the same way 3.14 is.

Bifurcation diagram relates to the logistic equation which clearly demonstrates the transition to chaos. Firstly, this was observed by Rober May - an Australian theoretical physicist, who in 1971 went to the Institute for Advanced Study at Princeton and began to drift into biology, or more exactly, mathematical biology. Biologists had developed an equation that predicted the populations of different kinds of insects, animals, fish, etc. reasonably accurately; it was called the logistic equation or logistic mapping. It was a simple difference equation - a quadratic equation similar to the type most students solve in high school: y= ax(1-x), where x is from the interval between 0 and 1. It is hard, in fact, to imagine a simpler nonlinear equation, but it seemed to work well, so biologists adopted and used it. One of the main parts of the equation is the parameter a. The equation itself is presented as a difference equation, which means that when one value is substituted into it you get the next, that is, x(n+1) = a x(n)[1-x(n)]. For example, you could substitute a quantity x(n) corresponding to the population of rabbits in a given year, and you would get x(n+1) corresponding to the population that would appear the next year. Continuing in this way you get a sequence of numbers.

What May was interested was the long-term behaviour of the system; in other words, what is going to happen to the population of rabbits over many years. For values of a between 0 and 3, the number of rabbits in each successive population doe not change. But if a is greater 3, say 3.2, things get more complicated - the system suddenly bifurcates undergoing a periodic oscillation between two population sizes: the number of rabbits fluctuates back and forth between two different populations. If we increase a a little more we get a period four cycle, that is, the number of rabbits fluctuates between four different populations . A little more gives a period eight cycle and so on. In each case we are going through a bifurcation. Soon everything breaks down and we get chaos. It sets in at the value of a equal to 3.57.

The strange thing, which May found in the bifurcation diagram was that within the region of chaos (for a greater than 3.57) there are "windows" of order. These are regions where, even a is still increasing, there is a periodic population, usually with a 3 or 7 year cycle. They only last a while, however, with increasingly alpha, then the bifurcations occur again with period doubling, and chaos takes over again.

One of the major things one can see in the diagram is that the bifurcation structure appears to duplicate itself on smaller and smaller scales. If we look closely we see tiny copies of the larger bifurcations. This is the self-similarity typical for any fractal structure.

Does bifurcations described by the logistic equation occur in nature?

Controlled studies have been made in the laboratory using small creatures such as blowflies and moths, and bifurcations do indeed occur, as the equations predicts. But there are always outside influences, and things are not quite as cut and dry as they are in mathematics. Nevertheless, there is reasonably good agreement.

In social context, we often refer to bifurcations as spontaneously occurring qualitative changes - when a 'normal' flow of social events suddenly burst into avalanches of social changes of various magnitude and scale.

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In physics, a critical point is a point at which a system changes radically its behaviour or structure, for instance, from solid to liquid. In standard critical phenomena, there is a control parameter, which an experimenter can vary to obtain this radical change in behaviour. In the case of melting, the control parameter is temperature.

Self-organized critical phenomena, by contrast, is exhibited by driven systems which reach a critical state by their intrinsic dynamics, independently of the value of any control parameter. The archetype of a self-organized criticality (SOC) is a sand pile. Sand is slowly dropped onto a surface, forming a pile. As the pile grows, avalanches occur which carry sand from the top to the bottom of the pile. At least in model systems, the slope of the pile becomes independent of the rate at which the system is driven by dropping sand. This is the (self-organized) critical slope.

As championed by theoretical physicist Per Bak, SOC is relevant to a large number of naturally occurring phenomena from avalanches, to prices on the commodities market, to the use of the English language in newspapers. SOC links the multitude of complex phenomena observed in Nature to simplistic physical laws and/or one underlying process - it is a theory of the internal interactions of large systems.

Specifically, SOC states that large interactive systems will self-organize (as awhole!) into a critical state governed by a power law. For example, in the case of sand pile, if one were to count the number of avalanches and the number of grains involved in each avalanche over a 24 hour period, one would find that there was 1 avalanche which involved 1,000 grains, 10 avalanches which involved 100 grains, 100 avalanches which involved 10 grains, and so on (exactly, as the power law predicts!).

What makes SOC so intriguing, however, is that it may actually be able to model many complex natural phenomena. The essential link is that these phenomena maintain power law distributions in what can be considered very noisy conditions, and SOC computer models successfully generate power law distributions for a variety of conditions as well.

Examples: distribution of earthquake magnitudes, monthly variation of cotton prices, light emitted from a quasar, ranking of cities by size, pulsar glitches (changes in star's rotational velocities), x-ray intensity from solar flares, avalanche sizes in Conway's Game of Life, extinction events in biological evolution, etc.

For more example, see: Bak, P., How Nature Works: The Science of Self Organized Criticality, Springer Verlag, 1996. Here is a short review of Bak's book: "Many seemingly disparate aspects of the world, from the formation of the landscape to the process of evolution to the action of nervous systems to the behavior of the economy, all share a set of simple, easily described properties. These properties are all so similar, Per Bak writes, that they make us wonder if they are all manifestations of a single principle. Can there be a Newton's law, an f=ma, of complex behavior? In "How Nature Wor" Bak argues that self-organized criticality, the spontaneous development of systems to a critical state, is the key to such a principle. While many theories have been proposed to describe individual complex systems, self-organized criticality is the first general theory of complex systems with a firm mathematical basis. "How Nature Works", written by the discoverer of self-organized criticality, describes for general readers a concept of increasing importance. Few books offer such a compelling glimpse into the science of the future as this one".

Some intersteing websites on self-organized criticality: ,,and .

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The sensitive dependence of initial conditions as a true sign of chaotic dynamics was firstly studied mathematically by E. Lorenz (1963) in his three-equation model of atmospheric flow. A slight change in parameter value of the model had generated significantly different behaviour. Lorenz labelled this phenomenon as "butterfly effect". He found also that dynamic trajectories described by the equations moved very quickly along a branched, S-shaped, three-dimensional attractor of a ëstrangeí form.

The "Butterfly Effect" is the essence of chaos; according to Lorenz' metaphor, one flap of a butterfly's wings would be enough to alter the course of the weather forever.

The 'classical' mechanics has already confessed that the large systems of particles governed by gravity are chaotic. The best example is our solar system. The gravitational attraction of the other planets to the Earth makes the location of the Earth in its orbit chaotic. That is, the shape and size of the Earth's orbit does not change very much, but the precise position of the Earth in its orbit is very unstable.

The implications of this chaos in Earth's orbit can be put dramatically. Suppose a butterfly decides to move one meter to go from one flower to another. The effect of this single movement by this single butterfly is to move the entire Earth from one side of the Sun to the other in 500 million years - see F. Tipler's book "The Physics of Immortality" Macmillan: London, 1995. In this book, Tipler says: "Although the butterfly can move the Earth, it probably won't, because another butterfly on the other side of the Earth can cancel out the motion of the first if it moves in the opposite direction. The chaotic effect will be cumulative only if the butterflies act together, which of course they won't". However, people can consciously synchronise their actions and achieve together what butterflies can't - we can dramatically change the conditions of life on our planet. Unfortunately, up to now our conscious togetherness has brought a lot of irreversible ecological, social and economic disasters and crises.

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