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SIGNIFICANT
POINTS IN THE STUDY OF COMPLEX SYSTEMS
by Yaneer Bar-Yam
In order to help establish a backdrop for the ICCS conferences, I have compiled (with much feedback and
contributions from others) a list of "significant points" in the study of complex systems.
These are supposed to
represent key conceptual insights coupled with mathematical tools for the analysis and discussion of
complex
systems in general. Feedback and additions are welcome. The points are provided only in brief. In general
some familiarity is assumed. Some controversial points are included (what field has no controversy?).
-
Multi-scale
descriptions are needed to understand complex systems. Relevant mathematical
tools are scaling laws, fractals & trees, renormalization, multigrid. These specific methods are
not
exclusive of the more general issue of relating finer scale descriptions to larger scale descriptions
(e.g.
which fine scale parameters are relevant on larger scales, etc.). Examples: weather - patterns on all
scales (cyclones, tornadoes, dust devils); proteins - secondary, tertiary, quaternary structure;
physiology - molecules, cells, tissues, systems; brain - hemispheres, lobes, functional regions, etc.;
economy/society - similar.
- Fine scales influence large scale behavior. Relevant
mathematical tools are nonlinear feedback
iterative maps, mathematics of deterministic chaos, amplification & dissipation. The specific methods
of deterministic chaos are not exclusive of more general issue of fine scale effects on large scale
behavior. Examples: weather - "butterfly effect", proteins - enzymatic activity is amplification,
physiology - neuromuscular control (a nerve cell action triggering a muscle), economy/society - the
relevance of individuals to larger scale behaviors (how many people watch Michael Jordan).
- Pattern formation: Prominent among
simple mathematical models that capture pattern formation are
local activation / long range inhibition models. e.g. Turing patterns, and the work of Prigogine.
Examples: weather - cells of airflow, protein - alpha and beta structure, physiology - processes of
pattern formation in development, brain/mind - various patterns of interconnection and pattern
recognition mechanisms (on-center off-surround), magnetic bubble memories, patterns of species in
phenome or genome space, economy/society - patterns of industrial/residential/ commercial areas.
- Multiple (meta) stable states: Small
displacements (perturbations) lead to recovery, larger ones
can lead to radical changes of properties. Dynamics on such a landscape do not average simply.
Mathematical models are generally based upon local frustration e.g.. spin glasses, random Boolean
nets. Attractor networks use local minima as memories. Examples: weather - persistent structures,
proteins - results of displacements in sequence or physical space, physiology - the effect of shocks,
dynamics of e.g. the heart, brain/mind - memory, recovery from damage, economy/society - e.g.
suggested by dynamics of market responses.
- Complexity - answer to question "How
complex is it?": There is much discussion of this
question. A general answer: The amount of information necessary to describe the system. There are
various important issues that require clarification. One of these relates to the use of inference to
obtain
the description from a seemingly smaller amount of information. This leads the concept of algorithmic
complexity. Another relevant point: The apparent complexity depends on the scale at which the
system is described, however, once a particular scale is chosen the complexity should be well defined
and bounded (at a particular instant) by the information necessary to describe the microstate of the
system (the entropy). Also note that complexity on a large scale requires correlations on a small scale,
which reduces the smaller scale complexity. Example: random motion (high small scale complexity)
averages out on a larger scale.
- Behavior (response) complexity: To
describe the behavior (actions) of a system acting in response
to its environment, where the complexity of the environmental variables are C(e) and of the action is
C(a), we often try to describe the response function f, where a=f(e). However, unless simplifying
assumptions are made, specifying the response to each environment requires an amount of
information that grows exponentially with the complexity of the environment (a response must be
specified for each possible environment). Specifically C(f)=C(a)*2^C(e). This is impossible for all
but
simple environments (e.g. less than a few tens of bits). This means that behaviorism in psychology,
or
strict phenomenology in any field, or testing the effects of multiple drugs, or testing computer chips
with many input bits, is fundamentally impossible.
- Emergence: Related to the dependence
of the whole on parts, the interdependence of parts, and
specialization of parts. This is directly relevant to questions about how we study systems both
theoretically and experimentally. Parts must be studied "in vivo". For example - "If
you remove a
vacuum tube from a radio and the radio squeals do not conclude that the purpose of the tube is to
suppress squeals." While studying the parts in isolation does not work, the nature of complex systems
can be probed by investigating how changes in one part affect the others, and the behavior of the
whole.
- 7+/-2 rule: This is related to the
interdependence of parts of a system. For a system divided into
components, looking at the dependencies between them we can ask when does the state / behavior
of one of the components depend on the state of each of the other ones, and not on an average. This
pertains to the question of when the central limit theorem applies to a number of independent
variables. The conclusion is that this number is approximately 7. Supported by the phenomenology of
substructure branching ratios in proteins, physiology, brain, and social systems (e.g. organizational
rules about the number of members of a committee).
- The relationship of descriptions and
systems: This is relevant to our understanding of theory and
simulations, the recognition of systems in their models, encoding and decoding (compression), and the
subject of algorithmic complexity. Specific applications are apparent in biological development
(genome vs. physiology), engineering design, and memory vs. experience.
- Selection is information (à la
Shannon theory): The amount of information necessary to specify a
system is obtained by enumerating the possible states and comparing them with the possible states of
the description e.g. a bit string, or e.g. English language (at about 1 bit / character). This enables
the
systems to be enumerated and one of them specified. Selection as information is relevant to the issue
of multiple selection: replication (reproduction) with variation, and comparative selection
(competition) as a mechanism for POSSIBLE increase in complexity. Consistent with modern
biological views of evolution it is essential to emphasize that selection does not have to increase
complexity.
- Composites: To form a new complex
system take parts (aspects) of other complex systems and
recombine them. For this to work parts must be partially independent. Examples - sexual
reproduction, creativity (e.g. seeing a person walking and a bird flying and imagining a person flying
by combining information of shape and motion represented in different parts of the brain), and
modular construction (building blocks) in artificial systems. The purpose of composites is to allow
rapid evolution.
- Control hierarchy: When (if) a single
component controls the collective behavior (not the individual
behaviors of all the components) of a system, then the collective behavior cannot be more complex
than the individual behavior. i.e. there is no emergent complexity. Examples: muscle (since the muscle
is controlled by a single neuron, its collective behavior is no more complex than the neuron behavior),
society/economy: corporate hierarchies/dictatorships/etc. (to the extent that central control is
exercised complexity of collective behavior is bounded by the complexity of the controlling
individual).
Modeling and
simulation: There are a number of simulation methodologies that have arisen as having
general application in the study of complex systems. These include: Monte Carlo, simulated annealing,
cellular
automata. Other methods have been mentioned above.
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