Architecture of Complexity
Introduction
• Abstractly states some of the common properties that
physical, biological, and social systems have in common
• Tries to cast some light on how complexity can be
found in natures without relating it to any particular
field of study
• Breaks the paper up into four parts:
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Hierarchy
Evolution of Complex Systems
Dynamic properties of Hierarchically organized systems
Relation Between Complex Systems and their descriptions
Hierarchical System
• “complexity frequently takes the form of hierarchy, and
that hierarchic systems have some common properties
that are independent of their specific content”
• Complex System in which each of the subsystems is
subordinated by an authority relation to the system it
belongs to.
• What are elementary units?
– Physics
– Social
Social Systems
• Formal Organization
– Families (Elementary unit)
– Villages, tribes
– Business firms, governments, universities
• Grouping is defined by “some measure of
frequency of interaction in this sociometric
matrix”
Biological and Physical Systems
• Cell as a building block
– Tissues
• Organs
– Systems of Organs
• Cell
– Cell Membrane
• Microsomes
– Mitochondria
» … etc
Biological and Physical Systems
• Elementary units or complex system?
– Gasses
– Diamonds
• Span of Control – number of subordinates
who report directly to the boss
• Flat hierarchy – having a wide span of control
at one level but not on the next level
• Physical and biological hierarchies are mostly
described in spatial terms
Symbolic Systems
• Book
– Chapters
• Paragraphs
– Sentences
» Words
• Music
– Movements
• Parts
– Themes
» Phrases
Further the point of hierarchy:
Higher Level Languages
• Java
– Primitives
• Functions
– Classes
» Groups of Classes with inheritance
• Programs
Evolution of Complex Systems
Watchmaker Parable
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There are two watchmakers, Tempus and
Hora.
Both make fine watches and are highly
regarded by their customers, who call
them constantly.
But Hora prospered, while Tempus went
out of business. Why?
Watchmaker Parable, cont.
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Each watchmaker had different ways of
constructing watches.
Tempus designed his watch so that if he had a watch
partially assembled and he had to put down the watch
and answer the phone, the watch fell apart.
Hora designed his watch so that he could put together
subassemblies of 10 parts. These subassemblies in turn
could be put together into 10 larger subassemblies,
which when put together made up the watch. If
interrupted by the phone, Hora only lost the
subassembly.
Which way was faster?
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p is the probability of being interrupted while adding a single
part.
For Tempus, the probability of completing 1 watch is
(1 –
p)^1000, i.e. a really small chance. On average, each
interruption costs the amount of time needed to assemble
1/p parts.
For Hora, he has to complete 111 subassemblies (100 10part subassemblies, 10 100-part subassemblies, 1 1000-part
watch). The probability of completing 1 subassembly is (1 – p
)^10. On average, each interruption costs the amount of
time needed to assemble 5 parts.
Which way is faster? cont.
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Assume p = 0.01.
Hora needs to make 111 complete assemblies, Tempus only
1. Tempus has 1/111 the amount of assemblies to complete.
But Tempus loses 100 parts for each interruption, Hora only
5. Tempus loses 20 times as much work.
Tempus completes (.99)^1000, or 44 watches per million
attempts. Hora completes (.99)^10, or 9 out of 10
subassemblies. Thus, Tempus needs to make 20000 times
more attempts to complete an assembly.
(1/111) * 20 * 20000 = 4000. Tempus takes 4000 times
longer to complete a watch.
Analogy to Biological Evolution
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Can't really make a numerical estimate, since there are so
many connections in living organisms.
An “estimate”: Assume there is a hierarchy of potentially
stable subassemblies, each level of the hierarchy with the
same span s. Then the time required for each subassembly
at each level ~ 1/(1 – p)^s
Thus, the time to assemble a system of n elements is logs(n)
~ the number of levels in the system.
The time to evolve multicelled organisms from single-celled
organisms is about the same as the evolution of single-celled
organisms from molecules.
Problems with the analogy
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Biology assumes that there is no teleological
mechanism, i.e. complex forms can arise from simple
forms by random processes like survival of the fittest.
Not all large systems are hierarchical. Example: most
polymers are linear chains of identical monomers.
Assumes nothing about changes in the total energy in
the system. Biological systems need an energy source
like the Sun.
Application to Problem Solving Process
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Proof of a difficult theorem: Analogous to searching
for a path through a maze. You start with axioms and
previously proved theorems, try various
transformations, hope to obtain new promising
expressions, and repeat until you find a solution
Requires exploration, trial, and error.
Not a random process, but requires you to be highly
selective.
Application to Problem Solving Process
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The cues signalling progress towards a solution are like the
stable intermediate forms in biological evolution.
Suppose you have a locked safe with 10 dials, each dial with
100 numbers on it. There are 100^10 possible settings, half
of which you need to examine to find the correct code (50
billion billion).
But if the dial clicks when it hits the correct number, finding
the correct code is much simpler, i.e. 100*10 possible
settings, half of which you need to examine to find the
correct code (500).
Problem Solving Examples
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Proving theorems.
Solving puzzles.
Making investments.
Balancing assembly lines.
Writing computer programs.
All of these require selective trial-anderror.
Sources of Selectivity
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One way: trial-and-error. Various configurations are
tried out. Promising configurations are kept and are
used to guide further trials. Stable configurations are
what matter.
The other way: use previous experience. Check if a
problem is similar to a previously solved problem, try
the paths that led to its solution. Reduces trial-anderror search. Biological terms: reproduction and gene
inheritance.
Application to Empires
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Philip of Macedon assembled the Macedonian Empire.
Alexander the Great inherited it and added the Persian
empire and other parts to it.
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When Alexander died, the empire did not
crumble, but fragmented into the major parts that
composed it.
When Lawrence of Arabia tried to organize the Arabs in
World War I, he was limited to the separate Arab tribes as
building blocks, and could not unite them.
Lesson: To build a long-lasting empire, your world should be
composed of large, stable political systems.
Summary
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Complex systems evolve from simple
systems more rapidly if there exist stable
intermediate forms.
These complex systems will be hierarchic.
The complex systems in our world are
hierarchic because they have had enough
time to evolve.
Nearly Decomposable Systems
Nearly Decomposable Systems
• Interactions among subsystems vs. interactions within
subsystems
• Ex:
o Formal Organizations
o Rare Gas: decomposable
o Perfect Gas: nearly decomposable
• def: nearly decomposable - interactions among subsystems
are weak, but not negligible
Two Proposition Summary
• (a) short-run behavior of each of the component subsystems
is approximately independent of the short-run behavior of the
other components
• (b) in the long run, the behavior of any one of the
components depends in only an aggregate way on the
behavior of the other components
Nearly-decomposable system example
Symbolic of a Dynamic System
• In the short run each room will reach an equilibrium
temperature (an average of the initial temperatures of its
offices) nearly independently of the others.
• each room will remain approximately in a state of equilibrium
over the longer period during which an over-all temperature
equilibrium is being established throughout the building.
• after the short-run, a single thermometer in each room will
adequately describe the dynamic behavior of the entire
system.
Examples
• In the natural world nearly decomposable systems are far
from rare.
• On the contrary, systems in which each variable is linked with
almost equal strength with almost all other parts of the
system are far rarer and less typical.
• Social Systems:
o economic dynamics
o formal authority in organizations
 departmental boundaries = walls in previous example
Hierarchic Span
• Strong vs. weak linkages
• Subsystems form until all capacity for strong interaction is
utilized in their construction.
• Then these subsystems will be linked by the weaker secondorder bonds into larger systems.
• Ex: Water has valence of zero but electric dipole permits
weak interaction between water and salts dissolved in it.
• Ex2: Social interaction - One cannot enact the role of "friend"
with large numbers of other people
Summary
• Hierarchies have the property of near-decomposability.
• Intra-component linkages are generally stronger than intercomponent linkages.
• This separates the high-frequency dynamics of a hierarchy
(the internal structure of the components) from the lowfrequency dynamics (interaction among components).
• Mr. Rivas will now discuss some important consequences of
this separation for the description and comprehension of
complex systems.
The Description of Complexity
Main Point #1
• When information is put in outline form, it is
easy to include information about the
relationships between the major parts and
information about the internal relations of
parts in each of the sub-outlines.
• i.e. when drawing a face, one usually starts in a
hierarchic fashion: first the outline of the face,
then add features such as the nose, eyes, mouth,
and then add more detail to those features such
as pupils, eyelids, lashes, etc.
Main Point #1 Cont.
• As such, detailed information about the
relations of subparts belonging to different
parts is not as important as the preservation
of information about hierarchic order.
Main Point #2
• Little information is lost about the dynamic
properties of nearly decomposable systems
when they are represented as hierarchies.
• Subparts belonging to different parts only
interact in an aggressive way and the details of
their interactions are not important.
Examples
• When studying the interactions between two
large molecules, we don’t consider in detail
the interactions of their respective nuclei or
atoms
• When studying the interactions of two nations
we don’t need to study the interactions of
each citizen of one nation with each citizen of
the other nation
Main Point #2 Cont.
• The fact, then, that many complex systems
have a nearly decomposable hierarchic
structure makes it easier for us to understand,
describe and even “see” such systems and
their parts.
• Vice versa, we may not be able to understand
such a system since such detailed knowledge
and calculation would be nearly impossible.
Main Point #3
• There is no conservation law that requires that
the description of a complex system be
composed of a complex structure of symbols.
Example
Original matrix composed of 8x8 symbols, reduced to a 4x4 by representing
redundant patterns with symbols.
Main Point #4.1
• Redundancy in 3 forms:
1. Hierarchic systems are usually composed of a
few different kinds of subsystems in various
combinations and arrangements.
i.e. protein varieties arise from arrangements of 20
different amino acids; ~99 elements in periodic table
provide all kinds of building blocks needed for an
infinite variety of molecules.
Main Point #4.2
2. Hierarchic systems are often nearly
decomposable. Hence, only a tiny fraction of
all possible interactions needs to be taken
into account to understand the greater
picture of the system.
Main Point #4.3
3. By appropriate “recoding” the redundancy
that is present but unobvious in the structure
of a complex system can be made clear.
i.e. the structure of the sequence, 1 3 5 7 9… is
most simply expressed by observing that each
member is obtained by adding 2 to the previous
one.
The Architecture of Complexity
State Descriptions and Process
Descriptions
1. “A circle is the locus of all points equidistant from a
given point.”
2. “To construct a circle, rotate a compass with one
arm fixed, until the other arm has returned to its
starting point.”
Euclid
• If one carries out the Process Description in sentence
2, they will process an object that satisfies the State
Description in sentence 1.
State Descriptions and Process
Descriptions
State Descriptions
• A mode of apprehending
structure
• Pictures, blueprints,
diagrams, chemical
formulae
• Characterize the world as
sensed
• Provide the criteria for
identifying objects
Process Descriptions
• A mode of apprehending
structure
• Recipes, differential
equations, equations for
chemical formulae
• Characterize the world as
acted upon
• Provide the criteria for
producing objects
State Descriptions and Process
Descriptions
• An organism must make connections between goals in
the sensed world, and actions in the Process world.
• An organism can make these connections consciously
and that is called Means-end analysis:
• Given: A desired state, and an existing state
• Task of an adaptive organism: find the difference
between those 2 states, and then find the correlating
processing that will erase the difference
• Human problem solving is a form of means-end
analysis
The Description of Complexity in SelfReproducing Systems
• Evolution of complexity does not imply selfreproduction
– If evolution of complexity from simplicity is sufficiently
probable, it will occur repeatedly.
– If the existence of a complex object increased the
probability of another one, the equilibrium between
the complexes and the components could be greatly
altered in favor of the former.
– Most simplest possibility: The complex system to
serve as a description of itself.
• DNA, double helix unwinds to allow half of the helix to serve
as a template on which the new matching half can form.
Ontogeny Recapitulates Phylogeny
• A well known generalization in Biology in
which an individual organism, in its
development, goes through stages that
resemble some of its ancestral forms.
– Human embryo develops gill bars and then
modifies them
• Ontogeny recapitulates only the grossest
aspects of phylogeny.
– Simplest forms are modified to more be complex
Summary: The Description of
Complexity
• How complex or simple a structure is depends
upon the way we describe it.
• Most complex structures found in the world are
incredibly redundant; we can use this redundancy
to simplify their description.
– To achieve this simplicity, we must find the right
representation.
• The correlation between State description and
Process description is basic to the functioning of
any adaptive organism and to its capacity for
acting purposefully upon its environment