Download
1 / 47
470 likes | 609 Views
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
E N D
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: • 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
Watchmaker Parable • 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. • 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? • 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 10-part 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. • 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 • 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 • 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 • 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 • 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 • Proving theorems. • Solving puzzles. • Making investments. • Balancing assembly lines. • Writing computer programs. • All of these require selective trial-and-error.
Sources of Selectivity • 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-and-error search. Biological terms: reproduction and gene inheritance.
Application to Empires • Philip of Macedon assembled the Macedonian Empire. Alexander the Great inherited it and added the Persian empire and other parts to it. • 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 • 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 • Interactions among subsystems vs. interactions within subsystems • Ex: • Formal Organizations • Rare Gas: decomposable • 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
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: • economic dynamics • 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 second-order 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 inter-component linkages. • This separates the high-frequency dynamics of a hierarchy (the internal structure of the components) from the low-frequency dynamics (interaction among components). • Mr. Rivas will now discuss some important consequences of this separation for the description and comprehension of complex systems.
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: • 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 • 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 • 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.
State Descriptions and Process Descriptions • “A circle is the locus of all points equidistant from a given point.” • “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 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 • A mode of apprehending structure • Pictures, blueprints, diagrams, chemical formulae • Characterize the world as sensed • Provide the criteria for identifying 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 Self-Reproducing Systems • Evolution of complexity does not imply self-reproduction • 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
