Coarse-Graining Of Complex Systems

The query of the human relationship betwixt micro-level in addition to macro-level is merely every bit of import inward physics every bi...

The query of the human relationship betwixt micro-level in addition to macro-level is merely every bit of import inward physics every bit it is inward sociology. Is it possible to derive the macro-states of a scheme from information nigh the micro-states of the system? It turns out that in that place are to a greater extent than or less surprising aspects of the human relationship betwixt micro in addition to macro that physical systems display. The mathematical technique of "coarse-graining" represents an interesting wrinkle on this question. So what is coarse-graining? Fundamentally it is the thought that nosotros tin hand notice supplant micro-level specifics amongst local-level averages, without reducing our powerfulness to calculate macro-level dynamics of demeanour of a system.

H5N1 2004 article yesteryear Israeli in addition to Goldenfeld, "Coarse-graining of cellular automata, emergence, in addition to the predictability of complex systems" (link) provides a brief description of the method of coarse-graining. (Here is a Wolfram demonstration of the means that coarse graining industrial plant inward the land of cellular automata; link.) Israeli in addition to Goldenfeld also furnish physical examples of phenomena amongst what they refer to as emergent characteristics. Let's run into what this approach adds to the theme of emergence in addition to reduction. Here is the abstract of their paper:

We report the predictability of emergent phenomena inward complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) every bit an example, nosotros present how to build local coarse-grained descriptions of CA inward all classes of Wolfram's classification. The resulting coarse-grained CA that nosotros build are capable of emulating the large-scale demeanour of the master systems without accounting for small-scale details. Several CA that tin hand notice endure coarse-grained yesteryear this structure are known to endure universal Turing machines; they tin hand notice emulate whatever CA or other computing devices in addition to are thence undecidable. We thus present that because inward exercise 1 entirely seeks coarse-grained information, complex physical systems tin hand notice endure predictable in addition to fifty-fifty decidable at to a greater extent than or less bird of description. The renormalization grouping flows that nosotros build cause a hierarchy of CA rules. This hierarchy agrees good apparent dominion complexity in addition to is thence a practiced candidate for a complexity stair out in addition to a classification method. Finally nosotros debate that the large scale dynamics of CA tin hand notice endure real simple, at to the lowest degree when measured yesteryear the Kolmogorov complexity of the large scale update rule, in addition to moreover exhibits a new scaling law. We present that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity every bit 1 goes to increasingly coarser scales. We translate this large scale simplicity every bit a designing formation machinery inward which large scale patterns are forced upon the scheme yesteryear the simplicity of the rules that principle the large scale dynamics.

This paragraph involves several interesting ideas. One is that the micro-level details create non affair to the macro effect (italics above). Another related thought is that macro-level patterns are (sometimes) forced yesteryear the "rules that principle the large scale dynamics" -- rather than yesteryear the micro-level states.

Coarse-graining methodology is a menage unit of measurement of computational techniques that permits "averaging" of values (intensities) from the micro-level to a higher bird of organization. The computational models developed hither were primarily applied to the properties of heterogeneous materials, large molecules, in addition to other physical systems. For example, consider a two-dimensional array of atomic number 26 atoms every bit a grid amongst randomly distributed magnetic orientations (up, down). H5N1 coarse-grained description of this scheme would endure constructed yesteryear taking each 3x3 foursquare of the grid in addition to assigning it the up-down value corresponding to the bulk of atoms inward the grid. Now the information nigh ix atoms has been reduced to a unmarried slice of information for the 3x3 grid. Analogously, nosotros mightiness consider a metropolis of Democrats in addition to Republicans. Suppose nosotros know the affiliation of each household on every street. We mightiness "coarse-grain" this information yesteryear replacing the household-level information amongst the bulk representation of 3x3 grids of households. We mightiness lead maintain to a greater extent than or less other stride of aggregation yesteryear considering 3x3 grids of grids, in addition to representing the larger composite yesteryear the bulk value of the factor grids.

How does the methodology of coarse-graining interact amongst other inter-level questions nosotros lead maintain considered elsewhere inward (emergence, generativity, supervenience)? Israeli in addition to Goldenfeld connect their function to the thought of emergence inward complex systems. Here is how they depict emergence:

Emergent properties are those which arise spontaneously from the collective dynamics of a large assemblage of interacting parts. H5N1 basic query 1 asks inward this context is how to derive in addition to predict the emergent properties from the demeanour of the private parts. In other words, the cardinal number is how to extract large-scale, global properties from the underlying or microscopic degrees of freedom. (1)

Note that this is the weak shape of emergence (link); Israeli in addition to Goldenfeld explicitly postulate that the higher-level properties tin hand notice endure derived ("extracted") from the micro bird properties of the system. So the calculations associated amongst coarse-graining create non imply that in that place are system-level properties that are non-derivable from the micro-level of the system; or inward other words, the success of coarse-graining methods does non back upwards the thought that physical systems possess strongly emergent properties.

Does the success of coarse-graining for to a greater extent than or less systems lead maintain implications for supervenience? If the states of due south tin hand notice endure derived from a coarse-grained description C of one 1000 (the underlying micro-level), does this imply that due south does non supervene upon M? It does not. H5N1 coarse-grained description corresponds to multiple distinct micro-states, so in that place is a many-one human relationship betwixt one 1000 in addition to C. But this is consistent amongst the fundamental requirement of supervenience: no divergence at the higher bird without to a greater extent than or less divergence at the micro level. So supervenience is consistent amongst the facts of successful coarse-graining of complex systems.

What coarse-graining is inconsistent amongst is the thought that nosotros demand exact information nigh one 1000 inward club to explicate or predict S. Instead, nosotros tin hand notice eliminate a lot of information nigh one 1000 yesteryear replacing one 1000 amongst C, in addition to however create a perfectly satisfactory chore of explaining in addition to predicting S.

There is an intellectual wrinkle inward the Israeli in addition to Goldenfeld article that I haven't yet addressed here. This is their connectedness betwixt complex physical systems in addition to cellular automata. H5N1 cellular automaton is a simulation governed yesteryear simple algorithms governing the demeanour of each jail cellular telephone inside the simulation. The game of Life is an illustration of a cellular automaton (link). Here is what they say nigh the connectedness betwixt physical systems in addition to their simulations every bit a scheme of algorithms:

The occupation of predicting emergent properties is most severe inward systems which are modelled or described yesteryear undecidable mathematical algorithms[1, 2]. For such systems in that place exists no computationally efficient means of predicting their long fourth dimension evolution. In club to know the system’s land subsequently (e.g.) 1 meg fourth dimension steps 1 must evolve the scheme a meg fourth dimension steps or perform a computation of equivalent complexity. Wolfram has termed such systems computationally irreducible in addition to suggested that their beingness inward nature is at the root of our apparent inability to model in addition to sympathise complex systems [1, 3, 4, 5]. (1)

Suppose nosotros are interested inward simulating the physical physical care for through which a pot of boiling H2O undergoes abrupt turbulence before long before 100 degrees C (the transition betoken betwixt H2O in addition to steam). There seem to endure 2 large alternatives raised yesteryear Israeli in addition to Goldenfeld: in that place may endure a fix of thermodynamic processes that permit derivation of the turbulence direct from the physical parameters acquaint during the brusk interval of time; or it may endure that the entirely means of deriving the turbulence phenomenon is to furnish a molecule-level simulation based on the fundamental laws (algorithms) that principle the molecules. If the latter is the case, in addition to so simulating the physical care for volition examine computationally impossible.

Here is an extension of this approach inward an article yesteryear Krzysztof Magiera in addition to Witold Dzwinel, "Novel Algorithm for Coarse-Graining of Cellular Automata" (link). They depict "coarse-graining" inward their abstract inward these terms:

The coarse-graining is an approximation physical care for widely used for simplification of mathematical in addition to numerical models of multiscale systems. It reduces superfluous – microscopic – degrees of freedom. Israeli in addition to Goldenfeld demonstrated inward [1,2] that the coarse-graining tin hand notice endure employed for uncomplicated cellular automata (CA), producing interesting interdependences betwixt them. However, extending their investigation on to a greater extent than complex CA rules appeared to endure impossible due to the high computational complexity of the coarse-graining algorithm. We demonstrate hither that this complexity tin hand notice endure substantially decreased. It allows for scrutinizing much broader score of cellular automata inward damage of their coarse graining. By using our algorithm nosotros constitute out that the ratio of the numbers of uncomplicated CAs having coarse grained representation to “degenerate” – irreducible – cellular automata, strongly increases amongst increasing the “grain” size of the approximation procedure. This rises main questions nigh the formal limits inward modeling of realistic multiscale systems.

Here K&D seem to endure expressing the persuasion that the approach to coarse-graining every bit a technique for simplifying the expected demeanour of a complex scheme offered yesteryear Israeli in addition to Goldenfeld volition neglect inward the instance of to a greater extent than extensive in addition to complex systems (perhaps including the pre-boil turbulence illustration mentioned above).

I am non certain whether these debates lead maintain relevance for the modeling of social phenomena. Recall my before give-and-take of the modeling of rebellion using agent-based modeling simulations (link, link, link). These models function from the unit of measurement bird -- the bird of the individuals who interact amongst each other. H5N1 coarse-graining approach would perchance supplant the individual-level description amongst a fix of groups amongst homogeneous properties, in addition to and so seek to model the likelihood of an outbreak of rebellion based on the coarse-grained bird of description. Would this endure feasible?