The Multiscale thinking may be thought as an extension of the renormalizationg group (RG)thinking. In both cases one blocks together the degrees of freedom of the fine scale into coarser collective objects. RG is postulating that the form of the effective interaction between the coarse objects has the same form as the interaction between the fine scale objects.
This severe limitation is not deliberate: it originates in the mathematical techniques used by the RG. Clearly, the situation in which a qualitatively new interactions arrise at some scale is not easyly and naturally tractable this way.
The multiscale approach, relying on more direct, numerical treatment can in principle construct collective objects which are qualitatively different from their components. However, the multiscale paradigm still assumes that the dynamics of the collective objects effective dynamics is a generic result of the interactin between their different parts.
Collective objects which are the result of a fine-tuned, tightly bound conglomerate of fine scale objects are not appropriately described by the multiscale procedure. In particular, if the collective objects are no naturally expressible by weakly interacting parts, or are highly sensitive to small changes in the fine scale objects/parameters, then the multiscale algorithm fails.
In many systems the failure of this "naive" multiscale paradigm signals problems with the definition of the system or a certain irrelevance to the natural reality: fine-tunned systems have generically a low probability to appear and highly integrated systems are usually "artificial" (often man-made) and untypical.
Yet many complex systems are found lately to be "self-organized". More precisely, the amount of systems with high integration and between their parts is much larger in nature than expected from generic stochastic estimations. It often happens that even though the range of parameters necessary for some nontrivial collective phenomenon to emerge is very narrow (or even an isolated single point out of an continuum infinite range), the phenomenon does take place in nature.
The explanation of the generic emergence of systems which are non-generic from the multiscale point of view (multiscale irreducible systems) seems to be related to self-catalysing dynamics (see document on Scaling and Power Laws) but the main point for the present document is that whatever the explaining mechanism is, it might lead to collective objects whose properties are not explainable by the generic dynamics of their components.
As an example: the properties of water for the metabolism of living systems depends very finely on the value of some specific electron energy levels: in fact the small change in these levels implied by using heavy water is enough to completely dishable its proper function.
The example indicates the very important characteristic of the multiscale-irreducible systems: the emergence of such a non-generic systems, disturbing the automatic multiscale procedure is associated with the borders between sciences: once certaing biological properties (as above) not explainable by the generic chemical and physical properties of their parts, it is natural that the students of such systems will consider those biological properties as a datum and try to concentrate their understanding efforts to their consequences.
The reduction of biology to chemistry and physics is not invalidated here by the intervention of new, animistic forces, but by the mere irrelevance of our reductionist generic reflexes to a non-generic fine-tunned situation.
As suggested, above, the arrousal of non-generic situations in self-catalysing systems is not so surprizing: Consider a space of all possible systems obtainable form cetrtain chemical and physical parts: even if a macroscopic number of those systems are not self-reproducible and only a very small number are reproducible, after enogh time, one of the self-reproducible systems will eventually arrise. Once this happens, the self-reproducible systems will start multiplying leading to a final (or far-future) situtation in which those - a priory very improbable systems - are "over-represented" compared with their "natural" occurence probability.
By its very dismissal at these irreducible-complex systems, the Multiscale paradigm is offering science one of its most precious gifts. By (re-)tracing the natural conceptual-frontiers between the various scientifc disciplines, multisalale is teaching us when a reductionist approach is worth launching and where are the limits to which it should be pushed. Where the generic multiscale dynamics has to be applied and wher one has to accept as elementary working elements specific object with "fine-tunned" fortuitous properties.
The multiscale-irreducible systems are labeled usually at the algorithmic level by the fact that whatever algorithms are used, their effective dynamics turns out to be very slow (it takes macroscopic time spans). The phenomenon by which the effective dynamics of the collective objects takes macroscopic times is called Critical Slowing Down (CSD).
This shows up also in the artificial dynamics related to the computation of the properties of those systems.
More precisely, in mulitiscale reducible systems, the local algorithms display CSD, while the multiscale algoriths don't. In the multiscale-irreducible systems, the dynamics/computatins related with the internal changes in the irreducibly complex objects is always CSD.
One can explicitely take care of these degrees of freedom and then apply multiscale to acclerate the process. The identification of the slow modes which have to be treated separately is in fact the identification of the multiscale irreduceble degrees of freedom.
One could look at the necessity to give up the extreme reductionism (going with the reduction below the first encounteres non-generic object) as "a pity". Yest one has to understand the emergence of these notrivial thresholds as the the very salt which gives "taste" to the world as a WONDERfull place.
Moreover one should be reassure that the fundamental "in principle" reduction of macroscopic realty to the fundamental microscopic laws of the material reality is not endangered.
It is instructive to consider the following example in which the reduction is guaranteed by construction, yet completely ineffective. Consider a program running on a PC. In principle one can reduce the knowledge of the program to the knowledge of the currents running through the chips of the computer. Yet such a knowledge is not oly diifficult to achieve, validate and store, but it is also quite irrelevant for what we call "understanding". The right "elementary", "irreducible" approprite for understanding the algorithm is the flow chart of the program, and in any case coarser than the "assembler" instructions of the machine.
In the same way, the problem of reducing mental activity to neuron fireings is not so much related to the issue of whether one needs in addition assumptions of a "soul" which is governed by additional, transcendental laws, but to whether the generic non-fine-tunned dynamics of a set of neurons can explain the cognitive functions. In fact, after milions of years of intensive selection by survival pressure, it is reasonable to assume that the system of neurons is highly non-generic and therefore a reductionist approach to its understanding is going to be quite inneffective.
In the best case the situation is similar with being given the map of a complicated labitinth: one can have the knowledge of each wall, still it would take highly non-trivial effort to find the way out.
Even upon knowing way out would not mean understanding the system: any small addition or breaking of a small wall would expose its illusory, unstable nature of this knowledge by generating a totally new situation which would have to be solved from scratch.
At a more general level,the computational difficulty is one of the main characteristics of complex systems and the time necessary for their investigation and/or simulation grows very fast with their size \cite{par94}. The systematic classification of the the difficulty and complexity of computational tasks is a classical problem in computer science \cite{kir1}.
The emergence of large time scales is often related to random fluctuations of multiscale spacial structures within the system. Long range and long times scale hierarchies ({\bf Multiscale Slowing Down}) are usually related to collective degrees of freedom ({\bf macros}) characterising the effective dynamics at each scale.
Usually, it is the dynamics of the {\bf macros} during simulations which produces the Multiscale Slowing Down and reciprocaly, the slow modes of the simulation dynamics project out the relevant macros \cite{blessing,martin}.
Therefore, a better theoretical understanding of the multiscale structure of the system, enables one to construct better algorithms by acting directly on the relevant macros. Reciprocally, understanding the success of a certain algorithm yields a deeper knowledge of the relevant degees of freedom of the system. (e.g. hadrons in the theory of quarks and gluons, Cooper pairs in superconductors, phonons in crystals, vortices in superfluids, flux tubes, instantons, solitons and monopoles in gauge theories, etc.).
One can entertain the hope that many complex systems in biophysics, biology, psychophysics and cognition display similar properties and may be some kind of multiscale universality generalizing the universality classes and scaling of the critical systems. Such a situation would have a significant unifying effect on a very wide range of phenomena spreading over most of the contemporary scientific fields.
In the absence of a rigorous theoretical basis for such a hope, its investigation relies for the moment mainly on the use of computers.
The present paper implements this point of view into the study of general random complex systems and gives some examples.
Reducible vs. Irreducible Complexity
Certain complex systems with high level of frustration and connectivity present a certain hierarchy in their energy landscape which is responsible for the hierarchy of time scales characterizing their multiscale slowing down. This "rugged energy landscape" is also the origin of ultrametric (UM) properties of their ground states space.
One could hope to make some relation between the ultrametric hierarchy and the existence of an effective representations of the dynamics in terms of a spacial multiscale hierarchy of macros. This in turn would become the basis of an efficient MCA.
It turns out and it was rigorously proven in \cite{nath} that the case is exactly the opposite: the ultrametric hierarchy {\bf insures} the {\bf in}existence of a representation of the effective macroscopic dynamics of the complex system in terms of their macroscopic disjoint sub-sets. \footnote{ I.e. a complex ultrametric system is irreducible to a set of interacting sub-systems This might have something to do with phenomena like catastrophic forgetting in neural networks.}.
In conclusion, in ultrametric systems it is ruled out that various regions of the system can be treated as independent collective degrees of freedom ({\bf macros}). This picture can be extended to finite but small temperatures with the help of the "pure state" concept \cite{par94}.
This failure of separability of the whole into (almost) independent parts has conceptual implications in the sense that one cannot "understand" the complex system by "analyzing " it into its parts. In this sense an ultrametric system is conceptually irreducible to simpler entities. We will see that optimal global algorithms reduce in fact a system to its "irreducible" core.
One is tempted to conclude that the entire discussion of reductionism can be reformulated in terms of "irreducible complex systems". I.e. in place of {\bf assuming} ultrametricity and deducing the inexistence of independent dynamical sub-objects, one can propose this {\bf dynamical inseparability} as the fundamental property underlying irreducible complexity.
The situation can be compared with having to find one's way in a labyrinth in the phase space: each small local change in the position of the potential energy labyrinth walls determines large unpredictable changes of the solution route depending on details scattered across the entire phase space.
Consequently, we are discerning 3 main complexity cases:
Conclusions
The macros appearing in complex systems can be multiscale reducible i.e. can be iteratively broken into smaller tighter macros. However, in many cases there might exist complex irreducible cores. While such irreducible macros might have fortuitous characteristics, lack generality and present non-generic properties, they might be very important if the same set of cores appears recurrently in biological, neurological or cognitive systems in nature.
In such situations, rather than trying to understand the macros structure, dynamics and properties on general (multiscale, analytic) grounds as collections of their parts, one may have to recognize the unity and uniqueness of these macros and resign oneself in just making an as intimate as possible acquaintance with their features.
One may still try to treat them by the implicit elimination method \cite{blessing,martin} where the complex objects are presenting, isolating and eliminating themselves by the very fact that they are projected out by the dynamics as the slow-to-converge modes.
E.g. finding the exact configuration of a molecule might involve very complicated quantum mechanical computations. Yet, for deducing the volume-pressure relation of a dilute gas of such mulecules one can treat them as irreducible elementary objects.
This kind of decisions may iterate: in deciding the shape of the molecules, the structure of the nuclei of the atoms may be considered as irreducible elementary objects. Their composing protons and neutrons (in turn composed of quarks and gluons) can be ignored.
Of course this is not the case at nuclear reactions energy and length scales where the nucleons interactions are crucial.
Identifying the relevant degrees of freedom for each phenomenon is therefore both crucial and non-trivial.
The implicit elimination method: identifying the slowly converging modes and their associated macros, is therefore of great practical and conceptual value.
For such general systems, some of the notions usually associated to criticality might become inapplicable. Yet, one needs criticality in order to insure the emergence of the universality properties which make the MicRep method reliable at macroscopic scales.
We propose to use the very emergence of CSD in the dynamics as a criterion for the legitimate use of universality. We turn the tables and transform the CSD from a curse into a blessing in disguise. We use CSD as the label which isolates the relevant Macro's. We use the RCSD algorithms as operational proofs that the relevant Macro's were efficiently identified and expressed algorithmically. We use the Macro's in order to visualize and understand the emergence of the collective dynamics, in order to relate the salient complex phenomenology to the simple underlying microscopic causes \cite{computercenter}.
We propose to use this understanding in the task of formulating and studying MicRep models for basic problems in a wide range of fields. We are claiming that CSD is the key to characterize, build, identify and study such models.