Chapter 5

Modeling and Simulation of Complex Systems: Pitfalls and Limitations of Interpretation 1

5.1. Introduction

This chapter does not aim or claim to go into mathematical detail concerning the wide variety of models currently used in simulation. Several works have been published on this subject, covering general formalisms and particular instances and providing illustration of a multitude of applications. In this chapter, we shall discuss the way in which certain formal aspects inherent in modeling techniques can, and must, be considered when interpreting the results of a simulation, particularly in relation to their generality and generalization, that is, their validity outside of the domain initially considered. As any experienced user of simulations of fairly complex systems knows intuitively, or as a result of (often painful) failures, great care is required when using simulation results, especially when considering global (or globalizing) properties.

It is thus interesting to return to certain fundamental points, such as non-linearity, computability, and continuous/discontinuous dialectics, and to look at them from the angle of their impact on activities in the field of complex systems engineering. This allows us, among other things, to understand why certain decisions, for example, in the financial sector or in the field of information and communications systems, have a chance (or no chance) of being effective. It also helps us see why it is difficult to understand and explain to the “man in the street” why certain things happen and why certain responses may be used – although these responses are not always the right ones, not because of the incompetence of decision makers but because of the harsh non-linear reality of the situations considered. This difficulty does not stem from the concepts handled, but from their “unnatural” nature: we are not a priori particularly familiar with them. This is certainly due to the nature of the information diffused, but it can also be explained on a deeper level. In physiological terms, to promote the survival of the human species, our sensory-cognitive-motor heritage is based on a priori linear representations of situations which are sufficient for reactive adaptation to the environment, although the physiological substrate is fully able to handle non-linearity. The consideration of non-linear aspects may be a luxury left to the cognitive occupations of animals with no fears for their immediate survival; however, we shall go no further down this route, as it would lead us away from the main theme of this study.

Moreover, in this case, we are interested in applications resulting from artificial complex systems, that is, developed and created by humans and not by Mother Nature. Curiously enough, those interested in formal properties (stability, sensitivity to certain initial conditions or limits, genericity, and so on) often look at these problems with certain natural ecosystems in mind, from termite mounds – architectural masterpieces, with a thermo-regulated internal environment, capable of rapid reconstruction in case of disaster, a necessary factor in ensuring the survival of the colony – to the way in which shoals of fish instantly avoid predators by collective movements. Other examples include hives, the flight of migratory birds, “fairy rings” (strange circular patterns created by fungi growing in certain fields), and living organisms, magnificent multi-functional edifices which maintain an unvarying form in highly changeable environments and when under attack from all manner of sources. Surprisingly, when considering road networks, intelligent transport systems, or healthcare systems in which patient services are integrated with a card for accessing a digital file and accompanying financial services, we no longer find the same formal rhetoric, and the mathematical artifacts handled are not the same. We may wonder if there is some hidden specificity which attributes complexity to nature and relative simplicity to human constructions. Might this be a planned, or desired, attribute, insofar as the engineer developing a complex system would be able to limit or master, if not identify, degrees of complexity? Or – more crudely – is this just due to ignorance of the complex nature of models?1

5.2. Complex systems, models, simulations, and their link with reality

In this section, we shall discuss the need for a common vocabulary and demonstrate the value of introducing the issue from an original angle.

5.2.1. Systems

The main norms used in systems engineering (from MIL-STD-499B to the more recent ISO/IEC 15288, the latest norm, from 2008, via the EIA/IS-632, ISO-12207, and SE-CMM norms) define a system as “an integrated group of elements – personnel, products, processes – which are interlinked and interconnected with the aim of achieving one or more defined goals”. This definition will be important later when discussing the modeling and interpretation phases. As of now, note that a system is an identified form acting in space and time – hence, the importance of the notions of topology (form of the subjacent space) and dynamics (evolution in time) – made up of a group of interacting elements which change state and interactions – hence the crucial nature of the internal layout of the system, but also of the evolution of this layout over time.

Note that the internal structure of a system is eminently recursive, which naturally leads to stratifications and relationships between component parts. These stratifications may be either vertical (in a social system, we find levels made up of individuals, groups, or companies) or horizontal (in the same social system, this would correspond to the relationships between businesses or political parties). We must therefore find formalisms that can either integrate these two notions of structure and dynamics or be able to simultaneously handle different formalisms which are better adapted to one notion or the other2.

A system is interesting to observe, study, and handle because of the properties it presents, which give meaning to the planned use in a particular context. From an epistemological perspective, we find two contradictory (or rather complementary) points of view: reductionism and holism. For the reductionist, all properties – including emergent properties, which we will discuss later – can be explained by properties of the level below, down to the lowest level. On the other hand, the holistic approach considers that these properties give the system its “ontological” totality and individuality. These two points of view are certainly extreme, but reflect two different approaches to design, one by extension (based on the enumeration of attributes that characterize the object studied) and the other by intension (or by comprehension, i.e. the object is circumscribed by a collectivizing formula along all potential objects). An engineer working on a system is thus confronted by two conflicting points of view: one, analytical, deconstructs the system into simpler elements, which, when brought back together, recreate the system. The other approach, which is teleological and functional, looks at the stability of the object and its reactions to external stimuli. These two ways of seeing the system guide different approaches to modeling. They can be associated directly with the two following principles:

– Control principle: developed in the field of synergetics and in its mathematical form by Hermann Haken [HAK 00, HAK 04], this principle affirms that in the majority of complex systems, certain parameters, known as order parameters, govern the dynamics of the system in the sense that other variables are subject to them, that is, express themselves as a function of the order parameters. These order parameters present “slow” dynamic variables, as opposed to “fast” variables which rapidly attain their equilibrium value if the system is disturbed. This means that we can, on the one hand, consider the variation of slow variables to be negligible when looking at the transitory dynamics of fast variables and, on the other hand, only looking at the equilibrium values of fast variables when studying the dynamics of slow variables. When considering the equations, we may express “fast” variables as a function of the dynamic variations of the “slow” variables (i.e. they are subject to them, as mentioned above); the dynamics of the system may also be resolved using only functions affecting “slow” variables, that is, the order parameters (this boils down to a principle of simplification of the same kind as Occam’s Razor).

– Reflexivity and coherence principle: if each individual element contributes to the appearance of a collective entity and if, reciprocally, each individual element is subject to the influence of this collective entity, then this cyclic relationship between cause and effect must be stable and self-organizing. The self-organization phenomenon is found in physics (e.g. when neighboring particles create a collective field which depends on their respective quantum states, and the quantum states depend on the collective field), ecological systems (e.g. the dynamic balance between predators and prey, where a break in the cycle can potentially lead to the extinction of a species), or social systems (democracy allows each person to develop political activities, and in return, the presence of this activity allows democracy to exist; the breakdown of this reflexive process of contribution is a classic sign of a dictatorship).

The first principle shows the plurality and the hierarchy of scales of dynamic evolution within global couplings, directly affirming the stability of this hierarchy. The second principle is based on the stability of the cyclical structure of the form (in the Kantian sense of the term) of the system. We note, indirectly, the opposition between a dynamic hierarchy, which naturally leads to a reductionist approach, and a fundamental circularity, which promotes a holistic vision of the system. This is hardly surprising, as all these principles and points of view are connected to the centuries-old conflict between nominalism and universalism.

5.2.2. Complexity

We shall now look at the notion of complexity, which gives meaning to the whole issue: without complexity, systems behave in a manner which is a priori controllable and does not require the use of specific identification and control tools such as simulation. In etymological terms, “complex” is clearly differentiated from “complicated”: “complex” contains the root “ple”, meaning “to braid”, whereas “complicated” contains “pli”, meaning “to fold”. The key idea is thus one of entanglement and interweaving of relationships within a complex object.

However, René Thom defines it [THO 90a, THO 90b]:

the notion of complexity is terribly ambiguous. Too often, it is the subject of foggy rhetoric; too often, when faced with a system which we are unable to understand or master, “complexity” serves as an excuse for intellectual laziness. Outside of strictly formalized situations (…), the complexity of an object cannot be evaluated numerically; we can only use qualitative evolutions, which are fundamentally subjective and depend essentially on the perspective from which we view them. (…) when a Boeing in flight crashes to the ground accidentally, throwing out large quantities of formless debris, we would conclude that the complexity of the system has increased dramatically because of this catastrophe. According to the usual finalist point of view, however, the intact airplane in flight is considered more “complex” than all of the debris. (…) Even in pure mathematics, the notion of complexity raises problems (…). The null function f = 0 is the simplest of all functions; however, from a structural stability standpoint, the null function has “infinite codimensions” in the function type space, and so is therefore of infinite complexity.

The notion of complexity in a form is not, therefore, intrinsic and is always relative to the nature of the formalism used to describe it: thus, topologically simple forms may be very complex in algebraic terms (e.g. any form obtained by continuous transformation of a sphere, without tearing or gluing, which is therefore topologically equivalent to the sphere but for which the equation may, arbitrarily, be difficult to describe3). It is therefore practically useless to discuss simplicity or complexity without defining the nature of the formalism. This linguistic aspect, however, is not the only way of relativizing the notion.

Certain seemingly complicated things are extremely simple. In contrast, apparently simple things may prove to be very complex [KLU 08]: a factory or refinery, with multiple buildings, reservoirs, and tubes snaking across hectares of land through clouds of malodorous smoke, for example, may be considerably less complicated than a pot plant with its micro-hydraulic networks and finely regulated metabolism. A colony of ants may seem more elaborate than certain crowds, and a toy shop may seem harder to manage than certain much larger companies. In short, complexity is a hard concept to define precisely. Take another example: a star in the night sky may fascinate us: I personally remember spending several minutes admiring the Milky Way one night in May in a national park in South Africa, far from all civilization, the somewhat frightening silence of nature interrupted only by the wild music of unknown creatures. On the other hand, a goldfish in an aquarium is quickly forgotten by a child who, only moments before, had begged his parents to buy one. The first example – the star – is a cosmic machine based on fairly rudimentary fusion mechanisms4, whereas the second is a subtle arrangement of cells working in harmony within multiple interlinked systems: circulation, skeleton, optics, neurology, hematology, audition, respiration, enzymes, biomechanics, behavioral, social, and so on.

What, then, is simplicity? What is complexity? We clearly cannot find them in ordered, robust, and stable structures, such as a crystalline network; nor do we find them in unordered, random, unstable structures, such as the gas in a room. They must therefore exist between these two extremes [GEL 94]. This would suggest a characterization based on a combination of structural (grammatical or algorithmic complexity) and entropic measurements, but a characterization of this kind includes the implicit hypothesis that complexity is more or less stationary (i.e. does not change over time). However, certain systems clearly do appear more or less complex depending on the length of the observation period: this can be measured by the Deborah number5, used in rheology to characterize the “fluidity” of a material and defined as the ratio of a relaxation time, characterizing the intrinsic “fluidity” of the material and the period of observation. This illustrates the difference between apparently unchanging solids, and fluids, which flow. If the period of observation is long or if the relaxation time of the observed object is short, the observer sees the object evolve and, in a manner of speaking, “flow” like a fluid.

Another characteristic of complex systems is the potential for ambiguity in defining their boundaries. A boundary, as the distinction between interior and exterior environments and the principal location of exchanges between these areas, may appear to be relatively well defined. However, it may fluctuate or be somewhat indistinct, in the sense that although some aspects are clearly inside or outside the system, there is a zone which could belong to either category. Thus, for an ant colony, the “interior” part of the system is not limited to the ant hill – which does not even exist for certain non-sedentary species, such as African army ants – but may, in theory, extend up to the edges of the area visited by the insects in search of food. However, a definition of this kind, based on the geometric envelope of trajectories of members of the colony, is either completely unstable, if we consider the specific envelope of trajectories at a given moment, or irrelevant when considering exchanges between the colony and the external environment if we include all trajectories of all ants throughout the lifespan of the colony (which is itself difficult to define, even using the lifespan of the queen as a reference). We might think that the question of boundaries would be easier to consider in the case of artificial complex systems, due to the use of rigorous engineering processes, but this is not the case: how, for example, would we define the boundaries of the gas supply system of a European country?

Clearly, these boundaries cannot be fixed at the political borders of the country in question, as shown by the 2009 natural gas crisis between Russia and the Ukraine, which had an impact on other European countries. Should we, then, fix the boundary at the limits of responsibility of the gas provider to the final user or of the entire supply chain, including sub-contractors or even those with a potential influence on component parts (e.g. physical components or financial management)? The systems considered differ greatly depending on our response to this question, with different problems in terms of design and maintenance.

The notion of a border is interesting because, as a location for exchange, it is the place where the capacities for adaptation of a complex system can be best observed, via the dynamic response to environmental pressures and other more or less expected disturbances, or via changes to the exchange flow which may have a knock-on effect on modifications of internal fluxes.

One last feature, which is somewhat mysterious in nature and tends to attract media attention, is the emergence of structures and behaviors from the actions of the system and its interactions with its environment. We do not claim to present an explanation for emergence and direct the interested reader to the abundant literature published on the subject. Starting from the principle that each reader has their own understanding of the subject, we shall simply provide a few illustrative examples of emergence, to suggest the potential challenges it poses in terms of simulating this kind of system. Our first examples are taken from the natural world: the way a shoal of fish avoids a predator and the appearance of paths near to watering holes in the African savanna. A shoal of fish is particularly fascinating and may suddenly change direction to avoid a predator, behaving as a single body: this raises interesting questions on methods of collective control. Various models can produce this type of behavior, and it is mathematically possible to pass from some models to others in spite of their apparent differences. The appearance of definite paths near watering holes is due to the fact that large number of animals follow the same itinerary; each new animal following this itinerary reinforces the path. The same phenomenon can be observed on certain lawns, particularly in cases where it is clearly quicker to reach a given point by walking across the grass than by following the path (seen as an imposed detour). Other examples found in artificial complex systems show that it is difficult to model these phenomena a priori: we shall consider the examples of commercial hubs in the air transport system and interest groups in social networks. The concentration of air traffic around hubs is essentially due to economic logic (regrouping of a number of aircraft to facilitate maintenance) and security considerations, particularly for entry into a territory. But the real emergent phenomenon has been the appearance of major commercial zones around these air transport hubs, as much in the United States as in Europe6 (e.g. London Heathrow) or in Asia. The presence of large number of shops allows passengers forced to pass through these hubs to kill time in a pleasant manner, and increased sales encourage the installation of further shops, and so development continues by a process of mutual influence (known, in technical terms, as positive retroaction). In a similar way, interest groups emerge on the Internet within social networks, without really understanding the conditions which led to their appearance and development (much to the displeasure of publicists!).

The emergence of behaviors or structures, as illustrated in the two cases given above, shows an important characteristic of complex systems which some consider crucial in defining complexity: self-organization. Whether this is manifested by a modification in the system architecture or by observed behaviors, it is a source of innovation and of new capabilities. The interest of mastering these details, as well as the conditions in which it appears, is evident: a clear understanding of these details leads to possible new system properties, while an understanding of conditions opens up the possibility of controlling these properties, by finding the means either to create them or modify them depending on external constraints, or to avoid them in the aim of guaranteeing operational security. This is the point where modeling comes in, providing the means of inverting the model to try and control emergence, and simulation can provide heuristics with this aim. Note, however, that these ideas are currently only a potential research subject, except in certain specific cases.

To conclude this overview of complex systems, we shall give a few more examples of artificial complex systems, “systems of systems”, for which we use the following definition7: a system of systems is an assemblage of systems which could potentially be acquired and/or used independently, for which the designer, the buyer, and/or the user wish to maximize performance of the global value chain, at a given moment and for an imaginable group of assemblies. In this definition, we use the general definition of “value chain” popularized by Michael Porter: the set of interdependent activities, the pursuit of which allows the creation of identifiable and, where possible, measurable value (i.e. what the client is prepared to pay for the product or the service). The examples developed in greater detail in [LUZ 08a] include intelligent transport systems, which not only consider the availability of transport vehicles but also provide information services (cartography, meteorology, real-time information on traffic conditions) to passengers, or even the possibility of passing from one transport service to another in a transparent manner in terms of payment, such as the Navigo pass used in and around Paris, or other “single ticket” initiatives taken by regional centers operating several methods of transport. Another example is Global Earth Observation System of System (GEOSS), the international system of systems for observing the Earth, which brings together sensors, processing resources, databases, and access portals; one last example is the global financial system, the complexity of which became apparent in 2008 with the emergence of an economic crisis.

5.2.3. The difficulty of concepts: models, modeling, and simulation

The meaning of the word “model” when used in reference to a phenomenon is usually common: the representation – that is, an organized and formalized set of opinions, beliefs, and information – in a language specific to the phenomenon. “Modeling” is used to refer to the process of finding and establishing this representation. However, we have a tendency to forget the limitations and constraints of this definition, which are clearly set out in three specific points: we are dealing with a “representation”, in a “particular language”, and the object to which these apply is a “phenomenon”. We shall cover these three points individually.

5.2.3.1. Representation

As we can see from the composition of the word itself, a representation is a “re-presentation”, that is, a new presentation of the object to the subject (the observer). The term also contains an aspect of “replacement”: the initial object is replaced with a new object of interest, with the observer’s accord and/or active participation: the observer becomes modeler. The model, by metonymy, itself becomes a sort of creation. In the same way, the link between the model and the reality is not immediate8. As Alain Badiou highlights in his work Le concept de modèle [BAD 07], “The model is not a practical transformation of the reality: it belongs to the register of pure invention and possesses a formal irreality”.

To support his affirmation and in a departure from his usual domain of formal logic, Alain Badiou makes reference to Lévi-Strauss and his work Anthropologie Structurale, where the model is said to be “constructed” or even “knocked together”:

for Lévi-Strauss, the formal, the constructed, the artifact, is a model relative to a given empirical domain. In positivist semantics, the model is an interpretation of a formal system. It is therefore the empirical, the “given”, which are models of the syntactic artifice. Thus, the word “model” takes on a sort of reversibility. (A. Badiou).

To continue our reflection on the relationship between the model and the real system at this level of abstraction9, we should note that a model is only a partial representation, at a given moment. Models should be approached with a critical spirit; not to do so demonstrates ignorance on two levels: first linked to the a priori incompleteness of the representation of reality and second to a posteriori incompleteness of knowledge. “The model, technical moment or ideal figure takes its place, at best, in the domain of scientific practice. Note that, as a transitory adjuvant, it is only destined for deconstruction, and the application of the scientific process does exactly that” (A. Badiou). We may also cite the example of Bohr’s planetary model of an atom, used as a useful image in teaching, which has now been replaced by a probabilistic quantum model allowing precise calculations based on the behavior of the atom. The permanent re-evaluation of a model is necessary as knowledge advances; worse, “stopping at a given model turns the model into an epistemological obstacle” (A. Badiou).

We should keep these precautions in mind over the following sections, where we shall discuss different types of phenomenological models, not the analytical models used for the most part by engineers.

The criteria used to suggest a scale for classifying representations include exhaustiveness and simplicity: the fact of considering all observed phenomena and the “elegance” of the model. We could consider that this attention to esthetics stems from the Kantian or Platonic logic subjacent to our own culture, which considers that the laws of nature must be simple as they are universal. We might also see these considerations in a practical light, as savings in terms of resources are almost always interesting if the model has no particular ontological status in connection with that which it represents. In addition to these criteria, a third property is linked to model usage and is directly relevant to the end-user of a model: credibility. A model is, in fact, a “fabrication of a plausible image” (A. Badiou). Chapter 3 details this essential characteristic. Moreover, as an artificial object, the model is controllable: we may anticipate the way in which it will react if one of its elements is modified. “This precision, in which we find the theoretical transparency of the model, is clearly linked to the fact that it is mounted as a whole, so that the opacity linked to the real system is absent” (A. Badiou).

5.2.3.2. Particular language

Here lies the source of the “richness” of models: their variability. A model may be analytical (a set of equations, following a reductionist approach), synthetic (guided, e.g. by the principle of optimality, symmetry, or conservation, in which case the approach followed is more holistic), quantitative (as in the previous examples, with the fundamental objective of calculating values), qualitative (focused on a given interesting characteristic, without necessarily aiming for digital adequacy), behavioral, economic, and so on.

For example, depending on the specific case considered, a cloud might be seen as a convection phenomenon or as a homogeneous group of water droplets suspended in air saturated with humidity. This shows the level of diversity in terms of modeling, which depends on the predictive quality of the desired model, in connection with other modeling idealizations made for connected systems and systems interacting with the main object of the model, in this case, the cloud.

In the same way, a model of a system of systems such as the European global satellite navigation system (the Egnos and Galileo programs) could concentrate on contractual aspects linking clients, suppliers, partners, and users, or on international flows between different physical components, or on the physical trajectories of satellites, or on economic aspects (the business model of the associated services), and so on.

A large variety of formalisms may therefore be used, each with its specificities, strong points, limitations, and interpretational difficulties. However, over and beyond this range of possible formalisms, we must – as pointed out by Alain Badiou – move beyond the position based on the Hegelian perspective of “the word as the murder of the thing”, where the model would “kill” any aspect of intuition in the approach to the phenomenon it mathematizes. This point demonstrates all the ambiguity of modeling, between an arbitrary characteristic of the model type and an intuitive leaning toward the use of a particular model.

5.2.3.3. Phenomenon

The usage of the term “phenomenon” places the user in a privileged position. It is not the “abstract” system which interests us, but a set of observed or observable behaviors.

This leads us to distinguish between emulation and simulation of a system: simulation is a dynamic use of models, whereas emulation is the quest for the most perfect analogy possible between calculated and observed behaviors. Simulation, then, is mostly based on the activity of the model builder, and its specificity is linked to the use of mainly computerized (but also physical) resources for model usage. In extreme cases, simulation may be seen as a “simple” complicated calculation effectuated over successive states of the model.

Beyond the limits intrinsic to the models’ subjacent formalisms, we find limits imposed by the computer usage, limits which are not just linked to processing time or memory size issues, but which are based on the actual impossibility of certain mathematical tasks. We shall return to these intrinsic limits of computability later.

5.3. Main characteristics of complex systems simulation

5.3.1. Non-linearity, the key to complexity

The interest of the previous, relatively general considerations lies in the existence of various modeling theories based on different epistemological principles and providing tools for mathematical formalization, explanation, or prediction. These often provide quantitative information on the phenomenon under consideration.

Whatever formalization is used, it highlights non-linear aspects, a cause of richness and difficulties of interpretation of phenomena. This will be the subject of discussion in the following sections, although we do not aim to cover the topic exhaustively10; for pedagogic reasons, we shall avoid mathematical development, and the reader may consult various works listed in the bibliography for greater detail.

5.3.1.1. Decomposition and reconstruction versus irreversibility

A linear system is defined as the behavior produced by the sum of two systems is the sum of the behaviors of their sub-systems, also known as the principle of superposition. Decomposition is therefore direct, and reconstruction is simple, with no difficulties except for that of recombining components. A non-linear system, on the other hand, does not have this property11. Thus, the cooking time of a big cake is not twice the cooking time of a cake half the size – unless we want our cake to be burnt. In general, the dynamic interconnection of elements tends to introduce aspects of non-linearity.

Another factor that indicates non-linearity is irreversibility, that is, the fact that we may determine the arrow of time by observing a phenomenon. Although the propagation of a wave without dampening is a reversible phenomenon, the same cannot be said of the propagation of heat through a metallic plate. Irreversibility is an interesting property as it is often linked to diffusion or energy loss effects (through friction or relaxation), which may be a sign of long-term smoothing effects where certain disturbances may be “forgotten”. A number of physical systems show pattern emergence phenomena as a result of this irreversibility.

5.3.1.2. Local to global transition versus chaos

One characteristic that greatly simplifies dealings with linear systems is that by understanding the system at local level (more precisely: in an open neighborhood of any regular point); we gain an understanding of the system as a whole. If something is seen to happen under particular conditions, we can deduce what happens under other conditions by analogy. This is not the case in a non-linear system, which may present singularities around which system behavior may be extremely strange. Singularities themselves may be arranged in an incredibly bizarre fashion, with the potential for creation of “strange”, or “chaotic”, attractors.

As dangerous as identified non-linearities can be, they may also present certain advantages. In aeronautics, for example, the existence of unstable non-linear modes allows the maneuverability of an aircraft to be greatly increased under certain conditions. However, this “richness” of possible behaviors, where regions of instability may appear suddenly and can be difficult to control, can cause serious problems. It is currently fashionable to claim (after the event …) that the financial crisis of 2008 was predictable. The same individuals who make such claims propose very different solutions to the problem, none of which appear a priori to enable us to leave the domain of instability. These are all clear indicators of non-linearity.

In a similar manner, the phenomena of hysteresis and bi-stability (existence of several local optima with alternate “hops” from one to the other depending on disturbances) only occur in non-linear situations. These phenomena may be found in various social or financial models (see the passage on catastrophe theory). Sudden hops, sudden divergences, and regions of inaccessibility are all part of the panoply of situations only found in non-linear systems.

5.3.1.3. Predictability versus unpredictability

Unpredictability, a consequence of the properties discussed above, is frequently encountered in non-linear systems. On the other hand, in a linear system, knowledge of behaviors over a certain finite time period enables full understanding of the global system, this is not the case in non-linear systems (which is a shame, as otherwise it would be very easy to make a fortune on the stock market).

One of the properties on which most literatures have been written, and which is particularly present in literature aimed at a non-specialist public, is sensitivity to starting conditions. Let us illustrate this using a very simple non-linear system: the “left shift” application. This application connects a positive real number x with the fractional part of 2x, that is, the difference between 2x and the nearest natural integer using values less than 2x (or more prosaically, the number made up of the digits after the decimal point when the number in question is developed in base 2). The dynamic system known as “left shift” is defined simply as follows: the initial state value at instant 0 is any number from within the unit interval (so from 0 to 1 inclusive: if we develop it in base 2, we have 0 to the left of the decimal point and any sequence of 0 and 1s after the decimal point). The system state value at instant k + 1 is given by the “left shift” application, applied to the system state value at time k. With a number developed in base 2 where there are no digits other than 0 to the left of the decimal point, this reduces to “forgetting” the first digit after the point and moving all the following digits one place to the left, hence the name of the application. Let us take two arbitrarily close numbers, differentiable only after the first n digits after the decimal point, where n may be arbitrarily large. The system trajectories are impossible to tell apart until we reach iteration n; as from this moment, each follows its own path as set out by the following digits. Clearly, this is a simple textbook illustration, but it illustrates sensitivity to initial conditions in that even if we “know” these conditions to an arbitrary level of precision, it is possible that trajectories diverge at some point. In a linear system, on the other hand, the error between trajectories would have been kept arbitrarily small via an increasingly precise knowledge of the initial conditions.

To digress a little from our central theme, we shall now consider the conditions of appearance of non-linearities in a system which is a priori linear: these may be the result of interaction or regulation loops. As a textbook example, let us take the discrete system xk+1 = 2xk+uk, which represents a system known via its state xk throughout sample time periods, with an entry uk acting as a regulation signal. The system is a priori linear if uk is not dependent on xk (if it is an exogenous entry independent of the system history) or if uk is dependent in a linear fashion (e.g. via a homeostatic state return corresponding to a reaction loop uk = −axk, with a “spring” constant a). If, however, the regulation shows non-linearity, the looping system becomes non-linear: consider, for example, uk = [−2xk], where the operator [.] is the integer part, that is, the closest integer using inferior values. For a state from 0 to 1 inclusive, and for the preceding linear system, uk may take the values 0 and −1, which corresponds to a switch or diode type behavior, and the looped system is none other than left shift, the non-linearity of which has already been demonstrated. This situation is far from exceptional in that all phenomena of sudden commutation, saturation, and mechanical wear and tear produce consequences of this kind.

Another consequence of non-linearity with an effect on predictability is the non-transitive propagation of knowledge. This is expressed by a theorem developed in 1972 by Kenneth Arrow, an economist at Stanford University, which illustrates why it is difficult to make a preferential choice when faced with three or more options. Consider the example of three users, each preferring a particular solution to a problem. User 1 classes three solutions following their order of preference: 1, 2, and 3. User 2 classes the solutions in the order 3, 1, and 2. User 3 chooses 2, 3, and 1. Given the choice between two solutions, the majority of users prefer solution 1 to solution 2, solution 3 to solution 1, but also solution 2 to solution 3. This goes against intuitive reasoning, which is based on transitivity, that is, the linear propagation of a relationship. In a similarly counter-intuitive manner, if we must choose between all solutions simultaneously in the example provided, no majority preference can be detected. This shows that we cannot easily extend the field of application of a sub-problem to a more general problem, another effect of non-linearity.

5.3.1.4. Sensitivity to disturbances

Sensitivity to disturbances is an extension of the property described above. In a linear system, the size of a disturbance is directly linked to its effect; this is not necessarily the case in a non-linear system. This is unfortunate when the disturbance is a sign of incomplete knowledge of the signal, as in a linear system a small disturbance only produces a small effect, making the system naturally robust. It is even more problematic when faced with major disturbances, which in a linear system would produce a strong reaction potentially able to deal with the disturbance. In a non-linear system, two other reactions are possible: small disturbances may produce major effects, or large disturbances may only produce small effects. The second situation is not of great interest for our purposes, but the first is more relevant and is known – even outside the domain of popular literature on the subject – as the “butterfly effect”12: in this illustration, the simple fact of a butterfly beating its wings could provoke a storm at a distant point on the globe. The principle of this idea is to illustrate, using a memorable image, the fundamentally non-linear character of weather systems (and more precisely of equations used to model weather systems, something which we shall discuss in greater detail later on). Another example used to illustrate the same issue might be the gunshot that killed the Archduke Franz Ferdinand, triggering the First World War and contributing, albeit indirectly, to the major transformations of the 20th Century.

Note that the famous butterfly effect is not necessarily negative or a symbol of powerlessness; it also shows the possibility of gaining considerable benefits from minor investments, if we can define the right utility function! Different economic stimulus actions attempted to activate this phenomenon to provide a way out of the 2008–2009 financial crisis. The figures involved seem enormous – billions of euros or dollars – but are, in fact, negligible when compared with the financial exchange flows across global markets (the difference is of several orders of magnitude, and furthermore, stimulus programs operate over several months, whereas the flows discussed apply on a daily basis13). This is, then, a clear example of a non-linear system in which we attempt to provoke significant effects by actions which are small relative to the dynamics of the system in question.

At the risk of appearing contradictory (although this only serves to highlight the difficult nature of any overly general prediction created by excessively early application of certain principles), we wish to raise an additional point concerning the butterfly effect: over the last decades, meteorologists have noted that after a period of 1–2 days, the growth of disturbances ceases to be exponential14, but becomes proportional to time. This is not particularly surprising when considered from the viewpoint of theoretical physics. In fact, statistical mechanics, the theory which allows long-term forecasting of certain statistical quantities which often correspond to average values of physical variables, undermines the dogma of the butterfly effect: even if the system is not microscopically predictable, the visible macroscopic aspects will be predictable. For example, gas molecules within a room take paths which cannot be followed beyond the first few collisions, but we can define a macroscopic quantity characterizing their average paths, that is, temperature, and we can calculate results in terms of this new variable, for example, if we open a door to another room or outside. It would be absolutely impossible to know what would happen if we remained at molecular level. To summarize, non-linearity produces unpredictability at a small scale, but it may give way to linearity at a higher scale level. These last two considerations should be considered in complex systems with a large number of degrees of freedom and which may be described using macroscopic quantities, that is, statistical methods. An example of this would be network-centric systems, where we might define the quantities linked to information and its diffusion within the system.

5.3.2. Limits of computing: countability and computability

The use of computing resources requires prior programming of the code needed to run these calculations. All programming of computer code is done via a symbolic language, no matter how complex the instruction sets available. Anyone with the slightest experience of programming languages knows that there are a finite number of available instructions; moreover, a program is itself finite, even if it may be arbitrarily long (sometimes reaching tens of millions of lines), even without adding useless lines! To summarize, if we count all the programs that may be written in all imaginable programming languages over a finite time period, the resulting number of programs will itself be finite. This set of programs is at most countable, that is, at most, it has the cardinality of the set of natural integers15. This is important as it sets a fundamental limit on what is accessible by informatics, as we shall see in the following sections.

We could raise the objection that the reasoning above is conditioned by a model of computation based on the use of computing languages. This is true, but in spite of human ingenuity, all other formalisms16 invented to model the intuitive idea of computation using a mechanical procedure (which therefore operate following a discrete chronology – we can separate, number, and order different instants in the calculation procedure – and give a result in finite time, if a result is given) have reached the same result. More precisely, all these formalisms have been shown to be equivalent in that they define the same class of mathematical objects (up to isomorphism)17.

We could also say that a computation depends on the initial conditions provided. This, again, is true, but we should not forget that this initial data input must follow the same constraints of mechanical use, that is, the quantity of data must be finite, although it may be arbitrarily large, and coded as a finite collection of symbols which may also be arbitrarily large. At most, we have a countable quantity of initial conditions which may be coded using computers. The set of calculations and the set of imaginable conditions are the product of two countable sets and are therefore always countable.

It is, in fact, possible to escape from this “tyranny of countability”. We must “simply” allow transfinite inferences (i.e. the time taken to obtain a result – in this case, a clear numerical result, and not an absence of result as in a looping program – may be infinite, which, we must admit, is not particularly realistic for a practical application!) or pair the computational resource with an object generating an initial condition non-mechanically, which could then be used and transformed before passing the deciding step of computer encoding (a real challenge, admittedly). This last suggestion only works if the physical world really leads to such objects, which is no longer obvious with the recent development of several theories of discrete space and time which question their continuous aspect, that is, fundamentally question their varying in a space with the cardinal of the continuum (like the set of real numbers or any differentiable real variety), thus immeasurably more than countability. In short, this excludes any reasonable computational model.

Using a computational model, we can define a variety of notions, such as recursivity, decidability, recursive enumerability, and semi-decidability. Without going into mathematical detail, note that these notions are applicable either to sub-sets of natural integers, or to problems, or to formulae expressed in a certain formal language. Although these domains of application seem different, they are brought together mathematically via a recursive isomorphism (i.e. for each of these applications, we can find a coding algorithm usable by a computer so that the final codes are all the same). The above notions can therefore be defined, informally, as follows: “recursively enumerable” or “semi-decidable” means that we can respond affirmatively, by a computation (if, of course, the response is affirmative18) to questions such as whether a given integer belongs to a specific set, whether a given class of problems has a solution, or whether a set of formulae can be demonstrated in a finite time period based on axioms and interference rules. The notions of recursivity and decidability, on the other hand, imply that the calculation allows a positive or negative response in finite time, something which is much more difficult. This avoids waiting eternally for a response which never appears. These notions are interesting insofar as we can prove that computability, as described above, is equivalent to the notion of “recursive enumerability”. Moreover, we can prove that the notion of computability is well defined, in that, for example, sub-sets of natural integers exist which are recursively enumerable, but others exist which do not have this property.

We shall provide an example to illustrate what has been said above, both in terms of cardinality and computability. Let us return to the “left shift” application defined earlier. The application is clearly computable, as, intuitively, it is easy to write a program which will do this. The sequence of output values is the sequence of figures showing the development of the initial condition in base 2. An initial value is therefore computable if a program can unpick the figures which constitute its development in base 2. Note that there is a clear one-to-one map between all the numbers of the unit interval (i.e. all possible initial conditions) and all possible subsets of the set of natural integers: I is in the sub-set considered if, and only if, the figure in position i is a 1. However, the computable initial values only correspond by this one-to-one map to recursively enumerable sub-sets, which only constitute accountable part of the unit interval, that is, a null set. In other words, this simple example19 shows that the trajectories accessible to any computability model only form a negligible set within all the trajectories possible within the system.

That said, we should remember that any sufficiently regular solution to a problem expressed by differential equations may be approximated in a computable manner with an arbitrarily small precision20. The hypothesis of regularity is obviously important, and this is where we find ambiguity and the necessity of caution when using computer tools: in numerous situations, the limits discussed above have no particular influence, but in specific cases (chaos, non-regularity of applications, and so on – in short, a number of situations which may be encountered among complex systems), computability constitutes a fundamental obstacle to the illustration of certain properties. Thus, using our earlier example of left shift (for generalizations of this example showing a degree of genericity in relation to dynamic systems, see [LUZ 93]), any computerized simulation will give, at most, periodic behaviors due to the finite nature of the encoding of any initial condition, while these behaviors are, in fact, negligible among chaotic behaviors (the trajectory “fills” all the unit interval), which are by their very nature inaccessible to computerized simulations.

This section, covering rather technical aspects of the subject, can be summarized as follows: from the point of view of mathematics, all modeling of computer processing resources a priori introduces limitations of complexity in comparison to what might happen in the real environment21.

5.3.3. Discrete or continuous models

This question is ever-present in epistemology, and the opposition of discrete and continuous model has been a subject of reflection since the Ancient Greeks, whether in terms of acceptance relating to cardinal multiplicity (discreteness being linked to counting in whole numbers, continuity to large cardinal numbers) or to geometric divisibility (e.g. discreteness being linked to the notion of points and continuity to that of a line). The apparently paradoxical reflections of Zeno of Alea demonstrate this. Which of the two is “superior” is an unsolvable philosophical question, one not even helped by language. In English, “continuous” is opposed by “discontinuous”, whereas in Russian, we find “” and “”; negation, expressed by “dis” in the English example and “He” in Russian, is applied to roots with opposite meanings.

To show that the question itself is badly posed, we return to the example of digital computers, used earlier to show the necessity of studying discrete entities. If we refer to the laws commonly used in physics to analyze this type of system (the Maxwell equations), we obtain a continuous model based on differential equations, from which we could conclude that the state of a computer of this kind must, in fact, vary continuously (e.g. no sudden changes in power). However, if we analyze the components of these computers more closely – for example, capacities – we see that the same models used in physics are based on the movement of a finite number of charges, suggesting, this time, using a reductionist approach, that a discrete model would be appropriate. We might then consider the electron from the point of view of its wave function and use Schrödinger’s equations, which send us back to a continuous model. The solutions to this group, from the perspective of quantum mechanics, are quantized, and different particles may be broken down into discrete components, taking us back to a discrete model. These components may themselves be modeled, using quantum field theory, by continuous fields, producing a continuous model, which itself would eventually be discretized, depending on the theory of quantum gravity used.

It is as easy to pass from a discrete representation (in cosmology, galaxies are seen as elementary components) to a continuous modeling (the set of galaxies is assimilated to a fluid to apply Einstein’s equations and discover the evolutionary dynamics of the universe) as it is to pass from a continuous representation (in hydrodynamics, a “tube” of fluid) to a discrete model (the “tube” is split into slices, each with a simplified behavior, and the coupling of partially derived equations is brought down to a discrete total of interactions in the form of borderline conditions between the supposedly “perfect” elements, the evolution of which may be explicitly calculated).

To show that the hermeneutics of the relationship between discrete and continuous is not limited to the physical sciences, but also find a fundamental way in mathematics, we only need to look to the fierce debates between proponents of the holistic and constructivist visions, opposing the collectivization used in the Zermelo–Fraenkel set theory, for example, with the requirements of constructability and temporal modularity expressed by Brouwer [SAL 91, SAL 92]22. Note that the concept of continuity is fundamentally relative and depends on the logic used in the formalization of the structure considered (something may be continuous from an internal perspective but not from an external point of view23). In the same way, depending on the topological viewpoint, continuous models may be able to deal with an idea of slow and not locally observable deformation, which discrete models cannot do; however, this is to forget that composition, the basic operation in a discrete approach, is itself a continuity, simply defined in an adequate topology (this is, in fact, the formalization of what slow and non-observable deformation actually is!).

In an attempt to conclude this debate, it seems to us that the basic question is not whether the model should be continuous or discrete, but which model is the most suitable to the scale at which we are working and for the conclusions we wish to obtain.

5.4. Review of families of models

The two main opposing approaches in mathematical modeling techniques are equational and computational approaches. This is linked to the traditional opposition between descending and ascending approaches. Equational approaches start from general principles – conservation of certain values such as energy or the quantity of information, symmetry (i.e. invariance when subjected to sets of transformations, such as translations or rotations, or more complex applications, such as those found in gauge theory), or maximization of certain quantities, for example, a defined action based on an energy – from which we deduce synthetic formulae, the resolution of which shows the desired trajectories of the variables studied. However, these equations often require complex theoretical tools to solve, and an explicit and global solution (not just for the neighborhood of a given point, or over a limited time period) may be difficult, or even impossible, to find.

Computational models, on the other hand, do not have this synthetic character, and they are based on local future state construction mechanisms, which start from a knowledge of the current state and, sometimes, the immediate environment. The whole of the trajectory can thus be obtained by successive step-by-step execution of the computational principle. These two approaches are very different. The first is based on a body of mathematical work on resolution techniques developed over two centuries; the second became popular and well known with the appearance of computers and intensive calculation. However, the two are not necessarily independent: it can be demonstrated that certain multi-agent systems or cellular automata, which will be discussed later in this chapter in the context of computational approaches, in fact produce solutions to equations – which can be partially derived by applying the principle of conservation of energy. Or from local information exchanges from spatially neighboring states used to calculate the following future state, it is often easy to produce – by carrying out probability density calculations on these exchanges – a “master” equation, which we may then attempt to solve and which, under certain hypotheses, boils down to some of the examples given later.

Each of these approaches presents certain advantages and disadvantages. Equational methods profit from their formal character and the theoretical background attached, allowing us to study the characteristics of stability, sensitivity, and robustness. At the modeled system level, this pays dividends in terms of reliability and operational safety. The major drawback of this approach, leaving aside potential mathematical difficulties, is that we are only dealing with a model, and so the property obtained by solving equations – often after considerable effort – is only a property of the model and not necessarily of the modeled system. Computational methods have the advantage of a priori simplicity, and their illustrative character is attractive, but they often require considerable processing resources and are subject to the intrinsic limits of algorithmic complexity, or even computability, of the phenomena being modeled. Moreover, as with the equational approach, the results obtained cannot be considered valid except for a partial representation of a system.

The real choice of an approach to consider in a given context, therefore, depends more on the availability of modeling and resolution tools rather than on an a priori analysis of different approaches.

The “art” of the modeler and interpreter of a simulation lies in the ability to profit, at the same time and in a complementary manner, from different models, to exploit this multi-scale (in terms of both time and space) and multi-objective approach, and to assign an index of truth and validity to each interpretation, possibly proposing extrapolations but with a clear understanding of the risks attached to this practice.

5.4.1. Equational approaches

Equational approaches can be split into two classes based on the systems considered: conservative and dissipative systems. The fundamental difference between the two is that the first may be considered to be isolated as far as the total energy of the system is concerned, whereas the second may not. Typical examples are the propagation of waves without dampening (the vibration of an ideal violin string) for conservative systems and diffusion (propagation of heat through a metal plate heated at any specific point) for dissipative systems. Historically, these two classes of systems form part of two different scientific views of physics: the Newtonian and the Aristotelian viewpoints. Aristotle considered movement as a struggle against environmental inertia (if two objects are placed in a viscous liquid, the heavier object falls more quickly), whereas Newton suggested that movement was invariant in the absence of external forces (e.g. objects, unaffected by wind, falling through a non-viscous environment, such as air: two objects fall at the same speed independently of their weight). In fact, from a strictly mathematical point of view, we pass from conservative system equations to dissipative system equations by the simple addition of nonlinear terms, following the example used earlier to illustrate non-linearities that may be created during the regulation of a system. However, among this continuum of systems, we shall direct our interest to certain particular representative cases. We shall, therefore, pay specific attention to waves and solitons, followed by reaction–diffusion and convection mechanisms.

Waves and wave propagation equations are well known, and their occurrence in systems has been mastered: studies have been conducted on potential resonance (which creates destructive “pumping” effects, as when a car is subjected to a pitching movement by turning the steering wheel to the right and to the left alternately at the right frequency) that must be mastered at interface level because, for example, mechanical vibrations might be transmitted to an electronic sub-system where they would cause damage. Solitons, on the other hand, are less well known, although they were first identified as early as 1834 by John Scott Russell, naval engineer and mathematician, who observed and followed a wave created by the sudden halt of a barge in the Union Canal (which links Edinburgh with Fort Clyde) over a considerable distance. Russell’s curiosity was awakened by the fact that the shape and speed of the wave did not change throughout its propagation. In the second half of the 19th Century, H. Bazin and H. Darcy, hydrodynamic engineers, reproduced this phenomenon on the Canal de Bourgogne to analyze it. The equational formulation and explicit resolution were carried out in 1895 by D.G. Korteweg and G. de Vries. The interest of these solutions lies in the fact that they apply to single waves of high amplitude and reduced speed and shape (when compared with “traditional” waves which might have formed in the same circumstances). When two solitons meet, no interference occurs, unlike standard waves which would be modified: we have all, at some point or another, jumped into a pool with the aim of creating a magnificent trail of waves, only for these waves to become formless ripples as soon as they hit the side or another wave created by someone else jumping in. This property of non-interference in solitons makes them interesting as a means of information propagation, a fact validated in the late 1980s by the use of fiber optic communications over distances of several thousand kilometers, and again in 1998 using combinations of solitons of different wavelengths. The first commercial equipment using solitons to transport real traffic over a commercial network appeared in 2001. In the natural world, phenomena that can be modeled by solitons are encountered in tidal bores: under certain conditions, tides create waves of this kind which travel up rivers, an interesting phenomenon to observe but one which is potentially dangerous to shipping. In France, tidal bores of this kind can be found on the Dordogne and Garonne rivers, and up until the 1960s along the Seine at Caudebec-en-Caux, but this natural curiosity – which, on occasion, proved fatal to overly adventurous tourists – disappeared following terracing works and dredging of the river. The most powerful tidal bore in the world, in China, produces a wave of 9 m tall which travels up the Hangzhou Bay at almost 40 km/h. Certain cloud formations, “rolls” of several hundred kilometers in length which move at high speeds, can be modeled by solitons. The Tacoma Bridge accident on 11 July 11, 1940, can also be modeled in this way. The same applies to the legendary rogue waves of over 20 m in height which tankers sometimes encounter on certain shipping routes. The ability to model all these phenomena is, therefore, useful in infrastructure or naval engineering, for example.

Let us now look at reaction–diffusion equations, which form a particular type of transport equations. Just for information purposes, and to show the variability of behaviors produced by different non-linearities, we shall give a family of equations of this kind: . Depending on the form of the non-linear term R(u), different phenomena are produced, enabling the modeling of a variety of situations. If R(u) = 0, we obtain pure diffusion (the “heat equation”), as in the classic example of a metallic plate which heats up as a whole if it is heated locally. If R(u) = u(1 − u), we obtain a simple model of population evolution with migratory phenomena. If R(u) = u(1 − u2), we have Rayleigh–Bénard convection. This convection phenomenon is that observed in a pan of boiling water heated under certain conditions: rolls of water circulation appear beneath the surface. The same model is applied to magma circulation when modeling plate tectonics; it also explains certain atmospheric circulation phenomena. If R(u) = u(1 − u)(u − a) when 0 < a < 1, we obtain the Zeldovich combustion model.

These different equations show three mechanisms: diffusion, reaction, and convection. The diffusion mechanism tends to homogenize concentrations of the elements concerned and is characterized by an irreversible transport phenomenon caused by the migration of elements (hence it is used when modeling this kind of situation). Note that borderline conditions, that is, the form of the edge of the system, are propagated “instantly” in that, at any given moment, each point is influenced by all other points. The reaction mechanism, on the other hand, is the manifestation of a local influence, a function of the current state. The convection mechanism shows a functional influence of the flow of speeds, considerably more important than local speed, creating large-scale circulation phenomena.

It is the combination of these mechanisms that creates complex behaviors using large-scale spatio-temporal correlations, hence, for example, the modeling of Turing patterns, such as stripes or spots (on a zebra, leopard or similar) which are also found in the Belousov–Zhabotinsky chemical reactions with oscillation dynamics, and in nature with the appearance of fairy rings (truffle-hunters will understand!), and the propagation of epidemics (to measure the speed of diffusion which is used to identify the peak of the epidemic, something seen every year with the seasonal attacks of Influenza, or even gastroenteritis).

Models of this kind may be used (in competition with the computational models described later) for modeling crowd behavior, useful in designing infrastructures to allow rapid evacuation in crisis management situations (e.g. fire, accident, or terrorist attacks; see [ZOL 08]). They are also used to model the propagation of information or even beliefs within a population: we can then attempt to stop or limit this propagation, for example, as a counter-insurgency measure (see end of this chapter). At equation level, this plays out as the introduction of additional terms with the effect of creating other solutions without diffusion. The interesting part of the exercise is to then find the real-world mechanisms, which correspond to these terms in the model. We see, then, why it is useful to look closely at equations and their resolution, in that this process may provide clues for “inversion of the model”, as it is described in command theory. Modeling thus leaves the sphere of pure description and becomes a genuine tool for the architecture of possibilities and assistance with design choices. A detailed knowledge of the equations and the mathematical nature of their solutions allow an appreciation of their behavior depending on the situation, giving the user all possible means of making the right choice.

5.4.2. Computational approaches

The first, and simplest, computational models that come to mind are finite state automata: these are made up of a finite number of objects, known as states, and transitions between these states which are carried out by reading the associated tag. Certain specific types of states can be identified: initial and final states. The execution of the automaton consists of starting from an initial state and arriving at a final state, and the list of tags obtained during this series of transitions determines the word retained for this particular journey. For a given finite automation, the set of potentially recognizable words makes up a language; if we consider all possible finite automata, we obtain the class of regular languages, widely studied in computing and used in practice, often unknowingly, for example, when using command-line languages (either under Unix or under Linux, or in the “invited command” box in Windows). It is interesting to complexify the initial finite automaton model: if, for example, we add the possibility of adding a symbol to the memory (known as a stack) with each transition, or of removing the last symbol added to the stack, the class of recognized languages is that of computer languages – known as context-free languages in theoretical informatics. If, instead of removing the last symbol of the stack, we add the possibility of taking something from the memory at random in a linear manner (i.e. proportionally to the length of the recognized word up to the transition in question at which point we allow ourselves to look in the memory), the class of recognized languages contains the natural languages – also known as context-sensitive languages in theoretical computer science. If we relax the constraints on access to the memory (giving ourselves the possibility of picking from a finite, but arbitrarily large, memory), we obtain the class of recursively enumerable languages mentioned above. The computability model obtained in this way is, in fact, the most general model possible, and we see that it may be obtained through a number of progressive extensions24.

Models using finite state automata, and some of the generalizations presented above, are often exploited due to their ease of use. Other extensions of finite state automata, such as Petri automata or networks, are also widely used in simulation: in this last case, transitions happen when particular events occur, adding a temporal notion to the interpretation of the process – but from a strictly theoretical point of view, this is a tag like any other – and a token mechanism is added, by which certain transitions are only authorized if the correct token is present to “validate” the transition. In this case, the token is also transmitted to the next state. This mechanism is what gives Petri automata powers of expression higher than those found in finite state automata and means that such models are particularly widespread in the analysis of usage scenarios for complex systems.

By following the same basic idea of state transitions, but using multi-dimensional states, or non-deterministic transitions (several possible source states or spontaneous transitions), we can model a wide variety of situations with (among other things) structured handled data.

This is the case for cellular automata, models used widely for the simulation of urban or agricultural developments, or of the propagation of pollution, fire, or epidemics. The starting point is a very simple idea and can be explained using Conway’s Game of Life: we start with cells which may take the value 0 or 1, which we juxtapose to form a line. The line created by the initial values of all the cells determines the initial state of the cellular automaton. We then apply eight rules that allow us to calculate the value of each cell at the next moment, depending both on its current value and on the value of each immediately neighboring cell. The line made up of these calculated values determines the next instant of the cellular automaton. If we represent the temporal evolution of the cellular automaton graphically, as an image where the top line is the initial state, the second line the following state, and so on, we obtain a black-and-white image showing the trajectory obtained starting from the initial state. By varying the eight rules characterizing a particular cellular automaton, we can obtain very different images using the same starting conditions; we can also observe stable patterns or patterns which are mobile in two dimensions (obviously from top to bottom if we place lines successively in this way). To go from this example to a model for epidemiology, we simply need to interpret the state 1 as the fact that a given person is healthy and 0 as the fact that they are infected. The rule then indicates how the infection is propagated locally depending on the state of a person and of their immediate neighbors. The image produces a “film” or propagation based on a starting population. Clearly, to produce an interesting model, we must simply complicate the algorithm, for example, by considering a neighborhood in two dimensions rather than one, meaning that an image corresponds to an entire population, and the film is a stack of images. The rules may also be modeled to follow certain constraints on the spatial aspect of neighborhoods (an epidemic does not spread in the same manner under different geographic conditions), and they may consider temporal constraints (someone who has already been infected may become immune, or someone not yet infected in a certain environment may have natural immunity, and so on), meaning that the calculation of an image may depend on the calculation of previous images.

The immediate interest of these models is that first they are easy to implement on a computer and that they produce graphical representations, a precious aid in simulation, and second their power of expression; in fact, the entire group of cellular automata, after simple generalization of the preceding mechanisms (increase the size of neighborhood to consider when determining the next value of each cell), supplies yet another universal computing model, which in theory allows the implementation of any computable function.

If we increase the computational capacity of each individual cell by making it compute functions instead of simply considering information coded as a bit (e.g. as in black/white, infected/not infected) and if we do the same thing by using information from neighboring cells, while preserving the key paradigm of locality, we naturally end up with multi-agent models. In theory, this model class is no more powerful than the previous class, but from an implementation point of view, it is considerably more practical in terms of algorithmic level translation of the local behaviors we wish to model. Used to represent the collective movement of shoals of fish or of flocks of birds [SCH 03], multi-agent models can also be used to model crowds, and this method is used, for example, in designing infrastructures where rapid evacuation needs to be possible, as mentioned above. The interest of these models is the ease with which a complex situation can be modeled: we must simply be able to specify how each individual agent evolves from one instant to the next based on interactions with a certain number of neighbors. Following this, step-by-step execution, that is, simulation, shows the descriptive capacities of the model by displaying possible global behaviors.

Let us now consider the last family of models of interest to us: networks, also known as graphs in mathematics. Their definition is simple, and reuses the aspects used previously for automata: a finite group of nodes (known as vertices on a graph) and a finite group of connections between nodes (the edges of the graph). The nodes correspond to the entities considered and the connections to the existence (or non-existence) of interactions between two entities. Models of this type can be found in both natural and artificial systems: food chains, river networks (in this particular case, loops are rare, except in the case of occasional more or less stagnant “dead” water courses), road and rail networks, telephone networks, electric networks, Internet networks, social networks, and so on. The existence of interactions, and more generally of looped chains of interactions, is what makes these structures interesting: in cancerology, graphs are used where the vertices are genes or proteins and the edges are the interactions of regulation between vertices. The resulting general behavior is a combination of all interactions, either direct or induced by cycles in the graph.

Moreover, graphs may be oriented, in the sense that an edge joining two nodes may have a “departure” and an “arrival” node. This notion is useful in showing positive or negative dependences between these vertices, as the orientation has an influence: in the first case, the quantities associated with the vertices vary in the same direction, and in the second case, one value decreases as the other increases. This type of model is widely used in qualitative physics [KUI 86, PUC 85], where, using a schematization of dependences between variables, we may determine the type of behavior shown (convergence to states, cycles, divergence, and so on) by the system thus modeled.

Viewed from the perspective of graph theory, we are interested in dynamics of assembly – preferential attachment to poorly connected nodes, for example – and disassembly – the targeted removal of certain edges – of graphs, and in the influence of certain operations applied to graphs on different measurements of their performance. We may distinguish three broad families of graphs, which have different properties and toward which we may wish to tend: random graphs, scale-free graphs, and small-world graphs. In random graphs, the distribution of degrees of connection (the degree of connection of a node is the number of nodes to which it is connected) follows a random law. Graphs of this kind are, therefore, robust when faced with the random deletion of edges; on the other hand, they are not particularly effective in terms of transmitting information between two given nodes. In scale-free graphs, the distribution of degrees of connection follows a power law . Certain nodes, therefore, have a large number of connections, but the majority will have few connections. This can be observed in air transport networks, with hubs on the one side and regional airports on the other, where several changes are required to travel from one regional destination to another, but direct flights are available between major airports. Social and biological networks are also often of this type. Scale-free graphs are robust when faced with random deletion (the cancellation of a flight from a regional airport does not disturb traffic at hub level), but they are vulnerable to targeted attacks on nodes with a large number of connections (following terrorist threats to Heathrow Airport, all traffic passing through this hub was seriously affected). In terms of system design, this means that particular protection should be accorded to this type of node: this demonstrates the kind of immediate results obtained from topological analysis of a network. Finally, in small-world networks, each pair of nodes may be connected by a relatively short path: in some ways, this group is somewhere between a random graph and a regular graph. Food chains and certain social networks work in this way. These graphs are often characterized by their Rényi number, which is the average length of the shortest routes between any two vertices, something found in “popular” science under the name “degrees of separation”: for a given person, we might try to see if, in his acquaintance, we can find someone who knows a given actor or politician or someone who knows someone who knows that person, and so on. Experiments of this kind have been conducted, and the degree of separation within the human population is around 6, a number that intuitively seems remarkably small [BAR 03]. On the Internet, the degree of separation is closer to 19.

Let us take the example of system of systems introduced at the beginning of this chapter, within which the socio-technical component (i.e. the human, and especially the organizational, dimension) is a key factor for success or failure. Certain views of this type of system can be modeled using networks, and the analysis of these networks becomes an effective means of identifying points for action in terms of maximizing the value chain within complex organizations: how might we improve the circulation of information through different levels of the company hierarchy? How can we optimize competences within the network and identify critical people or functions? How can we obtain scale effects in the proposed products or services? In particular, depending on chains of causality between the final user, their direct suppliers, and the various third parties that enable the supplier to provide that particular product or service, an organization will have different levels of performance, flexibility, and adaptability to, for example, the disappearance of one of these third parties or suppliers. By studying different networks from the point of view of their topology (degree of connectivity, study of transformation of topology when edges are removed, and so on), it is possible to identify both strong and weak points of organization and to suggest targeted means of improvement for certain properties.

[CRO 09] shows how the analysis of social networks within a company, particularly in innovation, production, and sales teams, and of their environment reveals keys to improved performance. This systemic analysis of network topology aims to provide answers to certain questions. Are teams being influenced by the right people? Are they connected in an appropriate manner for the task in hand? Have they established adequate relations with the exosystem? Are value-added collaborations taking place within the network of teams? Does the quality of relationships within teams promote efficient collaboration (in terms of optimization of competences and access to the necessary expertise at the right moment)? Does the organizational context encourage real collaboration (complementarity of expertise and abilities, collaborative technology at company level, flexible processes?) The pitfalls of certain modes of managing staff and abilities are also analyzed in terms of the topology of their subjacent structure. An example of this kind of analysis is given in [MCK 06] where, in the framework of complex contracts, qualitative influences are studied within contractual relationships between suppliers and sub-contractors, between multiple suppliers and even between competing suppliers in acquisition plans.

Note that the topological study of graphs does not provide quantitative information on these modeled systems, contrary to other models presented earlier. It does, however, provide a structural complement which gives additional information concerning certain measures of theoretical complexity.

5.4.3. Qualitative phenomenological approaches

The models presented in the following sections are different from those previously described in that they do not have the same power of prediction, but they present an original approach to the explanation, comparison, and classification of systems which is useful in delimiting, if not mastering, the complexity of these systems.

5.4.3.1. Scaling laws

Scale is a property of both space and time. It relates either to spatiotemporal resolution (a field of ripe wheat is a yellow polygon seen from an airplane or a more or less ordered group of stalks seen from the edge of the field) or to the extent of coverage of an analysis (we might look a few meters in front of us or look at several places and consider all of these points of view). It is obvious that depending on the scale, we will not be interested in the same aspects of distribution of sources of interest, nor in the same level of interactions between elements. Depending on the scale, then, we have a hierarchy of elements and interactions and look for invariants depending on the level considered and for interactions between different scale levels, which may have a negative effect (homeostatic loop leading to local equilibrium, something often looked for when regulating a system) or a positive effect (auto-amplification processes, such as the greenhouse effect in climatology). Isometric and allometric analysis can be used to identify certain scaling laws, based on similar relationships between parameters at all levels considered, or on the contrary on instances where this symmetry is broken (i.e. discontinuities25), as they may be the result of a particular physical phenomenon which could be beneficially exploited.

An example of a scale law of this kind was provided in the 1930s by Max Kleiber, a Swiss chemist studying the relationship between the mass of an animal and its metabolism, who proposed a universal formula: the quantity of energy burnt by unit of weight is proportional to the mass of the animal raised to the power of three quarters. In 1932, George Kingsley Zipf, a linguist at Harvard University, formulated the law that carries his name, another power law: for example, the number of towns with a certain number of inhabitants is expressed approximately as the inverse of the square of the number of inhabitants (this is why there are few very large cities, but a certain number of large towns and a very large quantity of small towns, as this distribution is linear on a logarithmic scale). The same applies to the presence of words of a certain length in the lexicon of a language and so on.

The scaling laws cited above characterize the appearance of a phenomenon based on the scale of observation or appearance; there are other situations where the phenomenon is repeated identically from one scale to another in a way which could be said to be recursive. This is known as a fractal phenomenon, a concept first introduced under that name by Benoît Mandelbrot in 1976. Like the fern or the cauliflower, which present an analog structure when we zoom in on one of their details, we find these phenomena in both artificial and natural systems. The internet, for example, may be modeled as a fractal if we look at connection topology; the same applies to certain development models used in urban areas, but also to frontline evolution models in wartime [MOF 02].

5.4.3.2. Application of catastrophe theory

A well-known result of the work of the Italian geometrists of the 19th century is the knowledge that if we project a surface immersed in 3D space and defined by a polynomial equation onto a plan, we obtain a curve known as the visible surface (this is the principle at work in the shadow of an object projected onto the ground by the sun). This curve possesses singularities which almost always belong to a finite catalog: double points, cusps, and so on. This result, demonstrated by Whitney in the 20th Century and then generalized, led to the emergence of catastrophe theory. René Thom stated in his inaugural conference of the Colloque de Cerisy on “Logos and Catastrophe Theory” in 1982:

we have an environment which is the location of a process, of any nature, and we distinguish two types of points, regular points and catastrophic points. Regular points are points where local phenomenological analysis reveals no particular accidents; variations may be observed, but these variations are continuous. On the other hand, we have points where phenomenological analysis reveals brutal and discontinuous accidents, and particularly observable discontinuities, which I call catastrophic points. (…) Under the hypotheses of genericity for dynamics, catastrophic points are not necessarily a bad thing; their structure is passably regular, for example locally polyhedral ….

Mathematically, classification theorems can be obtained by particular dynamics and low dimension spaces: this is what René Thom called elementary catastrophe theory. The generalization to other dynamics and other dimensions may be interesting from an epistemological viewpoint, but is not yet based on exhaustive classification theorems, calling on the excessively abundant and complex theory of bifurcations of dynamic systems, limiting its functional applications. Moreover, modeling via catastrophe theory is qualitative and does not provide functional tools for quantitative prediction: using catastrophe phenomenology, we in fact reconstitute the dynamic of minimum complexity which may produce the observed set of catastrophes, and this is done without the application of isomorphisms, hence the absence of quantitative thresholds. However, looking more closely, this is what happens in other supposedly qualitative models, for example, the modeling of turbulence or the aerodynamics of rotary or beating wing engines.

That said, the use of catastrophe theory fits into an invariant-seeking approach under so-called structural stability hypotheses – the idea that the global aspect of a system does not change as a result of small disturbances – and has a strong relationship with modeling approaches for complex systems (illustrations of this may be found in sociology, linguistics, plate tectonics, robotics, and so on; see [BER 77, GIL 81, PET 95, PET 98, POS 96, LUZ 00]). This paradigm can be characterized as “emergential” [BAR 96] in that it recommends the use of theoretical and practical results demonstrating the existence of emergent morphological and qualitative structures, by an organizing dynamic process, in physical substrates. This identifies an intermediary morphological level between the physical and symbolic levels. From the perspective of modeling, one consequence is that the same mathematical tools are used to model the organizational process of physical substrates and the process of organization at the highest level. Morpho-dynamic models are therefore used as models of the categorization process in a way described in the following section.

Consider a complex system, with attractors and their basins of attraction, that is, regions where, when a trajectory begins, it is attracted by the attractor. A result of the theory of dynamic systems affirms that almost all trajectories in the system will converge toward one of these basins. Let us discuss each basin as a particular category. Category changes are therefore expressed, from the viewpoint of system dynamics, as a bifurcation phenomenon controlled by physical parameters, intrinsically linked to the structural instability properties of the system.

In addition to categorization of a space at state level then in its globality, this approach also allows local categorization, that is, (more precisely) a classification of the systems which depend on the local behavior of these categories. Starting with an equational model which fits data in a certain domain of validity, we observe what happens around a point of operation which is of particular interest from the point of view of the application. At local level, we look at the local form of the equations which define the associated reference catastrophe. This gives an idea of the behaviors that are most likely to occur locally depending on the evolution of different variables. To give an analogy, this is what is often done around a regular point in sufficiently smooth systems, where each system may be assimilated locally, at the point considered, to its linear approximation. The analogy to a catastrophe is, in this case, the constant function, and we can effectively say that all the systems are locally the same at such a point, meaning that such regular points cannot be used to classify different systems (it is for this precise reason that catastrophe theory was developed and that it applies particularly to critical points).

The morpho-dynamic approach is attractive, but subject to certain limitations. First, the underlying mathematical tools are highly non-constructive, and second, morphology can be formalized in topological and geometric terms, but not, a priori, in logical terms, a fact which raises certain issues when using it in conjunction with other models. An interesting subject for research [BAR 96] would be to try and develop a scientific theory which could link formal structures at symbolic level – which, in one way or another, are logical forms – to the dynamics which govern the physical level.

5.4.4. Qualitative structuralist approach: application of category theory

First, remember that category theory aims, among other things, to provide a common language and a set of unifying concepts for different branches of mathematics: thus, an application between sets and a homomorphism of algebraic structures, or a continuous transformation between topological spaces, are united by the common concept of morphism between the corresponding categories (in this case, the category of sets, that of the algebraic structures in question, that of topological spaces, and so on). Due to these unifying concepts, analog results are, in fact, represented by a single result, allowing deeper understanding of the problems in that the subjacent mechanisms are revealed; these were initially hidden by technical details linked to particular structures for which results had been set out. One of the cornerstones of category theory is the systematic adoption of a relational viewpoint: everything may be defined as an arrow (a relationship or a transformation or, more precisely, a morphism) between objects. These objects themselves may be described by exclusive use of arrows (the preconceived conceptual difference between an object and an arrow is simply that arrows can be combined with other arrows and therefore have a “from” and a “to”, but an object may also be defined as an arrow for which the “from” and “to” coincide).

Here, the contribution to systems theory is clear: the systemic paradigm and category theory meet in the pre-eminence they accord to relational ontology, on both a dynamic and a static level, with the establishment of the evolution of relational structures. This is, moreover, where we find the main mathematical difference between category theory and set theory. Although the latter characterizes mathematical objects by describing their internal structure, that is, by separating them into different parts and elements, the former takes the opposite point of view and characterizes an object by its connections to other objects. It is interesting to look at the development of systemics in connection with the emergence of category theory. Both reached maturity in the mid-20th Century, as a logical progression of currents of structuralism, and underwent a “golden age” in the 1960s and 1970s. Both then took a back seat during a period where totally analytical approaches were favored, accompanying the generalized reductionist movement in the sciences (promoted by the hope of a take-off in computer processing resources), only to return to favor in recent years as a methodological complement providing global (diachronic and synchronic) vision. Category theory is, moreover, used as much in theoretical computing (modeling information systems, databases, multi-processor languages, and so on) today as in theoretical physics, where it is used in attempts to define a unified framework of physics26.

A relational vision of this kind has two consequences. First, an object is only described up to isomorphism, that is, independently of a particular implementation. Second, the strictly representational aspects of an object are removed in favor of the structure of inter-object relationships, a process which reminds us of the key notions of architecture in systems engineering. Moreover, one aim of category theory is to provide a precise definition of “naturality” or so-called universal relationships between properties (remember the basic results, such as the uniqueness of the base of a vector space up to isomorphism, or the use of representatives of equivalence classes to define operations independent of the particular representative, used to define addition on rational numbers and on real numbers as an extension of addition on natural integers). This naturality is used, among other things, to define natural transformations, a demonstration of the intuitive idea that complicated things may be transformed into other complicated things if we modify the corresponding sub-structures correctly and if the transformation is not too ad hoc in the sense that it may be defined by a general mechanism applying to every object being considered. Furthermore, category theory is particularly suited to consider certain representations frequently found in computing, such as automata, tree diagrams and graphs, and object-oriented structures. In this last case, it provides a practical means of dialog between specifications and implementations [FIA 05].

In fact, category theory has another role to play, albeit one of lesser interest to us in our current discussion: it is used in the search for a coherent language which would allow us to establish the foundations of mathematics, effectively the quest for the Holy Grail of meta-mathematics.

We shall now provide a rapid overview of certain basic concepts involved in this theory and provide illustrations of its contribution to the modeling of complex systems. A category is specified by a class of objects and a class of arrows, also known as morphisms. An identity arrow exists for each object, and for each arrow at the end of which a second arrow begins, we can define the way in which these two arrows fit together. Moreover, three arrows which may be put together do this in an associative manner, that is, the first two may be put together with the third or the first may be combined to the last two. A category may therefore be seen as an oriented graph, where the vertices are objects and the edges are the arrows. By applying definitions recursively, we can consider a particular category of which the objects are themselves categories: in this case, the morphisms between these composite objects are known as functors. If we clarify the action of a functor, we notice that a functor between categories may be defined as mapping objects of the first category to objects of the second category, so that a morphism between two objects of this first category is mapped onto a morphism between object images. Thus, a functor is a transformation which conserves, in a certain way, the base structure between categories considered. If the objects correspond to system components, and morphisms to links between components (the interpretation of which depends, for example, on the type of architectural view being considered), a functor is a translation between two different architectural views.

Taking a step back, consider the category of which the objects are functors: in this case, the morphisms are the natural transformations discussed above. Once again, we must define a property between “individual” objects of the categories concerned. The general idea is not to transform a category into another category by observing how the subjacent relational categories interact (this was the role of functors) but to see how this transformation may be parametered at the level of the objects which make up the category. In this way, we model the situation where a global transformation between complex architectures is carried out through local transformations at different levels of zoom.

We shall now introduce one last notion, a particular universal notion, that of limit. Let us take a diagram (i.e. objects and arrows between them): an object is “limit” if there are morphisms from this object toward each object in the diagram and if, in addition, each object satisfying this same property is such that there is a single morphism between it and the limit object. Let us consider how this principle can be applied to the modeling of complex systems: consider the basic elements of a complex system as objects and the relationships between them (energy flow, channels of transmission, proximity relationships, transition functions, and so on) as arrows, producing a diagram within a category on the condition that the minimum requirements in terms of mathematical properties are verified. A limit object, then, in some way subsumes the primitive elements, in that it represents a sort of invariant associated with the diagram, which may then be seen as its internal organization. If we apply this kind of approach recursively, we can define a hierarchy where, at each level, the objects are limits of objects at the lower level. A representation of this kind is coherent with the idea that complexity is a relative notion and depends on the level of observation. Applied to different views of the systems which make up a system of systems, for example, it also provides a means of describing the links and hierarchies between these different visions.

The above description gives only a broad overview of the use of category theory in modeling complex systems. To go further, we could enrich objects in terms of structure, particularly to consider aspects of dynamic evolution; see [LUZ 98a, LUZ 98b]. An approach of this kind seems to be able to provide some answers to the questions raised at the end of the previous section.

The above sections have dealt with the use of a category-based approach to the modeling and simulation of large complex systems (“large” due to the spatial, structural, and/or dynamic extension of the components involved). Although this approach may seem strange at first, this is essentially due to the way in which mathematics is taught. For this approach to be useful in terms of the results obtained, a certain “philosophical” vision is required – involving an understanding of the relational environment of the object studied, and therefore a holistic approach with more significant investment – along with a level of technicity guided by abstractional appetence, or a representational mindset, as the quest for and demonstration of results involves much diagram chasing. This is not, however, too different from the approach used by architects who, far from a specific implementation, must be able to take on board and connect different points of view of a system, or by simulation experts who must be able to quantify these points of view to give life to all useful representations of the system.

5.5. An example: effect-based and counter-insurgency military operations

Effect-based military operations have been “in fashion” in different doctrines over the last decade, although the concept was already in use during the Vietnam War and even during certain phases of WWII. An effect is defined in document JP5-0 (Joint Operation Planning, from the US Department of Defense) as “the physical and/or behavioral state of a system which results in an action, a set of actions, or another effect”. The “traditional” vision involves the conversion of desired outcomes into objectives then into missions which can be broken down into tasks and intentions; the Effect-Based Operations (EBO) vision, on the other hand, starts from the final state desired and translates it into missions with objectives and effects. Leaving aside semantic quarrels and doctrinal debates, it is interesting to note that the document Air Force Doctrine Document 2 (AFDD2 – Operations and Organization, published in 2008 by the US Air Force) highlights the fact that conflicts result from the interaction of “adaptive complex systems” and that war is “complex and non-linear”. Conflict resolution may be represented in a two-dimensional space, with means on one axis (classified from destructive methods to means of influence) and ends on the other (from physical to psychological). We can then classify different conflicts by interactions and couplings, particularly in terms of the relationship between cause and effect. [JOB 07] distinguishes four different classes in this way (simple, complicated, complex, and chaotic) and represents them using the two-dimensional plan mentioned above, from the resolution of a simple conflict by a pairing of means of destruction and physical results, up to a chaotic conflict where resolution is attained through means of influence with psychological results. This grid places an emphasis on the non-linear nature of military operations and attempts to reach conclusions on how they should be conducted, using analogies with the theory of complexity. This is far from the old world of schemas with linear relationships between cause and effect, the concrete form of which was confrontations between armies, with increasing lethality as the relationship between the forces present grew.

This context provides a potential field of application for complex system modeling and simulation techniques. It is particularly relevant for military strategists at the present time as current conflicts appear to have been planned without the necessary consideration of their complex character, particularly the interweaving of different dimensions (physical, psychological, political, and cultural) and the principle of non-linearity (a local action does not necessarily produce only local results, results which are not necessarily proportional to the action, and may work with a certain time delay and in the opposite direction to that expected). We do not claim to offer answers here, but we wish to highlight the importance of using the correct paradigms when reasoning to increase the chance of finding a possible solution.

The domain of counter-insurgency operations (sometimes encountered under the acronym COIN) has also been the subject of renewed interest in recent years in the context of conflicts in the Middle East. It is interesting to tackle this problem from a systemic viewpoint in attempting to provide elements of a response27. We are, in fact, dealing with various groups (factions, networks, and so on) that sometimes cooperate, but may potentially be in competition; these groups have various aims but are all opposed to an authority constituted for the control of a territory and a population. Note the heterogeneity, pairings between elements and the environment, the dynamic nature of pairings and the different dimensions of the problem and of motivations, and the impact of external pairings with the environment on internal structure (counter-insurgency operations attempt to play on these external pairings to destabilize the internal structure of the system as far as possible). Counter-insurgency consists of organizing and leading all military, paramilitary, political, economic, psychological, and civic actions at government level to destroy an insurrection. This is a clear illustration of an attempt to produce systemic effects.

Simple Lancaster-style models of attrition do not allow us to consider the dynamics of these linked systems: the insurgents do not work using a closed system – which would mean they had no external support – and the elimination of insurgents is not enough to stamp out the insurgency completely. Moreover, conflicts of this kind are not a simple matter of two identified opposing forces, but involve at least three groups (insurgents, the authorities, the population) with more than one highly permeable frontier between participant groups (the population is more or less neutral and may join either camp depending on circumstances, and there may be defections from insurgent ranks and from the official troops). For this reason, exchanges between these participants cannot be ignored. The situation therefore begins to resemble an ecosystem, where several species are in competition for a set of resources, considering the phenomena of migration and reproduction [DRA 08, SOL 06]. [DRA 08] considers decision trees as in game theory, with values which may be variable; [SOL 06] considers coupled evolution equations with occasional use of macro-variables for a posteriori analysis of the behavior of certain equations around equilibrium values (see the principle of subjection). In both cases, the results obtained directly depend on the intrinsic limits of the model being used. The combination of these methods may be useful in understanding and then trying to control systems with the aim of obtaining a particular trajectory (usually one which results in the neutralization of the insurrection with the least possible negative impact on the population).

The idea is therefore to make use of various models, to find analogies between modeled systems (in particular concerning certain dynamic regimes, in the hope of obtaining the regime seen as presenting the greatest performance in one of the models at system level) and critical analysis of certain strategies so that if, once the analogy is established, the effect observed is not that shown by one of the models, doubts can be raised as to its use for that system. Evidently, these considerations are all based on the fundamental possibility of access to data to support the models; otherwise, all extrapolation is pointless. The above example also shows the richness of trajectory behaviors (the ecological models in question have the various characteristics of complexity discussed throughout this chapter), and it is therefore sensible to apply all the precautions mentioned above so as to avoid planning useless actions.

5.6. Conclusion

Throughout this chapter, we have attempted to underline certain fundamental characteristics which must be considered in the modeling and simulation of complex systems, whether natural or artificial. These characteristics may appear to be based on theoretical considerations, even though we took great care not to go into excessive mathematical detail, and the reader would be right to wonder whether what we presented is really motivated by practical considerations and relevant to the field of application of the majority of engineering work.

Our response to this question is, of course, yes, and this approach is not unique to us: other recent articles in technical journals use this same focus [SHE 06, SHE 07], although we must acknowledge that there is currently little interest for the subject in the engineering community as a whole. This is mainly due to the fact that theoretical developments require sophisticated mathematical tools, the use of which is reserved to particular system classes and which are not particularly easy to adapt to the kinds of systems handled by engineers. This difficulty of use in conjunction with practical applications is clearly a major obstacle. We should also suggest that in the majority of applications developed up until now, conditions are such that the use of these tools and the management of these limitations could reasonably be left aside (although this is not to say that they would have been useless). In an engineering logic which was principally oriented toward industrial products28, the key words were control, stability, and predictability. To achieve these aims, engineers applied the principle of systematic decompositions to the object being studied, advocating the independence of components as far as possible. For this, fairly long temporal cycles were sufficient, and necessary for an approach based on the sequence of expression of need, design, and production, use of the product and, later, periodic updates. For as long as the conditions for success of this kind of approach were met, there was, a priori, no need – except in particular cases – to worry about the potential consequences of uncontrolled complexity. However, in recent times, conditions have been changing. The field of engineering has turned toward information services29 seeking flexibility, adaptability, and a logic different from that of total control (illustrated by the use of “total quality” principles): interest is now directed toward providing capacities and seeking value (following the successive introduction of principles of “quality assurance” and “maturity” of supply of products or services, we are now in a phase of “value assurance”). The breakdown into independent components is replaced by a linking and combination of existing blocks, tangible or otherwise – that is, off-the-shelf products or services – for economic reasons: first, it would be unthinkable to redo all the preliminary work for each new need, and second, the time taken to launch a product responding to the expressed need must be as short as possible. This tendency is reinforced by evolutions in society, with increased focus on the user and on immediate and personalized satisfaction. The reuse and combination of elements, including (or especially) outside of the domain in which they were initially guaranteed to function, is becoming increasingly common, a situation which often necessitates forays into the science of complex systems. We find ourselves in a phase of paradigmatic change, and use of this new logic of combination of services on demand is experiencing exponential growth in extremely varied domains of application. This is even the case in those domains, such as defense and transport, which seemed the least susceptible to use this kind of logic, as it places local user satisfaction above control of risk and operation. There have already been a number of real-world demonstrations of the effective risks and dangers of a lack of management of complexity, including breakdowns in the energy supply systems in the United States and Italy and the recent financial crisis. Let us hope that problems of this kind will not emerge in the defense sector (the kind of problem often seen in disaster films, where information, communications, and control systems develop emergent capacities for autonomous decision-making and suchlike!) nor in the field of air transport (which has proved surprisingly resilient for the moment, with the capacity to tackle crises without uncontrolled propagation, e.g. the cancellation of all flights to and across the United States on and after September 11, 2001, or the sudden changes in security procedures following threats to London Heathrow in August 2006, which caused enormous queues, were dealt with without causing major repercussions elsewhere).

Faced with these changes, traditional engineering problems, based principally on hypotheses involving closed, linear, decomposable, predictable systems with a clear understanding of initial conditions and disturbances, are transformed: the systems under consideration are mostly open, non-linear, and interdependent, with a finite horizon of predictability and little understanding of initial conditions and disturbances. The engineer is therefore confronted with complexity and cannot ignore fundamental characteristics, as they leave the field of ad hoc examples to join that of concrete applications. What, then, can we learn from these pitfalls and limitations which we have touched upon in the preceding sections?

First, great care is necessary in not reaching conclusions too quickly, either by intuition or by analogy with other situations. We have already seen the pitfalls created by non-linearity.

Second, great importance should be accorded to interactions and thus to the place of man in the global system. From questions of computability of human reason notwithstanding, it is clear that it follows mechanisms which we cannot reliably reproduce: capacities of induction, reliability when faced with errors, but also unpredictable generation of errors (to err is human, whether this be due to fatigue, stress, or a sudden access of incompetence) make human reason a key element of the global architecture, with corresponding strengths and weaknesses. Human behavior, which is fundamentally complex, should be considered as far as possible in the engineering progress. We should remember that the human factor has resulted in the loss of battles which were considered to be won in advance and, conversely, is capable of producing success against all odds.

Finally, we should make flexibility and agility an aim from an early point in the engineering cycle to favor adaptability and resilience, and we should be able to question the system architecture if the planned use requires it. We should focus on the traceability of options and decisions made during the decomposition and integration of existing blocks and components developed for the specific project, in order to be able to go back if necessary. We should aim for a reasonable mastery of unplanned occurrences rather than permanent control, with the ability to identify and evaluate in permanence and to react rapidly.

How, then, can we deal with these challenges? The solution involves model-driven engineering, accompanied by judicious interpretation of these models, following the example of the living world, where this representation and imagination-based approach (construction and use of models to choose the action to carry out a posteriori) have been the key to the survival and evolution of mammals and humans in particular. By mastering the tools of complexity, we avoid a certain number of crises that would inevitably be provoked by the complexity we aim to apply at all costs.

5.7. Bibliography

[BAD 97] BADII R.M., POLITI A., Complexity: Hierarchical Structure and Scaling in Physics, Cambridge Non-linear Science Series 6, Cambridge University Press, Cambridge, 1997.

[BAD 07] BADIOU A., Le concept de modèle (new edition), Fayard, Paris, 2007.

[BAR 96] BARTHELEMY J.-P., DE GLAS M., DESCLES J.-P., PETITOT J., “Logique et dynamique de la cognition”, Intellectica, vol. 23, pp. 219–301, 1996.

[BAR 03] BARABÁSI A.-L., Linked, Penguin Group, New York, 2003.

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1 Chapter written by Dominique LUZEAUX.

1 In all honesty, the very fact of posing the question gives certain elements of a response. However, to exonerate those who may be targeted by our criticisms, note that their approach is a priori defensible – why not aim for simplicity when we can develop a system from its foundations? – but unfortunately vain when faced with the pitiless formal statement.

2 Note that although these different affirmations appear more or less obvious, they are not always integrated into existing models. Theoretical developments do not always follow the basic principles of common sense.

3 Planet Earth is an example of this: the algebraic complexity of its form is such that it is sometimes described as “geoid”, a clear admission of powerlessness when faced with the tyranny of geometric nominalism.

4 At least for those with a clear understanding of relativist hydrodynamics.

5 This number was introduced by Professor Markus Reiner in an article published in Physics Today in 1964. Its name refers to the prophetess Deborah, who, alongside Barak, led the Israelites to victory against the Canaanites (Judges 4 and 5 in the Old Testament). This passage includes the Song of Deborah, with the verse “the mountains melted before Yahweh”, a possible allusion to the fact that, with infinite time for observation, God might see changes in objects which man sees as changeless due to the brevity of his lifespan. Leaving aside this religious interpretation, a scientist might interpret these events as a geological or meteorological event provoking major landslides, carrying away the enemy forces.

6 Note that this phenomenon can now also be observed in railway stations experiencing a high level of traffic: in addition to the ever-present newspaper stands and refreshment areas, we now find chain stores selling clothing, media, and so on.

7 See [LUZ 08a, LUZ 08b] for a discussion of this definition and its links to numerous bibliographical references, along with a certain number of examples, and methods and tools for mastering complexity in these systems of systems.

8 In the etymological sense of the term, that is, direct, without intermediary.

9 Again, in the etymological sense of the term, that is, ab-trahere, pulled from. It is interesting to note that the idea of separation is subjacent to most of the terms usually used in this context.

10 The domain has grown so much in the last decades that it would be futile to try and cover it in its entirety. Moreover, the mathematical theories used are manifold – and themselves complex – as shown by the following partial list of key terms: chaos, turbulence, attractors, fractals, cellular automata, multi-agents, emergence, “small world” graphs, critical phenomena, borderline effects, bifurcations, catastrophes, pattern formation, morphogenesis, diffusion, percolation, phase changes, synchronization, shock waves, solitons, and so on.3

11 It may seem strange to define the concept by negation, especially as, mathematically, almost all systems are non-linear. Furthermore, note that all non-linear finite dimension models of the type x′ = f(x) can be transformed into linear models, but with infinite dimensions using the Perron–Frobenius equation: . We shall not, however, go down this path, as it is largely based on formalism and its word-games.

12 The image was popularized in the 1960s by the meteorologist Edward Lorenz, also famous for the strange attractor notion.

13 Some statistics: in 2007, a little more than 100 billion dollars circulated on the financial markets every day, a little more than 1,500 billion on the exchange markets, and approximately 4,000 billion dollars in terms of derived products. This gives a daily flow of approximately 5,500 billion dollars per day for the global financial economy. The total of effective and promised government assistance in early 2009 was around 3,000 billion dollars – a ratio of less than 0.2%.

14 The divergence of trajectories, whether in sensitivity to starting conditions or the butterfly effect, is often due to a relative error on the state of the demarcated system, mathematically expressed by an equation of the type x′(t)/x(t) = a, where the state derivative x(t) is a time derivative. After integration, we obtain x(t) = x(t0)eat, hence the a priori exceptional behavior.

15 Mathematically, each program may be associated with an integer so that two different integers correspond to two different programs. This is done following Gödel’s numbering technique, for example. We thus obtain an injection of the set of programs into the set of natural integers.

16 For example, turing machines, post systems, recursive functions, Chomsky’s formal grammars, cellular automata, lambda calculus, and combinatory logic.

17 This is what Church’s thesis affirms: that any one of these formalisms is the right one to model the intuitive idea of mechanical calculus. An affirmation of this kind clearly cannot be demonstrated as it identifies an intuitive notion with a formal notion. The affirmation is connected to the epistemological paradigm.

18 If the response to the question was actually negative, the computation is authorized to loop, potentially infinitely.

19 Despite its simplicity, this example is not ad hoc: its genericity and its use in commanding robotic systems evolving in a limited environment with obstacles are illustrated in [LUZ 93, MAR 94, LUZ 95, LUZ 97, LUZ 98c], demonstrating its limitations and characterizations of classes of programs which may be used to control systems of this kind.

20 If the solution is not too pathological – continuous, for example – we need to simply apply the theorem of approximation of continuous functions by simple functions and then make use of the fact that the set of real numbers have the rational numbers as a dense subset, allowing an approximation of simple functions by simple functions using rational numbers, which are, obviously, computable.

21 Except for fans of The Matrix, for whom the real world is simply an algorithmic simulation … but we shall leave this subject in case a certain Mr. Smith decides to show up!

22 The relative relationship between discrete and continuous is studied from the angle of non-standard models in [SAL 99] where, for example, the Harthong–Reeb model represents the set of real numbers in a non-standard model of the set of relative integers, in a way compatible with order, showing how discrete calculus and geometry can be carried out through representations of this kind.

23 This refers to the Löwenheim–Skolem theorem, for example, which states that any finitely axiomatizable theory with an infinite model can be represented by a countable model.

24 We have simply shown the calculus models associated with the particular language classes which make up Chomsky’s hierarchy.

25 One example of discontinuities can be found in groups of sociable insects, including the division of ants into groups of workers and soldiers, where no indications of this division are given initially (i.e. at the egg stage); the future attribution of an ant, while it is still an egg, is determined by the global state of the group at the time.

26 On this last point, see the various publications of I. Raptis, A. Döring, C.J. Isham, J. Butterfield, A.K. Guts, and so on available online.

27 Note that the strategies of this combat method, used by T.E. Lawrence against the Turks, Mao Tse-Tung (China), Vo Nguyen Giap and Ho Chi Minh (Vietnam), the Sandinists (Nicaragua), then during the Intifada and al-Aqsa Intifada (Israel/Palestine), and most recently al-Qaeda, themselves use a systemic approach in their texts inciting insurrection. Further reflection on this can be found in [HAM 06], which provides a posteriori justification of the use of systemics as an analytical tool for fighting insurgency.

28 To re-cite the definitions provided by the EIA632 or the IEEE1220 norm, the product is the result of an activity or a process, usually considering a material form.

29 According to the AFNOR-ITIL/ISO20000 norm, a service is a composable immaterial provision, manifested in a perceptible manner and which, in predefined conditions of use, is a source of value for the consumer and the supplier.