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A number of original concepts and methods have been developed or are of special importance in CST:

Scaling

Scaling laws can be considered as a first level in modeling; they relate two quantities by rules such as: Q is proportional to L to the alpha where L is a parameter (e.g. a length or a number of agents), Q a dynamic quantity (e.g. a period, some degree of complexity) and some real number.

The concept is pervasive in all sciences but its importance in complex systems is due to two reasons:

• since understanding complex systems is so difficult, knowing how things depend on each other is already an important result;
• scaling laws are generic; genericity is itself an important concept which states that some results are robust and do not depend upon details.

Universality, genericity.

Because the full description of complex systems is very complicated (in other words of large algorithmic complexity, necessitating a ong description), a modeler cannot a priori be assured that the necessary simplification of his model would not change the behaviour predicted by the model. Rather than try to reproduce all the details of a real systems one models classes (of universality) of system which share the same set of properties: the generic properties.

Attractors

Most often, the modeling effort concentrates on the description of the attractors of the dynamics, i.e. those system configuration that are reached at large times.

The properties of the attractors (periods, number, size of the attraction basin etc.) of the dynamics are generic and robust when parameters are changed.

Regimes, transitions

The above assertion (robust when parameters are changed) has to be re-stated more precisely: there are large parameter regions where the same dynamical regime is observed (with slowly varying dynamical properties).

The dynamical regimes are often separated by abrupt phase transition.

The general picture is then: global dynamical properties, such as attractors are stable with respect to perturbations and robust with respect to changes of parameters.

These properties here expressed in terms of dynamical systems reflect what some call organisation or emergence

Networks

The concept of an interaction network is central to complex systems. The study of the static structure of the interaction network was recently renewed by the ideas of small world (Watts and Strogatz) and of scale free networks (Barabasi and Albert).

Many empirical studies on real world nets, whether biological (from food webs to gene regulatory networks) and social (from small communities to the Internet and WWW) were done in recent years;new characterisations of networks have been proposed and in some cases a bridge was re-established between network sociologistsand complex system scientists in the search for small motifs in relation with functional properties for instance.

Dynamics and functionnal organisation studies are far more difficult to achieve; they also require larger data sets.

The predictions of Vespignani etal on virus infection and immunization strategies in scale free networks are a good example of the importance of dynamical studies.

Other concepts have been proposed which pertinence can be less general such as percolation, fractal, chaos, self-organised criticality etc.

These general concepts guide research on any particular complex system: they are part of a theory of complex systems as the concepts of energy, temperature, entropy, state functions ... shape thermodynamics.

In some sense, as stated by Stuart Kauffman ("Waiting for Carnot") complex systems theory can illuminate research about most real complex systems as does thermodynamics in physics, chemistry and even biology.

Methods

A theory of complex systems also include a well developed set of common methods. These few paragraphs are certainly not a manual on how to use these methods, in the same way as the previous section was more a list than a set of formal definitions.

In analogy with quantum mechanics where common methods such as symmetry groups, variational methods, perturbation techniques contribute to solving Schroedinger equation, methods in complex systems mostly deals with attractors.

Optimisation, satisfiability

In a number of cases (not all!) complex systems dynamics are driven by the min(max)isation of some quantity: minimization of energy in physical systems, optimization in many applications, but also according to some vision of economics (utility maximization) or biology (fitness maximization).

The issues in complex systems are closely related to combinatorial optimization considered as a hard problem in computer science. But complex systems have developed a set of techniques often inspired from their original field of research.

To quote only three important methods:

• Monte Carlo methods are inspired from statistical physics. A big success of 1983 was the simulated annealing algorithm proposed by Kirkpatrick etal who demonstrated that a probabilistic search algorithm would give interesting sub-optimal results in acceptable computer time, far shorter that the theoretical exponential time necessary to reach the absolute optimal.
• "Genetic algorithms"(Holland). These probabilistic algorithms are inspired by models of biological evolution. The probabilistic "move" in the search space involve "cross-overs" such that local partial solutions are recomposed.
• "Replica" methods Directly inspired from statistical physics (Parisi) uses some mathematical tricks to formally compute the Z function of the system. First used for the solution of the spin glass system (a model of disordered system), the replica method is systematically applied to system which can be recast in a magnetism Hamiltonian formulation. Such cases have been surprisingly numerous and fruitful: e.g. the computation of the attractors in neural nets a la Hopfield (1985) or the solution of the K-SAT problem. The replica method is not only a trick: it introduces new important notions especially concerning the distance between attractors in the configuration space.
  Specific algorithms are used in conjunction with replicas methods to solve compute solution: the cavity method inspired several version such as the survey propagation algorithm.

The renormalisation group approach was among the first significant success in complex system approach and it considerably helped in establishing the scene. The idea is that in the neighborhood of a transition, the dynamical properties of a system are invariant under a change of scale. This property which reminds of fractality translates into a predictive method in the physics of phase transitions and also in more general problems involving for instance percolation and social sciences models inspired by physical models. Furthermore, from a fundamental epistemological perspective, it introduced the idea of classes of universality: the scaling laws obtained from the renormalisation group computation are valid for a class of models; they do not depend upon details. For physical systems, they only depend upon the dimensions of the physical system (in general 3) and of the order parameter (density in liquid gas transitions, or magnetisation, are examples of order parameters).

Simpler methods such as damage propagation methods (or the equivalent Derrida distance method 1985) are inspired from the idea of Liapounov exponents.They allow to predict dynamical regimes and their transitions.

Of course, as in the study of any non linear system linearisation in the vicinity of equilibria teaches a lot about system stability.A large set of problems and methods relate to inverse dynamics and learning algorithms(JPN).

The present list is far from exhaustive, but it should make clear that numerical simulations are certainly not the exclusive approach to complex systems models.

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