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Suppose one has the following "economic model":

• people are spread at different (fixed) locations in space.
• At each time step the wealth of each of them is increased by a (different) random growth factor (all extracted from the same common distribution). We will call such a dynamics auto-catalytic.
• each of the individuals spreads a fraction of its wealth among its neighbors.

It is found that in spite of the fact that macroscopically / statistically the growth factor is distributed uniformly among the individuals, the resulting overall distribution of wealth will be characterized by a spatial distribution with very wealthy neighborhoods separated by a very poor background.

We will find repeatedly that the emergence of such an inhomogeneity invalidates the averaging / representative agent / continuum approaches and transforms the Multi-Agent Modeling in a central research tool of such systems. (cf  Anderson' 1997 :

• Real world is controlled 'by the exceptional,
• not the mean; 'by the catastrophe,
• not the steady drip; we need to free ourselves from 'average' thinking.'

It is for this reason that in our examples we stress the spatio-temporal inhomogeneity / localization effects.

Consequently, the Multi-Agent Paradigm is in a two-ways reciprocal relation to the stochastic auto-catalytic systems:

- the Multi-Agent structure is the one which allows the localization to emerge and reciprocally, in turn
- the emergence of localization selects the Multi-Agent method as the appropriate research formalism. 

We will see below that this is the core mechanism in many complex systems spreading over many different fields.

Moreover, the spontaneous emergence of inhomogeneity/ spatio-temporal localization in a priory homogenous noisy conditions is induced generically by auto-catalysis.

A couple of questions and answers about the specific choice of examples:

Why are we concentrating on microscopically auto-catalytic systems?

Because usually, in the absence of auto-catalytic interactions the microscopic systems do not reach macroscopic dynamical relevance in terms of spatio-temporally localized complex / adaptive collective objects.

Why do we study those localized macroscopic objects as examples of Multi-Agent Modeling?

Because the homogenous systems do not need usually the Multi-Agent Simulation approach: they can be treated by averaging over the homogenous masses of similar microscopic objects (mean field, representative agent).

Thus, one of the key concepts underlying the emergence of complex macroscopic features is auto-catalysis. We therefore give at this point a provisory definition of it:

'auto-catalysis' = self-perpetuation, /reproduction, /multiplication'
As opposed to the usual stochastic systems in which the microscopic dynamics changes typically the individual microscopic quantities by additive steps (e.g. a molecule receiving or releasing a quanta of energy), the auto-catalytic microscopic dynamics involve multiplicative changes (e.g. the market worth of a company changes by a factor (index) after each elementary transaction). Such auto-catalytic microscopic rules are widespread in chemistry (under the name of auto-catalysis), biology (reproduction / multiplication, species perpetuation), social sciences (profit, returns, rate of growth).

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