This topic is crucial for those who come from physics and engineering, and it is useful for everybody.

This notion refers to any system property which cannot be directly inferred from the knowledge of its constituents.

Properties at different levels interact – it this is not simply a bottom-up story.

This argument has been widely addressed in the literature, so we will only list the concepts which seem most interesting

A complex system cannot be properly described at a single level, like for example in the case of systems where microscopic interactions give rise to macroscopic patterns.

While classical cybernetics emphasized homeostasis and negative feedback, this is a key ingredient in every interesting phenomenon.

This notion (which is intimately tied to that of frozen accidents) refers to the fact that the future of a system may depend upon accidental events which occur in its evolution.

An effective representation of many complex systems (actually, of many systems) is in network terms: the entities are represented by nodes, and the interaction by links between pairs of nodes.

This is a really intriguing property, and perhaps one of the major reasons why one should be interested in the study of complex systems.

In classical hierarchies, one element at level L “lies below” a single element at level L-1.

Adaptation, a pretty well studied concept, refers to the process whereby a system (whether natural or artificial) undergoes modifications in order to cope with its environment.

Natural, social and artificial systems had better be robust with respect to malfunctioning of some elements (this is a major concern for designing organizations as well!).

In many cases the studies of evolving populations are limited to the case of a fixed external environment, or of a slowly changing environment whose time evolution is not affected by the changes in the population itself.

This is a powerful concept, which is based on the fact that, in many cases, one sees that the same system can be found in different dynamical regimes, depending upon the value of one or some parameters.

This is not a Complex Systems Science concept, but it may be a very useful approach to introduce it.