This course provides an introduction to complex networks and systems with the aim of understanding their key features and behavior. Basic mathematics and tools, with emphasis on the geometric intuition, will be used to explain the possibly complex and sometimes unexpected behavior of dynamical systems occurring in many real life examples, from electronics to biology to social science and even the telecommunication networks and the www. The complexity of a system is discussed in terms of its structure and its dynamical behavior (the so-called "chaos" lies in the latter). The synchronization among interacting systems and the physical mechanisms responsible for this phenomenon will be briefly discussed as well. Any idea is illustrated by some application to science or engineering and simple tools or Matlab routines.

The following paper is suggested:
http://www.nature.com/nature/journal/v410/n6825/pdf/410268a0.pdf
(Steven H.Strogatz, "Exploring complex networks", Nature, Vol. 410, Mar. 8, 2001)

Ability to predict and model the behavior of complex systems, i.e., the interconnected network of nonlinear dynamical systems.

Basic knowledge of mathematics and physics

* Introduction to complex networks: the origin of complexity, examples.

* Structural complexity: basic concepts of graph theory and network models (regular structures, Erdös-Rényi random graphs, small-world and scale-free features); Characteristics of real networks (e.g., www, routes of airplanes,...).

* Discussion of practical applications (e.g., design of a sensor-placement scheme capable of detecting all possible contamination events for a water distribution system, error and attack tolerance of complex networks,...)

* Nonlinear dynamics

. Discrete-time systems: one dimensional maps (logistic, tent). Fixed points, periodic points, stability,bifurcation diagrams, sensitive dependence on initial conditions, Lyapunov exponents and chaos.

. Continuous-time systems: generalities (state equations, existence and uniqueness, phase plane and phase portraits). Equilibrium points and stability, limit cycles. Examples: Van der Pol oscillator, Naďve Internet, model of an epidemic.

* Coupled dynamical systems: properties and synchronization mechanism (the discussion is based on simple examples).

Revisions / Exam: Written exam