Networks and complex systems

For over a century, modelling of physical as well as non-physicalsystems and processes has been performed under an implicitassumption that the interaction patterns among the individuals of theunderlying system or process can be embedded onto a regular andperhaps universal structure such as a Euclidean lattice. Anotherwidespread hypothesis assumes that all the entities composing asystem can freely interact with each other, without any particularrestriction.In late 1950s, two mathematicians, Erdös and Rényi, made a stepforward in the classical mathematical graph theory: they described anetwork with complex topology by a random graph. Their workinitiated a cascade of innovations in network theory, followed byintensive studies in the next 40 years and even today. Althoughintuition clearly indicates that many real-life complex networks areneither completely regular nor completely random, the random graphmodel was the only sensible and rigorous approach that dominatedscientists’ thinking about complex networks for nearly half of acentury. This fact is due essentially to:• the absence of super-computational power• the absence of detailed topological information about verylarge-scale real-world networksIn the past few years, the computerisation of data acquisition andthe availability of high computing power have led to the emergence ofhuge databases on various real networks of complex topology. Thepublic access to the huge amount of real data has in turn stimulatedgreat interest in trying to uncover the generic properties of differentkinds of networks. In this endeavour, two significant recentdiscoveries are the small-world effect and the scale-free feature ofmost complex networks.The discovery of these effects has led to dramatic advances in thefield of complex networks theory. In particular, it has led to theconviction that in order to correctly analyse a real system scientistsìhave to take into consideration not only the feature they areaccustomed to (the dynamics), but also a new aspect: the systemunderlying topology.