The Architecture of Complex Systems

At the present time, the most commonly accepted definition of a complex system is that of a system containing many interdependent constituents which interact nonlinearly. Therefore, when we want to model a complex system, the first issue has to do with the connectivity properties of its network, the architecture of the wirings between the constituents. In fact, we have recently learned that the network structure can be as important as the nonlinear interactions between elements, and an accurate description of the coupling architecture and a characterization of the structural properties of the network can be of fundamental importance also in understanding the dynamics of the system. In the last few years the research on networks has taken different directions producing rather unexpected and important results. Researchers have: (1) proposed various global variables to describe and characterize the properties of realworld networks and (2) developed different models to simulate the formation and the growth of networks such as the ones found in the real world. The results obtained can be summed up by saying that statistical physics has been able to capture the structure of many diverse systems within a few common frameworks, though these common frameworks are very different from the regular array, or capture the random connectivity, previously used to model the network of a complex system. Here we present a list of some of the global quantities introduced to characterize a network: the characteristic path length L, the clustering coefficient C, the global efficiency E_glob, the local efficiency E_loc, the cost Cost, and the degree distribution P(k). We also review two classes of networks proposed: smallworld and scale-free networks. We conclude with a possible application of the nonextensive thermodynamics formalism to describe scale-free networks. Watts and Strogatz [17] have shown that the connection topology of some biological, social, and technological networks is neither completely regular nor completely random. These networks, that are somehow in between regular and random networks, have been named small worlds in analogy with the smallworld phenomenon empirically observed in social systems more than 30 years ago [11, 12].