Abstract:
One of the most fascinating discoveries of the last years is that numerous complex systems, although very different among them, as belonging to physical, biological, chemical, informational, social or other environments, exhibit common properties as the scale invariance (i.e. the system properties appear identical over a large range of scales). Moreover, these properties significantly control their dynamics: for instance, it is well known that the scale-invariance of complex networks implies their robustness under random deletion of elements. In such systems, the basic characteristics are not described by the nature of their constitutive elements, but rather by their topological properties (i.e. relations among the system elements). This theory schematizes an interconnected system by a graph, defined as a mathematical set of N nodes (elements of the system) connected by links or edges (relations among the elements). Indeed the node-links schematization allows an effective descriptions for an extremely varied class of phenomena: social networks, as scientific collaboration networks, informatics systems, as the WEB and internet, biological systems, as protein-protein interactions networks and metabolic networks, technological systems, as electronic circuits, and so on. Here, we discuss how such systems self-organize themselves into a steady scale-free structure. In particular, we show that the power-law is the most probable distribution that both nodes and links, in a reciprocal competition, assume when the respective entropy functions reach their maxima under mutual constraint. The proposed approach predicts scaling exponent values in agreement with those most frequently observed in nature.
