On Necessary Conditions for Scale-Freedom in Complex Networks with Applications to Computer Communication Systems

Many complex networks exhibit a scale-free, power-law distribution of vertex degrees. This common feature is a consequence of two generic mechanisms relating to the formation of real networks: (i) Networks tend to expand over time through the addition of new vertices, and (ii) New vertices attach preferentially to those that are already well connected. We show that for many natural or man-made complex networks possessing a scale-free power-law distribution with the exponent gamma > 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order log N. Our results expose another necessary characteristic of such networks. Furthermore, our method has the benefit of relying only on conditions that are static and easily verified for arbitrary networks. We use the preceding results to derive a closed-form formula approximating the distance distribution in scale-free networks. Such distributions find extensive applications in computer communication networks and software architecture.