Scale-free networks with an exponent less than two.

We study scale-free simple graphs with an exponent of the degree distribution gamma less than 2. Generically one expects such extremely skewed networks--which occur very frequently in systems of virtually or logically connected units--to have different properties than those of scale free networks with gamma>2: The number of links grows faster than the number of nodes and they naturally possess the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation.