Classical Random Graphs with Unbounded Expected Degrees are Locally Scale-Free

A common property of many, though not all, massive real-world networks, including the World-Wide Web, the Internet, and social networks, is that the connectivity of the various nodes follows a scale-free distribution, P(k) proportional to k(-alpha), with typical scaling exponent 2 <= alpha <= 3. In this letter, we prove that the Erdos-Renyi random graph with unbounded expected degrees has a scale-free behaviour with scaling exponent 1/2 in a neighbourhood of expected degree < k >. This interesting phenomenon shows a discrepancy from the Erdos-Renyi random graph with bounded expected degree, which has a bell shaped connectivity distribution, peaking at < k >, and decaying exponentially for large k.