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) ∝ k^−α, with typical scaling exponent 2 ≤ α ≤ 3. In this letter, we prove that the Erdős–Rényi 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 Erdős–Rényi random graph with bounded expected degree, which has a bell shaped connectivity distribution, peaking at 〈k〉, and decaying exponentially for large k.