Scale-free Networks

The scale-free regime of a network is defined for graphs whose degree distribution is $p(k) \propto k^{-\gamma}$ for $\gamma \in (2, 3)$. Which of the following properties about the resulting networks is true as $\gamma$ approaches 2?

a) They become indistinguishable from an Erdős–Rényi random network, because their second moment, $\langle k^2 \rangle$, decreases.

b) Vertices with extremely large degrees become more common.

c) The expected distance between nodes only increases.

d) The tail of $p(k)$ becomes heavy-tailed, indicating a decrease in the average degree, $\langle k \rangle$.

e) None of the above


Original idea: Daniel Gardin

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