# All scale free networks are sparse

Scale free networks have a degree distribution which follows a power law, $P(k) \sim k^{-\gamma}$ This letter explains why $\leq < \gamma \leq 2$ cannot occur (for large networks).

Note that $\gamma <0$ implies a very dense network (and hence not really “scale free” in the traditional sense), and $\gamma >2$ implies a sparse network.

The core of their argument comes from looking at the scaling of the largest and the lowest degree nodes. For $\gamma \leq 2$, the number of nodes with the largest and second largest degree is $\O(N)$, while for $\gamma >2$, the number of such nodes grows sublinearly with N. Also, the number of nodes with degree 1 (or order 1) grows linearly with N. Thus, if $\gamma \leq 2$, one needs $\O(N)$ degree one nodes to associate with the highest degree node, leaving no way to place edges for the second highest degree node.

Note these results are asymptotic. One could have a finite network with $0 \leq \gamma \leq 2$. To grow such networks, however, requires either a cutoff on the allowed node degree, or that $\gamma$ must increase over 2.

http://prl.aps.org/abstract/PRL/v107/i17/e178701

Charo I. Del Genio, Thilo Gross, and Kevin E. Bassler
Phys. Rev. Lett. 107, 178701 (2011)
All Scale-Free Networks Are Sparse