Scale free networks (power law)

Question

I'm working with a dataset, of which I'm analysing the degree distribution. I'm finding that it obeys the famous power law/scale free degree distribution $\propto k^{-\gamma}$, but the value of $\gamma$ is quite unusual, only about 1. Now, from all the online resources I find on networks, it is said that most networks have $2 < \gamma < 3$. I'm trying to figure out what the implication is of having a gamma that is so much lower than that, and so far I have not been successful. Does anyone know (a reference) about the implications of various power law exponents?

Answer 1

Are you sure that it is supposed to be a scale free distribution? Social networks are commonly scale free networks and are hybrids of totally random networks and preferential attachment networks. There are a few consequences of having such a degree distribution.

One is that if some sort of percolation process is done on the newtork, the giant component will be very hard to destroy. This is because the percolation threshold has a factor of $\langle k^2 \rangle - \langle k \rangle $ (the average squared degree minus the average degree) and this will not converge for that degree distribution. You might be able to find something here: http://www.barabasilab.com/