Can I call this distribution a power-law distribution? I have some data about the number of nodes in thousands of networks, and I plotted them to understand what distribution it may follow. As the figure shows, I thought it might be a power-law distribution, so I used Gillespie's poweRlaw tool to estimate the $\alpha$, and the p-value is 0.109 in this case.

However, I am confused that my figure is quite different from typical power-law distribution, especially the "fat" tail that indicates some extreme values.
So does it follow a power-law distribution? If not, how can I describe this kind of distribution?
I am not good at math and this is the first time I ask a question here, I will be very appreciated for the help of this community.
 A: Hello and welcome to MSE.
As often, saying that a degree sequence fits a specific law distribution is open to debate. This will depends on the consequence you want to draw. In your case, given that the $p$-value is not great, and more importantly the absence of high values, in my own opinion I would say that it does not. Have you tried other distributions? I would test non-scale free laws, mainly the log-normal distribution, or maybe exponential.
Regarding Power-Law degree sequence, I highly recommend reading :


*

*The article Scale-free networks are rare, available on arxiv. 

*This webpage summarising the current dispute over power law sequences



The fact is for years, network scientist have claimed that "real-world networks are typically obeys a power-law degree sequence". In this article, Broido and Clauset tested the omnipresence of scale-free structures by applying statistical tools to a large number of network data sets drawn from social, biological, technological, and informational sources. Their finding is that across domains, scale-free networks are in fact rare.

Maybe you could try to follow their approach to your specific data set?
