Scale-free networks are rare

Question

[The Last Word]

I recently read the paper titled "Scale free networks are rare" (reference). This gives me heart because my very own network clusters had a degree exponent value ranging from 0.8 to 1.2 when ideal scale free networks are said to have a slope value between 2 and 3. I have two questions.

1. In the paper, the authors state that many biological networks are not scale free, He has also classified the networks from weakest to strongest based on their scale free nature. However, he does not make this distinction based on slope values. I fit my network with a power law line using cytoscape and it does exhibit a heavy tail. One of the criteria that the author uses to classify scale free networks is also based on the number of nodes in the tail but again, I can't find any information on how to classify a node as belonging to the tail?

2. The author further states that " if p ≥ 0.1, then we deem the degree sequence to be plausibly scale free, while if p < 0.1, we reject the scale-free hypothesis.

The equation being, p(k) = C*(k^−α), let us take the example, C = 39.489, −α = -0.965 , R^2 value is 0.598 and the correlation value is 0.494. How do I calculate the value of p?

Kindly advice.

[Jean-Karim Heriche]

Note that the generality of the scale-free notion and power law distribution of degrees have long been called into question, see for example The powerful law of the power law and other myths in network biology and Power-law distributions in empirical data. I'll have to read this paper but it looks just like the nail in the coffin of this idea (i.e. that scale-free networks are everywhere).
There's also indication that scale free-ness can be seen in random networks when looking at incomplete data, see Effect of sampling on topology predictions of protein-protein interaction networks and Sampling Biases in IP Topology Measurements. Conversely, subnets of scale-free networks are not scale-free.
I am just wondering what kind of biological knowledge you hope to gain by this kind of analysis.

[The Last Word]

I was incidentally trying to prove that my network has scale free property and is not just a random collection of nodes.

[Ram]

Hi, I've changed the equation to be compliant with plain text arithmetics. Please let me know if I got it right.

I changed:

p(k) = C k−α (raised to)

to

p(k) = C*(k^−α)

[The Last Word]

yes you got it right.