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I'm reading the following paper:

https://academic.oup.com/bioinformatics/article/32/1/122/1743683

and in Figure 3 (Section 4.4) the authors have shown some vertex degree distributions:

enter image description here

The above are vertex degree distributions $p(k)$ for some biological networks.

The authors claim that because the distribution of the measured degree distribution is a Gamma distribution and not a Poisson, thus the network is not random. But what's the reason behind the validity of this claim?

Is this claim from the paper correct: "A sufficient proof for the non-randomness of a graph is the significant deviation of the vertex degree distribution from the corresponding Poisson distribution."

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  • $\begingroup$ The paper in question is not mathematical. Also, posts here should be mainly self-contained. Please edit it accordingly or, if that is impossible, please delete it. $\endgroup$
    – Iosif Pinelis
    Commented Aug 25 at 13:32
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    $\begingroup$ As Iosif Pinelis mentioned, this question might be more suitable for another site, like stats.stackexchange.com . But to answer your question: both Poisson and Gamma distributions are random. The authors claim that the observed distribution of degrees sequences is not Poisson distributed (as one might naively expected from, e.g., an Erdos-Renyi model for a graph) but rather Gamma distributed. They believe this because of the remarkably better fit as shown in the figures you attached. (Their claim is visually obvious, so they don't bother with a statistical test for model selection.) $\endgroup$
    – Bill Bradley
    Commented Aug 25 at 13:58
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    $\begingroup$ Seems to be a cross-post of stats.stackexchange.com/questions/653265/… $\endgroup$
    – kjetil b halvorsen
    Commented Aug 25 at 18:10
  • $\begingroup$ @Statisticsstudent : It's not unusual to hear mathematicians using the word "random" to mean "uniformly distributed," although that's not what it means in standard definitions of the term "random variable." $\endgroup$
    – Michael Hardy
    Commented Aug 25 at 20:09
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    $\begingroup$ @Statisticsstudent : The mode of the Poisson distribution with expectation $\lambda$ is $$ \begin{cases} \lambda-1 & \text{unless $\lambda$ is an integer,} \\ \text{both } \lambda-1 \text{ and } \lambda & \text{otherwise.} \end{cases}$$ The mode of the gamma distribution proportional to $(\lambda x)^{\alpha-1} e^{-\lambda x} (\lambda \, dx)$ for $x>0,$ which has expectation $\alpha/\lambda,$ is $(\alpha-1)/\lambda.$ So for both distributions, the mode is near the expectation. Now notice how different the modes are in the graphs that you see. $\endgroup$
    – Michael Hardy
    Commented Aug 26 at 17:47

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