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I know that random networks have a bell curve (Poisson distribution) in their degree distribution and scale free networks have a straight line in their log-log plots. However, how do the log-log plots of random networks look like. How do I differentiate between the log-log plots of real world and random networks

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You can just test how this looks like: random networks have Poisson distribution (I have never heard it being called 'bell curve' though), so it follows Poisson density. R code to see log-log plot of it may be (using mean intensity 10)

curve((function(x) dpois(10, x))(x), 0.1, 100, log="xy")

which gives you

Poisson density on log-log scale

Most notable feature here, compared to the scale-free networks, is the rapid, concave, fall in the density function.

This may be good enough for a quick visual inspection, if you want to categorize your distribution into one of these categories, I'd recommend you to run a formal test.

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