First off, I'm not a statistician. However, I have been doing statistical network analysis for my PhD.
As part of the network analysis, I plotted a Complementary Cumulative Distribution Function (CCDF) of network degrees. What I found was that, unlike conventional network distributions (e.g. WWW), the distribution is best fitted by a lognormal distribution. I did try to fit it against a power law and using Clauset et al's Matlab scripts, I found that the tail of the curve follows a power law with a cut-off.
Dotted line represents power law fit. Purple line represents log-normal fit. Green line represents exponential fit.
What I'm struggling to understand is what this all mean? I've read this paper by Newman which slightly touches on this topic: http://arxiv.org/abs/cond-mat/0412004
Below is my wild guess:
If the degree distribution follows a power law distribution, I understand that it means there is linear preferential attachment in the distribution of links and network degree (rich gets richer effect or Yules process).
Am I right in saying that with the lognormal distribution I'm witnessing, there is sublinear preferential attachment at the beginning of the curve and becomes more linear towards the tail where it can be fitted by a power law?
Also, since a log-normal distribution occurs when the logarithm of the random variable (say X) is normally distributed, does this mean that in a log-normal distribution, there are more small values of X and less large values of X than a random variable that follows a power law distribution would have?
More importantly, with regards to network degree distribution, does a log-normal preferential attachment still suggest a scale-free network? My instinct tells me that since the tail of the curve can be fitted by a power law, the network can still be concluded as exhibiting scale-free characteristics.