Week 2 · Real degree distribution & benchmark graphs
Degree distribution vs. random models
The real week-1 network is a 303-node Marvel superhero graph with 1,434 undirected edges, a maximum degree of 106 (Spider-Man), 17 isolates, and a giant component covering 91.4% of all nodes. Against that benchmark, we compared the empirical CCDF with two null models: a matched G(n,m) random graph and a Barabási–Albert model tuned to m = 5 new edges per node (the closest integer match to the real edge count), each pooled over 200 realizations.
What the CCDF shows
On a log–log CCDF, the real degree distribution sits much closer to the tuned BA model than to the random-graph baseline. The BA curve reproduces the long tail and the hub scale much more naturally, which is exactly what one would expect from a preferential-attachment process. The main structural miss is also informative: BA can never produce isolated nodes, while the Marvel graph has 17 true singletons. That is a real, nontrivial gap between the synthetic model and the empirical network.
CCDFs for the real network, a matched random graph, and a tuned BA model. The fitted power-law line is optional and helps compare the tail shape against a strict scale-free hypothesis.
What we conclude
The shape is clearly much closer to BA than to random. However, a formal power-law comparison still rejects a pure power-law description in favor of a log-normal alternative. So the most defensible summary is: the network is heavy-tailed and BA-shaped, but it does not satisfy a strict power-law fit. That is a stronger, more honest conclusion than claiming a confirmed exponent where the statistical test does not support it.
powerlaw package gives
α ≈ 2.10 (xmin = 5). The rigorous test is
powerlaw.distribution_compare: it prefers the log-normal model over a pure power law
with R = −17.86, p = 0.0005. Power law vs. exponential is a closer call
(R = −15.07, p = 0.07) — not significant, so we can't rule out exponential either. This is the correct
statistical story, even though the network still looks much more BA-like than random in a visual sense.