One hero gone. Who loses their way?

A null model supplies a baseline. We compare the observed network with networks that preserve selected properties and randomize the rest.

We measured twelve quantities on the full 303-article snapshot and on 1,000 draws from each of two null models.

What changes under the shuffle

Some quantities change little under degree-preserving rewiring.Degree assortativityWhether well-linked nodes tend to link to other well-linked nodes (above 0) or to sparsely linked ones (below 0, called disassortative). is −0.098 in the data versus a null mean of −0.120 (\htmlClass{x-z}{z} = +2.0, empirical \htmlClass{x-p-empirical}{p} = 0.024). The observed network is slightly less disassortative than this baseline. The result is exploratory: these one-sided p-values are unadjusted across twelve quantities.

Mean distance is 2.674 in the largest observed component versus 2.631 under the shuffle (\htmlClass{x-z}{z} = +2.9, empirical \htmlClass{x-p-empirical}{p} = 0.005). The difference is small but not zero. Short paths alone do not establish unusual wiring; the choice of baseline matters.

Part of that gap is the shuffle merging islands into the giantThe largest component: the biggest group of nodes that can all reach each other through links.. Swaps cannot link the 17 isolates, since a degree-zero article stays degree zero, which leaves 286 linked articles (303 minus 17). In all 1,000 draws those 286 land in a single component: the nine-character Morituri island is absorbed every time. In 962 draws the giant is exactly those 286 nodes; in the other 38, an unrelated two-article pair happens to split off elsewhere, leaving a giant of 284. A like-for-like check that holds the population fixed instead, shuffling only the 277-article core used in Screen Test, gives 2.674 against a null mean of 2.601 (\htmlClass{x-z}{z} = +5.2, from 400 shuffles at 10 swaps per link, fewer than the 1,000 draws at 20 swaps per link used elsewhere on this page).

Clustering is 0.307 versus a shuffle mean of 0.143 (sd 0.008); no draw reached the observed value. The network has 1,839 triangles versus 1,158 on average under the shuffle. Links among characters from the same teams or eras are one possible explanation, not a mechanism established by this test.

The same number, two nulls, two conclusions

Clustering is about ten times the \htmlClass{x-gnm}{G(\htmlClass{x-n-nodes}{n}, \htmlClass{x-m-links}{m})} mean (0.032), but about 2.2 times the degree-preserving mean (0.143). Holding degrees fixed reduces the apparent excess without eliminating it. The two nulls answer different questions; \htmlClass{x-gnm}{G(\htmlClass{x-n-nodes}{n}, \htmlClass{x-m-links}{m})} does not control for hubs.

The observed 19 components compare with 18.04 under degree-preserving rewiring (\htmlClass{x-p-empirical}{p} = 0.039) and about 1.02 under \htmlClass{x-gnm}{G(\htmlClass{x-n-nodes}{n}, \htmlClass{x-m-links}{m})} (\htmlClass{x-z}{z} = +131.6). The shuffle retains the 17 isolates, and it absorbs the nine-character Morituri island into the giant component in every one of the 1,000 draws: the largest component averages 285.9 nodes versus the observed 277 (\htmlClass{x-z}{z} = −23.3). The draws that reach 19 components (38 of 1,000) do so because a different, unrelated pair of articles splits off elsewhere, not because the shuffle keeps the Morituri island separate. Treat the component-count p-value as exploratory: matching the real count of 19 does not mean the null reproduces the island structure.

Pick a Marvel character with at least one link. On average they have 10.0 links. Now pick one of their friends at random: that friend has 24.2. Three times out of four (75.3%) the friend is at least as linked as the character you started from. Nobody is lying; picking a friend is degree-biased samplingPicking in a way that favours nodes with more links: following a link lands on a hub far more often than picking a node at random does., because a hub is a friend to many and therefore keeps getting picked.

For a uniformly selected linked article followed by one random neighbour, the observed mean is 24.22 versus a shuffle mean of 24.19 (\htmlClass{x-z}{z} = +0.1, \htmlClass{x-p-empirical}{p} = 0.47). This quantity can change with rewiring. The edge-sampled mean, \htmlClass{x-edge-degree}{\htmlClass{x-k2-avg}{\langle k^2 \rangle} / \htmlClass{x-k-avg}{\langle k \rangle}} = 22.08, is different: it depends only on degrees and is exactly preserved.

The paradox itself follows from the degree sequence alone: scrambling who links to whom leaves the average friend's degree at 24.19, almost exactly the observed 24.22. Its per-character form tells a different story. The share of linked articles whose random friend is at least as linked as they are is 75.3% in the real network against a shuffle mean of 76.3% (\htmlClass{x-z}{z} = −3.5) — below every one of the 1,000 draws. By this measure Marvel is slightly less paradoxical than degree-preserving rewiring predicts.

It does not need hubs to exist, only variance. Under random \htmlClass{x-gnm}{G(\htmlClass{x-n-nodes}{n}, \htmlClass{x-m-links}{m})}, where the biggest article has around 19 links, the paradox weakens but refuses to die: 10.44 for a random friend against 9.47 for a random character. A Poisson distributionThe spread of link counts a fully random network gives: most nodes near the average, few far from it. still has a variance, and a variance is all the paradox asks for.

The two characters nobody out-links

The paradox has an edge case worth chasing: is there anyone whose friends are never better connected than they are? Across all 286 linked articles, exactly two.

Spider-Man is the largest hub, with 106 links. Radian has six links on the nine-character Morituri island, where no neighbour has more. Both are local degree maximaNodes with at least as many links as every one of their neighbours., despite their very different degrees.

The shuffle produces 1.26 local maxima on average versus the observed two (\htmlClass{x-z}{z} = +1.3, \htmlClass{x-p-empirical}{p} = 0.205). \htmlClass{x-gnm}{G(\htmlClass{x-n-nodes}{n}, \htmlClass{x-m-links}{m})} gives 16.7 on average. These comparisons do not establish that the observed pair is exceptional.

This is a readable overview of the 16 best-connected articles (four tie at 27 links for the last two places; we keep the first two alphabetically, Emma Frost and Hank Pym). Select a line to follow its actual links. The full journey planner below includes every article.

Positions are not geography. Line colours are drawing aids, not detected communities. The map intentionally omits smaller stations; no full-network conclusion is inferred from this overview.

The planner respects the active closure after its result is revealed. Restore service to compare. It uses breadth-first search; every hop is a real link in the selected graph.