Who matters, how far apart we are, and what breaks when they're gone

Four ways to ask "who's important," a general shortest-path explorer, a node-removal fragmentation test, and the tightest cliques in the network — every number checked against a degree-preserving-shuffle null before it gets to call itself a finding.

"Important" is not one thing. A character can be important because a lot of other characters link to them, because they sit close to everyone else, because they're the bridge that holds two otherwise-unconnected parts of the story together, or because the important characters happen to be the ones linking to them. Those are four different, precisely defined questions — and on this network they don't always agree on an answer.

Everything below runs on the same 303-character, 1,784-edge frozen snapshot from data/week1/ that Week 1 and Week 2 used. No new data, no team or decade labels — just paths, four centrality measures plus PageRank, degree-preserving-shuffle null models, node removal, and maximal cliques, computed offline in analysis/build_week3.py and loaded frozen from data/week3/week3_summary.json. The three big interactives below recompute nothing expensive live — they look numbers up and draw them; the one live computation left in the browser is the shortest-path search in the Six Degrees explorer, which is cheap enough to run per click.

• Giant component277 of 303
• Avg. shortest path2.674
• Diameter6
• Largest clique8 members

Five measures, five different definitions of "matters." Switch between them below — the ranking changes every time, because each one is answering a different question. Every measure can be computed on two graphs: the undirected giant component (277 characters, treating a link either direction as a connection) and the full directed network (303 characters, following Wikipedia's links exactly as written). Use the Measure and Scope switches together to compare.

Wikipedia links have a direction, and treating the network as undirected throws that away. Spider-Man's article is linked from 106 other characters' pages but itself links out to only — a huge asymmetry that changes the answer once "importance" has to follow arrows.

Recomputing betweenness with arrows respected — a shortest path only counts if it can actually be walked in the link direction — reshuffles the ranking. Wolverine and Hercules both move ahead of Spider-Man:

That second list is PageRank's actual mechanism showing through: Spider-Man links out to only characters, so each one inherits an unusually large slice of his vote — which is how characters like Black Cat and Mayday Parker outrank characters with a higher raw in-degree.

Pick any two characters from the 277-character giant component (the other 26 sit outside it — see Week 1) and find the shortest chain between them. Switch between treating links as mutual or following them exactly as written; the two modes can disagree, including about whether a path exists at all.

High centrality and structural importance aren't automatically the same thing — a character can rack up connections without their removal actually breaking anything. To test it directly: remove characters one at a time, in order of degree, in order of betweenness, or at random, and watch the giant component shrink.

A clique is a group where everyone is directly connected to everyone else — for k members, that's k(k−1)/2 links, all of them present. A maximal clique is one that can't be grown any further: no other character is connected to every single member. NetworkX's find_cliques() only ever returns maximal cliques, so picking "4-cliques" below means groups that max out at exactly 4 — never a 4-member slice carved out of a bigger group (a slice like that could always be grown back, so it would never come back as its own maximal clique).

Observed:

• Degree, closeness, harmonic and betweenness agree on a core of seven characters in all four top-10s (Spider-Man, Hulk, Wolverine, Doctor Strange, Deadpool, She-Hulk, Phoenix) — but disagree below that, and diverge sharply once direction is respected: Wolverine and Hercules both out-rank Spider-Man on directed betweenness.
• Against 200 degree-preserving shuffles, Spider-Man's betweenness (z ≈ 3.7) and Hercules's (z ≈ 3.5) both sit meaningfully above what their degree alone predicts — genuine brokers, not degree artifacts. Rockman's near-identical betweenness, by contrast, lands almost exactly on the shuffle mean (z ≈ 0): it's fully explained by sitting between one well-connected neighbor and one degree-1 pendant, a structural fact about that shape that survives literally any degree-preserving rewiring.
• Removing Spider-Man alone — the top character by both degree and betweenness — fragments the giant component more than removing anyone else, confirming the two rankings agree on the single biggest structural point of failure here. But the effect is modest: 277 characters drop to 271, a 2% shrink, with only 13 articulation points in the whole giant component. This network does not hinge on one hub the way its centrality numbers alone might suggest.
• The largest cliques are all 8-member, all drawn from the same X-Men-associated core (Betsy Braddock, Cyclops, Emma Frost, Jean Grey, Rachel Summers, Wolverine common to every one) — a real, nameable team, visible from the character names alone without needing any team-affiliation data this dataset doesn't have.

Interpretation:

Different centrality measures are not competing estimates of one true "importance" — they are different, equally valid questions, and a network with a heavy-tailed degree distribution will always show large disagreements between them precisely where that distribution is most extreme (its handful of hubs). Directedness is not a footnote here: it changes who counts as a broker, because Wikipedia's own linking behavior is itself asymmetric (see Spider-Man's 9 outgoing links against 106 incoming).

Limitations:

• This is still a citation network built from one encyclopedia's editorial choices, not the comics' actual social structure — an untouched hero with a short article will always look structurally unimportant here regardless of their role in the fiction.
• Null-model z-scores test against a degree-preserving shuffle specifically — they say a value is unusual given the degree sequence, not unusual in any absolute sense, and a heavy-tailed degree sequence already constrains how assortative or disassortative the network could possibly be.
• No team or decade attribute data was added for this analysis — the clique section's "X-Men" label is a read of the character names, not a tested homophily claim.