Who matters in the underworld, and what breaks when they're gone

The same Week 3 playbook — five centrality measures, directed vs. undirected rankings, a general shortest-path explorer, a node-removal fragmentation test, and the tightest cliques — run over Wikipedia's Marvel supervillains network instead of the heroes: 370 nodes where 43 villains nobody links to sit outside a 323-node giant component, and Purple Man — not the biggest hub — turns out to be the single worst point to disappear.

"Important" is not one thing. A villain can be important because a lot of other villains link to them, because they sit close to everyone else, because they're the bridge that holds two otherwise-unconnected parts of the underworld together, or because the important villains 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 370-villain, 1,136-edge frozen crawl from data/Marvel_Comics_supervillains/ that the Week 2 supervillains post used — no new data, no team or decade labels. The giant component is a stretchier world than the heroes': the category articles mostly share one Rogue's Gallery each (Venom's page links to a lot of Spider-Man antagonists; Apocalypse's to X-Men antagonists), so hubs sit on both sides of the hero/villain divide — Hulk, Deadpool and Doctor Doom loom large here too, alongside genuine villains like Norman Osborn and Magneto. Just paths, four centrality measures plus PageRank, degree-preserving-shuffle null models, node removal, and maximal cliques, computed offline in analysis/build_week3_villains.py and loaded frozen from data/week3/week3_villains_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 component323 of 370
• Avg. shortest path3.552
• Diameter10
• Largest clique6 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 (323 villains, treating a link either direction as a connection) and the full directed network (370 villains, 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. Hulk's article is linked from 58 other villains' 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. Norman Osborn overtakes Hulk for the top spot, and three specialists surge: Ultron (9 → 5), Super-Adaptoid (13 → 6) and Thunderbolt Ross (43 → 10):

That second list is PageRank's actual mechanism showing through: Hulk links out to only villains, so each one inherits an unusually large slice of his vote — which is how villains like Thunderbolt Ross (PageRank #5 against an in-degree rank of #16), the Beyonder and Dracula outrank villains with a higher raw in-degree.

Pick any two villains from the 323-villain giant component (the other 47 sit outside it — see Week 2) 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 villain can rack up connections without their removal actually breaking anything. To test it directly: remove villains 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 villain 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 six villains in all four top-10s (Hulk, Norman Osborn, Venom, Deadpool, Black Cat, Apocalypse) — but disagree below that, about who sits at the top: Norman Osborn owns closeness, harmonic, directed degree and directed betweenness, while Hulk only owns raw degree and undirected betweenness.
• Against 200 degree-preserving shuffles, Deadpool's betweenness (z ≈ 4.3), Purple Man's (z ≈ 4.2) and Hulk's (z ≈ 3.9) all sit far above what their degree alone predicts — genuine brokers, not degree artifacts; no shuffled network in 200 draws produced a higher betweenness for any of them. Norman Osborn's (z ≈ 2.6) is more above-the-null than extreme. The closeness nulls point the other way: Hulk's and Norman Osborn's real closeness sits below the shuffle mean (z ≈ −7.1 and −5.8, extreme 200/200) — their link counts come from a tight local cluster, so shuffled networks spread their degree around and get closer to everyone faster.
• Removing Purple Man alone — matching neither the top-degree (Hulk) nor the top-betweenness (also Hulk) character — fragments the giant component more than removing anyone else: 323 villains drop to 317, splitting off 2 extra villains (3 pieces total, a 1.5% shrink), and it's the network's most extreme broker under the null (z ≈ 4.2) despite being nowhere near the degree top. The effect itself is modest — the giant has 45 articulation points among its 323 villains, and taking the top 5 by degree or betweenness only shaves it to 310 — but who it is doesn't match either centrality ranking, the one finding that most cleanly separates "structurally load-bearing" from "merely famous."
• The largest cliques are all 6-member, five of them, all drawn from the same Spider-Man rogue-gallery core (Black Cat, Eddie Brock, Mysterio, Norman Osborn, Tinkerer, Venom present in every one) — a real, nameable shared neighborhood, visible from the villain names alone without needing any team-affiliation data this dataset doesn't have.
• This underworld is a looser network than its heroes counterpart: diameter 10 against the hero network's 6, average shortest path 3.55 against 2.67, and a 6-clique against the heroes' 8. Fewer shared-neighborhood connections per villain means longer chains.

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 hands the crown to Norman Osborn, because Wikipedia's own linking behavior is itself asymmetric (see Hulk's 11 outgoing links against 58 incoming). And the network's biggest structural weakness is its own villain — Purple Man — invisible in the degree and PageRank rankings until you a) ask about shortest paths in edge direction or b) actually delete him.

Limitations:

• This is still a citation network built from one encyclopedia's editorial choices, not the comics' actual social structure — an untouched villain 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 "Spider-Man rogue gallery" label is a read of the villain names, not a tested homophily claim.
• 43 of the 370 villains are complete isolates — no link in or out, 12% of the network — all structurally invisible to every measure above; they only exist in the node list.