The hub that isn't a person

Running the Week 2 test suite on the 57 pages of the List of Middle-earth characters — and the network's biggest in-degree winner is a category page. The null models have opinions about that.

Not a category — a list page

Fourth dataset, same four tests. Its source isn't a Wikipedia category: we seeded List_of_Middle-earth_characters, the wiki's own index page, and took the 57 distinct pages reachable from it after resolving redirects (crawled 2026-09-10). That seed choice has a consequence the rest of the page keeps meeting: the "node set" contains a few metadata pages alongside actual character articles. With 790 directed edges between them (an edge A→B when B's article appears on A's), this is our densest small dataset: mean in- and out-degree 13.9, no isolates, a single weakly connected component of all 57 nodes.

The in-degree ranking makes the problem plain: Elves in Middle-earth — itself a list-page, not a person — leads with 41 citations; Sauron 39; Aragorn 38; Gandalf 34; Elrond 29. Minimum degree 0 means someone is cited by nobody, and the in-degree maximum is nearly 3× the mean. So the Week 2 question splits in two: is the real structure distinctive, and how much of the headline hub is even a character? Same order as before: chance, degree-preserving chance, growth, and the individual level.

Could chance alone produce this?

Erdős–Rényi with the same 57 nodes and 790 edges. Across 300 offline draws the biggest hub's in-degree ranged 18–27 (mean 21.4, sd 1.6). The real max is 41 — roughly 12 standard deviations above the random mean and 52% beyond the most extreme of 300 draws. Sauron's 39 and Aragorn's 38 also clear the ER ceiling; even Gandalf's 34 does.

One visible complication: the "Elves in Middle-earth" hub is partly an artifact (every character page listing its elves links to it), so some share of the distinction belongs to Wikipedia's structure rather than Middle-earth's story. But the same caution applies in reverse: four of the top five are genuine characters, and all four beat chance by well over 7 standard deviations. Draw your own random network below — its biggest hub will never come close.

What survives once degree is held fixed?

Degree-preserving directed shuffle throughout: chains a→b, b→c, c→d rewired to a→c, c→b, b→d with every node keeping its exact in- and out-degree. On the undirected projection (an edge if either article links to the other; 550 undirected edges here):

• Clustering coefficient — average local clustering on that projection, all 57 nodes, degree-<2 contributing 0. Real value: 0.5984.
• Reciprocity — mutual directed edges ÷ total directed edges ≈ 480 ÷ 790 = 0.6076.

Offline, 300 shuffle realizations (3,000 swaps each). This is the first dataset where both statistics are distinctive. Clustering: null mean 0.5623 (sd 0.0103) vs. real 0.5984 — z ≈ 3.5, beyond every shuffle (p ≈ 0). Holding everyone's link counts fixed and re-dealing them lowers the triangle rate: some of Middle-earth's clique structure isn't explained by degree at all, and a likely suspect is the artifact page again — "Elves in Middle-earth" links to a huge part of the network, while a different shuffle gives its 57-in-degree to scattered mid-degree nodes who form fewer triangles. The honest statement: the clustering signal is real, but some of it is one unusual node's link count being helped by a hub whose out-links co-reference each other.

Reciprocity is bigger: null mean 0.3394 (sd 0.0165) vs. real 0.6076 — z ≈ 16.2, beyond all 300 shuffles (p ≈ 0). Sixty percent of Middle-earth links run both ways against a degree-only 34% — the second-largest reciprocity gap per sd we've seen behind only Westeros's.

Run the same test yourself: pick a statistic, run some shuffles, and watch the null build up.

Growing a network that favors the popular

Directed preferential attachment, Price model, unchanged from the Silmarillion page: each new node joins with m outgoing links chosen with probability proportional to the target's current in-degree. Mean out-degree is 13.9, so we ran trials at m = 13, 14 and 15 (50 each).

Price(m=13) trials reached max in-degree 35–43 (mean 38.2), Price(m=14) 35–41 (mean 37.6), Price(m=15) 33–40 (mean 37.3) — the real top of 41 fits inside that spread without tuning at m = 13–14. This is the hub closest to a comfortable Price-model fit we've measured on the smaller datasets: the growth model neither overshoots (Marvel, Street Fighter) nor scrapes under (Silmarillion, DS9). But remember the artifact: a hub of 41 in a 57-node network would have to be generated by a model whose world is this list page's link pattern — a genuinely pleasing consistency for the Price hypothesis, and a genuine worry for anyone reading the number as a character-level claim.

Move the slider and regenerate to see for yourself.

A Middle-earth born all at once

Preferential attachment depends on history: characters arrive one at a time, and each newcomer is more likely to connect to whoever's already popular, so an early lead can snowball into a hub like "Elves in Middle-earth." But what if this corner of Middle-earth was born all at once?

If all 57 pages existed simultaneously, there'd be no established popularity for a newcomer to follow — everyone would start with the same shot at new connections. That's the counterfactual below: a second model, the same 57 pages and roughly the same edge budget (m = 14 wires about 798 edges against the real 790), that links nodes uniformly at random instead of preferring the well-connected. It's not a claim about how the Middle-earth article network actually came to exist — it's a way of asking how much of the list page's 41-link hub could come from history alone.

Run each model a few times at the same m. Big Bang spreads connections fairly evenly — across 200 offline runs its biggest hub ranged 19–26 (mean 21.8), right where the ER draws land and nowhere near the real 41. Sequential growth at the same m lands higher and tighter (35–41, mean 37.6), bracketing the list page's real hub. The gap between the two models is the point: the same link budget, spent without preference, produces a hub barely half the real one — so whatever built "Elves in Middle-earth," it behaved like a node that other pages kept preferring, not like a lucky draw.

Which is the uncomfortable part: the mechanism that reproduces the artifact hub best is precisely rich-get-richer growth acting on Wikipedia's structure (every elf's article links the list page), not on any character's fame. The model doesn't know the hub is metadata — it just sees popularity and feeds it. That's the node-level caution from the growth section, rendered visible.

Are your friends more popular than you?

Exact definitions, same as everywhere else: undirected projection (a link either way counts one connection). A character's degree is their number of distinct neighbors; their average neighbor degree is the mean degree of those neighbors. Network average ⟨k⟩ = 19.30 against the edge-weighted mean neighbor degree ⟨k²⟩⁄⟨k⟩ = 23.24 — 20% higher. Marvel's same gap was 133%; Silmarillion's 15%. Middle-earth sits in the mild band, consistent with a construction built for reachability rather than star power.

Individually: 44 of 57 (77.2%) have neighbors on average better connected than they are; 13 (22.8%) hold their own; zero ties. Two characters have degree ≥ every neighbor's: Aragorn — and Elves in Middle-earth, the artifact page whose own degree (41) is hard to beat because most of the network links to it. This is the only dataset where a list page is a paradox champion, which is the cleanest illustration of why it should read as data caution: who the "popular friend" is depends on what counts as a "character."

Search for a character below to see their own numbers.

The same closing protocol the Marvel page used, run on the Middle-earth crawl: the in-degree CCDF P(K ≥ k) against two nulls — an ER random graph, whose in-degree is exactly Poisson with λ = 790/57 ≈ 13.86, and a 200-seed ensemble of directed Price graphs (N = 57, m = 14 — edge-matched), with the ensemble maximum marked. A discrete power-law fit (Clauset–Shalizi–Newman MLE, via the powerlaw package) then summarizes the tail: exponent α̂ with its standard error, and a self-selected cutoff k̂_min.

• Mean in-degree ⟨k⟩13.86
• Max in-degree41 · Elves in Middle-earth
• Fitted exponent α̂2.58 ± 0.25
• Fitted tailk ≥ 9 · 40 of 56

Not demonstrably non-random at the top. At λ = 13.86 on 57 pages, the Poisson null stays broad deep into the tail: in-degree ≥ 41 is expected about 0.02 times per draw — rare, but a far cry from Marvel's 10⁻⁸⁹ scale — and 13 pages clear in-degree 20 where 4.0 were expected. Like ASOIAF, this is a dense dataset, and density keeps the Poisson null alive into the region where the real hubs live.

Right on the Price ensemble — at an uncomfortably exact ceiling. The 200-seed ensemble at m = 14 produced max in-degrees of 34–41 (mean 37.6): the real hub's 41 is the ensemble's own maximum, matched to the digit. After the artifact caution above, the coincidence reads two ways: the growth story fits the hub perfectly, and it fits a list page's Wikipedia wiring just as well as any character's fame.

Neither, strictly. The fit rests on 40 tail points (k ≥ 9), the standard error is ± 0.25, and the same density that keeps the Poisson null alive caps how far any tail can extend — nobody can be cited by more than 56 of 57. "Scale-free-looking tail" is as far as this data goes.

What we can't conclude (for this question):

• The mechanism. The Price ensemble matches the hub exactly, but the hub is a list page — the fit may be absorbing Wikipedia's category structure, not a fame mechanism, and the frozen crawl has no timestamps to test Price's growth story at all.
• The exact exponent. α̂ comes from 40 tail points; everything below k = 9 is outside the fit. Treat it as a descriptive summary, never a model verdict.
• A like-for-like Price test. The comparison is a shape comparison against in-degree, not a fitted-model likelihood verdict.

What the models get right, and what they miss

Pulling the four sections together:

• Chance is rejected, hard. 300 ER draws maxed at 27 against a real hub of 41 — a gap of ~12 standard deviations, and four separate characters clear the ceiling. On hubs, the list-page dataset is closer to Marvel's verdict than to the Silmarillion's.
• For the first time, clustering itself survives the shuffle. Real 0.5984 against a null of 0.5623 (z ≈ 3.5) — small, but beyond all 300 realizations. Reasonable suspect: the artifact page's out-neighbors all co-reference each other. Reciprocity (real 0.6076 vs null 0.3394, z ≈ 16.2) repeats the pattern we keep finding: mutual citation is the hardest thing for a degree-only null to fake.
• Preferential attachment fits the hub without tuning. Price trials at m=13 and m=14 bracket the real hub of 41 — for once the growth story is consistent with both the density and the extremes. It still can't speak at the node level: "Elves in Middle-earth" isn't a person, and no named character here is ever a hub the way Spider-Man is.
• The friendship paradox holds, mildly. ⟨k²⟩⁄⟨k⟩ (23.24) exceeds ⟨k⟩ (19.30) by 20%; 77.2% of pages live it. Aragorn and the list page are its only never-out-ranked nodes. The mild gap keeps the theme from the Silmarillion: dense small worlds are self-similar near the top — structural inequality is all in the artifact pages and true superstars, not in the rank-chasing of the individual level.

So: Middle-earth's Week 2 verdict is "distinctive hub, distinctive reciprocity, and — uniquely in our collection so far — a clustering coefficient that no degree sequence can fully account for." The data story hidden in the numbers is about metadata: the dataset's biggest hub isn't a character, its most central resolver is a list page, and a fair reading of this network's null models has to remember that both happened by seed choice.