Is Spider-Man an accident?

Testing the Marvel network against three kinds of "nothing special going on here," then growing a network that tries to build its own Marvel.

Week 1 found a heavily skewed network: 303 characters, mean in-degree 5.89, and one hub — Spider-Man, cited by 106 others, almost double the runner-up (Hulk, 64). Is a hub that size just what falls out of scattering 1,784 links across 303 nodes — or does it take something more deliberate, an actual rich-get-richer mechanism?

No new data this week: everything below runs on the same 303 nodes and 1,784 directed edges from data/week1, computed live in your browser, across four questions. Could pure chance produce this? What survives once every character's link count is frozen and only who connects to whom is randomized? Can a growth model that favors the well-connected reproduce it — and where does it break? And what does this shape mean for an individual hero's own connections?

Could Spider-Man's 106 in-links just be chance? The simplest null model says every possible directed pair of characters is equally, independently likely to link: an Erdős–Rényi (ER) random graph with the same 303 nodes and 1,784 edges as the real network. If scattering that many edges at random can produce a hub that size, an ER draw should occasionally come close.

It doesn't. Across 300 offline ER draws, the biggest hub's in-degree ranged from 11 to 18 (mean 13.9, standard deviation 1.2) — Spider-Man's 106 is about 6× larger than the largest draw, and roughly 74 standard deviations above the mean. Draw your own random network below and see where its biggest hub lands.

ER throws away how many links each character has, too — a strong null, maybe too strong. The tighter question: keep everyone's exact in- and out-degree fixed, scramble only who links to whom. Does the network still look distinctive?

The shuffle keeps every character's degree the same but changes who connects to whom: repeatedly take a chain a→b, b→c, c→d and rewire it to a→c, c→b, b→d. Two statistics, defined precisely:

• Clustering coefficient — the average local clustering coefficient on the undirected projection of the network (an edge exists between two characters if either one's article links to the other's), averaged over all 303 nodes; degree-<2 nodes contribute 0. Same projection as the friendship-paradox section below. Real value: 0.3075.
• Reciprocity — mutual directed edges ÷ total directed edges = 700 ÷ 1,784 = 0.3924.

Offline, across 300 shuffles (3,000 swaps each): clustering's null mean was 0.180 (sd 0.009) — the real value sits about 14 standard deviations above it, beyond all 300 shuffles (empirical p ≈ 0). Reciprocity's null mean was 0.069 (sd 0.007) — the real value sits about 44 standard deviations above that, again beyond every shuffle. Degree alone doesn't explain this network's triangle-heavy structure or its mutual-citation rate.

The chart below starts pre-populated with shuffles, so the null distribution is visible immediately — pick a statistic, or run more yourself.

Preferential attachment: grow to 303 nodes one at a time, and give each newcomer m outgoing links to existing characters — chosen with probability proportional to how many in-links they already have. It's the classic rich-get-richer mechanism, in its directed citation-network form (the Price model), which fits our directed network better than the usual undirected Barabási–Albert setup. Out-degree is exactly m for every non-seed node by construction (the growth ingredient); in-degree accumulates by preference (the rich-get-richer ingredient).

This is a generative hypothesis tested against the data — not a claim about how the Wikipedia article network actually got written.

At m=6 (6×303 ≈ 1,818 edges, close to the real 1,784), 50 offline trials all produced a bigger hub than Spider-Man: max in-degree 116–166 (mean 141) against the real 106 — preferential attachment doesn't just reproduce a big hub here, it overshoots. Lowering m brings the hub down (m=3: 70–137, mean 99, much closer; m=2: 44–125, mean 79) but pulls the edge count further from 1,784 — no single m matches both density and hub size. And at every m, every trial: one connected component, zero isolates — versus the real network's 17 isolates and a 9-character island. A model where everyone who joins immediately attaches can't produce someone nobody ever mentions.

Move the slider and regenerate to see for yourself.

A Marvel universe 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 Spider-Man. But what if the Marvel universe was born all at once?

If all 303 characters 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, same 303 characters and roughly the same edge budget, that links nodes uniformly at random instead of preferring the well-connected. It's not a claim about how Marvel's Wikipedia network actually came to exist — it's a way of asking how much of the hub structure could come from history alone.

Run each model a few times at the same m. Big Bang spreads connections fairly evenly — most nodes end up a similar, modest size, and the largest hub barely varies run to run. Sequential growth, at that same m, reliably produces one or two stand-out characters, because whoever gets ahead early keeps attracting a disproportionate share of new links. That's the mechanism in miniature: Spider-Man's 106 real in-links aren't just abstract "popularity" — in a growing network, being popular early is itself an advantage that compounds. In a universe born all at once, there's no "early" for that advantage to attach to.

None of this shows preferential attachment is what actually produced the real Wikipedia network — the CCDF comparison above already found PA, at the edge-matched m, reliably overshoots Spider-Man's real in-degree. What it does show is why network history can matter: a heavy-tailed degree distribution is consistent with a growth-plus-preference story, but consistent isn't proof — this counterfactual just isolates what history specifically contributes.

Definitions: on the undirected projection (same one used for clustering above), a character's degree is their number of distinct neighbors, and their average neighbor degree is the mean degree of those neighbors. Network-wide, ⟨k⟩ = 9.465, while the edge-weighted mean neighbor degree ⟨k²⟩⁄⟨k⟩ = 22.08 — more than double. That gap is the friendship paradox. It's worth stating plainly that ⟨k²⟩⁄⟨k⟩ is a pure function of the degree sequence, so it's mathematically guaranteed to survive a degree-preserving shuffle — not something this page needs to test.

What isn't guaranteed is what it looks like character by character. Of the 286 connected characters, 250 (87.4%) have neighbors who are on average more connected than they are; the other 36 (12.6%) out-rank their average neighbor. Only two are never out-popularitied by any neighbor at all: Spider-Man, and, inside the isolated Morituri island, Radian.

Search for a hero below to see their own numbers.

We plotted the in-degree CCDF P(K ≥ k) on the frozen Week 1 snapshot (303 characters, 1,784 directed links) against two nulls: an ER random graph, whose in-degree is exactly Poisson with λ = 1,784/303 ≈ 5.89, and a 200-seed ensemble of Barabási–Albert graphs (N = 303, m = 3 — mean degree 5.94, matching ours), shown with a 5–95% band. 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⟩5.89
• Max in-degree106 · Spider-Man
• Fitted exponent α̂2.22 ± 0.12
• Fitted tailk ≥ 5 · 112 of 245

Not random. Under ER, a character with in-degree 20 or more should essentially never exist — expected count 0.0012, real count 21. A degree of 106 is 1 × 10⁻⁹¹ off the Poisson scale: the gray curve exits the chart by k ≈ 18, while the real network still has 22 characters beyond it.

Heavier than BA. From k ≈ 7 on, the real CCDF runs above the BA ensemble's 95% band and stays there; the fitted exponent (α̂ = 2.22) sits well below BA's theoretical 3.0 (smaller exponent = heavier tail), and Spider-Man's 106 clears BA's ensemble max of 77. Forced to choose between random and BA: BA, easily — it just understates Marvel's skew.

Neither, strictly. The power law describes the tail, not the whole network: only 112 of the 245 cited characters sit in the fitted region, and a network this small carries wide uncertainty on α̂. "Scale-free-looking tail" is as far as this data goes.

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

• The mechanism. A heavy tail doesn't prove preferential attachment — many processes produce one, and the frozen snapshot has no timestamps to test BA's growth story at all.
• The exact exponent. α̂ comes from just 112 tail points; everything below k = 5, including the 58 never-cited characters, is outside the fit.
• A like-for-like BA test. BA is undirected while our measure is directed in-degree: a shape comparison, not a fitted model verdict.

Pulling the four sections together:

• Independent random edges (ER) are rejected outright. 300 draws topped out at max in-degree 18 — nowhere near the real 106.
• Degree alone doesn't explain the clustering or reciprocity either. Both sit far outside the shuffle's null distribution (z ≈ 14 and z ≈ 44) — real relational structure (families, teams, mutual citations) beyond what link count alone predicts. The one thing that stayed fixed under the shuffle — the giant-component/island split — is invariant by construction, not a discovery about the network.
• Preferential attachment gets the mechanism right, the calibration wrong. It does produce extreme hubs — every trial, at every tested m, beat Spider-Man's real in-degree. But no single m matched both the real edge count and hub size, and the model structurally can't produce isolated characters or a disconnected island: everyone who joins, connects.
• The friendship paradox holds, strongly, for two different reasons. Network-wide, ⟨k²⟩⁄⟨k⟩ (22.08) more than doubles ⟨k⟩ (9.465) — guaranteed by the degree sequence alone. Individually, 87.4% of connected characters personally experience it — but who plays "the popular friend" for whom depends on the network's actual wiring, not just link counts.

So: no, Spider-Man isn't an accident of scale or density. Something like rich-get-richer growth fits the real hub structure far better than any random baseline — but that's a hypothesis this data is consistent with, not something these four tests prove happened, and even its best-fitting version can't explain the 17 characters nobody bothered to link to.