Loading the frozen snapshot…
Week 1 · Networks
Why the same heroes keep turning up.
Can you collect a complete set of Marvel article cards? Each pack gives you five cards, but some are much easier to draw than others. Open a few packs, then compare what happens when every card gets the same chance.
We turned 303 Wikipedia articles about Marvel superheroes into cards. A link is a reference from one of these pages to another, not a sign of friendship or fame. The cards use those reference counts; they do not rank the characters’ strength.
Open a pack
Five cards per pack · you can get repeats
Our rule: the more articles mention a character, the more often its card appears. Every card still has a chance. This is a game rule, not a measure of fame.
Open a pack to see your next five cards.
What we found
58 articles receive no incoming links.
Within our set of 303 Wikipedia articles, Spider-Man receives references from 106 pages, while 58—including Baymax—receive none. We give each card one chance per incoming link, plus one extra so every card can appear. Spider-Man therefore gets 107 chances for every one Baymax gets. That is why collecting the last few cards takes so long.
Being mentioned and mentioning others are different things. Spider-Man’s page points to nine other articles. Betsy Braddock’s points to 28, but only seven point back to her. The direction of a link matters.
What if every card had the same chance?
Keep the same 303 cards and five draws per pack. Change only the odds.
At 20 packs, equal odds give about 15 more different cards on average. Repeats still happen under both rules.
Calculated averages across repeated collections, not a prediction for your next pack. This comparison starts from an empty collection and does not change your saved cards.
How are these averages calculated?
For a card with draw probability p, the chance of seeing it at least once in d independent draws is \htmlClass{x-seen}{1 - \htmlClass{x-miss}{(1 - \htmlClass{x-p-draw}{p})^{\htmlClass{x-draws}{d}}}}. Add that chance over all 303 cards. Equal odds use \htmlClass{x-p-draw}{p} = 1/\htmlClass{x-n-items}{303}; our rule uses \htmlClass{x-p-draw}{p} = (\htmlClass{x-in-links}{\text{incoming links}} + \htmlClass{x-plus-one}{1})/\htmlClass{x-weight-total}{2{,}087}.
Spider-Man and Baymax: the exact odds
These are probabilities per draw, not a promise about the next pack.
What we asked
Which articles get linked to, which ones do the linking, and who is left out entirely? And how much longer does a collection take when a card's odds follow its incoming links instead of being equal?
What we did
We loaded all 303 roster entries before adding a single edge, so the 17 isolates survive. Then we plotted in-degree and out-degree on linear and A chart where both axes are logarithmic: each step along an axis multiplies the value by the same factor, so small and huge counts fit on one chart., drew the whole roster to find what sits outside the The largest component: the biggest group of nodes that can all reach each other through links., and built a five-card pack rule where each draw picks an article with weight (\htmlClass{x-in-links}{\text{incoming links}} + \htmlClass{x-plus-one}{1}) / \htmlClass{x-weight-total}{2{,}087}. Finally we asked Wikipedia's API to re-derive three articles' out-links and checked them against the snapshot.
What surprised us
The articles that link out most are not the ones that get linked to most: Betsy Braddock links out to 28 articles but only 7 link back to her, while Spider-Man alone is linked by 106. A second surprise was structural: a nine-character island from a single series, Strikeforce: Morituri (Marvel, 1986–1989), that cites itself 22 times and the rest of the roster never.
Under our rule the A node with far more links than most of the others. are the easy cards by construction: a card's odds grow with its incoming links, so those same 106 links make Spider-Man common while 58 articles, including all 17 isolates, share the lowest drop rate. Finishing the set takes about 1,945 packs on average, about five times the 382 packs it would take if every card had equal odds: the rare cards set the pace, honestly reflecting a rule we chose, not a measure of fame.
THE TAKEAWAY
The last few cards take the longest.
The collection time is a consequence of our chosen odds—not evidence that Wikipedia links measure fame. Check the estimate and its limits.
Curious? Go deeper.
Charts, data and methods for anyone who wants to check the details.
Explore the explanations & evidence
See the link counts on linear and log scales
Who gets mentioned most—and how did we check?
THE POST · WEEK 1 · NETWORKS
Who gets linked, and who does the linking?
In-degree against out-degree
Why those are different people
Incoming links count references from other articles; outgoing links count references in the article itself. Familiarity and article length could help explain the different rankings, but this snapshot does not measure editors’ decisions or establish those causes.
Spider-Man receives 106 links and links out to 9; Betsy Braddock links out to 28 and receives 7. Five articles tie for tenth place in outgoing links; counting all of them, Deadpool, Doctor Strange, She-Hulk and Wolverine appear in both top-ten lists. Across all 303 articles, in- and out-degree correlate (Spearman 0.70, A correlation computed on the values themselves: 1 when the points lie on a rising straight line, 0 when there is no straight-line relation. 0.50, both \htmlClass{x-p-value}{p} < 0.001), despite the different leaders.
Does the snapshot still match Wikipedia? (we checked)
The snapshot was frozen on 26 August 2026, and the edges were harvested from the raw wiki-source, where an internal link appears as [[Page name]]. So we re-derived a corner of it ourselves: fetch the wiki-source through the API, extract the [[…]] targets, keep the ones that resolve to a roster article, and compare with the frozen edge list.
| Article | Live | Snapshot | Agree | Only live | Only snapshot |
|---|---|---|---|---|---|
| Betsy Braddock | 28 | 28 | 28 | 0 | 0 |
| Radian (Morituri) | 6 | 6 | 6 | 0 | 0 |
| Baymax | 0 | 0 | 0 | 0 | 0 |
Exact agreement on all three, including the two interesting edge cases: the roster's biggest out-linker, and an isolate whose zero is a real zero rather than a loading mistake. Three articles out of 303 is a spot check, not a validation of the whole snapshot — but it does confirm we understand how the edges were built. One practical note for anyone repeating this: Wikipedia answers 403 unless the request carries a User-Agent header naming your client.
What does the whole network look like?
The whole roster, drawn
303 ARTICLES · 19 PIECES · UNDIRECTEDIgnore link direction and the roster is not one network but nineteen. A giant component of 277 articles holds 91.4% of the roster. Then there is one nine-article island, and 17 articles with no link in either direction.
The islands outside the giant component
The nine-article island is a single series, Strikeforce: Morituri (Marvel, 1986–1989). Its characters cite each other 22 times and never once cite anyone else in the roster — and nobody else cites them. That is the shape a self-contained corner of a canon makes in a link network: not a weak connection to the centre, but no connection at all.
The 17 isolates are a different phenomenon. Baymax, Super Rabbit, Miracleman, Yo-Yo Rodriguez and thirteen more sit on the roster because a category page says they are Marvel superheroes, yet no article in the category links to them and they link to nobody. They are the reason we added every node before any edge; load the edge list alone and 17 of the 303 characters silently disappear.
The island, drawn on its own
Components are computed on the undirected graph, so “island” means unreachable in either direction. Following arrows only, the roster splits further: 66 Groups in which every node can reach every other by following links only in the direction they point., the largest holding 229 articles.
How uneven are the links? Explore the charts
How the draw changes what you see
Most articles receive very few links. One gets 106. The rare cards are the ones tucked away in the tail of our collection, not the celebrities.
Both series use percentages, but they answer different questions: how common is a degree among articles, versus how often did your weighted draws find it? Log view uses \htmlClass{x-degree}{\text{degree}} + \htmlClass{x-log-shift}{1} so the 58 zero-incoming articles remain visible. Empty bins are omitted.
Read the distribution as a table
Sketch your own distribution
Drag across the sketch to set a rough shape, or enter counts for seven degree bins. Your sketch is an unscored thought experiment.
Why does collecting everyone take so long?
The expected wait is about 9,721 individual draws. Each minimum-rate card has a \htmlClass{x-weight-min}{1} / \htmlClass{x-weight-total}{2{,}087} chance per draw. Isolates matter, but they are not the whole bottleneck: 41 other articles share their drop rate.
Requiring the 17 isolates adds about 694 draws to the expected wait compared with collecting only non-isolates under the same probabilities. That is about 7.1% of the full expected wait.
Try estimating the number of packs
Browse all 303 cards
Your card index
Card art is typographic. Every name, degree and source link comes from the roster.
Explore: can Baymax get home?
Bonus: can Baymax get home? ↗What can this dataset tell us?
What the data means
303 article entries and 1,784 directed links from the course’s frozen 26 August 2026 snapshot. A link means one article links to another within this roster. It does not measure friendship, hero strength, popularity or the whole Marvel universe.
Download the snapshot and metrics (JSON)Limits of the experiment
The drop rule is designed for learning. It does not model purchases or a real trading-card product. A long-tailed plot alone does not establish a power law. Collection progress is saved in this browser and may be lost when storage is cleared.
Technical methods & code
The directed roster gives in-degree and out-degree. Five independent draws per pack allow duplicates. The unequal The coupon collector's problem: how many random draws, repeats allowed, it takes on average to collect every item of a set. expectation is calculated by integrating the probability that at least one card is still missing, checked against 20,000 independent A way to simulate collecting: each item gets a random waiting time until it first appears, shorter for likelier items, and the set is complete when the last item has appeared. simulations. Expected whole packs lie between 1,944.19 and 1,944.99; approximately 1,945 is a rounded presentation, not an exact value. See analysis/week01_packs.py and the downloadable week01_packs.json.
Before building on the snapshot we recomputed the figures the course briefs quote for it, and every one comes out the same at the precision the briefs print: 303 characters, 1,784 links, 1,434 linked pairs (350 of them both ways), an average degree of 9.5, 17 isolates, one island of 9, Spider-Man's 106 incoming links, Betsy Braddock's 28 outgoing, and in the largest component an average distance of 2.67 and a diameter of 6. See analysis/course_reference.py.
Analysis and source code on GitHub. All experiments run locally in your browser; the frozen data is never edited.
AI use and authorship
Built with AI assistance for interface design, programming, explanation drafts and test construction. Metrics come from the provided snapshot and reproducible analysis; the games and metaphors are authored presentation choices. Tests check the browser's collection model against exact enumeration and the page's numbers against the analysis output.