The Letter That Changed Sociology
In 1967, a psychologist in Cambridge, Massachusetts had an idea so simple it sounded like a party trick: pick a stranger across the country and try to reach them using only personal connections. The results changed how we think about the structure of human society.
Stanley Milgram—yes, that Milgram, the one with the obedience experiments—recruited 296 volunteers in Omaha, Nebraska and Wichita, Kansas. Each received a packet with a simple instruction: get this to a specific person in Boston. A stockbroker named Mr. Sharon. You can't mail it directly. You can only pass it to someone you know on a first-name basis, and they do the same. A chain of handshakes from the Great Plains to New England.1
The packets that made it through arrived in an average of about six steps. The press loved it. "Six degrees of separation" entered the lexicon, was turned into a play by John Guare, and eventually became a game involving Kevin Bacon.
But here's the thing Milgram's experiment proved, and the thing it didn't.
Only 64 out of 296 letters actually arrived. That's a completion rate of 22%. The other 232 chains broke—someone got the packet, shrugged, and threw it in a drawer. The average chain length of six came from only the letters that made it.
Think about that. The chains that survived were, almost by definition, the short ones. Longer chains had more opportunities to break. If every link in the chain has, say, a 75% chance of forwarding the letter, then a 6-link chain has a (0.75)6 ≈ 18% chance of completing, while a 12-link chain has only a 3% chance. The long paths never reach the finish line—not because they don't exist, but because they can't survive their own length.2
This is survivorship bias wearing a sociological costume. You're measuring the winners and concluding everyone's a winner.
So was Milgram wrong? Not exactly. The number six was accidentally close to right—but for deeper mathematical reasons that wouldn't be understood for another thirty years.
A World of Shortcuts
Imagine arranging everyone in the world in a circle. Each person holds hands with the people next to them—their neighbors, their coworkers, the barista they see every morning. This is what mathematicians call a ring lattice: a tidy, orderly network where everyone connects to their nearest neighbors and nobody else.
In a ring lattice, the world is huge. If you want to get a message from one side to the other, you have to pass it person by person, all the way around. The average path length grows linearly—O(N). In a world of 8 billion people, that's a lot of handshakes.
Now here's the magic trick. Take this perfectly ordered world and add a few random connections. Not many. Just rewire a small percentage of the existing links so they point to random people elsewhere in the ring. Maybe 1%. Maybe even less.
A regular ring lattice (left) vs. the same network with a few random shortcuts (right, dashed red). The shortcuts collapse distances without destroying local clustering.
What happens is spectacular. The average path length collapses. It drops from being proportional to N (the number of people) down to proportional to log(N). In a world of 8 billion, that's the difference between 8,000,000,000 and about 23. A few random shortcuts turn an impossibly vast network into a cozy neighborhood.3
Meanwhile—and this is the crucial part—the clustering barely changes. Your friends still mostly know each other. Your neighborhood still feels like a neighborhood. The local structure stays intact even as the global structure transforms. You get the best of both worlds: the intimacy of a village and the reach of a planet.
This is the Watts-Strogatz model, published in Nature in 1998, and it's the mathematical backbone of six degrees of separation.3
Small World Network Builder
Start with a ring lattice. Drag the rewiring probability slider and watch how a tiny fraction of random shortcuts collapses path lengths while clustering stays high.
The Bacon Number and the Erdős Number
Mathematicians, naturally, got there before Hollywood did. Paul Erdős published over 1,500 papers with more than 500 co-authors. Your "Erdős number" is your distance from him in the collaboration graph: if you wrote a paper with Erdős, your number is 1. If you wrote with someone who wrote with Erdős, you're a 2. Most active mathematicians have an Erdős number of 4 or 5.
Then came Kevin Bacon. In 1994, three Albright College students—Craig Fass, Brian Turtle, and Mike Ginelli—were watching TV and noticed that Bacon seemed to have been in a movie with everyone. They invented the "Six Degrees of Kevin Bacon" game: pick any actor, and try to connect them to Bacon through shared film credits.4
The remarkable thing is that Bacon isn't even the most connected actor in Hollywood. That honor goes to people like Dennis Hopper and Samuel L. Jackson. But Bacon's average distance to every other actor in the Internet Movie Database is about 2.9. The network is so well-connected that almost everyone is reachable in three hops.
And if you happen to be both a mathematician and an actor—like Danica McKellar, who played Winnie Cooper on The Wonder Years and has a published math theorem—you have an Erdős-Bacon number. Hers is 6 (Erdős number 4, Bacon number 2). Natalie Portman's is 7.5
Four Degrees, Not Six
In 2016, Facebook's data science team did what Milgram could only dream of. They analyzed the entire Facebook social graph—1.59 billion people—and measured the actual average distance between every pair of users. The answer: 3.57 degrees.6
Four handshakes. Not six. The world had gotten smaller, or more precisely, we'd gotten better at revealing how small it always was. Digital networks didn't create the small-world property; they made it visible and measurable.
Facebook found that 92% of user pairs are connected in 5 steps or fewer. The average is 3.57. Mean = ~3.57.
Dunbar's Ceiling and Granovetter's Bridge
If the world is so small, why doesn't it feel small? Why don't you bump into your high school classmate at a café in Tokyo?
Part of the answer is Robin Dunbar's number: 150. That's roughly the maximum number of meaningful social relationships a human brain can maintain. You can have 5,000 Facebook friends, but you can only genuinely track about 150 lives. Dunbar derived this from primate brain sizes—bigger neocortex, bigger social group—and the number has held up remarkably well across hunter-gatherer tribes, Roman military units, and corporate org charts.
But here's the paradox: if everyone only knows 150 people, and those 150 people overlap heavily (your friends know each other), how do chains of connection ever escape your social bubble?
The answer came from sociologist Mark Granovetter in 1973, in one of the most cited papers in social science history: The Strength of Weak Ties.7
The Strength of Weak Ties
Your closest friends inhabit the same world you do. They know the same people, hear the same gossip, see the same job postings. It's your acquaintances—the people you see occasionally, the friend of a friend at a party—who connect you to entirely different social clusters. These weak ties are the bridges in the network. Remove them and the world shatters into isolated islands.
Granovetter studied how people found jobs. The majority found them not through close friends but through acquaintances—people they saw less than once a week. Your best friend already knows everything you know. It's the person you met at a conference two years ago who will tell you about the opportunity you'd never have heard of otherwise.
This is why LinkedIn works. It's not monetizing your friendships. It's monetizing your weak ties. Every "2nd connection" recommendation is Granovetter's 1973 paper dressed up in a blue interface.
Why Small Worlds Are Terrifying
Everything that travels through networks—information, innovation, disease—is governed by small-world topology. And the math cuts both ways.
Consider an epidemic. In a regular lattice, a virus spreads slowly, wave-like, from neighbor to neighbor. It's containable. But add those random shortcuts and suddenly an infected person in one cluster can seed an outbreak on the other side of the network. COVID-19 didn't spread along neat geographic lines; it hopped from Wuhan to Milan to New York through exactly the kind of long-range connections that define small-world networks.
Information cascades work the same way. A tweet doesn't go viral by being passed from neighbor to neighbor in some orderly chain. It goes viral because a handful of weak-tie bridges carry it across cluster boundaries. One retweet from someone in a different social world can expose your message to an entirely new audience, each of whom has their own set of bridges.
Weak ties (dashed red) bridge tightly-knit clusters. Remove the bridge and the clusters become isolated worlds.
Innovation diffusion follows the same pattern. Ideas don't spread within echo chambers—everyone there already thinks the same way. Ideas spread when someone carries a concept from one cluster to another. The physicist who attends a biology conference. The designer who reads economics papers. These cross-pollinating weak ties are the engine of creativity.
Find the Path
Now it's your turn. Below is a small-world network. Two nodes are highlighted: your start and your target. Click through connections to find a path between them. Can you match the shortest path?
Six Degrees Game
Click connected nodes to build a path from red to green. Try to find the shortest route!
The Lesson
The mathematics of small worlds tells us something genuinely surprising about the universe we live in. Order and randomness aren't opposites—they're collaborators. A perfectly ordered network is vast and slow. A perfectly random network has no structure, no community, no trust. But a network that's mostly ordered with a dash of randomness? That's where the magic happens. That's our world.
Milgram's letters, Watts and Strogatz's model, Facebook's global graph, Granovetter's job seekers—they're all telling the same story. The world is small not because everyone knows everyone, but because a few people know someone unexpected. The accountant in Omaha who happens to play chess with a professor in Boston. The friend of a friend who went to school with someone who worked with Kevin Bacon.
Every time you talk to a stranger at a party, you're potentially creating one of those shortcuts. You're rewiring the human lattice, just a little bit. And as the math shows, a little bit is all it takes.
"We are all much closer to each other than we think. The trick is noticing."