The problem
A handful of giant hubs, everyone else a leaf — that shape isn’t what networks naturally become. It’s what they’re designed to become. Extro is a different design.
Scale-free is a design, not a destiny
The networks we live on are games built by people, and the rules of the game decide its shape. Today those rules almost always produce a scale-free network: a power-law where a few nodes hold nearly all the connections. The mechanism is preferential attachment — new links pile onto whoever is already popular — and the platform’s design is what makes it linear, even superlinear.
Why? Because on those networks you don’t control how you’re reached, and you don’t control your algorithm. You can’t set a price on your attention, can’t refuse a feed tuned for someone else’s revenue, can’t choose who gets through. Strip individuals of that control and reach concentrates — by design, not by nature. Change the rules and you change the shape.
Free reach
scale-freeDesigned so popularity compounds and no one can refuse reach, links pile onto whoever already has the most. A power-law forms: a few giant hubs, everyone else a leaf.
Reach that costs
single-scalePut a real, rising cost on reach and attachment turns sublinear. No power-law tail — a homogeneous mesh where degree clusters around a mean and no node runs away.
Three ways a network can grow
How hard a popular node pulls the next connection is the attachment rule, Π(k) ∝ kα — the pull as a function of a node’s current degree k. The exponent α decides everything:
Make reach cost more as a node grows and the pull bends sublinear — popularity stops compounding.
- Superlinear (α > 1): the pull accelerates — one node captures a finite fraction of everything. Winner-take-all.
- Linear (α = 1): the pull tracks popularity exactly — the scale-free, power-law regime above.
- Sublinear (α < 1): the pull flattens as a node grows — the power-law tail disappears. The network becomes single-scale.
What that does to the shape
The attachment rule writes itself into the degree distribution — how many nodes have each number of connections.
Scale-free: a few nodes hold almost all the links. Homogeneous: almost everyone sits near the average.
A scale-free network has a long power-law tail: most nodes tiny, a few enormous. A homogeneous network has a Poisson-like distribution instead — degree clusters tightly around a mean, so almost every node is close to average. That’s the shape of random and spatially-embedded networks, where physical resources bound how many links any one node can hold. No runaway hubs — a graph that rewards exploration over rich-get-richer.
Extro’s move
Extro bends the attachment rule with economics. Reaching out has a real cost, and that cost is only sustainable when it’s balanced by reach coming back: a node that broadcasts far more than it’s welcomed pays steeply and runs dry. So you can’t buy your way to the top by volume, and you can’t cash in by following everyone — the other side has to accept you. The quality of who reaches you and who you reach has to clear, like any market.
Because every connection is mutual and priced, the pull a popular node exerts grows sublinearly instead of linearly — popularity stops compounding for free. Reach settles into an equilibrium bounded by what each node can actually spend, and the degree distribution tightens toward the homogeneous, single-scale shape above. No one grows alone.
The thesis, precisely. A rising price on reach turns linear/superlinear preferential attachment into sublinear attachment — a single-scale, non-scale-free network with no power-law tail. The mechanism that does it is economic, and it’s spelled out in how 402 works.