"Network effects" is one of those phrases that can begin to sound explanatory merely because we've heard it often enough.
A service gains users. More users make the service useful to more people. That attracts additional users. Eventually the network becomes difficult for a competitor to reproduce because the existing relationships are themselves part of what people value.
That's real.
It also leaves something out.
Not every additional participant contributes the same thing, and not every increase in possible connections produces an equivalent increase in value. A network can become larger while becoming noisier, harder to navigate or less useful to the people already inside it.
So I've become interested in a slightly different question.
What becomes possible when another point is added?
Geometry gives us a useful analogy, provided we don't ask it to prove too much.
Two distinct points determine a line. Add a third point that doesn't lie on that line and now those points can define a plane. Add a fourth that doesn't lie in the same plane and another spatial dimension becomes necessary to describe their positions.
The important condition is independence.
A third point placed on the original line doesn't create a plane merely because there are now three points. A fourth point placed in the same plane doesn't create volume merely because there are four.
Something has to be added that can't already be represented by the structure we had.
Networks aren't geometric spaces in this simple sense, but I find the distinction useful.
Suppose a community directory contains one hundred restaurants and we add the hundred-and-first. That's valuable if someone needs another restaurant, but it may not fundamentally change what the directory can represent.
Add the first hospital and something different happens.
Add public meetings.
Add local organizations.
Add events.
Add the people who write about those things and the places mentioned in their stories.
Now the collection hasn't merely become larger. New kinds of relationships can be represented.
A restaurant can sponsor an event. An organization can meet at a particular place. A journalist can write about a town board whose meeting appears on the calendar. A business mentioned in an article can have a persistent profile containing information that remains useful after the article leaves the front page.
The value isn't contained entirely in any one of those objects.
Some of it appears because they can now be related.
That's closer to what interests me about network effects.
The usual mathematical intuition is that the number of possible pairwise relationships grows faster than the number of participants. With nn participants, there are n(n−1)/2n(n-1)/2 possible pairs.
That doesn't mean all of those relationships exist.
It certainly doesn't mean they're all valuable.
A network of a thousand people doesn't become useful merely because almost half a million pairings are mathematically possible. Most participants may never interact. Some connections may duplicate others. Some may actively make the network worse.
Possibility isn't value.
What matters is which relationships become meaningful for whatever the network is trying to do.
This becomes especially visible when we're representing something complicated.
Consider describing a person as simply good or bad.
We've compressed an enormous amount of information onto one axis. That may occasionally be enough for the question we're asking, but it won't tell us much about how the person actually behaves.
Introduce other distinctions and the description can become more useful. Someone can be generous and unreliable. Courageous in one setting and hesitant in another. Patient with children and impatient with adults. Honest about some things while remarkably good at deceiving themselves about others.
We haven't discovered the hidden geometry of morality.
We've given ourselves more distinctions with which to describe what we