"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're observing.

Some distinctions will turn out to be useful repeatedly. Others will overlap enough that maintaining both doesn't buy us much. Still others will work in one context and fail in another.

That's where the analogy to dimensions earns its keep.

Adding another descriptive category is useful when it lets us distinguish cases that previously looked the same.

If it doesn't, we've added vocabulary without necessarily adding information.

The same problem appears in public arguments.

Freedom versus security.

Individual versus collective.

Centralized versus decentralized.

These oppositions can be useful. Sometimes the tension between two alternatives really is the thing we need to examine.

The trouble begins when the axis becomes the entire map.

A decentralized arrangement may preserve local autonomy while making coordination difficult. A centralized institution may solve a coordination problem while concentrating authority. Another arrangement may centralize one function and distribute another.

Now "centralized or decentralized?" isn't wrong.

It simply isn't enough to describe the object.

Adding another distinction gives us somewhere else to stand.

This is one reason large networks don't automatically become wiser as they grow. Additional participants can introduce new experience, information and relationships, but they can also reproduce the same positions thousands of times.

Imagine a discussion in which ten thousand people are divided into two camps repeating nearly identical arguments.

The network is enormous.

The descriptive space may still be remarkably small.

One additional person who introduces a distinction neither side had considered can sometimes change the conversation more than another thousand participants repeating what is already represented.

That doesn't make the new perspective correct.

It makes another possibility visible.

Then the network has to do something with it.

Music provides another useful analogy.

Two notes create an interval. Add another and we can hear relationships that weren't available from either pair alone. Chords participate in keys, and what a particular note seems to be doing depends partly on what surrounds it.

The note hasn't changed.

Its context has.

I used to want to take that observation one step further and say that a field had appeared—that the relationships themselves had become a kind of higher-order object.

Maybe that's a useful description in some contexts.

I no longer think the analogy establishes it.

What I can say with much more confidence is that relationships can carry information that isn't available from examining elements independently.

A newspaper article provides a simple example.

Read by itself, it contains names, places and events.

Connect the person to an author profile, the organization to a directory entry, the place to other stories from the same community and the event to a calendar, and the article becomes easier to situate.

Nothing mystical appeared between the database rows.

We preserved more relationships.

Those relationships give a reader more ways to recover context.

That's enough.

It also explains why network growth can fail.

If every new object enters without meaningful relationships, we've built a larger catalog.

If every participant is pushed toward the same few interactions, we've built scale without much variety.

If relationships multiply faster than anyone can navigate them, additional connectivity can become a source of noise.

And if the network's categories are fixed too early, new things may be forced into distinctions that no longer describe them very well.

So there is no point at which enough nodes automatically produce emergence.

There is no magic number where a collection becomes a field.

Something new may become possible with the third participant.

It may require three thousand.

It may never happen at all.

The result depends on what the participants contribute, which relationships actually form, how information moves through them and what the network allows participants to do that they couldn't do before.

That makes the practical question considerably less glamorous than "How large is the network?"

If I'm building a directory, a publishing system, a community or a body of knowledge, I want to know what another addition allows us to represent or do.

Does it introduce information we couldn't express before?

Does it connect things that were previously isolated?

Does it let us distinguish situations we had been treating as identical?

Does it make existing information easier to find or understand?

Or did the number merely go up?

Sometimes scale itself is useful. A telephone network with three users has obvious limitations that one with millions doesn't. A marketplace needs enough buyers and sellers to make transactions reasonably likely. A community needs enough participation to sustain activities no individual could maintain alone.

Other times the important addition is not another instance of what we already have.

It's another kind of thing.

That distinction has become increasingly important to me while building software around a newspaper.

Ten thousand articles would make a large archive.

Articles connected to authors, businesses, organizations, places, events, print editions and readers begin to make a different kind of resource.

The value isn't that we've discovered some universal geometry underneath community life.

We've simply stopped throwing away as many of the relationships.

And once those relationships remain available, people can use them in ways we didn't have to specify in advance.

Someone finds an old story through a business profile.

Someone encounters a community organization through an event.

A printed page points into an archive.

A current article acquires meaning from something published years earlier.

A directory entry gets corrected by someone who knows the organization better than we do.

Those are network effects too.

They don't require exponential growth.

They require another connection to make something possible that wasn't possible before.

That's the test I'd keep.

Not whether the network is bigger.

Not whether we've crossed some threshold.

Not whether another point has created another dimension.

Ask what we can now see, distinguish or do that we couldn't before.

Sometimes the answer will be nothing.

That's useful information too.