One of the most important pieces of research for understanding the world we are dealing with today was published in 1998 by Duncan Watts and Steven Strogatz. *Collective Dynamics of ‘Small-world’ Networks* presents a simple model for considering the impact of adding “long-range connections” into a world that is structured more-or-less like the world we’ve found ourselves in: one in which the structure of space dictates that if **B** sits between **A** and **C**, then to go from **A** to **C**, I will have to pass through **B**.

### What space is for

This is the everyday space we inhabit: I can’t teleport from one location to another, I must pass through a series of intervening points. We take it as a given, but in many ways this kind of space is a very important feature of our reality. One way is that it acts like an insurance policy, making sure things don’t spread too far too fast, save that for another essay. It also binds us together into shared clusters: what is local to me is also local to my neighbor.

While we still can’t teleport, what we can do is leverage technology that speeds us up to the point that it is *as-if* we are teleporting. Nowhere is this more stark than in information-space. Near-instant communication at any scale is now cheap and ubiquitous.

This has had a major impact on our personal relationships. Many are now doing a much of their business and socialization online — things that used to take place in person. Of course one can have both a rich online and offline life, but we are all bandwidth constrained, and there is a tradeoff. Each online interaction or relationship is one that won’t take place locally, in meat-space.

So, the way the Watts-Strogatz model works is by beginning with a so-called lattice network. A lattice is just like what we have been describing above: to get from A to C I must pass through B. If A, B, and C are people, we might say that A and B are friends, and B and C are friends, and so B is a common friend of A and C.

To generalize the lattice a bit more, we may allow that each node has “friends” in a certain neighborhood, rather than only right next-door. For instance, perhaps person A is directly connected to both their closest neighbor, B, and the person one more house down, C. So A is friends with B and C. And if C is also friends with with both B and A, having the same size “friend neighborhood”. The important thing here is that it follows that A has two friends, B and C, and B and C are ALSO friends with one another. This is the essence of “clustering” — my friends are likely to be friends with one another, too.

At the same time, parts of the lattice that are not near me or my neighbors can be arbitrarily far away. The world is wide, it takes a very long time to get across it. To get from New Hampshire to New York I have to get all the way through Vermont — could take weeks. And there are states even beyond that! 1

It’s like life before mass transit and communication. We are likely to talk to those nearby, and by virtue of sharing nearby space, the people we talk to are likely to talk to one another, too. Others are simply too far away. So we cluster, and other cluster far from us.

The Watts-Strogatz small-world model begins with such a lattice, and then modifications to the lattice are made and their consequences are examined. Specifically, they take neighbor-connections, and with some probability **p**, “rewire” the connection randomly from one of the neighbors to some other node (“person” in our example) on the network. This is the introduction of *long-range connections*. This is teleportation. The relationships are no longer purely constrained by space. Someone can be “nearby” while also being “somewhere else”.

The rewiring process does a decent job modeling the property we claimed above: that each of us has finite bandwidth, so we can’t really add connections without removing some.

The famous result is that as you add long range connections this way, the average distance between any two nodes, counted by the shortest number of hops one must take through the network to reach a given node, abruptly and dramatically drops down to some very small value (a “phase transition”). In other words, the thing about *space* that made some clusters “far away” breaks down. No one is far away anymore. But everyone is still *locally* clustered. These two properties alongside one another, clustering, without any cluster being far from one another, yields the so-called “small-world” property. Locally, things look like they always have, but globally things have changed drastically.

Here I reproduce one of the graph’s from the famous paper. The horizontal axis, here in a logarithmic scale, represents the number of long range connections which is driven by an increasing probability of “rewiring” from left (pure lattice) to right (purely random connections). The blue line represents the average distance between any two nodes, so when it drops it means “no one is far away anymore”. The orange line indicates the strength of local clustering, with maximum clustering when the network is a pure lattice (left orange), and a breakdown of clustering in a totally random network (right orange).

The area in which the blue line (average distance) is low, and the orange line is high (local clustering) that’s the “small-world” sweet spot.

This is the same plot, but here with the horizontal axis in linear scale. Notice how the blue line (average distance between nodes) simply **plummets** straight away. That is masked a bit in the logarithmic scale, but the transition to small-world is * ABRUPT* and

**nearly-immediate****.**

It’s clear that we crossed the small-world threshold in human society long ago, both physically and informationally. And that has been the general tenor of those who discuss these sorts of things: that we are living in a world with the small-world property: low average distance between nodes (with huge implications we haven’t gone into), and local clustering of nodes (my friends are friends).

But I think what happens towards the right of this graph has been neglected in how we’ve mapped this model to our situation. On the right, we have the total breakdown of local clustering. No longer are my friends friends. No longer are my neighbors neighbors. When we “rewire” enough of our connections away from the local, everyone is somehow close, but we lack the substrate for community. I used to think we were in the small-world, but I believe now we are in the close-but-alien world. Close without community.

To be sure, with social media some cyberspatial clustering is emerging. But it is not at all clear what it means for the topology of community to be divorced from the physicality of location, and surely our physical clustering is being obliterated. Maybe our online friends are friends with our other online friends. Our neighbors still don’t know one another.

Without clustering, we lose a kind of natural overlap, a higher-order relationship among individuals. I don’t think we truly recover that online. And I have my doubts about online-first communities successfully localizing with one another (though I wish those who attempt this the best, and maybe it will work in some cases).

We won’t get back to the lattice-world, outside of a cataclysm. And perhaps we shouldn’t want to. But maybe we can at least get back to the small world, where local is local, where we have a cluster to call our own — and we should.

I recognize this is difficult to fathom.

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