Buttons and threads, or how to achieve criticality in a non-linear fashion

I learned something a long, long time ago, first in studying the origin of life, and then in studying the relationships within networks of genes, and now when thinking about basic epidemiology. Nothing is linear. It’s an idea that’s been discussed since at least the 1980s, when Stuart Kauffman applied it to the logic of the emergence of life on Earth. Here he is talking about the appearance of autocatalytic sets, that is, collections of interlinked enzymes (or ribozymes) that generate emergent properties, like a metabolism.

Now, the next question is how hard is it to get such systems? Does it take a careful crafting of a chemist, or can it arise by chance? The body of theory I’ve been working on now for more than a decade suggests that it’s not hard.

You see this with an analogy: suppose you take 10,000 buttons and spread them out on a hardwood floor. You have a large spool of red thread. Now, what you do is you pick up a random pair of buttons and you tie them together with a piece of red thread. Put them down and pick up another random pair of buttons and tie them together with a red thread, and you just keep doing this. Every now and then lift up a button and see how many buttons you’ve lifted with your first button. A connective cluster of buttons is called a cluster or a component. When you have 10,000 buttons and only a few threads that tie them together, most of the times you’d pick up a button you’ll pick up a single button.

As the ratio of threads to buttons increases, you’re going to start to get larger clusters, three or four buttons tied together; then larger and larger clusters. At some point, you will have a number of intermediate clusters, and when you add a few more threads, you’ll have linked up the intermediate-sized clusters into one giant cluster.

So that if you plot on an axis, the ratio of threads to buttons: 10,000 buttons and no threads; 10,000 buttons and 5,000 threads; and so on, you’ll get a curve that is flat, and then all of a sudden it shoots up when you get this giant cluster. This steep curve is in fact evidence of a phase transition.

If there were an infinite number of threads and an infinite number of buttons and one just tuned the ratios, this would be a step function; it would come up in a sudden jump. So it’s a phase transition like ice freezing.

Now, the image you should take away from this is if you connect enough buttons all of a sudden they all go connected. To think about the origin of life, we have to think about the same thing.

The pattern should also affect how we think about genes. We’ve got about 20,000 genes; each gene influences the expression of some set of other genes. You may think you know exactly which genes are directly affected by a gene you are interested in — you can do experiments and work out the connections, a process called epistasis — but because each of those genes also have multiple connections, you in effect have to consider that every single gene in some way influences the activity of every other gene. Tug on one, and every other gene in the system is affected. Each of us is a supercluster of interacting genes, being tugged on in various ways by the environment.

I’m not an epidemiologist, but this also how I think about the pandemic. I am a button. I’ve been alone for months; if I had gotten the disease, I would have suffered alone but I’d also have been a dead-end for the virus. Now my wife is home, another button, and we are tied together with a red thread such that if I get the disease, she almost certainly will, and vice versa. But also, she was living with my daughter, her husband, and my granddaughter for a few months, she was part of a four-button cluster, which I’ve now joined. If one of us had the virus, it would have readily spread within that group. But it would have ended there.

Unless…what if I cheated? I decided to go out to a bar and chat with ten friends. I’ve basically connected a red thread to each of their clusters, and increased my connectivity greatly. Maybe you think it’s still a manageable number, but that’s only because you don’t see all the red threads outside of your immediate group. The point of Kauffman’s analogy is that the expansion of the network is not linear, as you might naively expect, but jumps rapidly as the number of connections increases, and can undergo a phase transition, where just going out to a bar can achieve criticality, and suddenly you are connected to everyone in the country, and the virus has avenues to reach everyone.

So think of yourself as a button. Every time you touch someone, lean in close and breathe their air, you are tying a red thread to them, linking your fate to some degree to them. You can safely build a little network with close family, and you’re still OK — the threads tangle together just your small family unit. But if your child has a playdate with a neighbor’s kid…they have made a new thread that encompasses everyone in your family, and everyone in the neighbor’s family, and you’ll have no idea how many threads connect you all. And if you decide to take the whole family to that newly opened beach and mingle with thousands of other people, forget about it — the number of connections have shot up exponentially. You’ve lost all control.

The problem is that people don’t grasp the idea of exponential increases intuitively. I don’t. I’ve worked with enough models that I know that these kind of phenomena can produce surprisingly large effects rapidly, though, and that our current situation is a perfect example of that kind of phenomenon, and damn, stay home and stop stitching all those buttons together.


  1. kome says

    That’s a nice way to think about it. I think about it more like the “6 degrees of Kevin Bacon” thing. We’re all only a few degrees to several degrees of separation from each other. And given the number of people who are confirmed to be infected, in spite of active attempts to suppress or diminish reporting, I just don’t feel like risking things unnecessarily.

  2. anxionnat says

    Good analogy. Those of us alive during the AIDS epidemic of the early 80s found out about that in all-too-painful reality back then. When you’re talking about sick and dying people, it just gets so overwhelming, so fast…Reminds me of the story about the doctor who cured the king’s son and asked that, as his payment, he be given one grain of rice on the first square of the chessboard, 2 grains on the second square, four on the third square, and so on–thus bankrupting the kingdom.

  3. psychomath says

    Yes, but it does come with a cost. I’m in Honolulu in Hawaii. We have about 650 confirmed cases, and our unemployment is at 25-30%. We shut down everything tourists would want to do and instituted a mandatory 14 day self-quarantine for all visitors and returning residents. Our tourism dropped from about 30,000 people per day to a few hundred. All the hotels, the restaurants, the services for visitors are closed.

    I really don’t know how we are going to survive this. Unemployment will soon be running out, and the line of cars waiting to receive food from food banks are already literally miles long. I saw what was coming and started isolating before the lockdown because I care for a parent with conditions that make them very vulnerable, but I also have friends and family whose situation is about to become desperate.

    If we are going to insist that people restrict themselves for the good of society, as I definitely think we should, then society has to support all of those who are being forced into destitution because of it. Stop attacking the idiots who demand their haircuts, and start demanding that the government institute programs to support the people who are not allowed to even try to support themselves. This is a desperate situation for many millions of people leaving aside the real threat of Covid-19.

  4. says

    Ya I read this thing (the transcript) a while ago, about the wonders of network systems, basically. Energy efficiency and all that. [just re-reading it, they mention the complexity of the human genome there too!]. But then it also mentioned that all this wonderful network stuff also increases some risks, like spreading infectious diseases. Oh.

  5. unclefrogy says

    this is beginning to resemble one of the situation solutions George Orr came up with in Ursula K le Guin’s novel “Lathe Of heaven” solving the population problem with a plague.
    Stupidity, ignorance ,greed and hate we forge ahead of the cliff.
    (nothing but gloomy sundays)
    uncle frogy

  6. jack16 says

    PZ: Buttons and threads. What is the restriction in the real world that prevents threads to clusters? I’d guess that his bayesian probability would be small. What do you think?