Self-Organized Criticality: Just the Right Amount of Vibrancy

What is the ideal state for an organization, a group of people, or an individual?

You might think this isn’t a real question, that any state could be good — is there really a systematic explanation for it?

Yes, there is. Let me paint a few scenarios, and you can see what I mean.

A stagnant pond versus a pot of boiling water. A unit where no one speaks up in meetings, and everyone just agrees with the leader, versus a unit plagued by constant infighting, chaos, and an inability for anyone to lead. A person so idle they’re restless, unsure what to do today, versus someone so busy they suffer from insomnia, with a dozen things exploding in their mind simultaneously.

You’ll immediately realize that none of these are particularly good states. The ideal state should lie somewhere between a stagnant pond and boiling water. But the answer isn’t simply a lukewarm midpoint.

The best state is an edge — an “almost chaotic, on the verge of collapse” edge. It’s very close to chaos, so close that it could collapse at any moment. Yet, it’s precisely this position, “seemingly dangerous, but not entirely safe,” that makes it vibrant, agile, and full of life.

The mental model I’m presenting to you today is called “self-organized criticality.” This term sounds academic, but don’t you think it’s also a bit cool? It originates from physics, but it absolutely should be applied across all industries; it should become a fundamental part of our understanding.

What is Self-Organized Criticality? #

In 1987, three physicists at Brookhaven National Laboratory — Per Bak, Chao Tang, and Kurt Wiesenfeld — proposed the concept of “self-organized criticality” [1].

In physics, “criticality” typically refers to a special state where a system undergoes a phase transition. When approaching this state, correlations between different parts of the system can span many scales; small perturbations sometimes only affect local areas, but at other times can influence a much larger range. That research spoke of a pile of sand, but let’s put the sandpile aside for a moment and talk about something you’re more familiar with.

A collection of things, whether a group of people or a pile of sand, forms a system. What I’m concerned with now is, within this system, how many subsequent events, on average, a single event experienced by an individual will trigger.

Take, for example, rumor propagation. A colleague in your company learns a piece of gossip; how many people will they tell? And how many more people will each of those recipients tell? The average number of people each person passes the message to is called the “branching ratio.” The physicists’ insight is that the branching ratio represents the system’s propagation capability: how many subsequent actions, on average, a single action can generate.

If the branching ratio is less than 1 — everyone hears it, has a laugh, and can’t be bothered to pass it on — the news will disappear after a round or two. In this situation, we say the system is in a “subcritical” state: it’s very stable, but also very sluggish; no matter how big a disturbance you introduce, it won’t go far.

If the branching ratio is greater than 1 — everyone who hears it can’t resist adding their own embellishments and telling several more people — then it spreads like wildfire, reaching the entire company in half a day, and after that, there’s no containing it. This is called “supercritical”: such a system is easily startled and loses control at the slightest provocation.

However, when the branching ratio is precisely around 1, the system is in a “critical” state: a word you speak might stop after one person, or it might spread throughout the entire building; no one can predict it beforehand. In such a system, you’ll observe a large number of small events, a small number of medium events, and extremely rare, yet genuinely possible, large events.

Strictly speaking, event sizes in a critical state follow a “power law” distribution, which we previously discussed as a “heavy-tailed distribution” — very many small events, few large ones, no typical scale, and occasionally, black swans.

Criticality is a particularly ideal state. I’ll give you a scenario, and you’ll understand: imagine a group of people sitting together in a meeting —

In a subcritical meeting, people are generally reluctant to speak. Even if you stand up and offer a radical opinion, few will express an opinion, and no one will applaud. The meeting room is orderly, but nothing new is brainstormed.

In a supercritical meeting, everyone rushes to speak. Even a cough can incite a commotion from the whole room. The meeting room is extremely lively but full of noise, and still, nothing substantial gets done.

However, a critical meeting allows a good idea to travel across the room, sparking responses, additions, and extensions along the way; while unreliable ideas mostly fizzle out as soon as they’re uttered. It’s neither dull nor out of control. You can feel an excitement where “everyone is a little tense, yet still engaged.”

Do you see the advantages? Criticality is neither rigid nor chaotic, possessing both order and vitality; it can be described as a state of “unity, tension, solemnity, and liveliness.”

Three Characteristics of Critical Systems #

Critical systems possess many interesting properties. Translating them into plain language, they can be summarized as the following three points [2] —

First, and most importantly: information travels far.

In the meeting scenario I just described, a good idea can travel across the entire room — in physics terms, this means the “correlation length” is very large: two points far apart in the system can still influence each other; even a tiny disturbance at the periphery can reach the center.

However, it’s different from supercriticality — supercriticality means “spreads too violently,” with everyone overloading and burning out at the slightest disturbance; criticality means “spreads just right,” a signal can travel very far without setting the entire system ablaze.

Second, fluctuations can generate structure. In rigid systems, fluctuations are merely noise, small jitters around the average, smoothed out as soon as they appear. But in critical systems, fluctuations are no longer the enemy; instead, they become generators of structure — a tiny random perturbation has the opportunity to grow into new patterns of any scale.

Innovation is often generated in this avalanche-like manner.

Third, it is particularly sensitive. Even the weakest signal can be caught and amplified — for instance, a small demo or an ordinary debate could stir up a significant resonance in a critical system. But this sensitivity comes at a cost: critical systems propagate far but recover slowly; a disturbance can reverberate for a very long time, taking a long time to subside [3].

A critical system is like an intelligent and sensitive person: she can hear whispers from afar, yet cannot forget the recent disturbance. So, if such a system were to perform intellectual work, wouldn’t its efficiency be extremely high? Indeed.

For years, there’s been a debated theory in neuroscience called the “critical brain hypothesis,” which posits that our brain’s highly efficient working state is one near criticality.

Consider this: in 2003, two neuroscientists at the U.S. National Institutes of Health, John Beggs and Dietmar Plenz, observed a phenomenon in cortical tissue in a petri dish: the collective discharge of neurons varied greatly in size, much like a series of “neuronal avalanches,” with the size distribution approaching a power law [4]… Isn’t this criticality?

At least in some experiments and models, neural information capacity and transmission capabilities are higher in this state [5].

Think about it: your most creative, most perceptive moments, your “flow” state — where you’re highly alert, your attention is focused on a problem, yet tangential associations occasionally emerge; you feel you have a grip on this complex thing, rather than being driven by information — doesn’t that resemble criticality?

The same applies to organizations. Creativity flourishes in organizations where information flows freely — a small discovery at the grassroots can trigger a cascade of responses; an inconspicuous experiment can lead to a chain of innovations.

Typically, people describe a system as either too idle or too chaotic, unaware that between idleness and chaos lies a beautiful state called criticality.

The Secret of Self-Organized Criticality: Slow and Fast Variables #

So, if criticality is so beneficial, how can a system reach and maintain this state?

One way is to have a controller, for example, a meeting facilitator who continuously calls on people to speak when the meeting stalls, and controls speaking time or interrupts tangents when it gets too chaotic. This is called “tuned criticality”: regulated by external forces.

However, the “self-organized criticality” we’re discussing is a miraculous mechanism: the system possesses an inherent feedback loop that automatically pulls subcritical and supercritical states back towards criticality.

This brings us to Bak and his colleagues’ sandpile model.

Imagine slowly adding sand, grain by grain, to a table. The sandpile grows taller, and its slope becomes steeper. But it won’t continue to steepen indefinitely — once the slope reaches a certain degree, whoosh, an avalanche will occur, some sand will roll down the slope and spill off the edge of the table, and the slope will decrease again. Then you continue adding, it steepens again, and after reaching a certain degree, it collapses again… This cycle repeats over and over.

That relatively high, steep state is the sandpile’s critical state. But the sandpile doesn’t need anyone to tell it “what the critical slope is.” If the slope is too gentle, you add sand, and it steepens on its own, climbing towards criticality; if the slope is too steep, it collapses on its own, flattening the slope and returning to near criticality. That critical slope becomes the point to which the entire sandpile automatically reverts.

Earthquake activity, solar flares, and neuronal avalanches all bear the hallmarks of self-organized criticality [4,6]; even your own physical and mental states can be understood through this mechanism.

Maintaining self-organized criticality relies entirely on the interplay of two variables —

One is the “slow variable,” which is the thing that accumulates quietly: the slope of the sandpile, the stress on a fault line, dead branches in a forest, the fatigue in your body. It imperceptibly, little by little, pushes the system towards a dangerous edge.

The other is the “fast variable,” which is the thing that releases abruptly: an avalanche, an earthquake, a fire, a breakdown. In an instant, it discharges the accumulated load.

The slow variable is responsible for pushing the system towards criticality, while the fast variable is responsible for pulling the system back from the brink. With this push and pull, the system oscillates around criticality.

Personal Application: Three Principles #

This wisdom of self-organized criticality can be applied to ourselves. Let’s discuss three principles —

First, load slowly, release quickly.

Work, study, family, social relationships — adults face various tasks, and the busier they get, the more tired they become; fatigue accumulates on you like tiny grains of sand. Thus, you need to release, and like a sandpile avalanche, have a quick release. For example, completing a long-term project, taking a vacation to relax, or even having an emotional outburst with someone.

After the outburst, you’re unburdened and can begin the next round of slow accumulation.

Many people, however, do the opposite: either they’re usually very relaxed, and when they occasionally take on a particularly large task, they have no idea how to start; or they’re busy every day and never get any rest… These are either subcritical or supercritical states.

This reminds me of an anecdote from the Book of Rites. Zigong once disapproved of people indulging in revelry during festivals — much like some officials today who can’t stand to see people happy and always want to control them — but Confucius said that people should naturally be this way: “To be tense and never relax, even King Wen and King Wu could not endure; to be relaxed and never tense, even King Wen and King Wu would not allow; to be both tense and relaxed, that is the way of King Wen and King Wu.”

Many interpret “being tense and relaxed” as “appropriate moderation” or “balancing work and rest” — but that is not the way of King Wen and King Wu for highly efficient individuals.

“Zhang” (张) means to tighten, and “chi” (弛) means to loosen. Confucius meant for these two to alternate: be tense for a while, then relax; accumulate for a period, then release once. Confucius was describing the rhythm of the sandpile, saying one should not be like a highway toll collector performing the same action from clock-in to clock-out every day.

Second, leave outlets for release.

For the system not to keep accumulating until it bursts, sand must be able to spill from the sandpile’s edge. The “sand outlets” in life are sleep and rest, decisively abandoning hopeless projects, deleting useless information, admitting a failure, leaving whitespace in your schedule, and allowing bottled-up emotions to be expressed. Without these outlets, you will trend towards supercriticality.

Third, use a few local rules instead of grand self-management.

The grains of sand in a sandpile don’t need to understand the entire sand mountain. Each grain of sand only needs to obey one simple rule: if the slope is too steep, I slide. You are the same.

You don’t need to re-plan your grand life strategy every day; you just need to set a few local threshold rules for yourself: if the same problem appears three times, stop and solve it systematically; if a project makes no progress for two consecutive rounds, re-evaluate it; if your schedule is filled to a certain proportion, stop taking on new tasks…

As long as you consistently adhere to these rules, you will naturally develop a healthy order.

Beware of “High-Pressure Subcriticality”: Small Collapses Prevent Big Ones #

Hearing this, you might be tempted to say that while criticality is good, it still seems a bit dangerous, given the presence of black swans. You’re right, not everything should be critical — criticality amplifies good ideas, but it also amplifies rumors and infectious diseases — some things are better kept in a subcritical state.

However, please note that for things that are inherently alive and should be critical, if you forcibly suppress them into a subcritical state just for the sake of convenience, that would be detrimental.

For example, in an organization, if you demand absolute safety and stability, disallowing small failures, preventing bad news from surfacing, and suppressing minor conflicts from being released, then you’ll have to use various means to block every single channel for small breakdowns. You might achieve a considerable period of calm. But the accumulated load, which should have been gradually discharged by small breakdowns, won’t disappear. It will silently build up more and more underneath.

We might call this state “high-pressure subcriticality.” It doesn’t mean it won’t collapse; it’s accumulating, and when it does collapse, it will be a big one.

For instance, earthquakes are inherently a critical phenomenon. Tectonic plates slowly accumulate stress, which is then suddenly released at a certain moment. Small earthquakes on a fault line are countless in normal times, while large earthquakes are usually extremely rare. However, if a seismic fault experiences no small earthquakes for a long time, it’s often not a sign of peace, but rather a dangerous signal — geologists call this a “seismic gap.” The absence of small tremors doesn’t mean stress has disappeared; it means it’s tightly locked, building up more and more underneath until it unleashes a major one.

As we discussed when talking about antifragility, a forest where small fires have been suppressed for a hundred years will accumulate thicker and thicker layers of dead wood and fallen leaves, eventually leading to a massive blaze that consumes everything.

One spirit of criticality is not to fear incidents. Criticality cannot guarantee that major incidents won’t happen, but allowing small incidents to occur regularly helps to avoid major ones as much as possible.

Isn’t it the same with raising children? You neither want them to become a “powder keg” — which is high-pressure subcriticality, overly obedient due to suppression, eventually exploding all at once; nor do you want them to have a “glass heart” — which is supercriticality, losing emotional control over every minor issue. You want them to be in a critical state: allow children to make mistakes, throw tantrums, even talk back, so they learn to digest small conflicts and mend relationships themselves, which helps them handle bigger challenges!

Just the Right Amount of Vibrancy #

A 2021 study analyzed data from over thirty-five thousand people and conducted two experiments specifically to examine the relationship between an individual’s “discretionary free time” and happiness [7]. The conclusion was an inverted U-shaped curve: too little free time, constantly chasing tasks, leads to unhappiness; but too much free time, being idle and having an empty daily life, also leads to unhappiness.

Notably, the researchers found that the happiest point wasn’t located at the comfortable “neither idle nor busy” midpoint, but distinctly leaned towards the “busy” end — a state where one’s workload is consistently full, always feeling a slight push, yet not overwhelmed.

That is precisely the critical state.

You might be a little tired. But your reactions are very agile, you feel excited, you move quickly, you are full of motivation, you can both release and pull back, you’ve harnessed the power of chaos without surrendering yourself to it.

That is just the right amount of vibrancy.

You might say I’m dangerously probing the edge; I say I’m writing reality in places where order has not yet sealed off possibilities.

【As evidenced by the verse】

Stagnant water, no ripples; raging flames, wild. Preferring the cliff’s edge to find beauty. Frequent small collapses keep spring ever-young, Tension and relaxation are commonplace.

注释

[1] Bak, Per, Chao Tang, and Kurt Wiesenfeld. “Self-Organized Criticality: An Explanation of the 1/f Noise.” Physical Review Letters 59, no. 4 (1987): 381–384.

[2] Kadanoff, Leo P., Wolfgang Götze, David Hamblen, Robert Hecht, E. A. S. Lewis, V. V. Palciauskas, Martin Rayl, J. Swift, David Aspnes, and Joseph Kane. “Static Phenomena Near Critical Points: Theory and Experiment.” Reviews of Modern Physics 39, no. 2 (1967): 395–431.

[3] Hohenberg, P. C., and B. I. Halperin. “Theory of Dynamic Critical Phenomena.” Reviews of Modern Physics 49, no. 3 (1977): 435–479.

[4] Beggs, John M., and Dietmar Plenz. “Neuronal Avalanches in Neocortical Circuits.” Journal of Neuroscience 23, no. 35 (2003): 11167–11177.

[5] Shew, Woodrow L., Hongdian Yang, Shan Yu, Rajarshi Roy, and Dietmar Plenz. “Information Capacity and Transmission Are Maximized in Balanced Cortical Networks with Neuronal Avalanches.” Journal of Neuroscience 31, no. 1 (2011): 55–63.

[6] Watkins, Nicholas W., Gunnar Pruessner, Sandra C. Chapman, Norma B. Crosby, and Henrik J. Jensen. “25 Years of Self-Organized Criticality: Concepts and Controversies.” Space Science Reviews 198, nos. 1–4 (2016): 3–44.

[7] Sharif, Marissa A., Cassie Mogilner, and Hal E. Hershfield. “Having Too Little or Too Much Time Is Linked to Lower Subjective Well-Being.” Journal of Personality and Social Psychology 121, no. 4 (2021): 933–947.