emergent complexity modeling

A sheet of paper tears where tension exceeds strength. Epithelium, the cell sheet lining our organs, behaves similarly, but until recently models could not show how tears appear in it. The classical vertex model describes tissue as a network of polygons tightly packed together and forbids cells from detaching.

Picture a flock of birds gathering around a feeder that no one built. Each bird searches for food on its own, yet they all end up together. Usually, such behavior is explained by social instinct: birds want to be near their kin. But what if the flock is simply a side effect of greed?

Every winter, hospitals fill up, and by summer they seem to empty. Epidemics breathe like the tide. But why do waves of disease keep rolling in, even without new mutations? Mathematicians found an answer in a tug-of-war between two strains. Within a single epidemic, two virus variants often coexist.

A cluster of glowing dots drifts through the darkness, now compressing into islands, now spreading out in waves. These are not fish or birds: this is a mathematical model of active matter, where each particle spends its own energy on movement, and also is born, dies, and competes with neighbors for space.

A mosquito swarm resembles a living cloud — hundreds of bodies circle together, but no one lines up or mimics a neighbor's movements like starlings or fish. For a long time, it was unclear what held such unity together. The authors of a new study suggested that the insects communicate by sound.

A kitchen mixer runs at a single speed, but start jabbing the button and the mixture whips up differently. Chemical reactions behave much the same way when the rates of their individual steps are not constant but pulse over time.

A cloud of atoms cooled to near absolute zero suddenly starts behaving like a brain. One excited atom triggers a chain reaction — neighbors light up in turn, and an avalanche of activity sweeps through the gas. Physicists call this Rydberg facilitation: atoms are easier to excite when an excited one is nearby.

Usually, the complexity of systems, from satellite motions to heart rhythms, is assessed with averaged entropy (a measure of unpredictability, as understood by Boltzmann). But that's like evaluating the ocean only by wave height: the average figure hides the full richness of underwater currents.

The meditating brain is an orchestra, and attention is the conductor. When you're focused on your breath, all instruments play in harmony. But suddenly, a thought-cello plays a false note: that's mind wandering. An experienced conductor doesn't panic: they instantly catch the dissonance and gently guide the orchestra back to the main melody — the breath.

Power systems with solar panels and wind turbines are like a chess game: cloud cover and load changes are the opponent's moves, impossible to foresee. To maintain balance, operators used to run through dozens of scenarios (uncertainty) — that took too much time and computing power.

A robot swarm is like a cloud of fireflies: each moves on instinct to wherever is freer, and chaos turns into order. For a long time, it was a mystery how coherence emerges without communication or commands. The answer came with the creation of the Response Map — a tool that, like an X-ray, illuminates the hidden logic of the machine brain.

Evolution flows like water. In a wide river, changes are smooth and slow, while a mountain stream churns with every stone. Population size and mutation rate set the channel: broad and calm, or narrow and rapid-filled.

A desert where every step reshapes the terrain. Dunes settle, new slopes emerge, and the next step is dictated by what you've just trampled. That's roughly how "sandscapes" work — a mathematical model inspired by living systems.

Machine learning has long ceased to be just a tool — it is becoming an investigator, independently fishing out the laws of nature from data. But how does a silicon mind come to understand?

Imagine an orchestra where every musician plays their own part without listening to the others. Suddenly, without any visible command, neighboring musicians begin to sync perfectly, yet there is still no overall melody.

A voice recorder captures a city square. You hear the hum of a crowd, footsteps, snatches of conversation, the rumble of traffic. Somewhere in that sonic stew, a street musician plays a simple tune on a flute. Hearing it isn't easy: there's just too much mixed together.

At a party, guests don't just chat in pairs; they gather in circles of shared interests: three discussing movies, four debating sports. The usual acquaintance map, like a phone's contact list, can't capture that: it only sees one-on-one links.

If you toss a handful of coins into a box with dividers, each coin lands in its own compartment and stays there forever. Roughly the same happens to atoms in a special optical lattice when excited by a laser. These atoms are called Rydberg atoms: one electron in each is promoted to a very distant orbit, making the atom sensitive to its neighbors.

Picture a crowded hall where each person is a molecule, and their private conversation is the motion of electrons within it. Normally, everyone talks to themselves, and the overall buzz is chaotic. But if a bright spotlight is turned on in the center, it begins to conduct the crowd: people synchronize their movements, even though the light merely reflects off the walls.

Two musicians play a duet. In a perfect hall, they hear each other perfectly and gradually fall into resonance. But try playing in a room where someone keeps slamming the door. You'd think the rhythm would be thrown off. But no: sometimes that very random slam makes both of them sync up instantly.

A neural network learning something new is like a traveler in mountainous terrain. Each piece of learned knowledge is a depression in the landscape into which the traveler rolls and gets stuck. To escape and reach a new basin, energy is needed. But as the traveler gains experience, the walls of their pit become steeper, and each subsequent climb requires more effort.

A colony of cells is like a city that grows from its outskirts: new houses appear where there's more space. Inside, it's crowded, and construction stalls. Similarly, cells divide more often at the edge, where neighbors squeeze them less. Scientists modeled this without any chemical commands — only mechanics. In the model, each cell is a ball that presses on others.

A Network That Decides When to Stop A crowd grows: each new person joins the group with a probability that depends on how large the group already is. If the group is small, joining is easy; if it is huge, people start to hesitate, and growth slows down. This mechanism, called feedback, changes the entire behavior of the network.

A seven-metal alloy is like a team where every player is pulling the blanket their own way. Atoms of iron, cobalt, nickel, aluminum, titanium, tantalum, and vanadium can either fall into a strict pattern or scatter into random positions. This determines whether the material will be as strong as steel or as light as aluminum.

A neuron in the brain is like a person handing out business cards at a crowded conference. At first, making acquaintances is easy, but as the address book fills up, each new contact requires more effort: there's little room left. The authors based their model on this 'crowding' rule: the more incoming connections a neuron already has, the harder it is to form another one.

When you press your finger on a piece of soft putty, it obediently dents and then slowly returns to its original shape. But if you press on a window pane, it either holds or cracks — no smooth deformation. Both materials are amorphous: their atoms are not arranged in a strict crystal lattice like in metal. Yet they behave completely differently.

A student learned physics from a textbook and is taking an exam. The first two problems are solved brilliantly, but on the third, they err on something not covered in the book. That is roughly what the hybrid atmospheric model NeuralGCM faced when run through three levels of testing: cyclones, El Niño, and anomalous warming.

A quantum walk is like a drunk person who turns randomly at every intersection. Except instead of streets, there are graph nodes, and instead of steps, there are quantum transitions. Scientists have discovered that such a walk has a hidden measure — Krylov complexity.

In the ocean, sometimes a huge solitary wave appears out of nowhere and then vanishes. Such rogue waves are rare but dangerous. Similar rare and intense spikes occur in other systems, from financial markets to lasers. The problem is that they are hard to predict, let alone trigger on demand.

In a collider detector, each proton collision leaves a trail of dozens of particles, like the splash from a stone thrown into water. Making sense of that splash and understanding what exactly happened at the moment of impact is a task that humans can no longer handle alone.

In the everyday world, order preserves and disorder destroys. But in quantum computing, this logic is turned on its head. Consider a diamond: carbon atoms with their nuclei and electrons behave like tiny magnets capable of storing a quantum "whisper"—information encoded in the orientation of their spins.

In 1935, Erwin Schrödinger, peering into the equations of quantum mechanics, saw something strange: two particles could be linked so tightly that the state of one instantly determines the state of the other, even if they are separated by kilometers.

A density of four people per square meter is the threshold beyond which a crowd ceases to be a sum of individual pedestrians and becomes something else. In Pamplona's Plaza Consistorial during the San Fermín festival, local density reached nine people per square meter.

Every few years, a huge pool of warm water in the western Pacific Ocean shifts eastward, like water in a bathtub when you tilt it sharply to one side. That is El Niño. But each tilt is different: sometimes the water sloshes against the shores of Peru, sometimes it stalls in the middle, sometimes it pours out in a powerful surge, and sometimes it barely creeps.

The ocean doesn't always stick to the same schedule. Sometimes the warm El Niño current strengthens, sometimes it weakens, and each regime dictates its own reaction speed to the past. These hidden regimes are like shifts in currents that remember their path with different depths.

In a tropical forest, the dry season gives way to the rainy season, and with the weather, the rules of life change: some animals go into hibernation, while others come out of hiding. Likewise, in complex systems, from climate to economics, the links between processes are not set in stone. They depend on a hidden "regime" that itself switches from time to time.