Graphs

In a mountain range, there are trails that can be contracted to a point if you walk through a flat valley. But there are also loop routes that cannot be shrunk without crossing a ridge.

A cartographer who draws not mountains and rivers but invisible bonds between electrons: that's one way to describe the new neural network. Instead of atomic coordinates, it takes molecular orbitals and draws a graph where nodes are orbitals and edges are measures of their quantum entanglement.

A cartographer who draws not mountains and rivers but invisible bonds between electrons: that's one way to describe the new neural network. Instead of atomic coordinates, it takes molecular orbitals and draws a graph where nodes are orbitals and edges are measures of their quantum entanglement.

A cookbook without a shopping list: that's what a scientist faces when looking at a scheme of chemical reactions. The arrows and formulas on paper look neat, but once you start hunting in the lab for molecules that will actually behave that way, a real quest begins.

A stadium packed with fans hears the announcer’s voice for the first time, and instantly the whole place erupts. The message sounds again and again. Only a handful of people in nearby rows respond. That’s roughly how neural networks behave in brain organoids — tiny lab-grown replicas of the brain.

An ordinary road map with traffic is a heap of numbers. Sending them all as is takes too long. But you can reduce the volume without losing the gist. Researchers found a clever solution: first redraw the map so that streets become points, and intersections become bridges between these points.

A child exploring an unfamiliar apartment crawls from object to object and remembers what is visible from each corner. This is how the brain builds a mental map — an internal model of space. The gradCSCG algorithm works on the same principle: it receives a video stream from a camera and independently draws a map of the room, relying only on the changing images.

Data analysis on the internet or sensor systems has long resembled trying to conduct an orchestra where each musician is locked in their own booth, and the conductor cannot leave the podium.

A family tree becomes a tangle if you account for adopted children and remarriages. Evolution often resembles such a tangle: although the Standard Model, the traditional view of divergent evolution where species branch like a tree, paints a neat tree of species, in reality genes travel not only by inheritance but also directly between unrelated organisms — a process…

Recommendation systems that decide which post you’ll see next run on deep neural networks made up of thousands of operations — intricate computation Graphs (blueprints of operations on data). These systems are living, ever-evolving Dynamical Systems: the slightest architecture change cascades through the entire graph.

7 278 601 292 779 396 715. That’s exactly how many unlabeled chordal graphs, graphs with no induced cycles of length four or more, can be built on twenty vertices. This number isn’t a puzzle for a supercomputer; it’s the exact output of a new algorithm that treats symmetry as a tool, not an obstacle.

In the courtroom, a string is presented as input. Twelve jurors (computation paths) examine it symbol by symbol. Each can render a 'guilty' (accept) or 'not guilty' verdict. For a conventional nondeterministic automaton (NFA), one 'yes' vote suffices.

A huge sheet of paper lies on the table. A cartographer tries to fit all conceivable landscapes onto it: mountain ranges, tangled gorges, endless plains. Each landscape is a finitely generated group, and the map is its Cayley graph, a graphic portrait of the group, where vertices are elements and edges are connections between them.

Inside a crystal, an invisible ballet unfolds: the magnetic moments of atoms, like dancers, sometimes align in unison, sometimes scatter in antiphase. The choreography is dictated by quantum exchange — music sensitive to the geometry of bonds.

A neuron resembles an electrical circuit with batteries and variable resistors. The resistors are played by ion channels — protein pores in the membrane that open and close, letting charged particles through. Whether a nerve impulse fires depends on how fast and at what voltage these pores operate.

X-ray detectors work like honeycombs: particles flash here and there in hexagonal cells. Conventional algorithms stretch these combs onto a square grid, creating ghostly polarization artifacts. Scientists used a neural network that lives by the rules of the hive — it processes the hexagonal structure directly, preserving the natural pattern.

Finding the right molecule among billions of options is like finding a single grain in a sack of rice. To avoid sifting through everything by hand, chemists increasingly turn to neural networks. But large models require powerful computers, while small ones often lack the big picture. GLACIER tackles this problem like a bright student learning from several mentors at once.

A kitchen sponge soaked in syrup: in some spots there's more syrup, and your finger sticks right away; in others, it barely touches. That's roughly how a catalyst surface works. Molecules don't react everywhere but at especially sticky points — active sites.

Designing machines is like cooking: you need to know exactly where the 'dough' will tear. Old computer simulations check every point on a part for hours, like measuring the entire pot to make sure the handle doesn't fall off.

Spacetime is often pictured as a smooth fabric. But at the very microscopic level, this fabric might not be continuous; instead, it could consist of individual 'atoms'—events linked by causality. This picture of the world is called causal set theory.

A social network is like a city: there are giant neighborhoods and tiny alleys. When an algorithm looks for tight-knit groups, it often picks out a couple of huge clusters and many small fragments. But the client (say, a municipality) needs districts of roughly equal size to distribute clinics or schools evenly.

A city is a living system: it consumes resources, produces waste, and its 'circulatory system'—the road network — delivers people and goods to their destinations. But while engineers design transport arteries as rigid geometry of straight lines, nature has spent millions of years perfecting ideal distribution networks.

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.

After a game of Scrabble, the board looks like a mosaic: tiles are scattered but form a connected pattern — islands of words in a sea of empty cells. Such a pattern can be described mathematically, but usually that requires complex rules.

Every carbon atom in a molecule carries a passport — the energy with which its deepest electrons sit in their orbitals. This energy is sensitive to neighbors: add an oxygen nearby or a double bond, and the number in the passport changes. Chemists have long measured such energies with X-rays, but calculating each new molecule takes hours of machine time.

Studying the birth of galaxies requires simulating billions of years of dark matter growth and star formation — even supercomputers handle this slowly.

Each heartbeat can be drawn as a point on a sheet of paper. Thousands of these points merge into a cloud. The Mapper algorithm turns this cloud into a network: it connects points that behave similarly and colors them according to blood flow strength. The result is not just a pile of numbers but a colored map showing which parts of the heart work in sync and which are out of step.

A city map: intersections are nodes, streets are connections. For an algorithm to understand the structure, each node needs a label. The classical approach — standard model — assigns simple tags, like house numbers.

A mole on your skin looks like a mosaic of colored pieces. Each piece is a small patch of the image where color and texture barely change. Such pieces are called superpixels. If you connect them with lines, you get a network that looks like a subway map: stations are the pieces, and lines are the connections between them. The weight of each line shows how strongly two pieces are related.

Strike a tuning fork, and it rings with a single pure note. But if the fork is trembling and people are chattering nearby, the sound won't be perfect. Same with molecules: each one absorbs light of a specific color, but due to its own vibrations and the jostling of neighbors, that color gets smeared out. The authors of the paper figured out how to show this to students without complex formulas.

A dense social network: everyone is connected to almost everyone else, and in this web it's hard to see who really matters. Too many threads — computers choke, simulations stall. Scientists have figured out how to prune such networks without losing what's essential. The idea is to remove connections that don't participate in shortest paths.

Particles in an accelerator can oscillate in two transverse directions at once, and these oscillations are not always independent. If you give a pendulum a push in one direction and it starts swinging sideways as well: that's coupled oscillations. In accelerators, this happens due to magnet imperfections or special insertions, making particle motion complex.

Alzheimer's disease is like a tangled ball of yarn: thousands of genes are intertwined, and it's hard to tell which ones snap first. Previously, scientists looked for important genes by simply counting how many connections each node in the network had. But that approach is like judging a person by the number of acquaintances they have — the most sociable isn't always the most influential.

Inside every cell, work is constantly underway: proteins communicate, genes turn on and off, and glitches in this system lead to disease. To find the culprit behind an ailment, scientists used to manually comb through research papers and databases, like a detective interviewing witnesses one by one. But there are so many connections that no one can handle it alone.

A city can be studied in different ways: one person maps the streets, another tracks traffic flows, and a third maps population density. The maps don't match, but they describe the same place. A similar problem arises when biologists look at a cell through different instruments: one shows its shape, another the distribution of proteins inside. We need to understand how these data are related.

The fruit fly Drosophila is a tiny creature with a brain the size of a poppy seed. It contains about one hundred thousand nerve cells, and scientists have already mapped all their connections, like a city subway map where stations are neurons and lines are the links between them. This map is called a connectome.

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.

Imagine a city map where streets are replaced by car speeds and buildings by the metal content of stars. Such a map can tell you where the center is and where the outskirts are, even without knowing exact distances. This is precisely the approach astronomers took with our Galaxy, using longitude–velocity (ℓ–v) diagrams.

The cerebral cortex is like a crumpled sheet of paper: it has thickness, curves, and folds of varying depth. With age, this sheet changes unevenly — thinning in some places, losing the sharpness of its folds in others. Previously, doctors assessed brain age as a whole, as if looking at the sheet from a distance.

Graphs are considered identical if their 'terrain' matches. Imagine two maps with mountains and valleys: even if drawn differently, the heights and steepness of slopes can be the same.