Random graph
Named by 18 essays across 6 fields — each of them below, with the objects they name alongside it.
The moment everything joins up
Add edges to a set of points one chance at a time and the graph goes from dust to a single piece — not gradually, but over a window that narrows as the point count grows. The last obstacle is almost always a single point with no edge at all, and that is what fixes where the change happens.
The moment a giant appears
Raise the chance of an edge slowly and a random graph does nothing for a long time, then in a narrow window acquires a component holding a definite fraction of everything. The fraction is the root of an equation, and the equation says why the transition is where it is.
Two thresholds, not one
A random graph acquires a piece holding most of its points at average degree one, and is still not connected. Connectivity waits until the average degree reaches the logarithm of the size, and what holds it up is the very last isolated point.
Finding a threshold with two moments
Every monotone property of a random graph has a threshold, and locating one is nearly always the same two calculations — count what the property needs, and check the count does not concentrate on rare cases. The triangle is where the method is cleanest.
Nearly always, or nearly never
Toss a coin for every pair of points and ask whether the graph that results has some property. For a property a first-order sentence can state, the answer in the limit is never a genuine probability — it is zero or it is one, and the game is what proves it.
The window where the giant is born
Below the threshold the largest piece is a few dozen points, above it a definite fraction of everything. At the threshold it is neither, and the size it does take — the two-thirds power — is an exponent with no elementary derivation that a measurement finds immediately.
Sharp, or merely a threshold
Every monotone property of a random graph has a threshold. Some of them turn on over a range that shrinks relative to the threshold as the graph grows, and some do not — and which kind a property is turns out to be decided by whether it is about a local structure or about the whole graph.
The graph that coin tosses always make
Take infinitely many vertices and toss a coin for every pair to decide whether they are joined. The result is random in every detail — and, with probability one, it is always the same graph. The same graph can be written down without any coins, by joining two numbers when one binary digit of the larger is a one.
The shape a random ball grows into
Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.
The squares that answer every request
Rado's graph has, for any finite sets U and V, a vertex joined to all of U and none of V. A finite graph can only answer the small requests, and the Paley graphs — residues modulo a prime, joined when their difference is a square — are the classic way to build one. Checking every request exactly finds the thresholds: 13 residues answer every request of two, 29 every request of three, 89 every request of four. The theorem that guarantees it asks for 64, 576 and 4,096. Coin-toss graphs of the same sizes almost never manage it.
Almost every tree can be turned over
Cayley's n^(n−2) counts trees with labels on their points. Take the labels off and the count has no formula, because a symmetric shape absorbs labellings: the star on seven points can be labelled only seven ways, the one asymmetric shape 5,040. The bookkeeping that reconciles the two counts says something unexpected — almost every tree, labelled or not, can be turned over onto itself, where almost every graph cannot.
Half the neighbours forces a tour
Deciding whether a graph has a closed tour through every point is one of the hardest problems computers are asked to solve. Gabriel Dirac found a condition that settles it in one glance: if every point is joined to at least half the others, a tour exists. The proof is a counting argument on a single path, it runs as an algorithm, and two small graphs show that the half cannot be lowered — while random graphs show how much stronger than necessary it is.
Every light on, from every light off
In Lights Out, pressing a light toggles it and its neighbours. On a five-by-five board only one pattern in four can be reached from all off — and yet all on is always one of them. The same is true on every board and on every network of lights whatever, provided the influence runs both ways. The proof is a single observation about symmetric matrices over a field with two elements, and it fails, as a three-arrow example shows, the moment influence runs one way.
Half the chance for every silent set
In the lights game where a press toggles only the neighbours, a random network can be lit about 56 times in a hundred when it has an even number of vertices and 44 when odd. The numbers are 0.560562 and 0.439438, they add to one, and they come from a chain: each new vertex raises or lowers the dimension of the silent sets by one, and each dimension halves the chance.
The order that keeps elimination narrow
Eliminating letters one at a time costs whatever the widest step costs, and the least width any order can achieve is the treewidth of the formula's graph. Finding it exactly is NP-hard, but a search over sets of letters instead of sequences finds it for small graphs, and a greedy rule that adds the fewest new edges finds it on 566 of 600 random graphs and misses by more than one only once. On Tseitin's 6 × 6 grid that rule forms twenty-four times fewer resolvents than sweeping row by row. What it cannot do is prove that its order is the best.
Triangle-free, and two-sided eventually
Every graph with no triangle and the most edges possible is bipartite. Erdős, Kleitman and Rothschild proved that almost every triangle-free graph is bipartite too — and yet among the 246,348,115 triangle-free graphs on nine labelled vertices, fewer than half are. The share falls before it rises, and the reason is that small graphs are sparse.
The cheapest tree costs ζ(3)
Join n towns by every possible road, give each road a cost drawn at random between nought and one, and find the cheapest network of n − 1 roads that connects them all. However many towns there are, the bill tends to 1 + 1/8 + 1/27 + 1/64 + … = 1.2020569. The proof watches the cheap roads arrive one at a time and turns the cost into the area under a curve that counts the pieces of a growing random graph.
No graph is poorer in squares than chance
Take any graph with a fixed share of its possible edges and count how often four points chosen in order close up into a ring. However the edges are arranged — in cliques, in two-sided blocks, at random — the count never falls below what a random graph of the same density gives. Sidorenko conjectured that the same holds for every pattern without an odd cycle. For a triangle it is false; for a path or a ring of four it takes two lines; for K₅,₅ with a ten-cycle removed nobody knows.
Named alongside it
The objects these essays reach for when they reach for this one.
ThresholdExhaustive searchExpectationPhase transitionBipartite graphComponentConnectivityCounting two waysExtension propertyFinite fieldFirst momentGreedy algorithm