The Cascade That Needs a Crowd

Two ways a thing spreads

The drop before last took a square grid, sprinkled a few live cells across it, and let a rule fill in any cell with at least two live neighbours. On the lattice, whether that flood reaches across the whole board depends on the grid's dimension. This drop keeps the exact same rule, switch on once r of your neighbours are on, and runs it somewhere else: not on a tidy grid but on a random graph, a scatter of nodes wired to each other at random. The rule is the same. The behaviour is not.

Start with the difference between two kinds of spreading. A cold is simple contagion: one infected person breathing near you is enough. On a network that makes a cold unstoppable, one case infects a neighbour, who infects a neighbour, and the infection walks the entire connected web it started in. A single spark takes the whole component. There is no such thing as "not quite enough" cold.

But most things that move through a crowd are not colds. You do not join a protest, buy the expensive new phone, adopt a risky farming method, or pull your savings out of a wobbling bank because one acquaintance did. You wait until several of the people around you have. Damon Centola and Michael Macy named this complex contagion in 2007: adoption that needs social reinforcement, more than one neighbour, before it takes. The sociologist Mark Granovetter had written down the mechanism decades earlier, in 1978, as a threshold model of collective behaviour: each person carries a private number of neighbours they need to see act before they will act too. Set that threshold to one and you are back to a cold. Set it to two or three and, as we are about to see, the entire character of the spread changes.

Physicists came to the same rule from a different door. In 1979 Joseph Chalupa, Paul Leath and Gary Reich, modelling how magnetism switches on in a dilute, disordered material, defined bootstrap percolation: seed some sites active, then activate any site with at least r active neighbours, and iterate. It is the same rule as Granovetter's, and it is the exact activation mirror of the k-core from drop #047: the k-core deletes every node with fewer than k surviving neighbours until a cascade of deletions stops; bootstrap percolation adds every node with at least r active neighbours until a cascade of activations stops. One peels a network down. The other floods it up. And both, on a random graph, do the same startling thing at their threshold: they jump.

Light the first ones

Module 01 is the cascade, live, on one fixed graph of 140 nodes. Every node carries a fixed random threshold, so sliding the seed density only ever adds sparks, never removes them. The initial seeds glow ember; every node the cascade then switches on lights up in ice; dormant nodes stay faint. Pick how demanding the rule is with r, set the seed density, and let it settle, or step it one round at a time to watch the front travel.

At r = 1 there is nothing to see but plumbing: each seed fills its own connected patch and stops, and adding seeds just lights more patches. Switch to r = 2 and drag the seed dial slowly upward. For a while, almost nothing happens, a 5% seed, a 7% seed, and the board stays mostly dark, because most dormant nodes never manage to collect two active neighbours at once. Then, between a 7% seed and an 8% seed, the board flips. Seven percent fizzles, staying around seven percent. Eight percent floods, a chain reaction that lights 93% of the graph (130 of the 140 nodes) in a handful of rounds. One extra percentage point of seed is the entire difference between a few dead embers and a network-wide takeover.

That is the phenomenon in one board: below a knife-edge the cascade is a local fizzle, above it a global flood, and there is essentially no seed density that gives you a modest partial spread. As ever, the sharp way to see it is to stop staring at one small graph and average over many.

The cliff

Module 02 builds thousands of fresh random graphs at a fixed average degree of five, seeds each at a chosen density, runs the cascade to the end, and plots the final active fraction against the seed density. Three curves, for graphs of 250, 2,000 and 12,000 nodes, and a pale theory line behind them.

Choose r = 1 and the plot is almost boring: the active fraction is pinned near the ceiling at every seed density, because a single seed is enough to walk the whole giant component. Simple contagion has no seed threshold worth the name, it is already global from the first spark. Now choose r = 2. The curve lies flat on the floor, near zero, and then, at a seed density of only about 2%, it jumps almost vertically to nearly the whole graph. Choose r = 3 and the same thing happens later, at about 14%, and higher. Watch the three sizes as you do this: at 250 nodes the jump is a rounded shoulder, smeared by luck; at 2,000 and 12,000 it tightens into a genuine vertical cliff, a value that is near zero on one side and near one on the other with nothing in between. That is a first-order transition, a discontinuity, the same fingerprint the k-core's leap carried.

And there is a pattern in where the cliff sits: it marches to the right as r grows. The more neighbours you insist on seeing act before you will, the larger the initial crowd it takes to set the cascade off, and, as the plot shows, the more sudden the takeover when it finally comes. Demanding robustness in what will move you does not make the spread gentler. It makes it rarer and more violent.

The tipping point

Why a jump, and not a ramp? The answer is one self-consistency picture, and it is the seed-density mirror of the one that explained the k-core.

Track a single quantity: z, the probability that a given node ends up active. A node is active if it was one of the seeds, which happens with probability p, or if enough of its neighbours are active. The number of a node's neighbours is Poisson with mean c, each active with probability z, so the chance it collects at least r active neighbours is P(Po(c·z) ≥ r), and self-consistency demands

z = p + (1 − p) · P(Po(c·z) ≥ r),

with the final active fraction equal to that same z. Module 03 draws it: the curve y = g(z) = p + (1 − p)·P(Po(c·z) ≥ r) against the line y = z. Their crossings are the solutions, and the cascade, which starts from just the seeds and can only grow, settles at the lowest one.

For r = 1 the curve g(z) = p + (1−p)(1 − e^(−c·z)) leaves the y axis at height p and immediately climbs steeply, staying above the line until it meets it once, high up. There is a single crossing, and it slides smoothly as you move the seed. No jump, ever. That is simple contagion.

For r ≥ 2 the shape is different, and it is the whole story. The curve still starts at the seed floor p, but now it leaves the axis flat, hugging the floor, because collecting two or more active neighbours is a rare, second-order event when few nodes are active yet. So just past the axis the line y = z overtakes it, and the curve dips below the line before eventually rising back above it and saturating. That gives three crossings: a low one where the cascade comes to rest (a few local patches), an unstable middle one that is the true tipping threshold, and a high one the cascade cannot reach from below. Slide the seed density up and watch the picture: the low crossing and the unstable crossing drift toward each other, meet, and annihilate. Mathematicians call that a fold, or a saddle-node bifurcation. The instant it happens, the resting place the cascade was sitting in simply ceases to exist, and the only solution left is the high one. The active fraction leaps up to it.

That is the deepest way to say what the jump is. The flood is not the cascade suddenly growing faster. It is the low, stable, safe state, the fizzle, disappearing out from under it, so that the system has nowhere to sit but the top. A tipping point is not a place where a little more push gives a little more motion. It is a place where the ground you were standing on vanishes.

Why it is worth knowing

This one rule, activate at r neighbours, is the honest skeleton under a pile of things that look nothing alike from the outside. It is the physicist's model of a cascading failure in a power grid or a financial system, where each node fails once enough of its neighbours have and the failure can either die locally or take down the network. It is the sociologist's model of how a norm, a fashion, a protest, or a bank run tips from a fringe to a majority, and why the tip, when it comes, is abrupt rather than gradual. It is why marketers chase a critical mass of early adopters rather than a broad thin sprinkle, and why the same product can flop from a 7% launch and explode from an 8% one. And it is the warning the k-core carried, read forward instead of backward: a network can look stable and sparse right up to the moment a small increase in pressure flips it wholesale.

The threshold r is the knob that separates the two worlds. At r = 1 you have a cold, unstoppable and gradual. At r = 2 or more you have a norm, resistant until it is not. The number 2% was not looked up anywhere. It was measured, live, as the exact seed density at which a curve stops dipping below a line and a resting place for the fizzle winks out of existence. Nothing above was fetched: a random-number generator, a graph knit from its output, a rule that lights a node when enough of its neighbours are lit, and a count of how far the fire ran.