Simulation three of five

Diffusion, and the shape it runs through

Hold the contagion fixed, change only the pattern of who is connected to whom, and the same thing spreads three entirely different ways.

A rumour, a technology or a disease gets talked about as though its spread were a property of the thing itself; this model makes that habit uncomfortable.

Three networks of the same size and roughly the same number of connections carry one contagion at one transmission probability: a ring lattice where everyone knows their immediate neighbours and nobody else, a small world which is that same ring with a few connections moved somewhere far away, and a scale free graph where a few nodes have very many connections and most have very few.

Switch between them without touching the transmission probability.

01The three shapes

Each dot is an agent and each line a connection; the lattice and the small world are drawn as rings because that is what they are, and the scale free graph puts its heavily connected nodes near the middle because that is the fact about it that matters.

Figure one change the structure, leave the contagion alone

The network

Adoption, with the previous runs kept

02Why the ring is slow and the rewiring is not

On the ring a contagion moves one step around the circle at a time, so reaching the far side means crossing every agent in between, and the curve is close to a straight line: a front advancing at a constant rate.

Move a small handful of connections to random destinations: the average person still has the same number of neighbours and almost every connection is still local, but the number of steps between two randomly chosen agents collapses, because a few connections now cross the whole ring and the contagion uses them. Watts and Strogatz showed that a very small amount of that rewiring is enough to take the path length of a random graph while keeping the local clustering of a lattice, and that infectious things spread more easily in such a network than in the lattice it came from.

The scale free graph has hubs: once the contagion reaches one it is everywhere at once and the curve becomes a sharp step rather than a rise, so whether it reaches a hub early is close to the whole story of the run.

03The threshold rule changes the answer

Everything above assumes a simple contagion, one exposure can do it and each infected neighbour is an independent chance, which describes a virus well and most things people adopt badly.

Switch the model to the threshold rule: an agent now needs several neighbours to have adopted before it will, because the thing spreading is costly, risky or embarrassing and one person doing it is not enough. Centola and Macy call this a complex contagion and show that it does not behave like the simple kind at all.

Run the threshold rule on the small world: a single connection reaching across the ring delivers exactly one adopted neighbour to the agent at the far end, and one is not enough, so the long connection is wasted. A complex contagion needs a wide bridge, several connections arriving at the same place from the same adopted region, which the clustered lattice provides and random rewiring destroys.

So the ranking reverses: structures that carry a simple contagion fastest can be the ones that stop a complex one. Set the threshold to two and compare the lattice with the small world.

04Why this belongs to SIMEA

Because the network in this model is not a fact of nature: in almost every setting the programme cares about, who is connected to whom is decided by a system, who appears in a feed, who is grouped with whom, which messages are delivered and which are quietly not. A platform that changes its recommendation rule has changed the structure of the graph.

This is a larger power than the one in the previous simulation, where the system chose what an agent was shown: here it chooses the shape of the space anything can travel in at all, and the same contagion in the same population goes from unstoppable to contained without anybody's mind being changed.

05Where the model stops

Assumptions and limits

Nobody recoversAn agent that adopts stays adopted, with no immunity, forgetting or abandoning, so the curves can only rise and any real epidemic model would need at least one more state.
The graph never changesConnections are fixed for the whole run, while real networks change in response to what is spreading through them, often faster than the thing itself spreads.
Everyone is identicalNo agent is more susceptible, more persuasive or more exposed than any other, so the only thing that distinguishes them is their position in the graph.
The seeding is clusteredA complex contagion cannot start from a single adopter, so every run here starts from one agent together with its neighbours, a choice that favours the threshold rule. A single random starting adopter would make the threshold runs die almost always, which is a finding of its own and not the one this page shows.
These are three families, not three networksEach structure is drawn from a family with a seed, another seed gives a different graph with the same statistics, and single runs of a stochastic contagion vary a great deal. Press reset a few times before believing any one curve.
Not a public health modelNothing here is calibrated to any disease, technology or population, and no number on this page should be carried into an argument about a real one.

06Sources

The small world construction used here, in which a ring lattice is rewired with a small probability. Duncan J. Watts and Steven H. Strogatz, Collective dynamics of 'small-world' networks, Nature 393(6684), 4 June 1998, pages 440 to 442. nature.com/articles/30918 Checked 2026 08 22.
The threshold rule and the reversal it produces. Damon Centola and Michael Macy, Complex Contagions and the Weakness of Long Ties, American Journal of Sociology 113(3), 2007, pages 702 to 734. journals.uchicago.edu/doi/abs/10.1086/521848 Checked 2026 08 22.
The scale free graph is built by preferential attachment, in which each arriving node connects to existing nodes with probability proportional to how many connections they already have. Albert-László Barabási and Réka Albert, Emergence of Scaling in Random Networks, Science 286(5439), 1999, pages 509 to 512. science.org/doi/10.1126/science.286.5439.509 Checked 2026 08 22.