Simulation three of five

Diffusion, and the shape it runs through

Hold the contagion completely fixed. Change nothing about how infectious it is, how many people it starts with, or how many neighbours the average person has. Change only the pattern of who is connected to whom, and the same thing spreads three entirely different ways.

There is a habit of speaking about a rumour, a technology or a disease as though its spread were a property of the thing itself. Some things are catching and some are not. This model is here to make that habit uncomfortable.

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

Switch between them with the transmission probability untouched and watch the curve underneath change shape completely.

01The three shapes

Each dot is an agent. A line is a connection. Filled dots have it and hollow dots do not. The lattice and the small world are drawn as a ring because that is genuinely what they are. The scale free graph is drawn with its heavily connected nodes near the middle and its poorly connected ones near the edge, 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 can only move one step around the circle at a time. Reaching the far side means crossing every agent in between, and the curve that produces is close to a straight line: a front advancing at a constant rate.

Now move a small handful of connections to random destinations. The average person still has the same number of neighbours. 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 readout under the picture gives you both numbers to compare.

The scale free graph does something else again. It has hubs. Once the contagion reaches a hub it is everywhere at once, and the curve becomes a very sharp step rather than a rise. Whether it reaches a hub early is close to being 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. That is a reasonable description of a virus and a poor description of most things people adopt.

Switch the model to the threshold rule. Now an agent needs several neighbours to have adopted before it will, because the thing being spread 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 and watch what happens to the long connections that were so useful a moment ago. A single connection reaching across the ring delivers exactly one adopted neighbour to the agent at the far end, and one is not enough. The long connection is wasted. What a complex contagion needs is not a shortcut but a wide bridge: several connections arriving at the same place from the same adopted region. That is what the clustered lattice provides and what random rewiring destroys.

So the ranking reverses. Structures that carry a simple contagion fastest can be the ones that stop a complex one, and the reversal is not a subtlety at the margin. 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 is interested in, the answer to 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, and this page is what changing the structure of a graph does.

It is worth noticing that this is a larger power than the one in the previous simulation. There the system chose what an agent was shown. Here the system chooses the shape of the space in which anything can travel at all, and the same contagion in the same population goes from unstoppable to contained without anybody's mind being changed about anything.

05Where the model stops

Assumptions and limits

Nobody recoversAn agent that adopts stays adopted for ever. There is no recovery, no immunity, no forgetting and no abandoning, so the curves can only rise. Any real epidemic model would need at least one more state.
The graph never changesConnections are fixed for the whole run. 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. The only thing that distinguishes agents here 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. That choice favours the threshold rule. A single random starting adopter would make the threshold runs die almost always, which is itself a finding, and it is not the one this page is showing.
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. The paper reports that such networks are highly clustered like lattices yet have small characteristic path lengths like random graphs, and that infectious diseases spread more easily in them than in regular lattices. 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. The paper argues that complex contagions require social affirmation from multiple sources, that long ties can hinder rather than help their diffusion as the adoption threshold rises, and that what matters is the width of the bridges across a network rather than the length of individual ties. 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.