How Opinions Spread Through a Network

Diffusion research inherited its mathematics from epidemiology, and the inheritance has been productive and misleading in roughly equal measure. Epidemic models assume that contact with an infected individual carries a fixed transmission probability, independent of how many other contacts are infected. For information, that assumption is often reasonable. For opinions and behaviour, it is usually wrong.

Simple versus complex contagion

Simple contagion needs one exposure. Hearing a news item once is enough to know it. Complex contagion requires reinforcement: several independent contacts must adopt before a person does. Costly, risky or socially visible behaviours - changing a political position, adopting an unfamiliar practice, joining a protest - are complex contagions.

The distinction produces opposite predictions about structure, which is what makes it useful rather than merely descriptive.

Why long ties help one and hurt the other

A long tie connects otherwise distant parts of a network. For simple contagion these are decisive: they create shortcuts and collapse the time it takes something to cross the whole system. This is the standard small-world result and it explains why news reaches everywhere so quickly.

For complex contagion, long ties are close to useless. A single distant contact provides one exposure, and one exposure is below threshold. What complex contagion needs is wide bridges: several connections between the same two clusters, so that a person in the receiving cluster gets multiple independent confirmations. Clustered, locally redundant structure - the kind that slows simple contagion - is what makes complex contagion possible.

The practical consequence is direct. A campaign aiming to inform should target well-connected nodes with long ties. A campaign aiming to change behaviour should saturate a dense local cluster and let it spread outwards. Doing the first while intending the second is a common and expensive error.

Threshold models

The workhorse formalisation gives each individual a threshold: adopt when the fraction of contacts who have adopted exceeds it. Thresholds are heterogeneous - a few people adopt with almost no social proof, most need substantial reinforcement, some never adopt.

Two properties of these models are worth stating because they are counterintuitive:

• Cascades depend on the low-threshold minority being connected. Early adopters scattered across the network never reach critical mass locally. The same number clustered together can start a cascade. Composition matters less than placement.
• Fractional thresholds make well-connected nodes hard to convert. If the rule is "adopt when 30 per cent of contacts have", a node with 200 contacts needs 60 of them. A node with 10 needs 3. Highly connected nodes are excellent transmitters and poor early adopters - which is the opposite of how influencer strategies usually assume they behave.

Where opinion dynamics differ from adoption

Adoption is binary. Opinions are continuous and can move by degrees, which produces a different family of models. The most informative of these add bounded confidence: people are influenced only by others whose views are already reasonably close to their own, and ignore views beyond that range.

The behaviour of such models is sharp. When the confidence bound is wide, the population converges to a single consensus. When it narrows past a threshold, the population fragments into stable clusters that no longer influence each other at all. The transition is abrupt rather than gradual. This is the cleanest formal account available of how a society can move from disagreement to non-communication, and it connects directly to media, polarisation and misinformation, because platform curation directly affects the effective confidence bound.

The identification problem

Observing that connected people behave similarly does not establish influence. Three mechanisms produce the same correlation: influence, homophily - similar people connect in the first place - and shared environment. Separating them from observational data is genuinely hard, and a large early literature on social contagion substantially overstated influence by not doing so. Credible designs use randomised exposure, natural experiments in tie formation, or timing arguments that homophily cannot explain. The methodological standard is the same one described in validation and calibration.