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Social influence can lead to behavioural ‘fads’ that are briefly popular and quickly die out. Various models have been proposed for these phenomena, but empirical evidence of their accuracy as real-world predictive tools has so far been absent. Here we find that a ‘complex contagion’ model accurately describes the spread of behaviours driven by online sharing. We found that standard, ‘simple’, contagion often fails to capture both the rapid spread and the long tails of popularity seen in real fads, where our complex contagion model succeeds. Complex contagion also has predictive power: it successfully predicted the peak time and duration of the ALS Icebucket Challenge. The fast spread and longer duration of fads driven by complex contagion has important implications for activities such as publicity campaigns and charity drives.

There is a large body of evidence—which is increasingly quantitative—that the effect of social influence can be a significant driver of human behaviour. Improved understanding of this phenomenon should help to predict various phenomena of interest, for example how well public-health interventions will work, or the use of ‘nudges’ in public policy [

In particular, the work of Christakis and Fowler [

Models of social influence have taken three main forms: experimental generalisations, agent-based models, and compartmental models. Experimental generalisations take historical data on the spread of a behaviour and try to find functional forms which match that data. One of the first examples of this approach was by Bass [

Agent-based models take almost the opposite approach to the experimental generalisations mentioned above, in that they simulate all of the individual- (or ‘agent’-) level processes occurring and then try to calibrate the model by matching the aggregate behaviour to data [

Compartmental models put each individual in the population into one of a certain number of states, or compartments. Only the number of individuals in each compartment and the transitions between them are tracked, and hence the number of dimensions of the system can be much less than an equivalent agent-based model. This in turn allows a compartmental model to be fitted to data more easily than agent-based models, while remaining a mechanistic description of the underlying system. Treating social influence in this compartmental way has a long history, an example being Dietz [

Very few compartmental models for social influence modify the form of the infection term in the standard model. However, as shown in experimental studies [

While the work of Centola involved a controlled study to test for effects of complex contagion, if this is a strong effect in general then it should be possible to find evidence for it in observational data at the population level. In this paper, we set up simple and complex contagion models for populations, which we compare to search-interest data on photo fads—i.e. electronically mediated real-world behaviours—using maximum likelihood estimation and information theoretic model selection. We show using these methods that complex contagion is strongly favoured as a model of social influence, which can then be used predictively.

We propose here a general modelling framework based on a non-linear continuous-time stochastic process that enables us to capture most existing models of behavioural contagion as special cases. We start with a vector of non-independent integer random variables, _{1}, …, _{n}, _{i} represents the number of individuals engaging in the behaviour of ‘type’

We follow the broad mathematical approach of [

In terms of the ‘infectious’ classes that spread behaviour, we use two: _{1} = _{2} =

If we consider a large fixed population of size ^{−1/2}):

Our simple contagion model is a straightforward modification of the standard SIR model:

We can also now make our verbal argument above about ‘excitable’ models more quantitatively. Consider the special case of our models in which _{i} = 2 and _{i} = 1/

Our main data source was Google search volumes for a particular category of Internet meme: photo fads. These fads consist of participants uploading photos of themselves in a particular pose; descriptions of the fads are given in

Explanations of the nomination and photo fads (excluding those that are potentially offensive).

Neknomination | Consuming an alcoholic drink in one gulp. |

Icebucket Challenge | Pouring a bucket of iced water over the participant’s head and / or donating to ALS research. |

Cat Beard | Posing so that a cat’s face looks like the participant’s beard. |

Owling | Crouching like an owl. |

Cat Breading | Putting bread around a cat’s face. |

Bradying | Copying a widely publicised photograph of NFL player Tom Brady. |

Vadering | Recreating the |

Hadokening | Recreating the ‘Hadoken’ move from the video game |

Batmanning | Hanging upside-down like a bat. |

Lying Down Game | Lying rigidly in a public place. |

Leisure Diving | Appearing to carry out a leisure activity while diving into a swimming pool. |

Sleeveface | Holding a record sleeve that appears to replace the participant’s face. |

241543903 | Placing the participant’s head in a freezer. |

Perfect Splits | Performing the splits. |

Pottering | Appearing to fly on a broomstick as in |

Planking | Lying rigidly across a raised object. |

Skywalking | Posing on top of a high building. |

Tebowing | Copying the celebration stance of NFL player Tim Tebow. |

Teapotting | Holding the participant’s arms like a teapot. |

Dufnering | Copying a widely publicised photograph of golfer Jason Dufner. |

Stocking Planking | Recreating a stock photograph. |

Caught Me Sleeping | The participant pretends to be asleep while they are demonstrably photographing themselves. |

People Eating Money | Pretenting to eat banknotes. |

Playing Dead | Simulating a murder scene. |

Horsemanning | Pretending to be a headless ghost. |

Fitted simple and complex contagion models and data for search volumes as a percentage of peak, ordered by log-likelihood difference from best fit to worst. Fads with potentially offensive content are included for completeness, but without sketches.

Photo fads were chosen because they tended to have distinctive names, allowing them to be clearly identified in search data; they involved real-world behaviours that were spread by and reported on the Internet; and they were undertaken for no ostensive reason beyond their online popularity. These photo fads tended to be global phenomena, and hence took place in a population large enough to satisfy the assumptions of the ODE model.

To acquire these data, we visited the site

We avoided selection bias by taking all 37 Photo Fads listed on the website KnowYourMeme.com (a comprehensive source of information on internet memes). The search data was obtained from Google Trends, and consisted of search volumes quoted in terms of a percentage of the peak value, and aggregated weekly. We fitted models to the 26 fads that had sufficient (greater than 15) non-zero data points to allow the dynamics of behavioural contagion to be identifiable.

The data take the form of a set of real-valued Google Trends at discrete time points

For each set of fad data we calculated the Akaike Information Criterion (AIC) [

Some fads showed two clear peaks in the data. For each time series with more than one mode, we therefore fitted a model in which two separate sub-populations become infected, with the total infected fraction being the sum of infected in the sub-populations. The parameters for each population were fitted independently, except for the thresholds in the complex contagion model that were assumed constant. The AIC was again used to select between one-population and two-population versions of both contagion mechanisms.

The complex contagion model was used to predict the future spread of another fad, ‘ALS Icebucket Challenge’. This was a charity campaign that spread in a viral manner, with friends directly nominating each other to take part. A previous fad, ‘Neknomination’, had spread in a similar way, and so we used the parameters fitted from that fad to predict the future spread of ‘ALS Icebucket Challenge’. We made a verifiable prediction at the start of the campaign, shown in

Prediction of search volume for Icebucket Challenge, based on data available at the time (circles) and compared to the final volume (crosses). Top plot: complex contagion model; bottom plot: simple contagion model.

Of these fads, 22 of 26 showed significant evidence that complex contagion was a better model for the data than simple contagion. The fitted timeseries for all fads are provided in

The complex contagion model’s threshold for social influence allows it to capture the fast increase in popularity seen in most of the trends. The linear force of influence in the simple contagion model, however, means that it is slower to build to peak popularity. After the peak, the simple contagion model has a constant rate for individuals leaving the fad, leading to exponential decay in popularity. The complex contagion initially shows a fast drop in popularity as individuals see that their contacts are already taking part in the fad, but once most of the population has stopped taking part the few individuals remaining take longer to give it up. This correctly captures the ‘long tail’ of popularity seen in the data.

For a minority of fads, the simple contagion model was also adequate, but this was typically linked to few datapoints and / or poor signal quality. In terms of values for the parameters, these were quite variable between fads, which would be expected given e.g. the differing levels of effort needed to participate in each fad. Full fitted parameter values are available in

The log-likelihood difference between the simple and complex contagion models. (***) is very strong evidence, (**) is strong evidence, (*) is positive evidence, (.) is no significant evidence for either model, (–) is strong evidence against. † means that AIC selected models with two peaks.

Photo Fad | Log-likelihood difference | AIC Evidence | |
---|---|---|---|

Sneaky Hat | 47.4 | *** | |

Cat Beard | 44.0 | *** | |

Owling | 39.3 | *** | |

Cat Breading | 38.9 | *** | |

Lynndie England | 27.3 | *** | |

Bradying | 26.6 | *** | |

Vadering | 24.6 | *** | |

Hadokening | 24.2 | *** | |

Batmanning | 24.2 | *** | |

Lying Down Game | 17.8 | *** | |

Leisure Diving | 16.8 | *** | |

Sleeveface | 12.1 | *** | |

241543903 | 11.7 | *** | † |

Perfect Splits | 11.7 | *** | |

Mamming | 10.2 | *** | |

Pottering | 9.5 | *** | |

Planking | 8.5 | *** | † |

Skywalking | 7.1 | *** | † |

Tebowing | 6.9 | *** | † |

Teapotting | 6.0 | *** | |

Dufnering | 4.7 | ** | |

Stocking Planking | 2.0 | * | |

Caught Me Sleeping | -0.0 | . | † |

People Eating Money | -1.7 | . | † |

Playing Dead | -1.9 | . | † |

Horsemanning | -3.2 | – |

Social influence, or the effect of others’ behaviour on our own, is important in understanding many aspects of human behaviour. Although several mechanisms have been proposed to model this influence, it has not so far been possible to distinguish between these mechanisms in observational data. Here we have shown that the observed spread of real-world behaviours linked to online trends can be explained using a complex contagion model, and demonstrate that this model provides a predictive modelling framework for real-world behaviours spread online.

Fitted parameter values for complex contagion models. The second sets of parameters, if present, are for two-peak fits. Plain text comma-separated values.

(CSV)

Fitted parameter values for simple contagion models. The second sets of parameters, if present, are for two-peak fits. Plain text comma-separated values.

(CSV)

We would like to thank two anonymous reviewers for their helpful comments, which have improved this manuscript.