In particular, the work of Christakis and Fowler [ 7 ] analysed longitudinal social network and health data from the Framingham Heart Study and showed that if an individual had a friend, sibling, or spouse who had become obese in a given time interval then that individual was significantly more likely to also become obese. Similar results were also found when studying the cessation of smoking [ 6 ]. This proved controversial; it has been shown that social influence cannot be distinguished from homophily, or the clustering of individuals who are similar, in observational studies [ 8 ]. Aral et al. [ 9 ] try to determine an upper bound for the importance of social influence for behaviour spread, and find that for the adoption of a particular social media app at least half of the observed adoption events can be attributed to homophily. This discussion highlights the difficulty of using observational data to distinguish the effect of individual-level factors, in the form of homophily, from social influence. This same difficulty is not present in experimental data, however. Bond et al. performed a randomised controlled trial over Facebook to find evidence for social influence on the decision to vote [ 3 ]. By sending direct messages to ‘seed’ nodes in a network, and then tracking the behaviour of their contacts, the experimenters showed that individuals were significantly more likely to vote if one of their close friends had received a message. In a study also related to electronically mediated real-world behaviour, Centola [ 5 ] placed individuals in an artificially-structured online community in which users were informed about the health activities of their assigned contacts. This experiment showed that social signals significantly increased the likelihood of an individual taking part in a behaviour, and that up to three additional social signals significantly increased this likelihood even further. Taken together, these studies show that while individual-level factors are significant, social influence is also important in determining health behaviours.

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 [ 1 – 7 ].

Previous models

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 [10], who created a model of product adoption based on the idea of innovators and imitators. More recent attempts include fitting a variety of statistical distributions to the popularity of Internet memes [11]. The main disadvantage to this approach is that it does not provide a mechanistic model for social influence, and hence does not provide much insight into individual-level processes.

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 [12, 13]. Agent-based models are useful tools for reproducing the complex phenomena observed in real systems, but it is extremely difficult to fit their parameters to data well.

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 [14] who developed a model for the spreading of rumours similar to models from epidemiology. In fact, much of the social influence literature using compartmental models has been based on the SIRS model of an epidemic. In the SIRS model there are three compartments: susceptible (S), infectious (I), and recovered (R). Susceptible individuals have not yet been infected with the disease, infected individuals currently have the disease and are spreading it, and recovered individuals have had the disease but are no longer spreading it. In the standard SIRS model used to model infections [15], individuals moving between these compartments are modelled by a continuous time Markov chain with events and rates (1) This standard model can be modified by changing the functions for the rates, and by adding or removing compartments. For models of social influence on behaviour, the ‘infectious’ compartment represents individuals taking part in a behaviour and spreading it, and ‘recovered’ means the individual is no longer influencing others to take part in the behaviour. Many previous studies of social influence modify the standard model by changing the rates at which at which individuals move between compartments. Isham et al. [16], for example, developed a model for rumours on a network based on the SIR model modified to include ‘stiflers’ who cause infectious individuals to recover at a faster rate. One important additional source of realism is to consider the impact of contact network structure on spreading dynamics, however if the degree distribution of the network is not too heterogeneous and other properties such as clustering, assortativity and path length are not too far from a random graph then dynamics such as Eq (1) should be a good approximation [17].

Very few compartmental models for social influence modify the form of the infection term in the standard model. However, as shown in experimental studies [5], there is significant evidence that the form of ‘infection’ in social influence is different to that in a biological epidemic. The important difference is the number of exposures to infection that an individual must receive before becoming infected: in biological infection only one source of infection is required for a non-zero probability of infection, whereas in social influence multiple sources are required. Dodds and Watts [18], for example, generalise the SIS model to allow for infection processes that require multiple exposures.