A Simple Model for BMI Change in a Social Network
The obesity epidemic is both highly known and highly visible, yet also little understood. In a landmark paper in the New England Journal of Medicine in 2007, Fowler and Christakis  presented evidence suggesting that obesity might spread socially. That is, an essential part of the epidemiology of obesity may involve the social interactions between people and the spread of new behaviors throughout a social network. Although such ideas are not novel for other public health crises, such as smoking or other drug use, given that obesity is dependent on a whole suite of different lifestyle patterns, this result was surprising to many. Working off of previous research studying the propagation of smoking via mathematical models of human behavior , Professor Abrams of the Northwestern University Department of Applied Mathematics, graduate student John Lang of the University of Waterloo Department of Applied Mathematics, and Professor Hans de Sterck of the University of Waterloo Department of Applied Mathematics have now turned their attention towards developing a useful and predictive model for the propagation of obesity. Joining them this summer I have written several programs in the MATLAB programming language which test a simplified version of this model and evaluate its predictions.
The model we used describes the evolution of the Body Mass Index (BMI) of all individuals in a social network over time. BMI is one of the simplest and most common ways of acquiring a height-adjusted measure of a person's weight. Because BMI adjusts for height, a high BMI is a better indicator for obesity than a high weight alone. Research also indicates that BMI is a successful proxy for body fat percentage.
We model the change in BMI for each individual within a social network as having two given components. The first component represents the effects of social influence or peer behavior. The second represents the effects of individual preferences. Note that in economic terms these two components together represent the total utility function for each individual with respect to BMI. Hence part of the utility (or benefit, usefulness) of having any given BMI is assumed to be due to matching one's lifestyle with that of one's closest peers, and another part of the utility of having any given BMI is assumed to be due to individual, innate preferences (perhaps reflecting what the individual's ideal BMI would be in a hypothetical enviornment devoid of social influences).
We assume that the rate of change of an individual's BMI is proportional to the rate of change, i.e. the derivative, of his or her utility function with respect to BMI. This is done to make individuals naturally tend towards BMI values which maximize their total utility.
In order to keep the model reasonably simple, we have made some assumptions for convenience which are worth mentioning. First, we assume that social networks are undirected; that is, every social connection goes both ways. If I am your friend, then you are my friend. In mathematical language, this translates to social networks being represented by undirected graphs, whose corresponding adjacency matrices are thus symmetric. Second, we assume no self-coupling takes place. In other words, individuals are not considered to be their own peers, inasmuch as their own BMI does not contribute to their social utility. Mathematically this translates into the diagonal of every adjacency matrix consisting only of zeroes. Third, we assume that all individuals are identical with respect to how they respond to social pressures and individual preferences. That is to say, the weighting of the social and individual components of the model is the same for everyone. Fourth, we use an unweighted network; that is to say, we do not take into consideration the type or intensity of each peer relationship; the entries of the adjacency matrix of the network have either the value 0 or the value 1. No intermediate values are possible. Finally, we only consider networks where every person has at least one peer relationship. This corresponds to degree matrices with no zero entries on the diagonal, something which is mathematically useful, as we will see later. Moreover, this last assumption makes sense on an intellectual level, since there is no purpose in studying individuals who are completely socially isolated in the context of a model examining social effects.
William E. Krinsman