Download Supplementary Material Source code

Supplementary Material
Source code
Both the source code and a self-installing compiled version of the model can be downloaded at
http://academic.regis.edu/dbahr/obesity.html. When running the model, select the “Obesity rule”
from the social applications folder.
Modeling methodology
Network-based interaction models have a rich history in the physical and social sciences.
Applications have ranged from analyses of rioting, to tipping behavior in restaurants, to the
spread of infectious diseases (e.g., 6, 8, 10-11, 15, 18). In all cases, the fundamental
assumption is that individuals interact with each other and that some quantity or parameter
associated with the individual changes as a result of these interactions (18). The quantity could
be the amount of money left in a tip, or in this case, the changing quantity is body mass index
(BMI).
Individuals interact with their network neighbors as specified by a network topology (e.g., Fig.
1). All of the results presented in the text were tested on a wide variety of topologies including
(1) square, triangular, and hexagonal regular lattices with nearest neighbors, (2) regular lattices
with next-nearest (and further) neighbors, (3) random Gaussian distributed neighbors (a regular
lattice where most neighbors are nearby but some are far), (4) entirely random networks, and
(5) small-world scale-invariant networks commonly observed and used in social network
analyses (12, 19). When using regular lattices, “wrap-around” conditions ensure that everyone
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has the same number of neighbors. In other words, in Figures 2-4 and Supplementary Figure 1,
individuals on the far right wrap around and interact with individuals on the far left; individuals
at the top interact with individuals at the bottom.
Selecting appropriate interaction rules
We only present results that are common to all of the tested rules and networks, but with an
infinite number of possible interactions, which ones should be selected? Based on empirical
observations, we required all tested rules to obey three important constraints. #1: The
probabilities of an individual transitioning to any BMI class must sum to 1. (In deterministic
rules, the probabilities of all but one possible transition will be 0; the remaining BMI transition
will have probability 1.) #2: As an increasing number of obese people are networked to an
individual, then the more likely it is that the individual will also become obese. #3: If an
individual has N network neighbors, then the influence of each of these neighbors will be
greater than when there are N+1 neighbors. In other words, with three network neighbors, each
one has a relatively large influence on the individual (for example, 33% of the total influence).
But with 100 network neighbors, each neighbor has a relatively small influence on the
individual (for example, 1% of the total).
From the above constraints, #1 is a mathematical necessity. Any rules that do not have
probabilities summing to 1 are not physically possible. Constraint #2 is a result of the
Framingham study (4) which empirically demonstrated such a relationship. We are more likely
to become obese as additional friends and family also become obese. Constraint #3 is a wellknown result from many psychology and sociology studies, most notably Latané’s theory of
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social impact (6). To describe this constraint, Latané uses an analogy with money. If you have
only one dollar, then one more dollar is a substantial increase; but if you have a million dollars
then an additional dollar is barely noticeable. Similarly, if you have one network neighbor then
the influence of one additional neighbor will be significant. However, if you are very well
connected with hundreds of network neighbors, then the addition of one more neighbor will
have a barely noticeable impact. In general, the impact of the Nth person should be greater
than the impact of the N+1st person.
Mathematically, these three important constraints can be summarized as follows (c.f., 15).
(1) Let PΔ be the probability of changing from the current BMI class to a different BMI class.
Then
0 ≤ PΔ ≤ 1
(2) The probability of changing to a higher (or lower) BMI should increase with the number of
neighbors N that are in a higher (or lower) BMI class. In other words, PΔ ( N ) ≤ PΔ (N + 1) for
all N, or equivalently,
∂PΔ
≥0
∂N
(3) The impact of the Nth neighbor is greater than the impact of the N+1st neighbor. In other
words, PΔ (N ) − PΔ (N − 1) ≥ PΔ ( N + 1) − PΔ ( N ) , or equivalently,
∂ 2 PΔ
≤0
∂N 2
3
We have only used rules that obey (or closely follow) these three mathematical constraints.
Obviously this eliminates many possible rules such as “Life”, a popular two-state interaction
rule known for its complex evolving shapes. The “Life” rule, which is useful in computer
science as a model of theoretical computation, cannot be used in medicine as a model of
obesity (or any other social impact theory) because it obeys neither of constraints #2 or #3. For
example, in “Life”, an individual with two obese neighbors would always become obese, but
an individual with four or more obese neighbors would always become normal weight – this
contradicts constraint #2 and obviously contradicts the empirical observations of the
Framingham study (4).
On the other hand, a large class of “voter” or “majority wins” rules obey all three constraints.
Generally speaking, an individual in a “majority” rule will change to become the same as the
majority of their network neighbors. Many wide variations of this majority interaction rule
were tested and are described in the Methods section. For all tested rules, the observed
behaviors were very similar (for example, clustering).
The only significant deviations in observed behavior happen with unrealistic parameter values
that decouple the individuals from their social network. At excessively high volatility, an
individual’s BMI becomes very transient, effectively random, and independent of the
neighborhood. Similarly, exceedingly large forces make an individual’s BMI predetermined
by the external influence and no longer a function of the neighborhood. Excessive noise makes
the model independent of all parameters, and the neighbors again become irrelevant. The
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Framingham study indicates that the network neighborhood does influence BMI, so none of
these deviations are realistic
Mathematical background and development
Details, motivations, and sociological/psychological justifications for the mathematical
underpinnings can be found in the references (8, 11, 15, and references therein). For this
analysis, we present only a brief statistical mechanical development of “majority probably
rules” generalized to include social forces and volatility. Forces and volatility may be added to
most interaction rules (including threshold and weight-biased interactions), but the “majority
probably rules” is used for concreteness. Although the mathematics applies to many social
network simulations, we have tailored the following description to the obesity epidemic.
Consider an individual surrounded by N neighbors (friends, family, acquaintances, etc.) on a
social network. The social network is topological rather than geometrical and therefore makes
no assumptions about physical distance. Neighbors may be next door or across the country.
Each neighbor has some BMI that falls into one of Q different classes; for example, the figures
in this paper have set Q = 4 with classes representing underweight, normal weight, overweight,
and obese. Let nk of those N neighbors have a BMI that falls within class k. Then the
probability that the individual will join BMI class k is given by
Pk =
nk
N
(1)
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Essentially, the likelihood of having a particular BMI is given by the fraction of neighbors with
that BMI.
Not all neighbors have the same influence. The quality, quantity, and frequency of the
interactions between two individuals determine the strength of their relationship and their
relative influence. Any link on the social network can be assigned a directional interaction
strength sij that describes the impact of individual i on another individual j. Note that sij may or
may not equal sij, and this corresponds to the bidirectional/directional ties observed in the
Framingham study (4). To include interaction strengths, equation 1 is modified as
N
Pk =
∑s δ σ
ij
j =1
k
j
(2)
N
∑ sij
j =1
where σj is the BMI class of individual j, and the Kronecker delta function δ just ensures that
only individuals with BMI class k contribute to the sum (δkσj = 1 when k = σj, and δkσj = 0
when k ≠ σj). In other words, the numerator represents a sum over all links to neighbors j with
a BMI given by k, and the denominator is a sum over all neighbors (15). If the interaction
strengths are roughly the same for all network ties, then equation 2 reduces to equation 1.
The probabilities Pk are different for every person and are a function of an individual’s network
neighbors. Using either equation (1) or (2), a model can update every individual in the
population based on their associated probabilities. The update assigns each person a new BMI,
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which in turn changes the Pk. The model can continue updates and observe changes and trends
over time, as seen in Figures 2 – 4, and Supplementary Figure 1. Time units could be assigned
to the updates but would amount only to a rescaling of the otherwise non-dimensional results.
Note that the summation of interaction strengths in equation 2 is a basic point of the paper and
is used in many other models (8, 11, 15, 17, and references therein). The more people around
us that are obese, then the more likely we will be obese as well. This is a fundamental
observation from the Framingham study (4), and the interaction rules summarized in equations
(1) and (2) obeys all three necessary constraints outlined above in the section “Selecting
appropriate interaction rules”.
Tipping points
Interventions are interested in converting individuals to a lower BMI. Of particular interest is
finding how many individuals m must be converted and pinned to a lower BMI in order to
convert c additional individuals to that lower BMI. Conversely, we might want to know the
“point of no return” or tipping point where some number of obese individuals will convert the
remainder of a population. Consider a social network or population of N neighbors with the
same BMI. This might correspond, for example, to the N members of one of the BMI clusters
shown in Figure 2 (if all members are well-connected to each other by their network topology).
Suppose that m of the N are pinned at a different BMI by long-term changes in diet and
exercise. Using equation 1 we can calculate the probability that some number c of additional
people on the network will interact with the pinned individuals and be converted to the same
BMI. A similar generalization with equation 2 (and a less connected topology) is possible
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when the law of large numbers is used (c.f., 15), but for simplicity is not presented here. The
results are identical.
From equation 1, the probability that an individual will change to BMI class k is
Pk =
m
N
(3a)
Therefore, the probability that C of the N-m individuals will be converted to BMI class k is
given by
⎛ N − m⎞
⎟⎟(Pk )C (1 − Pk )N −m −C
P(C ) = ⎜⎜
⎝ C ⎠
(3b)
Note that 3b is a binomial distribution. The probability that the number of converts c is greater
than or equal to C is
P(C ≥ c ) =
N −m
∑ P(i )
(3c)
i =C
where P(i) is from equation 3b. For large N, P(i) is approximated by a Guassian (central limit
theorem) and the summation can be approximated by an integral which leads to the so-called
error function. By standard arguments which integrate the Guassian approximation to a
binomial distribution,
P (c ≥ C ) ≈
⎛ C − m (N − m) N ⎞
⎛ N −m⎞ 1
1
⎟
Erf ⎜⎜
⎟⎟ − Erf ⎜⎜
⎟
2
2
(
)
2
2
m
m
N
m
N
−
⎝
⎠
⎝
⎠
(3d)
For large N, the first term always evaluates to 1/2. Therefore,
P (c ≥ C ) ≈
⎛ C − m (N − m ) N ⎞
1 1
⎟
− Erf ⎜⎜
⎟
2 2
⎝ 2m ( N − m ) N ⎠
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(3e)
(15). In other words, the likelihood of converting c additional people to the lower BMI is an sshaped “error function” (Erf) which has a transition from low to high values (similar to a
smoothed step function). When m is small, the probability of converting anyone is very low.
As m increases, there is a (frequently sudden) threshold or tipping point past which everyone is
converted.
This error function corresponds to the tipping point observed in the numerical simulations.
Create enough overweight and obese individuals in the population and everyone else will
follow. Similarly, if enough people diet, then many more people will be converted to the lower
BMI as well. We emphasize that the exact formulation of equation 3e is not as important as its
s-shaped error function shape which implies such a threshold behavior. In fact, the exact value
of the threshold or critical mass is best approximated numerically (15) and shows that a group
of m-1 people will convert roughly m additional people. This assumes that the m-1 people are
pinned at their BMI value over long time periods, and it assumes that the sum of interactions
strengths can be approximated by an average value (i.e., an application of the law of large
numbers) so that
N
∑s
j =1
ij
= Ns avg
(4)
in which case equation 2 is approximated by equation 1 as assumed in the derivations above
(15). This means that either the number of neighbors must be relatively large or the variation
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in sij must be small. The latter is true when none of the individuals are especially charismatic
or influential relative to the rest of the population.
However, more generally, if P(c = C) has a peaked distribution of any kind (binomial or
otherwise), then its integral P(c ≥ C ) will have an s-shaped distribution with a transition
similar to the error function. Because P(c = C) is the sum of many random variables for many
individuals (15), central limit theorems suggest that such a peaked distribution is likely. In
other words, threshold behaviors should be relatively universal for a wide variety of interaction
rules that obey the three constraints outlined above (see “Selecting appropriate interaction
rules”), and we should always expect a “tipping point” past which the population will slide
rapidly into obesity.
Social force, volatility, and noise
Social forces and social volatility are analogous to the quantities of force and temperature in
physics and will modify the form of equations 1 and 2. Using standard mathematical
arguments (commonly used in statistical mechanics and other physics), social forces and
volatility have a precise mathematical description (15). While these social quantities may
appeal to analogous physical quantities, any ambiguity in the analogy is resolved by their
precise definition and form in the following equations. In particular, let hk be a social force
that encourages or discourages a particular BMI class k. Let T be the social volatility. Then
Pk =
(
)
1
1 hk
e f (k , N ) T
z
(5)
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where f(k, N) is the right hand side of equation 1 or 2 (whichever is desired). z is a normalizing
factor that ensures the probabilities add to 1 and is given by
Q
(
)
z = ∑ e hk f (k , N )
1
(6)
T
k =1
In statistical mechanics z is called a partition function.
Note that social forces have an exponential form e hk . Small changes in the force can have a
dramatic influence on the probabilities. On the other hand, equations 1 and 2 vary linearly
with changes in the number of people that have BMI weight class k. Therefore, per unit of
change, social forces will have a substantially greater influence than pinning the BMI of a
person’s neighbor.
Social forces and volatility are equally applicable to any interaction rule f by replacing f(k, N)
in equations 5 and 6. This will work even for deterministic rules by setting f(k, N) = 1 for
some particular k, and 0 for all other k. In other words, obesity modeling results that include
social forces and volatility are generally applicable to a wide variety of interaction rules.
The same is true for random noise which can be added to simulate unknown, unpredictable,
and uncorrelated events and interactions. With unbiased noise, the simulation calculates an
individual’s new BMI and then randomly changes it to a new value. Our simulations also used
biased noise that allowed random changes in BMI but only up or down by one weight class at a
time (k to k+1 or vice versa).
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Mapping social networks for study and interventions
The appropriate identification and mapping of social networks has a long history in the social
sciences and can be reviewed in that extensive literature (e.g., 19). Briefly, we can imagine a
social environment like an elementary school with a network of student friendships. Each
student at the school can be polled to identify which other students they consider friends. Each
student in the school can now be pictorially represented as a stick figure as in Fig. 1, and then
lines can be drawn between the student and the other stick figures that they consider friends.
Lines may cross. The resulting grid of lines represents the social network. Note that the
students may have important friendships and relationships outside the school boundaries (e.g.,
parents). Depending on the desired study or intervention these external friendships may or
may not be relevant and should or should not be included in the network.
Consider an intervention which will target a random selection of the elementary school
students and pin their weight at a desired value (e.g., Fig. 4). A random subset (e.g., 1%, 2%,
5%, etc.) of the students would be selected from the social network with no consideration for
weight, BMI, or any other factors. These students are monitored and kept at the desired weight
by adjusting (and/or maintaining) diet and physical activity. According to the model, the
pinned student’s friends or “network neighbors” will be influenced and over time the average
weight of the network (all students) will change to match the weight of the pinned students.
The relative success of such a study would indicate whether or not the same strategy should be
tried at the community or national level.
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Supplementary Figure 1: Dieting with friends of friends is a more effective strategy than
dieting with friends. (a) The blue color indicates overweight BMIs. A single overweight
individual is selected near the border of the cluster, and eight surrounding friends of friends are
also selected (all shown in red). Each is pinned at the normal weight value (green). (b) After
40 time steps, the normal-weight cluster has expanded to enclose the dieting cohort. The
central red individual (the original dieter) is now safely ensconced in the normal-weight cluster
and can stop dieting. Dieting with friends has a similar effect, but places the individual closer
to the edge of the cluster where they are more susceptible to the moving boundary after they
finish dieting.
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