^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

Individual behaviors, such as drinking, smoking, screen time, and physical activity, can be strongly influenced by the behavior of friends. At the same time, the choice of friends can be influenced by shared behavioral preferences. The actor-based stochastic models (ABSM) are developed to study the interdependence of social networks and behavior. These methods are efficient and useful for analysis of discrete behaviors, such as drinking and smoking; however, since the behavior evolution function is in an exponential format, the ABSM can generate inconsistent and unrealistic results when the behavior variable is continuous or has a large range, such as hours of television watched or body mass index. To more realistically model continuous behavior variables, we propose a co-evolution process based on a linear model which is consistent over time and has an intuitive interpretation. In the simulation study, we applied the expectation maximization (EM) and Markov chain Monte Carlo (MCMC) algorithms to find the maximum likelihood estimate (MLE) of parameter values. Additionally, we show that our assumptions are reasonable using data from the National Longitudinal Study of Adolescent Health (Add Health).

Numerous studies have examined the role friends play in influencing behavior. Researchers have made exten- sive use of data from the Framingham Heart Study-Network Study (FHS-Net) [

In response to these concerns, the actor-based stochastic model (ABSM) was proposed by [

In ABSM, a continuous time finite-state-space Markov process was used to model the dynamic relationship between social network and behaviors. Three steps describe this process. The first step determines when the chance for the next change will occur. Let

buted with parameter

ful change. The second step defines which actor has the opportunity to make a change (either a network change or a behavior change). The probability of a network change taken by a particular actor

from data (in a analysis), and

If actor

evolution. Then the behavior objective function of actor i is

actor

In summary, the probability to change to a new set of value

To use ABSM, the behavior variable must be bounded and discretized. For continuous behavior variables, such as body mass index (BMI), time spent watching television, etc., the process of discretizing can be arbitrary and tricky. In Section 3 (Results), we show that the effect of average BMI similarity can be very different for integer and categorical BMI.

Based on the above considerations, we were motivated to develop a linear-based behavior evolution model. In our model, the network evolution is similar to ABSM. However, the behavior evolution is defined by a continuous Markov process, which is completely different from [

For illustration purpose, consider two waves of data that are collected at time 0 and T. The complete data during time period

write as_{s},

Here the observed data are represented in black ovals, missing behavior data in blue ovals, and missing network data in red ovals. The network evolution process is represented by red arrows and behavior variable by blue arrows.

The number of events

For now, we assume the chance of making a network change is the same for each actor. This assumption can be extended to be actor-specific if the data are informative enough.

Let

that the edge

before the next event is

Define_{1} to t_{2}. For any time

where

from each other or any other random variable, and

Note that when

Since behavior variables are accumulated over time, we would expect that when modeling behaviors, the distribution of change from time

where

As an example, assume that the

where

where

havior variables

Exponential random graph models (ERGMs) are commonly employed to test whether the presence of network ties (edges) differs from what would be expected in a random graph, given some set of network statistics [

where

Parameter

function using simulated samples and obtain

MLE of parameters

In the general multi-dimensional situation, assume that

with mean

and

Here

according to the normal distribution with mean

Remember that the observed data are

· Sample k: let d be the number of edges

¾ If d is even,

¾ If d is odd,

· Sample

· Sample

1) Sample

and evaluate the density function of the above normal distribution at the realized value

2) Sample

a) Define the important list to be

b) If

¾ If

¾ If

c) If

d) Denote the probability from the situlation of

3) Likewise, sequentially sample

4) Use the Metropolis Hastings algorithm to decide whether to accept the generated sample

where

We used the Add Health “saturation sample” data to check the reasonableness of our assumptions and to per- form simulation studies. First, we show results based on the ABSM model; next we compare these results with our co-evolution model.

The Add Health saturation sample data are based on adolescents in 16 high schools where all students in a given school were asked to participate. There are two waves (1 year apart) of friendship network data, including environmental variables and self-reported height/weight. We focus on one school called “Jefferson High” as in [

The results based on ABSM are in

The estimated network objective function is

where

For example, consider the behavior evolution for individuals who have no friends. The estimated behavior objective function becomes

The probabilities for BMI evolution are shown in

Using the Add Health data for the school of Jefferson High, we can draw the histogram of BMI change and

Function | Estimate | S.E. | P value | |
---|---|---|---|---|

Network dynamics | ||||

1. Rate: rate friendship | 12.2900 | (0.4620) | <1e−4 | |

2. Eval: outdegree (density) | −3.4228 | (0.0370) | <1e−4 | |

3. Eval: reciprocity | 2.3341 | (0.0664) | <1e−4 | |

4. Eval: transitive triplets | 0.4957 | (0.0268) | <1e−4 | |

5. Eval: same sex | 0.0580 | (0.0451) | 0.1984 | |

6. Eval: same grade | 0.5417 | (0.0501) | <1e−4 | |

7. Eval: BMI similarity | 0.3901 | (0.1807) | 0.0309 | |

Behavior dynamics | ||||

8. Rate: rate BMI period 1 | 4.1708 | (0.3577) | <1e−4 | |

9. Eval: BMI linear shape | 0.1571 | (0.0275) | <1e−4 | |

10. Eval: BMI quadratic shape | 0.0144 | (0.0066) | 0.0291 | |

11. Eval: BMI similarity | 13.9074 | (3.7561) | 0.0002 |

BMI | −1 | Same | +1 |
---|---|---|---|

14 | 0.000 | 0.522 | 0.478 |

15 | 0.359 | 0.330 | 0.311 |

17 | 0.340 | 0.330 | 0.330 |

18 | 0.330 | 0.330 | 0.340 |

30 | 0.224 | 0.316 | 0.460 |

45 | 0.124 | 0.270 | 0.605 |

46 | 0.309 | 0.691 | 0.000 |

screen time change between these two waves (

We also draw the scatter plot of individual’s BMI change versus average friends’ BMI change to check lin- earity assumption. The plot in

To simulate a realistic network with reasonable BMI values assigned to each individual, we randomly sampled 30 individuals (from the same school) in the Add Health data. The average BMI of selected individuals is

We specified that network and BMI would evolve for 60 days using the following parameters values:

Apply the EM procedure described in Methods section, we obtained the following parameter estimations (

The explanation of the above parameters are mostly straight forward. For example, the rate of events

Since our model cannot deal with a network as large as 624 individuals, we include only students in grade 11 in this application. The sample size here is 110. We first run the ABSM model using RSiena (

Compare results from

Description | Parameter | True value | MLE (S.E.) |
---|---|---|---|

Rate of events | 1 | 1.08 (0.17) | |

Coeff. of trend | 0.001 | 0.002 (0.004) | |

Network effect | 0.1 | 0.12 (0.03) | |

Standard deviation of noise | 0.1 | 0.07 (0.02) | |

Coeff. of outdegree | −3.43 | −3.72 (0.35) | |

Coeff. of reciprocity | 2.33 | 1.63 (1.22) | |

Coeff. of transitive triplets | 0.50 | 0.39 (0.79) | |

Coeff. of BMI similarity | 0.39 | 0.71 (1.26) |

duals, we got significant effect of selection

We have developed a joint social network and behavior evolution model. In our model, behavior changes are

Estimate | S.E. | P value | |
---|---|---|---|

Network dynamics | |||

1. Rate: rate friendship | 4.7385 | (0.4698) | <1e−4 |

2. Eval: outdegree (density) | −3.0952 | (0.1425) | <1e−4 |

3. Eval: reciprocity | 2.3959 | (0.2244) | <1e−4 |

4. Eval: transitive triplets | 0.5471 | (0.0800) | <1e−4 |

5. Eval: same sex | 0.2099 | (0.1517) | 0.083 |

6. Eval: BMI similarity | 0.5425 | (0.6607) | 0.206 |

Behavior dynamics | |||

7. Rate: rate BMI period 1 | 2.8444 | (0.5431) | <1e−4 |

8. Eval: BMI linear shape | 0.1737 | (0.0853) | 0.021 |

9. Eval: BMI quadratic shape | −0.0504 | (0.0225) | 0.013 |

10. Eval: BMI similarity | 1.6567 | (1.8186) | 0.181 |

Description | Parameter | MLE (S.E.) |
---|---|---|

Rate of events | 2.59 (1.22) | |

Coeff. of trend | 0.00 (0.00) | |

Network effect | 0.02 (0.05) | |

Standard deviation of noise | 1.20 (0.39) | |

Coeff. of outdegree | −3.25 (0.15) | |

Coeff. of reciprocity | 2.19 (0.38) | |

Coeff. of transitive triplets | 0.59 (0.07) | |

Coeff. of BMI similarity | 0.36 (0.41) |

consistent over time. That is,

The field of social network analysis is a relatively young field. However, useful contributions are being made today. The range of applications is vast, from the contagion of health behaviors described in this paper [

Our model does require intensive computation. However, we are confident that more efficient algorithms can be developed. Though our model requires specific assumptions, we have demonstrated that these assumptions are reasonably easy to satisfy using real data. Sensitivity analysis will ultimately be required to determine if our model works well when some of the assumptions are violated.