Group Project Report
Maka Yue et al
March 10, 2019
Part B - TERGM
Introduction
Social Network Analysis with TERGM was conducted on a professional network of 28 health professionals located in South Dakota. Each of these actors has a possibility of five connection types–how often they contact others (contact), the strength of their relationship (know), how often they receive information from others (receive), how often they send information to others (send), and how much they trust them (trust). The connections of interest in our analysis are receive and trust.
Attributes
The attributes for this network include the following:
Workgroup: 1 = Stanford model of Chronic Disease Self-Management Program, 2 = PN/CHW
Attendees: 1 = absent, 2 = present
Sex: 1 = male, 2 = female
Race/Ethnicity: 1 = white, 2 = Black/African American, 3 = AI/AN, 5 = Other, 9 = Prefer not to answer
Specialization: 1 = Cancer, 2 = CD, 3 = neither
Roles: Health Educator, CHW, Clinician, Lawmaker, Public/Government Employee, Researcher/academic, NGO/NPO prof, Private Sector, Insurer, Health administrator, Public Health Prof, and other.
Topic: Cancer; Nutrition, PA, obesity; Diabetes; Heart Disease; Tobacco & alcohol; Reproductive Health; Population Health; Access to health services; Mental & emotional well-being; Survivorship; Other
Questions to be Answered
The types of questions than can be answered about this network using SNA and TERGM are:
Do researchers interact only with those within the same role?
Does a researcher’s sex contribute to the formation of relationships?
Are relationships consistent from one year to the next?
What influences whether a researcher acquires, maintains, or drops a relationship over time?
Overall Findings
Homophily had a singificant impact on the probability of tie formation across both networks and years. The largest predictor of tie formation for both years in the trust network is conference attendance–if both actors attended the conference, the probability of a trusting relationship between them increased significantly. Reciprocality of ties was the largest predictor of tie formation among the receive networks. According to the tergm, a tie from one year to the next was more likely to form if the tie resulted in tie reciprocity. From one year to the next, the trust network communities became tighter and the receive communities expanded.The most trusted cancer specialist and two most trusted CD specialists are SD10 and SD1 and SD9, respectively.
Analysis Methods and Approach
To better understand this network, descriptive analysis was conducted on the attributes data to see what it can tell us about the network, such as if professionals with the same roles have the same attributes. Next, we calculated network-level metrics such as density to identify how many connections there actually are. Node level analysis and subgroup analysis was used to identify any particular actors of interest and if there are communities present. Lastly, ergm/tergm analysis was conducted to identify what variables, if any, impacted tie formation.
The SD Trust table for 2016 contained 28 vertices, but the SD trust table for 2015 contained only 21 vertices, so the missing vertices were filled in to make a symmetric data matrix to keep the two time series data in the same scale. The SD attributes table contains attributes for SD 1-21 for 2015 and SD 1-28 for 2016. Any missing values were replaced with 0 and attached to the graph object as needed. Additionally, attributes variables were updated so they could be used in the tergm analysis.
Descriptive Analysis
Networks
There are two social networks of interest among this community: receive and trust. In the receive network, respondents identified how often they received information from another, from a range of 1 = never to 6 = every day. For the sake of our analysis, those connections that were 3 or greater–received information once a quarter and more–were considered true connections while those 2 and under were not. The resulting receive networks for 2015 and 2016 can be seen below:
It is interesting to see how the network changed from one year to the next. In 2015, respondents appeared to have more receive connections, and overall there seems to be one main group who sends and receives information among themselves. This group predominately specializes in communicable diseases, with one actor whose specialization is unknown. The receive network of 2016 is less connected and clear communities are not present. There is also less clear of a distinction between specialties.
The second social network of interest is the trust network. Respondents identified which other actors they trusted, from a range of 0 = no trust to 5 = fully trust. The same method of binarization was used.
Overall, the trust network is much more connected than the receive network. For 2015, the general community among the Communicable Disease (CD) actors is the predominant group, but there are more connections among the isolates that were present in the receive network. Additionally, in 2016 the trust network was even more connecting, showing that trust may increase with time. SD10 appeared as the most trusted of the cancer specialists and SD1 and SD9 the most trusted of the CD specialists.
Actors
This network consists of a total of 28 professionals, 21 from 2015 and 22 from 2016. There were 15 individuals who attended both years. Of the total attendees, the majority were white females who specialized in CD. They are evenly split between the Stanford Model of Chronic Disease Self-Management Program and PN/CWH working groups.
The actors were predominantly health educators, clinicians, or involved with NGO/NPO.
The professionals who attended both years were predominately health educators or involved with NGO/NPO or Public/Government.
Network-Level Analysis
Metrics
| Network | Year | Vertices | Edges | Network Density | Transitivity |
|---|---|---|---|---|---|
| Trust | 2015 | 28 | 82 | 0.1045918 | 0.6217617 |
| Trust | 2016 | 28 | 93 | 0.1186224 | 0.4773960 |
| Receive | 2015 | 28 | 71 | 0.0905612 | 0.7200000 |
| Receive | 2016 | 28 | 41 | 0.0522959 | 0.4528302 |
- There are 28 vertices in this network.
- In Trust network, the density increased from 0.1046 to 0.1186 in the two years period from 2015 to 2016, indicating a growing trust relationship among those researchers.
- However, the information they receive from each other decreased about 42%, from a edges density of 0.0906 to 0.0523.
- The transitivity in Trust network and Receive network both went down.
Node-Level Analysis
Local Clustering Coefficient
The local clustering coefficient of a node in a graph quantifies how close its neighbours are to being a clique. When the value equals to 1, as is of node SD2 in trust 2015, every neighbour connected to it is also connected to every other vertex within the neighbourhood.
| Node | Network | Year | Clustering Coefficient | _______ | Node | Network | Year | Clustering Coefficient |
|---|---|---|---|---|---|---|---|---|
| SD2 | Trust | 2015 | 1.0000000 | _______ | SD4 | Receive | 2015 | 1.00 |
| SD4 | Trust | 2016 | 1.0000000 | _______ | SD1 | Receive | 2016 | 1.00 |
| SD1 | Trust | 2015 | 0.8055556 | _______ | SD23 | Receive | 2016 | 1.00 |
| SD4 | Trust | 2015 | 0.8055556 | _______ | SD4 | Receive | 2016 | 1.00 |
| SD5 | Trust | 2015 | 0.7857143 | _______ | SD19 | Receive | 2015 | 0.75 |
| SD14 | Trust | 2015 | 0.7857143 | _______ | SD24 | Receive | 2016 | 0.50 |
Degree Centrality
We choose to only focused on degree centrality this time, which measures the total connections a node has.
| Node | Network | Year | Degree | _______ | Node | Network | Year | Degree |
|---|---|---|---|---|---|---|---|---|
| SD9 | Trust | 2015 | 20 | _______ | SD9 | Receive | 2015 | 17 |
| SD15 | Trust | 2016 | 18 | _______ | SD1 | Receive | 2015 | 16 |
| SD1 | Trust | 2015 | 15 | _______ | SD16 | Receive | 2015 | 15 |
| SD3 | Trust | 2015 | 15 | _______ | SD2 | Receive | 2015 | 14 |
| SD4 | Trust | 2015 | 15 | _______ | SD3 | Receive | 2015 | 14 |
| SD5 | Trust | 2015 | 15 | _______ | SD5 | Receive | 2015 | 14 |
- The same node has the highest and second highest centrality in both Trust and Receive network in 2015 and 2016. This could indicate a internal influence between the trust and receive relationship. Either a researcher tend to lay more trust on another when he/she receives more information from this person or the other way.
- The most important person(the one with most connections) does not stay the same in 2015 and 2016, which implies a unstable relationship in both networks.
Subgroup Analysis
Walktrap Community Detection
The walktrap community detection algorithm utilizes random walks along edges in the graph to identify communities. Since only a few edges will lead outside of a community, these random walks can identify cliques within a network. The process is iterated until individual groups are merged into the final communities and a specified modularity is reached. 
This community detection method found two main communities across all networks and years. After the first year’s conference, a third community was established in the receive network. This could have been a result of meeting and exchanging information at the initial conference. The trust network saw a third smaller community from 2015 welcomed into one of the larger communities. This is likely related to greater trust being established as the number of times met increased.
Statistical Analysis
ERGM
Exponential Random Graph Models were utilized to identify which attributes, if any, had a statistically significant influence on the probability of tie formation within the trust and receive networks, and if those same attributes had a significant impact across both years.
Trust Network 2015
##
## ==========================
## Summary of model fit
## ==========================
##
## Formula: tr15net ~ edges + mutual + nodematch("Workgroup") + nodematch("Attendees") +
## nodematch("Sex") + nodematch("Race") + nodematch("Cancer")
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -6.1608 1.0299 0 -5.982 < 1e-04 ***
## mutual 2.0599 0.4650 0 4.430 < 1e-04 ***
## nodematch.Workgroup 0.4129 0.2461 0 1.678 0.093397 .
## nodematch.Attendees 2.9466 1.0553 0 2.792 0.005235 **
## nodematch.Sex -0.3853 0.2560 0 -1.505 0.132350
## nodematch.Race 1.1388 0.3333 0 3.417 0.000634 ***
## nodematch.Cancer 1.1772 0.2667 0 4.413 < 1e-04 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 1048.0 on 756 degrees of freedom
## Residual Deviance: 298.1 on 749 degrees of freedom
##
## AIC: 312.1 BIC: 344.5 (Smaller is better.)The change in the number of edges, mutuality, attendees, race, and cancer are significant predictors of tie formation, as the p-value is far less than the conventional significance cut-off of 0.05. The column for “MCMC%” is left as null, because this model didn’t require Markov Chain Monte Carlo estimation.
The coefficients, -6.16, 2.0599, 2.9466, 1.1388, and 1.1772, are expressed as conditional log-odds, such that:
\[logodds(trust_{2015}) = -6.16*edges + 2.06*mutual + 2.95*attendees + 1.14*sex + 1.18*cancer\]
The log-odds that these heterophilous ties will form in the network equals -6.16, equaling a probability of exp(-6.16)/(1+exp(-6.16)), or 0.002. Likewise, the log-odds of a homophilous relationship forming equals (-6.16+2.06+2.95+1.14+1.18+1.17), corresponding to a probability of 0.76, or exp(1.17)/(1+exp(1.17)). The greatest indicator of a tie forming was homophily of attendees–whether they attended the conference together.
Trust Network 2016
##
## ==========================
## Summary of model fit
## ==========================
##
## Formula: tr16net ~ edges + mutual + nodematch("Workgroup") + nodematch("Attendees") +
## nodematch("Sex") + nodematch("Race") + nodematch("Cancer")
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -3.63503 0.28860 0 -12.595 < 1e-04 ***
## mutual 1.19859 0.40398 0 2.967 0.00301 **
## nodematch.Workgroup 0.08826 0.23886 0 0.370 0.71175
## nodematch.Attendees 1.50072 0.33594 0 4.467 < 1e-04 ***
## nodematch.Sex -0.29476 0.26539 0 -1.111 0.26671
## nodematch.Race 0.74782 0.25756 0 2.904 0.00369 **
## nodematch.Cancer 0.83494 0.27986 0 2.983 0.00285 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 1048.0 on 756 degrees of freedom
## Residual Deviance: 452.8 on 749 degrees of freedom
##
## AIC: 466.8 BIC: 499.2 (Smaller is better.)The change in the number of edges, mutial, attendees, race, and cancer are significant predictors of tie formation, as the p-value is far less than the conventional significance .05 cut off. The column for “MCMC%” is null, because this model didn’t require Markov chain Monte Carlo estimation.
The coefficients, -3.64, 1.20, 1.5, 0.75, and 0.83 , are expressed as conditional log-odds, such that:
\[logodds(trust_{2016})=-3.64*edges+1.20*mutual+1.50*attendees+0.75*race+0.83*cancer\]
The log-odds that these heterophilous ties will form in the network equals -3.64, equaling a probability of exp(-3.64)/(1+exp(-3.64)), or 0.025. Likewise, the log-odds of a homophilous relationship forming equals (-3.64+1.20+1.50+0.75+0.83)= 0.64, corresponding to a probability of 0.65, or exp(.64)/(1+exp(.64)).
Receive Network 2015
##
## ==========================
## Summary of model fit
## ==========================
##
## Formula: re15net ~ edges + mutual + nodematch("Workgroup") + nodematch("Attendees") +
## nodematch("Sex") + nodematch("Race") + nodematch("Cancer")
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -6.2324 0.9455 0 -6.592 < 1e-04 ***
## mutual 3.0986 0.5418 0 5.719 < 1e-04 ***
## nodematch.Workgroup 0.3225 0.2542 0 1.269 0.20453
## nodematch.Attendees 2.6414 0.9673 0 2.731 0.00632 **
## nodematch.Sex -0.4799 0.2572 0 -1.865 0.06211 .
## nodematch.Race 0.8015 0.3305 0 2.425 0.01529 *
## nodematch.Cancer 1.4052 0.2936 0 4.786 < 1e-04 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 1048.0 on 756 degrees of freedom
## Residual Deviance: 245.6 on 749 degrees of freedom
##
## AIC: 259.6 BIC: 292 (Smaller is better.)The change in the number of edges, mutual, attendees, race, and cancer are significant predictors of tie formation, as the p-value is far less than the conventional significance cut off of 0.05. The column for “MCMC%” is null.
The coefficients, -6.23, 3.10, 2.64, 0.80, and 1.41 , are expressed as conditional log-odds, such that:
\[logodds(receive_{2015})=-6.23*edges+3.10*mutual+2.64*attendees+0.80*race+1.41*cancer\] The log-odds that these heterophilous ties will form in the network equals -6.23, equaling a probability of exp(-.6.23)/(1+exp(-6.23)), or 0.002. Likewise, the log-odds of a homophilous relationship forming equals (-6.23+3.10+2.64+0.80+1.41)= 1.72, corresponding to a probability of 0.85 , or exp(1.72)/(1+exp(1.72).
Receive Network 2016
##
## ==========================
## Summary of model fit
## ==========================
##
## Formula: re16net ~ edges + mutual + nodematch("Workgroup") + nodematch("Attendees") +
## nodematch("Sex") + nodematch("Race") + nodematch("Cancer")
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -4.7615 0.4344 0 -10.960 < 1e-04 ***
## mutual 2.8543 0.5952 0 4.795 < 1e-04 ***
## nodematch.Workgroup 0.4490 0.3056 0 1.469 0.141798
## nodematch.Attendees 1.2016 0.4834 0 2.486 0.012920 *
## nodematch.Sex -0.6385 0.3570 0 -1.788 0.073704 .
## nodematch.Race 0.5887 0.3412 0 1.725 0.084448 .
## nodematch.Cancer 1.2547 0.3643 0 3.445 0.000572 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 1048.0 on 756 degrees of freedom
## Residual Deviance: 229.1 on 749 degrees of freedom
##
## AIC: 243.1 BIC: 275.5 (Smaller is better.)The change in the number of edges, mutual, attendees, and cancer are significant predictors of tie formation, as the p-value is far less than the conventional significance cutoff of 0.05. The column for “MCMC%” is null.
The coefficients, -4.76, 2.85, 1.2, and 1.26, are expressed as conditional log-odds, such that:
\[logodds(receive_{2016})=-4.76*edges+2.85*mutual+1.20*attendees+1.26*cancer\]
The log-odds that these heterophilous ties will form in the network equals -3.64, equaling a probability of exp(-3.64)/(1+exp(-3.64)), or 0.025. Likewise, the log-odds of a homosexual relationship forming equals (-4.76+2.85+1.2+1.26)= .55, corresponding to a probability of 0.63 , or exp(.55)/(1+exp(.55)).
TERGM
Trust Formation
##
## ==============================
## Summary of formation model fit
## ==============================
##
## Formula: ~edges + mutual + transitiveties + cyclicalties + nodematch("Sex") +
## nodematch("Workgroup") + nodematch("Race") + nodematch("Cancer")
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -4.3510 0.3589 0 -12.123 < 1e-04 ***
## mutual 0.7128 0.3731 0 1.911 0.056044 .
## transitiveties 1.1679 0.3414 0 3.421 0.000623 ***
## cyclicalties 0.4374 0.2565 0 1.705 0.088115 .
## nodematch.Sex 0.5192 0.2615 0 1.986 0.047082 *
## nodematch.Workgroup 0.1780 0.2642 0 0.674 0.500586
## nodematch.Race 0.8431 0.3115 0 2.707 0.006789 **
## nodematch.Cancer -0.2837 0.3144 0 -0.902 0.366921
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 934.4 on 674 degrees of freedom
## Residual Deviance: 377.6 on 666 degrees of freedom
##
## AIC: 393.6 BIC: 429.7 (Smaller is better.)
##
## ================================
## Summary of dissolution model fit
## ================================
##
## Formula: ~edges + mutual + transitiveties + cyclicalties
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -2.4944 0.5459 0 -4.570 < 1e-04 ***
## mutual 2.6614 1.2426 0 2.142 0.032208 *
## transitiveties 1.9284 0.5506 0 3.502 0.000462 ***
## cyclicalties -Inf 0.0000 0 -Inf < 1e-04 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 113.7 on 82 degrees of freedom
## Residual Deviance: NaN on 78 degrees of freedom
##
## AIC: NaN BIC: NaN (Smaller is better.)With all other variables held constant, a relationship is much more likely to form if it will close a mutual pair. The conditional log-odds in this case are increased by 0.7, which translates to an increase in the conditional odds of 2.01. Note that the change statistic can equal only 0 or 1 in this case.
The reciprocity effect on persistence is also positive, but less strong and not statistically signifiant. The point estimate is an increase of 2.7 in the conditional log-odds of persistence, a bit more than double the conditional odds.
Transitive triads are positive and borderline significant. Cyclical triads have a statistically significant negative influence. Overall, the results suggest that hierarchy plays a role in tie formation and persistence in this model, and there is a consistent anti-egalitarianism dynamic.
Receive Formation
##
## ==============================
## Summary of formation model fit
## ==============================
##
## Formula: ~edges + mutual + transitiveties + cyclicalties + nodematch("Sex") +
## nodematch("Workgroup") + nodematch("Race") + nodematch("Cancer")
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -5.1140 0.4491 0 -11.387 < 1e-04 ***
## mutual 2.6331 0.7214 0 3.650 0.000262 ***
## transitiveties 1.9412 0.4374 0 4.438 < 1e-04 ***
## cyclicalties -0.4982 0.3056 0 -1.630 0.103045
## nodematch.Sex 0.7273 0.4127 0 1.762 0.078000 .
## nodematch.Workgroup -0.7159 0.4724 0 -1.516 0.129635
## nodematch.Race 0.6707 0.3937 0 1.704 0.088430 .
## nodematch.Cancer -0.4568 0.5253 0 -0.870 0.384488
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 949.6 on 685 degrees of freedom
## Residual Deviance: 173.1 on 677 degrees of freedom
##
## AIC: 189.1 BIC: 225.4 (Smaller is better.)
##
## ================================
## Summary of dissolution model fit
## ================================
##
## Formula: ~edges + mutual + transitiveties + cyclicalties
##
## Iterations: 2 out of 20
##
## Monte Carlo MLE Results:
## Estimate Std. Error MCMC % z value Pr(>|z|)
## edges -3.2251 0.6059 0 -5.323 <1e-04 ***
## mutual 1.6814 1.6493 0 1.019 0.3080
## transitiveties 1.8584 0.7273 0 2.555 0.0106 *
## cyclicalties -0.3321 0.7480 0 -0.444 0.6571
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Null Deviance: 98.43 on 71 degrees of freedom
## Residual Deviance: 43.48 on 67 degrees of freedom
##
## AIC: 51.48 BIC: 60.53 (Smaller is better.)## [1] 2.6## [1] 13.46374## [1] 1.7With all other variables held constant, a relationship is much more likely to form if it will close a mutual pair. The conditional log-odds are increased by 2.6 which translates to an increase in the conditional odds by 13.46374. Note that the change statistic can equal only 0 or 1 in this case.
The reciprocity effect on persistence is also positive, but less strong and not statistically signifiant. The point estimate is an increase of 1.7 in the conditional log-odds of persistence, a bit more than double the conditional odds.
Transitive triads are positive and significant, but cyclical triads, while negative, are not significant. Overall, the results suggest that hierarchy plays a role in tie formation and persistence in this model, and there is a weak though consistent anti-egalitarianism dynamic.
Network Simulation
Trust Network Formation
The following animation simulates how the trust network formed based on our model.
## caching 5 properties for slice 0## caching 5 properties for slice 1## caching 5 properties for slice 2## loading ndtv-d3 animation widget...Receive Network Formation
The following animation simulates how the receive network formed based on our model.
## caching 5 properties for slice 0## caching 5 properties for slice 1## caching 5 properties for slice 2## loading ndtv-d3 animation widget...