Intro to ERGMs

SNA Module 4: Code-Along

Overview

1. Prepare: A quick look at the study by Daly and Finnigan from whence our data was derived.

2. Wrangle: Learn how to convert matrices into more workable formats like a familiar edge-list.

3. Explore: View some basic network stats and make a quick sociogram.

4. Model: Learn to fit and interpret a very basic ERGM.

5. Communicate: Reflect on what we learned.

Speaker Notes:

Prepare

Guiding Research & the ERGM Package

Guiding Study

• Context
• Questions
• Data
• Findings
• Takeaways

This longitudinal case study examines reform-related knowledge, advice, and innovation network structures among administrators in a district under sanction for under-performance and engaging in district-wide reform efforts.

Ebb and Flow of Social Network Ties (Daly and Finnigan 2011)

This study involves quantifying and visualizing the ties and overall structure of informal networks to answer the following research questions:

1. In what ways have reform-related social networks between district and site leaders evolved over a 2-year period?

2. How do district and school leaders perceive the evolution of the social networks and its effect on the exchange of reform-related resources within the district?

To answer RQ1, Daly and Finnigan administered online survey consisting of network relations and demographic questions asking participants to:

• select the administrators in the La Estasis network that they interact with for the purpose of advice seeking, collaboration, knowledge exchanges,

• assess their relationships on a 5-point interaction scale ranging from 1 (no interaction) to 5 (1 to 2 times a week).

QAP

• weak correlation between knowledge and innovation and advice and innovation

• leaders with knowledge ties also likely to have advice relations

P-Star (SIENA)

• Leaders with more experience more likely to be selected for reform-related advice

• over time, leaders tended to create ties with those frequently identified as resources for innovative practices

Overall, the findings from SNA and qualitative interviews indicate that over time:

1. Networks increased the number of superficial interactions.

2. More frequent exchanges, however, remained unchanged.

3. The result was a centralized network structure.

Speaker Notes:

Guiding Question

• Research Question
• ergm 📦
• Your Turn

Does gender or some other individual attribute predict confidential exchanges between school leaders?

• ERGMs are the primary building blocks of statistically testing network structural effects.

• Uses simulation to determine the probability of network feature (e.g., reciprocity, clique, homophily)

• Evolved to allow actor attributes for covariates

• The {ergm} package is used to fit exponential-family random graph models (ERGMs) in R

Let’s start by creating a new R script and loading the following packages:

• tidyverse

• ergm

• statnet

• tidygraph

• ggraph

# YOUR CODE HERE
#
#

Wrangle

Import Data & Tidy Graph

Import Data

• Import Data
• Inspect Data
• Discuss
• Answer

Let’s import two Excel files from the Social Network Analysis and Education companion site. year_1_collaboration.xlsx

leader_nodes <- read_csv("data/school-leader-nodes.csv")
leader_edges <- read_csv("data/school-leader-edges.csv")

Using your R script, take a look at the node and edge lists we just imported:

# ADD CODE BELOW
#
#

Think about the questions below and be prepared to share your response:

1. What do you think the values for each tie represent?

2. What do you think the “trust” and “efficacy” variables represent?

3. What else do you notice about this data?

4. What else do you wonder about this data?

Hint: Take a look at the SNA in Ed companion site

• Edge values indicate the frequency of interactions leaders have with nominated individuals on a four-point frequency scale ranging from 1 (the least frequent) to 4 (1–2 times a week).

• efficacy measures a belief in their ability to have an effect derived from a scale that includes 18 items rated on a 9-point Likert scale ranging from 1 (None at all) to 9 (A great deal).

• trust measures perceptions of the level of trust their school and district is a composite score based on scale contains eight items rated on a 7-point Likert scale ranging from 1 (Strongly disagree) to 7 (Strongly agree).

Convert to “Network”

• as.network() function
• Convert to Network
• Check Class

The {ergm} package requires an object of the class “network” — the format that {statnet} uses to store and work with network data.

The as.network() function does just that. Run the following code in your console to pull up the help page for the as.network() function and let’s take a look:

?as.network

Run the following code in your R script to combine our edge and node lists into a single “network” object:

leader_network <- as.network(leader_edges,
                             vertices = leader_nodes)

Now let’s use the class() function to verify that it is indeed a network object/data structure:

class(leader_network)

[1] "network"

Explore

Network Stats & A Quick Sociogram

View Network

• Inspect Network
• Visualize Network
• Output

First, let’s take a quick look at our network by printing

leader_network

 Network attributes:
  vertices = 43 
  directed = TRUE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  total edges= 143 
    missing edges= 0 
    non-missing edges= 143 

 Vertex attribute names: 
    district_site efficacy male trust vertex.names 

No edge attributes

Now use the plot() or autograph() function to create a quick sociogram

# YOUR CODE HERE
#
#

Model

Model Specification & ERGM Parameters

Base ERGM

• Model Specification
• ERGM 1
• Interpret

The syntax for specifying a model in the ergm package follows R’s formula convention: my_network ~ ergm_term_1 + ergm_term_2 + ergm_term_3 and so forth.

In the Help tab, search for “ergm-terms” and select the ergm::ergm-terms help page to take a quick peek at the plethora of parameters you can include in your model.

Let’s start with with a simple model that contains the following ergm-terms

• edges a required term that represents the total number of edges in the network

• mutual examines the tendency for ties to be reciprocated, i.e. “mutuality”.

# ensure reproducibility of our model
set.seed(589) 

# fit our ergm model 
ergm_mod_1 <-ergm(leader_network ~ edges + mutual) 

# get summary statistics for our model
summary(ergm_mod_1)

Call:
ergm(formula = leader_network ~ edges + mutual)

Monte Carlo Maximum Likelihood Results:

       Estimate Std. Error MCMC % z value Pr(>|z|)    
edges   -3.1094     0.1247      0  -24.94   <1e-04 ***
mutual   3.0984     0.2825      0   10.97   <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 2503.6  on 1806  degrees of freedom
 Residual Deviance:  892.4  on 1804  degrees of freedom
 
AIC: 896.4  BIC: 907.4  (Smaller is better. MC Std. Err. = 0.9802)

ERGM 2

• Confidential Ties
• Interpret

Let’s now test for homophily – or the “birds of a feather” phenomenon – in our network. The ergm() function now includes a nodematch() term to answer the following question:

Are school leaders more likely to confide in colleagues of their own gender or who work at the same site level (e.g., school vs. central office)?

# fit our ergm model 
ergm_mod_2 <-ergm(leader_network ~ edges + mutual + nodematch('male') + nodematch('district_site')) 

# get summary statistics for our model
summary(ergm_mod_2)

Call:
ergm(formula = leader_network ~ edges + mutual + nodematch("male") + 
    nodematch("district_site"))

Monte Carlo Maximum Likelihood Results:

                        Estimate Std. Error MCMC % z value Pr(>|z|)    
edges                    -3.6211     0.1797      0 -20.152   <1e-04 ***
mutual                    2.9306     0.2968      0   9.874   <1e-04 ***
nodematch.male           -0.1223     0.1492      0  -0.820    0.412    
nodematch.district_site   0.9765     0.1811      0   5.393   <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 2503.6  on 1806  degrees of freedom
 Residual Deviance:  856.6  on 1802  degrees of freedom
 
AIC: 864.6  BIC: 886.6  (Smaller is better. MC Std. Err. = 0.9926)

ERGM 3

• Trust Ties
• Interpret

Are school leaders with higher levels of trust more likely to have confidential exchanges?

# fit our ergm model 
ergm_mod_3 <-ergm(leader_network ~ edges + mutual + nodematch('male') + nodematch('district_site') + nodecov('trust')) 

# get summary statistics for our model
summary(ergm_mod_3)

Call:
ergm(formula = leader_network ~ edges + mutual + nodematch("male") + 
    nodematch("district_site") + nodecov("trust"))

Monte Carlo Maximum Likelihood Results:

                        Estimate Std. Error MCMC % z value Pr(>|z|)    
edges                   -4.55696    0.74818      0  -6.091   <1e-04 ***
mutual                   2.92834    0.29576      0   9.901   <1e-04 ***
nodematch.male          -0.11155    0.14675      0  -0.760    0.447    
nodematch.district_site  0.93434    0.17363      0   5.381   <1e-04 ***
nodecov.trust            0.09920    0.07539      0   1.316    0.188    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 2503.6  on 1806  degrees of freedom
 Residual Deviance:  854.6  on 1801  degrees of freedom
 
AIC: 864.6  BIC: 892.1  (Smaller is better. MC Std. Err. = 0.8594)

What’s Next?

• SNA Case Study: Birds of Feather Lead Together

• Guiding Study: Ebb and Flow of Social Network Ties Between District Leaders (Daly and Finnigan 2011)

Acknowledgements

This work was supported by the National Science Foundation grants DRL-2025090 and DRL-2321128 (ECR:BCSER). Any opinions, findings, and conclusions expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

Daly, Alan J, and Kara S Finnigan. 2011. “The Ebb and Flow of Social Network Ties Between District Leaders Under High-Stakes Accountability.” American Educational Research Journal 48 (1): 39–79. https://www.jstor.org/stable/27975281.