centrality_factor {CurricularAnalytics}

Calculate centrality

Description

A helper function for calculating the centrality for each node.

Usage

centrality_factor(node_list, edge_list)

Arguments

node_list

Dataframe with an 'id' column for each node and a 'term' column specifying which term the course is to be taken in.

edge_list

Dataframe with two columns 'from' and 'to' specifying directed edges starting at 'from' nodes directed towards 'to' nodes. Entries must use node ids from node_list.

Details

A course is considered central if it has many requisite edges flowing in and out of the node. More formally it is the number of long paths that include the node. That is, consider a curriculum graph G_c and a vertex v_i. A long path is a path that satisfies the following conditions:

• v_i,v_j,v_k are distinct

• v_j \to v_i \to v_k

• v_j is a source node (in-degree zero)

• v_k is a sink node (out-degree zero)

Let P_{v_i}=\{p_1,p_2,\dots\} denote the set of all paths defined as above. Then the centrality of a node v_i is given by

q(v_i)=\sum^{|P_{v_i}|}_{l=1}\#(p_l)

More plainly this is the number of paths containing v_i of at least length 3 where v_i is neither a source nor sink node.

Value

A dataframe containing the centrality of each node

Author(s)

Daniel Krasnov

References

Heileman, Gregory L, Chaouki T Abdallah, Ahmad Slim, and Michael Hickman. 2018. “Curricular Analytics: A Framework for Quantifying the Impact of Curricular Reforms and Pedagogical Innovations.” arXiv Preprint arXiv:1811.09676.

Examples

edge_list <- data.frame(from = c(1, 3), to = c(3, 4))
node_list <-
data.frame(
 id = 1:4,
 label = c("MATH 100", "DATA 101", "MATH 101", "MATH 221"),
 term = c(1, 1, 2, 2)
)

cf_df <- centrality_factor(node_list,edge_list)
print(cf_df)
# Output:
#   id cf
#1  1  0
#2  2  0
#3  3  3
#4  4  0