Project Bipartite Network to One-Mode

Projects a two-mode (bipartite/incidence) network into a one-mode adjacency matrix. Row-mode projection yields a matrix of shared-column connections among row nodes; column-mode projection does the converse.

Usage

project_bipartite(x, mode = "rows", method = "sum", ...)

Arguments

x

An incidence matrix (rows = type 1 nodes, columns = type 2 nodes) where non-zero entries indicate connections. Can also be a data.frame with columns type1, type2, and optionally weight.

mode

Character. "rows" (default) projects onto row nodes (result: n_rows x n_rows). "columns" projects onto column nodes (result: n_cols x n_cols).

method

Character. Projection method:

"sum": Weighted projection: A %*% t(A) (rows) or t(A) %*% A (columns). Edge weight equals sum of shared connection weights.
"binary": Co-occurrence count: binarize A first, then compute overlap. Edge weight equals number of shared connections.
"jaccard": Jaccard similarity: shared / (total_i + total_j - shared) for each pair.
"cosine": Cosine similarity: dot product of row (or column) vectors divided by the product of their norms.
"newman": Newman's weighted projection (Newman 2001): each shared affiliation contributes 1 / (d_k - 1) where d_k is the degree of the shared node. Gives more weight to connections through exclusive affiliations.

...

Additional arguments (currently unused).

Value

A square adjacency matrix with row and column names preserved from the input. Diagonal is set to 0 (no self-loops).

Details

For the Newman projection, affiliations shared with only one node of the focal type (d_k = 1) are skipped, since 1 / (d_k - 1) is undefined. This follows the convention in Newman (2001).

References

Newman, M. E. J. (2001). Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Physical Review E, 64(1), 016132.

Examples

# Incidence matrix: 4 students x 3 courses
inc <- matrix(c(1, 1, 0,
                1, 0, 1,
                0, 1, 1,
                1, 1, 1), 4, 3, byrow = TRUE)
rownames(inc) <- paste0("S", 1:4)
colnames(inc) <- paste0("C", 1:3)

# Student co-enrollment (weighted)
cograph::project_bipartite(inc, mode = "rows", method = "sum")
#>    S1 S2 S3 S4
#> S1  0  1  1  2
#> S2  1  0  1  2
#> S3  1  1  0  2
#> S4  2  2  2  0

# Course overlap (Jaccard similarity)
cograph::project_bipartite(inc, mode = "columns", method = "jaccard")
#>     C1  C2  C3
#> C1 0.0 0.5 0.5
#> C2 0.5 0.0 0.5
#> C3 0.5 0.5 0.0

# Newman's weighted projection
cograph::project_bipartite(inc, mode = "rows", method = "newman")
#>     S1  S2  S3 S4
#> S1 0.0 0.5 0.5  1
#> S2 0.5 0.0 0.5  1
#> S3 0.5 0.5 0.0  1
#> S4 1.0 1.0 1.0  0