Citation, co-citation and bibliographic coupling networks with full or fractional counting (Batagelj, 2020).
Arguments
- A1
A citation network between works, where
A1[p, q] = 1if the work p cites the work q- A2
For
approach = "citation", the authorship matrix, with the works in rows and the authors in columns- approach
Character string, “citation”, “cocitation” and “bcoupling”
- fractional
Whether to use fractional counting (default) or full counting
- symmetric
For fractional bibliographic coupling, the symmetric similarity:
none(default, the proportion of shared references),average,minimum,maximum,geometric,harmonicorjaccard
Value
Return the citation network between authors, the co-citation network or the bibliographic coupling network.
Details
A1 is a citation network \(Ci\) between works, where A1[p, q] = 1 when the work \(p\)
cites the work \(q\). With fractional counting each work has a total weight of one, which is divided
equally among the works it cites: \(Cin = D \cdot Ci\), where \(D\) is the diagonal matrix of one over
the number of references of each work (one when it has none). Fractional counting prevents the works with
many references, such as reviews, from dominating the result (Batagelj, 2020).
approach = "cocitation": \(Ci^T Ci\), the number of works that cite both works, or with fractional
counting \(Cin^T Cin\), in which each citing work contributes a total of one.
approach = "bcoupling": \(Ci \cdot Ci^T\), the number of works cited by both works. Fractional counting
cannot be applied in the same way to bibliographic coupling (Batagelj, 2020: 631), so it gives
\(biC = Cin \cdot Ci^T\), the proportion of the references of \(p\) that it shares with \(q\), which is
not symmetric. symmetric turns it into a symmetric similarity: the average, the minimum,
the maximum, the geometric mean (the cosine of Salton), the harmonic mean, or the
jaccard index, the shared references divided by the references of either work.
approach = "citation": the citations between the authors of the works, given the authorship matrix
\(WA\) in A2 (works in rows, authors in columns): \(WA^T \cdot Ci \cdot WA\), the number of times the
works of an author cite the works of another. With fractional counting each work is divided equally among its
authors, \(WAn^T \cdot Ci \cdot WAn\), so that the total of the network is the number of citations.
Which count should be conserved guides the choice between the two protocols (Prathap and Mukherjee, 2020): full counting conserves the number of paths between the nodes, and fractional counting the number of nodes (works).
References
Batagelj, V. (2020). On fractional approach to analysis of linked networks. Scientometrics, 123(2), 621-633. doi:10.1007/s11192-020-03383-y
Batagelj, V. (2022). Analysis of the Southern women network using fractional approach. Social Networks, 68, 229-236 doi:10.1016/j.socnet.2021.08.001
Batagelj, V., & Cerinšek, M. (2013). On bibliographic networks. Scientometrics, 96(3), 845–864. doi:10.1007/s11192-012-0940-1
Prathap, G., & Mukherjee, S. (2020). Letter to the Editor: Comments on the paper of Batagelj—on fractional approach to analysis of linked networks. Scientometrics, 124(3), 2717–2722. doi:10.1007/s11192-020-03541-2
Examples
# Five works: w1 cites w2 and w3, w4 cites w2, w3 and w5, w5 cites w3
Ci <- matrix(c(
0, 1, 1, 0, 0,
0, 0, 0, 0, 0,
0, 0, 0, 0, 0,
0, 1, 1, 0, 1,
0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(Ci) <- colnames(Ci) <- paste0("w", 1:5)
# Authors of the works
WA <- matrix(c(
1, 1, 0,
0, 1, 0,
0, 0, 1,
1, 0, 0,
0, 0, 1
), byrow = TRUE, ncol = 3)
rownames(WA) <- rownames(Ci)
colnames(WA) <- c("a1", "a2", "a3")
fractional_approach(Ci, WA, approach = "citation")
#> a1 a2 a3
#> a1 0 1.5 2.5
#> a2 0 0.5 0.5
#> a3 0 0.0 1.0
fractional_approach(Ci, approach = "cocitation")
#> w1 w2 w3 w4 w5
#> w1 0 0.0000000 0.0000000 0 0.0000000
#> w2 0 0.3611111 0.3611111 0 0.1111111
#> w3 0 0.3611111 1.3611111 0 0.1111111
#> w4 0 0.0000000 0.0000000 0 0.0000000
#> w5 0 0.1111111 0.1111111 0 0.1111111
fractional_approach(Ci, approach = "bcoupling", symmetric = "geometric")
#> w1 w2 w3 w4 w5
#> w1 1.0000000 0 0 0.8164966 0.7071068
#> w2 0.0000000 0 0 0.0000000 0.0000000
#> w3 0.0000000 0 0 0.0000000 0.0000000
#> w4 0.8164966 0 0 1.0000000 0.5773503
#> w5 0.7071068 0 0 0.5773503 1.0000000