Degree of the nodes of a network of two or three levels, counting the ties within each level and the ties between the levels in different combinations.
Usage
multilevel_degree(
A1,
B1,
A2 = NULL,
B2 = NULL,
A3 = NULL,
B3 = NULL,
complete = FALSE,
digraphA1 = FALSE,
digraphA2 = FALSE,
digraphA3 = FALSE,
typeA1 = "out",
typeA2 = "out",
typeA3 = "out",
loopsA1 = FALSE,
loopsA2 = FALSE,
loopsA3 = FALSE,
normalized = FALSE,
weightedA1 = FALSE,
weightedA2 = FALSE,
weightedA3 = FALSE,
alphaA1 = 0.5,
alphaA2 = 0.5,
alphaA3 = 0.5
)Arguments
- A1
The square matrix of the lowest level
- B1
The incidence matrix of the ties between the nodes of first level and the nodes of the second level
- A2
The square matrix of the second level
- B2
The incidence matrix of the ties between the nodes of the second level and the nodes of the third level
- A3
The square matrix of the third level
- B3
The incidence matrix of the ties between the nodes of the third level and the nodes of the first level
- complete
Whether to return every column described in the details, instead of only the
multileveldegree- digraphA1
Whether A1 is a directed network
- digraphA2
Whether A2 is a directed network
- digraphA3
Whether A3 is a directed network
- typeA1
Type of degree of the network for A1, "out" for out-degree, "in" for in-degree or "all" for the sum of the two
- typeA2
Type of degree of the network for A2, "out" for out-degree, "in" for in-degree or "all" for the sum of the two
- typeA3
Type of degree of the network for A3, "out" for out-degree, "in" for in-degree or "all" for the sum of the two
- loopsA1
Whether the loops of the edges are considered in matrix A1
- loopsA2
Whether the loops of the edges are considered in matrix A2
- loopsA3
Whether the loops of the edges are considered in matrix A3
- normalized
Whether to divide each degree by the largest value it could take, as described in the details (Espinosa-Rada et al., 2021)
- weightedA1
Whether A1 is weighted
- weightedA2
Whether A2 is weighted
- weightedA3
Whether A3 is weighted
- alphaA1
The alpha parameter of A1 according to Opsahl et al (2010) for weighted networks. The value 0.5 is given by default.
- alphaA2
The alpha parameter of A2 according to Opsahl et al (2010) for weighted networks. The value 0.5 is given by default.
- alphaA3
The alpha parameter of A3 according to Opsahl et al (2010) for weighted networks. The value 0.5 is given by default.
Value
A data frame with one row for each node of every level, and the multilevel degree, or every column described in the details when complete = TRUE
Details
The levels are placed in a single meta-matrix. Level one has n nodes and the
ties A1, level two has m nodes and the ties A2, and level three
has k nodes and the ties A3. The incidence matrices join the levels:
B1 the first with the second (n by m), B2 the second with
the third (m by k), and B3 the third with the first (k by
n). A level that is not given has no ties within it.
Each column of the result emphasises a different section of the meta-matrix:
multilevel: every node counts the ties within its own level and its ties with
the other levels, i.e. A1 + B1 + B3 for the first level,
B1 + A2 + B2 for the second, and B2 + A3 + B3 for the third.
bipartiteB1, bipartiteB2 and bipartiteB3: the degree in each
incidence matrix, i.e. only the ties between two levels.
tripartiteB1B2, tripartiteB1B3, tripartiteB2B3 and
tripartiteB1B2B3: the degree in the union of incidence matrices, i.e. only the
ties between levels.
low_multilevel (A1 + B1 + B2 + B3), meso_multilevel
(B1 + A2 + B2 + B3) and high_multilevel (B1 + B2 + A3 + B3):
the ties within a single level, together with all the ties between levels. For the
nodes of the emphasised level they are the same as multilevel: the first level
in low_multilevel, the second in meso_multilevel and the third in
high_multilevel.
The rows are named n1, n2, ... for the first level, m1, m2, ... for the
second and k1, k2, ... for the third. Without complete = TRUE, only the
multilevel column is returned.
With normalized = TRUE, each degree is divided by the largest value it could
take: the other nodes of the same level plus the nodes of the levels it is tied to. For
the multilevel column this is (n - 1) + m for the first level
((n - 1) + m + k when B3 is given), (m - 1) + n + k for the
second level ((m - 1) + n with two levels), and (k - 1) + m for the
third level ((k - 1) + m + n when B3 is given). The bipartite degrees are
divided by the number of nodes of the other level (Borgatti and Everett, 1997). The
normalized values are only defined for binary matrices. All the values are rounded to
three decimals.
The ties within each level can be directed (digraphA1, typeA1, ...) and
weighted (weightedA1, alphaA1, ...), in which case the degree of Opsahl et
al. (2010) is used. The ties between levels are undirected.
References
Borgatti, S. P., and Everett, M. G. (1997). Network analysis of 2-mode data. Social Networks, 19(3), 243–269.
Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215–239.
Opsahl, T., Agneessens, F., and Skvoretz, J. (2010). Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks, 32(3), 245–251.
Examples
A1 <- matrix(c(
0, 1, 0, 0, 0,
1, 0, 0, 1, 0,
0, 0, 0, 1, 0,
0, 1, 1, 0, 1,
0, 0, 0, 1, 0
), byrow = TRUE, ncol = 5)
B1 <- matrix(c(
1, 0, 0,
1, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 1
), byrow = TRUE, ncol = 3)
A2 <- matrix(c(
0, 1, 1,
1, 0, 0,
1, 0, 0
), byrow = TRUE, nrow = 3)
B2 <- matrix(c(
1, 1, 0, 0,
0, 0, 1, 0,
0, 0, 1, 1
), byrow = TRUE, ncol = 4)
A3 <- matrix(c(
0, 1, 1, 1,
1, 0, 0, 0,
1, 0, 0, 1,
1, 0, 1, 0
), byrow = TRUE, ncol = 4)
B3 <- matrix(c(
1, 0, 0, 0, 0,
0, 1, 0, 1, 0,
0, 0, 0, 0, 0,
0, 0, 0, 0, 0
), byrow = TRUE, ncol = 5)
multilevel_degree(A1, B1, A2, B2, A3, B3)
#> multilevel
#> n1 3
#> n2 5
#> n3 2
#> n4 5
#> n5 3
#> m1 6
#> m2 6
#> m3 4
#> k1 5
#> k2 4
#> k3 4
#> k4 3
# \donttest{
multilevel_degree(A1, B1, A2, B2, A3, B3, normalized = TRUE, complete = TRUE)
#> multilevel bipartiteB1 bipartiteB2 bipartiteB3 tripartiteB1B2 tripartiteB1B3
#> n1 0.273 0.333 NA 0.25 0.333 0.583
#> n2 0.455 0.667 NA 0.25 0.667 0.917
#> n3 0.182 0.333 NA 0.00 0.333 0.333
#> n4 0.455 0.333 NA 0.25 0.333 0.583
#> n5 0.273 0.667 NA 0.00 0.667 0.667
#> m1 0.545 0.400 0.500 NA 0.900 0.400
#> m2 0.545 0.800 0.250 NA 1.050 0.800
#> m3 0.364 0.200 0.500 NA 0.700 0.200
#> k1 0.455 NA 0.333 0.20 0.333 0.200
#> k2 0.364 NA 0.333 0.40 0.333 0.400
#> k3 0.364 NA 0.667 0.00 0.667 0.000
#> k4 0.273 NA 0.333 0.00 0.333 0.000
#> tripartiteB2B3 tripartiteB1B2B3 low_multilevel meso_multilevel
#> n1 0.250 0.583 0.273 0.583
#> n2 0.250 0.917 0.455 0.917
#> n3 0.000 0.333 0.182 0.333
#> n4 0.250 0.583 0.455 0.583
#> n5 0.000 0.667 0.273 0.667
#> m1 0.500 0.900 0.900 0.545
#> m2 0.250 1.050 1.050 0.545
#> m3 0.500 0.700 0.700 0.364
#> k1 0.533 0.533 0.533 0.533
#> k2 0.733 0.733 0.733 0.733
#> k3 0.667 0.667 0.667 0.667
#> k4 0.333 0.333 0.333 0.333
#> high_multilevel
#> n1 0.583
#> n2 0.917
#> n3 0.333
#> n4 0.583
#> n5 0.667
#> m1 0.900
#> m2 1.050
#> m3 0.700
#> k1 0.455
#> k2 0.364
#> k3 0.364
#> k4 0.273
# }