Package 'netmem'

Description

This function transforms an adjacency matrix into an incidence matrix.

Usage

adj_to_incidence(A, loops = TRUE, directed = TRUE, weighted = TRUE)

Arguments

Value

A numeric matrix where rows represent nodes and columns represent edges. - In a **directed** network, a source node has a negative value (-weight), and a target node has a positive value (+weight). - In an **undirected** network, both nodes involved in an edge share the weight (positive values). - If 'weighted = FALSE', all edges have a weight of '1'.

Examples

# Define an adjacency matrix (directed and weighted)
A <- matrix(c(
  1, 3, 0, 0, 2,
  0, 0, 2, 0, 0,
  5, 0, 0, 0, 0,
  0, 0, 0, 0, 1,
  0, 4, 0, 0, 0
), byrow = TRUE, nrow = 5)

# Convert to an incidence matrix (directed, weighted)
(inc_matrix <- adj_to_incidence(A))

# Undirected, weighted graph
(inc_matrix_undirected <- adj_to_incidence(A, directed = FALSE))

# Directed, unweighted graph
(inc_matrix_unweighted <- adj_to_incidence(A, weighted = FALSE))

# Ignore loops
(inc_matrix_no_loops <- adj_to_incidence(A, loops = FALSE))

Description

Transform an adjacency list into a matrix

Usage

adj_to_matrix(A, type = c("adjacency", "incidence", "weighted"), loops = FALSE)

Arguments

Value

This function transforms an adjacency list into a matrix

Author(s)

Alejandro Espinosa-Rada

Examples

adj_groups <- rbind(
  c("a", "b", "c"), c("a", "c", NA),
  c("b", "c", NA), c("c", NA, NA),
  c("c", "a", NA)
)
M <- adj_to_matrix(adj_groups, type = "adjacency", loops = TRUE)
M

Description

Reduces a multiplex network to a single matrix, in which the actors are joined when they are connected in the layers (Battiston et al., 2014; De Domenico et al., 2015).

Usage

aggregate_layers(layers, method = c("sum", "binary", "mean"), l = NULL)

Arguments

Details

method = "sum" (default) adds the layers, so the value of a tie is the number of layers in which it is present, which Battiston et al. (2014) call the overlapping network. "binary" gives one to every tie that is present in at least one layer, and "mean" divides the sum by the number of layers.

The layers are given as a list of matrices, or as the supra-adjacency matrix of supra_adjacency() together with the number of layers l, in which case the coupling between the layers is ignored.

Aggregating loses the information of which layer each tie belongs to, and measures computed on the aggregated network can differ from the ones computed on the layers (De Domenico et al., 2015).

Value

This function returns a square matrix with the actors of the layers.

Author(s)

Alejandro Espinosa-Rada

References

Battiston, F., Nicosia, V. and Latora, V. (2014). Structural measures for multiplex networks. Physical Review E, 89(3), 032804. doi:10.1103/PhysRevE.89.032804

De Domenico, M., Nicosia, V., Arenas, A. and Latora, V. (2015). Structural reducibility of multilayer networks. Nature Communications, 6, 6864. doi:10.1038/ncomms7864

Examples

A1 <- matrix(c(
  0, 1, 0,
  1, 0, 1,
  0, 1, 0
), byrow = TRUE, ncol = 3, dimnames = list(letters[1:3], letters[1:3]))

A2 <- matrix(c(
  0, 1, 1,
  1, 0, 0,
  1, 0, 0
), byrow = TRUE, ncol = 3, dimnames = list(letters[1:3], letters[1:3]))

# The tie a-b is in both layers
aggregate_layers(list(A1, A2))
aggregate_layers(list(A1, A2), method = "binary")

# The same from the supra-adjacency matrix, which knows its layers
aggregate_layers(supra_adjacency(list(A1, A2)))

Description

Number of alters of each node in each category, for categories that can overlap (Everett and Borgatti, 2026).

Usage

alter_composition(A, B, proportion = FALSE)

Arguments

Details

The memberships B are made row-stochastic, so that each row gives the proportion of a node that belongs to each category (for instance, the proportion of time spent on each project). The alter composition is the product $AB$: for a partition, it counts the alters of each node in each category; with overlapping categories, it gives the extent to which the alters of each node belong to each category. The rows of $AB$ add up to the degree of each node. A need not be square: its columns should be the rows of B, as in a matrix of respondents and the alters they named.

Value

This function returns a matrix of the nodes (rows) and the categories (columns).

Author(s)

Alejandro Espinosa-Rada

References

Everett, M. G. and Borgatti, S. P. (2026). Alter composition with overlapping group memberships. Social Networks, 85, 80–88. doi:10.1016/j.socnet.2025.12.001

Examples

A <- matrix(c(
  0, 1, 1, 0,
  1, 0, 1, 1,
  1, 1, 0, 0,
  0, 1, 0, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- colnames(A) <- c("a", "b", "c", "d")

# A partition
alter_composition(A, c("x", "x", "y", "y"))

# Overlapping categories: hours spent on two projects
B <- matrix(c(
  10, 0,
  5, 5,
  0, 8,
  2, 6
), byrow = TRUE, ncol = 2)
alter_composition(A, B)

Description

Blau's heterogeneity of the alters of each node, for categories that can overlap (Everett and Borgatti, 2026).

Usage

alter_heterogeneity(A, B, normalized = FALSE)

Arguments

Details

The heterogeneity of a node is $1 - \sum_k p_k^2$, where $p_k$ is the proportion of its alters in category $k$, taken from the rows of alter_composition() (Blau, 1977). A node with a single alter has some heterogeneity when that alter belongs to several categories. The IQV divides the index by its maximum, $1 - 1/K$, where $K$ is the number of categories. The nodes without alters are NA.

Value

This function returns a vector with the heterogeneity of the alters of each node.

Author(s)

Alejandro Espinosa-Rada

References

Blau, P. M. (1977). Inequality and heterogeneity: A primitive theory of social structure. Free Press.

Everett, M. G. and Borgatti, S. P. (2026). Alter composition with overlapping group memberships. Social Networks, 85, 80–88. doi:10.1016/j.socnet.2025.12.001

Examples

A <- matrix(c(
  0, 1, 1, 0,
  1, 0, 1, 1,
  1, 1, 0, 0,
  0, 1, 0, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- colnames(A) <- c("a", "b", "c", "d")
B <- matrix(c(
  10, 0,
  5, 5,
  0, 8,
  2, 6
), byrow = TRUE, ncol = 2)

alter_heterogeneity(A, B)

Description

E-I index and Yule's Q of each node, for categories that can overlap (Everett and Borgatti, 2026).

Usage

alter_homophily(
  A,
  B,
  method = c("ei", "yule"),
  similarity = c("product", "minimum", "cosine")
)

Arguments

Details

The internal ties of a node $i$ are $I = \sum_j A_{ij} S_{ij}$, where $S_{ij}$ is the similarity of the memberships of $i$ and $j$, and the external ties are $E = D - I$, where $D$ is the degree. The E-I index is $(E - I) / (E + I)$ (Krackhardt and Stern, 1988): -1 when all the alters are in the categories of ego (homophily) and +1 when none is (heterophily).

Yule's Q also uses the nodes that are not alters, to take into account how many nodes of each category are available: $a = I$ and $b = E$ for the alters, and $c$ and $d$ are the same quantities for the other nodes. Then $Q = (ad - bc) / (ad + bc)$, which is positive for homophily, and zero when the ties do not depend on the categories.

The similarity of two memberships can be defined in three ways:

similarity = "product" (default), $S_{ij} = \sum_k B_{ik} B_{jk}$. It reduces to the usual indices when the categories are a partition. If the categories are, for instance, the proportion of time spent in each of several places, it is the probability that two nodes are in the same place.

similarity = "minimum", $S_{ij} = \sum_k \min(B_{ik}, B_{jk})$, the trait version ($E_s-I_s$ and $Q_s$). Two nodes with identical memberships are fully similar even when they split their time among categories, which fits categories that are traits such as skills or interests.

similarity = "cosine", the cosine of the memberships, which favours two nodes that concentrate in the same categories, such as two specialists with the same specialty.

The ties are binary and the loops are ignored. For a directed network the alters are the out-neighbours. Nodes without alters, and Yule's Q with $ad + bc = 0$, are NA.

Value

This function returns a vector with the index of each node.

Author(s)

Alejandro Espinosa-Rada

References

Everett, M. G. and Borgatti, S. P. (2026). Alter composition with overlapping group memberships. Social Networks, 85, 80–88. doi:10.1016/j.socnet.2025.12.001

Krackhardt, D. and Stern, R. N. (1988). Informal networks and organizational crises: An experimental simulation. Social Psychology Quarterly, 51(2), 123–140. doi:10.2307/2786835

Examples

A <- matrix(c(
  0, 1, 1, 0,
  1, 0, 1, 1,
  1, 1, 0, 0,
  0, 1, 0, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- colnames(A) <- c("a", "b", "c", "d")
B <- matrix(c(
  10, 0,
  5, 5,
  0, 8,
  2, 6
), byrow = TRUE, ncol = 2)

alter_homophily(A, B)
alter_homophily(A, B, method = "yule", similarity = "minimum")

Description

Betweenness centrality of Freeman (1977), computed with the algorithm of Brandes (2001).

Usage

betweenness_centrality(
  A,
  digraph = TRUE,
  weighted = FALSE,
  alpha = 1,
  normalized = FALSE
)

Arguments

Details

The betweenness of a node is the sum, over all pairs of other nodes, of the proportion of geodesics between the pair that pass through the node. For undirected networks each pair is counted once.

For valued matrices, the tie weights are treated as strengths and transformed into lengths as $1 / w^{\alpha}$ (Opsahl et al., 2010). If alpha = 0 the binary network is used.

Value

This function returns the betweenness centrality of the nodes.

Author(s)

Alejandro Espinosa-Rada

References

Brandes, U. (2001). A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25(2), 163–177. doi:10.1080/0022250X.2001.9990249

Freeman, L. C. (1977). A set of measures of centrality based on betweenness. Sociometry, 40(1), 35–41. doi:10.2307/3033543

Opsahl, T., Agneessens, F., and Skvoretz, J. (2010). Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks, 32(3), 245–251. doi:10.1016/j.socnet.2010.03.006

Examples

A <- matrix(c(
  0, 1, 1, 1, 0, 0, 0, 0, 0,
  1, 0, 1, 1, 1, 0, 0, 0, 0,
  1, 1, 0, 1, 0, 1, 0, 0, 0,
  1, 1, 1, 0, 1, 1, 0, 0, 0,
  0, 1, 0, 1, 0, 1, 1, 0, 0,
  0, 0, 1, 1, 1, 0, 1, 0, 0,
  0, 0, 0, 0, 1, 1, 0, 1, 0,
  0, 0, 0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 0
), byrow = TRUE, ncol = 9)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

betweenness_centrality(A, digraph = FALSE)

Description

Densities of the blocks given by a partition of the nodes, and the image matrix that summarises them (Lorrain and White, 1971; Wasserman and Faust, 1994).

Usage

block_density(A, partition, cutoff = "density", loops = FALSE)

Arguments

Details

A partition of the nodes divides the matrix into blocks. The density of a block is the proportion of the possible ties that are present, and the image matrix assigns a one to the blocks whose density is at least the cutoff. The usual criterion is the density of the whole network (cutoff = "density"), so a block is one when it is denser than the network as a whole.

The diagonal blocks contain the ties within a position, where the possible ties exclude the loops unless loops = TRUE.

Value

This function returns the density of each block, the image matrix and the density of the network.

Author(s)

Alejandro Espinosa-Rada

References

Lorrain, F. and White, H. C. (1971). Structural equivalence of individuals in social networks. Journal of Mathematical Sociology, 1(1), 49–80. doi:10.1080/0022250X.1971.9989788

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 0, 0, 0,
  1, 1, 0, 1, 1, 1,
  0, 0, 1, 0, 0, 0,
  0, 0, 1, 0, 0, 0,
  0, 0, 1, 0, 0, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

block_density(A, partition = c(1, 1, 1, 2, 2, 2))

Description

The function provide a normalisation provided by Bonacich (1972).

Usage

bonacich_norm(A, projection = c("rows", "columns"), normalisation = FALSE)

Arguments

Value

This function returns the Bonacich normalisation.

Source

Adapted from Borgatti, S., Everett, M., Johnson, J. and Agneessens, P. (2022) Analyzing Social Networks Using R. Sage.

References

Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology, 2: 112-120.

Examples

A <- matrix(
  c(
    1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0,
    1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0,
    0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0,
    1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0,
    0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0,
    0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1,
    0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1,
    0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1,
    0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0
  ),
  byrow = TRUE, ncol = 14
)
bonacich_norm(A)

Description

Power centrality of Bonacich (1987), where being connected to well-connected others can increase or decrease the centrality of a node.

Usage

bonacich_power(
  A,
  beta = 0,
  digraph = TRUE,
  weighted = FALSE,
  scale = c("none", "ssq")
)

Arguments

Details

The centrality is $x = (I - \beta A)^{-1} A 1$. When beta is positive, a node is central when it is connected to central nodes, as in the eigenvector centrality. When beta is negative, being connected to well-connected others reduces the centrality of a node, which describes bargaining situations. With beta = 0 the measure is the degree.

Value

This function returns the power centrality of the nodes.

Author(s)

Alejandro Espinosa-Rada

References

Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170–1182. doi:10.1086/228631

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

bonacich_power(A, beta = 0.1, digraph = FALSE)
bonacich_power(A, beta = -0.1, digraph = FALSE)

Description

Brokerage roles of Gould and Fernandez (1989), for categories that can overlap (Everett and Borgatti, 2026).

Usage

brokerage_roles(A, B)

Arguments

Details

A node $b$ brokers in a path $a \to b \to c$ when $a$ has no tie to $c$. The role depends on the categories of the three nodes: coordinator (all in the same category), gatekeeper ($b$ and $c$ in the same category, $a$ in another), representative ($a$ and $b$ in the same category, $c$ in another), consultant (also called itinerant; $a$ and $c$ in the same category, $b$ in another) and liaison (the three in different categories).

With overlapping categories, each path counts partly for each role (Everett and Borgatti, 2026: Eq. 5). The coordinator part is $\sum_j B_{aj} B_{bj} B_{cj}$, the joint membership of the three nodes in each category; the gatekeeper part is $\sum_j B_{bj} B_{cj} (1 - B_{aj})$; the representative part $\sum_j B_{aj} B_{bj} (1 - B_{cj})$; the consultant part $\sum_j B_{aj} B_{cj} (1 - B_{bj})$; and the liaison part is the rest. When the categories are a partition, each path counts for a single role, as in Gould and Fernandez (1989). The parts of each path add up to one, so the total of each node is the number of paths it brokers.

A symmetric matrix is treated as a directed network with ties in both directions, so each path is counted from both ends.

Value

This function returns a matrix with the score of each node (rows) in each role, and the total.

Author(s)

Alejandro Espinosa-Rada

References

Everett, M. G. and Borgatti, S. P. (2026). Alter composition with overlapping group memberships. Social Networks, 85, 80–88. doi:10.1016/j.socnet.2025.12.001

Gould, R. V. and Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89–126. doi:10.2307/270949

Examples

A <- matrix(c(
  0, 1, 0, 0,
  0, 0, 1, 1,
  0, 0, 0, 1,
  1, 0, 0, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- colnames(A) <- c("a", "b", "c", "d")

brokerage_roles(A, c("x", "x", "y", "y"))

B <- matrix(c(
  10, 0,
  5, 5,
  0, 8,
  2, 6
), byrow = TRUE, ncol = 2)
brokerage_roles(A, B)

Description

Interactions among the 18 people of a three-week course, the 1992 NSF Summer Institute on Research Methods in Cultural Anthropology, 14 participants and 4 instructors. The data were collected by Steve Borgatti, Russ Bernard, Bert Pelto and Gery Ryan. At the end of the second week, each person sorted cards with the names of the others by how much they had interacted with them since the beginning of the course.

Usage

data(campnet)

Format

A list with:

network

An 18 x 18 binary matrix, known as Campnet, where network[i, j] = 1 if j is among the three people with whom i interacted most.

ranks

An 18 x 18 matrix with the rank that each respondent (row) gave to each of the others (1 = most interaction, 17 = least).

attributes

A data frame with the attributes of the people:

gender

1 = woman; 2 = man

role

1 = participant; 2 = instructor

Details

The network gives the betweenness scores reported by Everett and Borgatti (2026: Table 5).

Source

UCINET datasets, Camp 92 (https://sites.google.com/site/ucinetsoftware/datasets/camp-92).

Borgatti, S. P., Everett, M. G. and Johnson, J. C. (2018). Analyzing Social Networks. Second edition. SAGE.

Everett, M. G. and Borgatti, S. P. (2026). Alter composition with overlapping group memberships. Social Networks, 85, 80–88. doi:10.1016/j.socnet.2025.12.001

Description

Centralization of Freeman (1979): how much a network is dominated by its most central node.

Usage

centrality_centralization(
  A,
  measure = c("degree", "closeness", "betweenness", "eigenvector"),
  digraph = FALSE,
  type = c("out", "in", "all"),
  loops = FALSE
)

Arguments

Details

The centralization is the sum of the differences between the highest centrality and the centrality of every node, divided by the largest sum that a network of the same order can have. For degree, closeness and betweenness the maximum is given by the star network, and for the eigenvector centrality by a network with a single tie, which gives $n - 2$.

The function is named centrality_centralization() so that it does not mask sna::centralization().

Value

This function returns the centralization of the network, the centrality scores and the theoretical maximum.

Author(s)

Alejandro Espinosa-Rada

References

Freeman, L. C. (1979). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215–239. doi:10.1016/0378-8733(78)90021-7

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

centrality_centralization(A, measure = "degree", digraph = FALSE)
centrality_centralization(A, measure = "betweenness", digraph = FALSE)

Description

Applies an exponential temporal decay to the arcs of a citation adjacency matrix and optionally normalises the result so that each citing paper distributes exactly one unit of citation influence among its references.

Usage

citation_decay(A, years, lambda = 0.2, normalize = TRUE)

Arguments

Details

As in traversal_weights, A[i,j] > 0 means that paper j cites paper i. The weight of the arc $i \to j$ is $\exp(-\lambda \cdot (y_j - y_i))$ where $y_i$ and $y_j$ are the publication years of $i$ and $j$ respectively. A larger $\lambda$ discounts older citations more aggressively. The normalisation divides each column, which holds the references of a citing paper, by its sum, so that differences in the length of the reference lists do not inflate the raw weights.

Value

A numeric matrix of the same dimensions as A containing the decay-weighted (and optionally normalised) citation arc weights. traversal_weights uses only whether each arc is present, so these weights do not change the search path counts; they can be combined with them, for instance by multiplying the two matrices element by element.

Author(s)

Alejandro Espinosa-Rada

References

Hummon, N.P. and Doreian, P. (1989). Connectivity in a citation network: The development of DNA theory. Social Networks. 11(1): 39-63. doi:10.1016/0378-8733(89)90017-8.

Examples

# P1 is cited by P2 and P3, which are both cited by P4
A <- matrix(c(
  0, 1, 1, 0,
  0, 0, 0, 1,
  0, 0, 0, 1,
  0, 0, 0, 0
), byrow = TRUE, nrow = 4)
rownames(A) <- c("P1", "P2", "P3", "P4")
colnames(A) <- c("P1", "P2", "P3", "P4")
years <- c(P1 = 2000, P2 = 2005, P3 = 2006, P4 = 2010)

citation_decay(A, years, lambda = 0.2)

Description

Maximal complete subgraphs of an undirected network, found with the algorithm of Bron and Kerbosch (1973).

Usage

clique_max(A, min = 2, max = NULL)

Arguments

Details

A clique is a set of nodes that are all adjacent to each other, and it is maximal when no other node can be added to it. Unlike clique_table(), which returns the triangles of the network, this function returns cliques of any size.

Value

This function returns a list with the names of the nodes of each maximal clique.

Author(s)

Alejandro Espinosa-Rada

References

Bron, C. and Kerbosch, J. (1973). Algorithm 457: Finding all cliques of an undirected graph. Communications of the ACM, 16(9), 575–577. doi:10.1145/362342.362367

Luce, R. D. and Perry, A. D. (1949). A method of matrix analysis of group structure. Psychometrika, 14(2), 95–116. doi:10.1007/BF02289146

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 1, 0, 0,
  1, 1, 0, 1, 0, 0,
  0, 1, 1, 0, 1, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 1, 1, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

clique_max(A)

Description

Exploration of a 3-cliques, as the maximum number of three or more actors who have all possible ties present among themselves

Usage

clique_table(A, list_cliques = FALSE, number = FALSE)

Arguments

Value

This function return an edge list of actors participating in 3-cliques.

If list_cliques = TRUE it also return the list of cliques per nodes. If number = TRUE the output returns the number of 3-cliques in the matrix.

Author(s)

Alejandro Espinosa-Rada

References

Luce, R.D. and Perry, A.D. (1949). A method of matrix analysis of group structure. Psychometrika, 14: 95-116.

Roethlisberger, F.J. and Dickson, W.J. (1939). Management and the Worker. Harvard University Press, Cambridge, MA.

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0, 0, 1, 0,
  1, 0, 1, 0, 0, 0, 0, 0, 0,
  1, 1, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 1, 1, 0, 0, 0,
  0, 0, 0, 1, 0, 0, 0, 0, 0,
  0, 0, 0, 1, 0, 0, 1, 1, 0,
  0, 0, 0, 0, 0, 1, 0, 1, 0,
  1, 0, 0, 0, 0, 1, 1, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0
), byrow = TRUE, ncol = 9)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)
clique_table(A, list_cliques = TRUE, number = TRUE)

Description

Closeness centrality of Freeman (1978) and its harmonic version (Marchiori and Latora, 2000; Rochat, 2009).

Usage

closeness_centrality(
  A,
  digraph = TRUE,
  type = c("out", "in", "all"),
  weighted = FALSE,
  alpha = 1,
  harmonic = FALSE,
  normalized = FALSE
)

Arguments

Details

Freeman's closeness is the inverse of the sum of the geodesic distances from a node to the others. When the network is disconnected, the sum only considers the nodes that can be reached, and the harmonic version is recommended, as it adds the inverse of each distance (an unreachable node adds zero).

For valued matrices, the tie weights are treated as strengths and transformed into lengths as $1 / w^{\alpha}$ (Opsahl et al., 2010). If alpha = 0 the binary network is used.

Value

This function returns the closeness centrality of the nodes.

Author(s)

Alejandro Espinosa-Rada

References

Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215–239. doi:10.1016/0378-8733(78)90021-7

Marchiori, M. and Latora, V. (2000). Harmony in the small-world. Physica A, 285(3-4), 539–546. doi:10.1016/S0378-4371(00)00311-3

Opsahl, T., Agneessens, F., and Skvoretz, J. (2010). Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks, 32(3), 245–251. doi:10.1016/j.socnet.2010.03.006

Rochat, Y. (2009). Closeness centrality extended to unconnected graphs: The harmonic centrality index. ASNA.

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

closeness_centrality(A, digraph = FALSE)
closeness_centrality(A, digraph = FALSE, harmonic = TRUE)

Description

Co‐occurrence matrix based on overlap function

Usage

co_occurrence(
  A,
  similarity = c("ochiai", "cosine"),
  occurrence = TRUE,
  projection = FALSE
)

Arguments

Value

This function returns the normalisation of a matrix into a symmetrical co‐occurrence matrix

Author(s)

Alejandro Espinosa-Rada

References

Borgatti, S. P., Halgin, D. S., 2011. Analyzing affiliation networks. In: J. Scott and P. J. Carrington (Eds.) The Sage handbook of social network analysis (pp. 417-433), Sage.

Zhou, Q., & Leydesdorff, L. (2016). The normalization of occurrence and Co-occurrence matrices in bibliometrics using Cosine similarities and Ochiai coefficients. Journal of the Association for Information Science and Technology, 67(11), 2805–2814. doi:10.1002/asi.23603

Examples

A <- matrix(
  c(
    2, 0, 2,
    1, 1, 0,
    0, 3, 3,
    0, 2, 2,
    0, 0, 1
  ),
  nrow = 5, byrow = TRUE
)

co_occurrence(A)

Description

Three classic ways of finding communities.

Usage

community_greedy(A, weighted = FALSE)

community_label(A, weighted = FALSE, max_iter = 100)

community_betweenness(A, weighted = FALSE)

Arguments

Details

community_greedy starts with every node alone and merges at each step the two groups that increase the modularity the most, keeping the partition with the highest modularity (Clauset, Newman and Moore, 2004).

community_label gives every node a different label, and then each node takes the label that most of its neighbours have, until no label changes. It is fast but the result depends on the order in which the nodes are visited, so it varies between runs (Raghavan, Albert and Kumara, 2007).

community_betweenness removes, one at a time, the tie with the highest edge betweenness, i.e. the tie through which most geodesics pass, as those ties connect groups rather than being inside them. The components that remain at each step give a partition, and the one with the highest modularity is returned (Girvan and Newman, 2002).

Value

These functions return the group of each node, the number of groups and the modularity of the partition.

Author(s)

Alejandro Espinosa-Rada

References

Clauset, A., Newman, M. E. J. and Moore, C. (2004). Finding community structure in very large networks. Physical Review E, 70(6), 066111. doi:10.1103/PhysRevE.70.066111

Girvan, M. and Newman, M. E. J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99(12), 7821–7826. doi:10.1073/pnas.122653799

Raghavan, U. N., Albert, R. and Kumara, S. (2007). Near linear time algorithm to detect community structures in large-scale networks. Physical Review E, 76(3), 036106. doi:10.1103/PhysRevE.76.036106

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 0, 0, 0,
  1, 1, 0, 1, 0, 0,
  0, 0, 1, 0, 1, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 1, 1, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

community_greedy(A)
community_betweenness(A)
set.seed(18051889)
community_label(A)

Description

Components of a network: the groups of nodes that can reach each other.

Usage

components_id(A, mode = c("weak", "strong"), bipartite = FALSE)

Arguments

Details

In a weak component the nodes are connected when the direction of the ties is ignored. In a strong component every node can reach every other node following the direction of the ties, so a strong component is contained in a weak one. For undirected networks both are the same.

For two-mode networks, the nodes of both sets are placed in the same network before looking for the components, so a component contains the nodes of the first mode and the ones of the second mode that they share.

Value

A vector with the component of each node, and the size of the components.

Author(s)

Alejandro Espinosa-Rada

References

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 1, 0, 0,
  1, 0, 1, 0, 0,
  1, 1, 0, 0, 0,
  0, 0, 0, 0, 1,
  0, 0, 0, 1, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:ncol(A)]
colnames(A) <- rownames(A)
components_id(A)

# In a chain of citations every node is in the same weak component,
# but each of them is its own strong component
B <- matrix(c(
  0, 1, 0, 0,
  0, 0, 1, 0,
  0, 0, 0, 1,
  0, 0, 0, 0
), byrow = TRUE, ncol = 4)
components_id(B)
components_id(B, mode = "strong")

Description

This function returns the relational composition of the given matrices. The compound relations define the paths and the social process flows of the given matrices (Pattison, 1993). However, those whom they link may or may not be aware of them. The compound relations allow us to identify "the possibly very long and devious chains of effects propagating withing concrete social systems through links of various kinds" (Lorrain & White, 1971: 50).

Usage

compound_relation(l = list(), comp = 3, matrices = FALSE, equate = FALSE)

Arguments

Value

This function provides the composition or concatenation of compound relations and the primitives of the matrices.

Author(s)

Alejandro Espinosa-Rada

References

Boorman, Scott A. and White, Harrison C. (1976) Social Structure from Multiple Networks. II. Role Structures. American Journal of Sociology. 81(6): 1384-1446.

Lorrain, Francois and White, Harrison C. (1971) Structural Equivalence of Individuals in Social Networks. Journal of Mathematical Sociology. 1: 49-80

Pattison, Philippa (1993) Algebraic Models for Social Networks. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 0, 0,
  1, 0, 0, 0,
  1, 1, 0, 1,
  0, 0, 1, 0
), byrow = TRUE, ncol = 4)
rownames(A) <- letters[1:NCOL(A)]
colnames(A) <- rownames(A)

B <- matrix(c(
  0, 1, 0, 0,
  1, 0, 0, 0,
  0, 0, 0, 1,
  0, 0, 1, 0
), byrow = TRUE, ncol = 4)
rownames(B) <- letters[1:NCOL(B)]
colnames(B) <- rownames(B)

cmp <- compound_relation(list(A, B), comp = 2, matrices = TRUE, equate = TRUE)
cmp$compound_relations
cmp$compound_matrices
cmp$equated

Description

Convergence of iterated correlations (Breiger, Boorman and Arabie, 1975), a partition of the nodes into positions of structurally equivalent actors.

Usage

concor(A, splits = 1, max_iter = 50, tol = 1e-08)

Arguments

Details

The rows and the columns of the matrix describe how each node relates to the others, so they are stacked into a profile. The correlations between the profiles of every pair of nodes are computed, and the correlations of those correlations are computed again and again. The matrix converges to a matrix of ones and minus ones, which splits the nodes into two positions. The procedure is repeated within each position, so splits divisions give at most $2^{splits}$ positions.

Value

This function returns the position of each node and the number of positions.

Author(s)

Alejandro Espinosa-Rada

References

Breiger, R. L., Boorman, S. A. and Arabie, P. (1975). An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling. Journal of Mathematical Psychology, 12(3), 328–383. doi:10.1016/0022-2496(75)90028-0

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 0, 0, 0,
  1, 1, 0, 0, 0, 0,
  0, 0, 0, 0, 1, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 1, 1, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

concor(A, splits = 1)

Description

Core-periphery model of Borgatti and Everett (2000): a group of nodes that are connected among themselves and with the rest, and a periphery of nodes that are connected with the core but not with each other.

Usage

core_periphery(
  A,
  method = c("discrete", "continuous"),
  digraph = FALSE,
  rep = 50
)

Arguments

Details

The discrete model looks for the partition of the nodes into a core and a periphery that maximises the correlation between the observed matrix and the ideal pattern, where a tie is expected when at least one of the two nodes belongs to the core. The search starts from the nodes sorted by degree, and from rep random partitions, and then moves one node at a time while the correlation improves. As the search can end in a local optimum, the result of the random starts depends on the seed.

The continuous model gives each node a coreness score instead of a class. The scores maximise the correlation between the observed matrix and the products of the scores of each pair, and are given by the leading eigenvector of the matrix.

Value

This function returns the fit of the model, and the partition or the coreness scores.

Author(s)

Alejandro Espinosa-Rada

References

Borgatti, S. P. and Everett, M. G. (2000). Models of core/periphery structures. Social Networks, 21(4), 375–395. doi:10.1016/S0378-8733(99)00019-2

Examples

A <- matrix(c(
  0, 1, 1, 1, 1, 0,
  1, 0, 1, 1, 0, 1,
  1, 1, 0, 1, 0, 0,
  1, 1, 1, 0, 0, 0,
  1, 0, 0, 0, 0, 0,
  0, 1, 0, 0, 0, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

core_periphery(A)
core_periphery(A, method = "continuous")

Description

Compares a statistic of the observed network with its distribution in random networks that share some of its features (Anderson, Butts and Carley, 1999; Wasserman and Faust, 1994).

Usage

cug_test(
  A,
  FUN,
  cmode = c("edges", "size", "dyad"),
  reps = 1000,
  digraph = TRUE,
  ...
)

Arguments

Details

The random networks are drawn from a uniform distribution conditioned on:

size: only the number of nodes, so every tie is present with probability one half,

edges: the number of nodes and the number of ties,

dyad: the dyad census, i.e. the number of mutual, asymmetric and null dyads (U|MAN).

The test says whether the statistic is higher or lower than expected once those features are taken into account. Conditioning on the dyad census, for instance, removes the tendency towards reciprocity before looking at the triads.

Value

This function returns the observed statistic, the mean and the standard deviation of the distribution, and the proportion of random networks with a statistic greater or equal, and lower or equal, than the observed one.

Author(s)

Alejandro Espinosa-Rada

References

Anderson, B. S., Butts, C. and Carley, K. (1999). The interaction of size and density with graph-level indices. Social Networks, 21(3), 239–267. doi:10.1016/S0378-8733(99)00011-8

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  1, 0, 1, 0, 0, 0,
  1, 1, 0, 1, 0, 0,
  0, 0, 1, 0, 1, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 1, 1, 0
), byrow = TRUE, ncol = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

set.seed(18051889)
cug_test(A,
  FUN = function(x) trans_coef(x, method = "global"),
  cmode = "edges", reps = 100, digraph = FALSE
)

Description

Cumulative sum of matrices

Usage

cumulativeSumMatrices(matrixList)

Arguments

Value

This function returns the cumulative sum of matrices

Author(s)

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1,
  0, 0, 0,
  0, 1, 0
), byrow = TRUE, ncol = 3)
B <- matrix(c(
  0, 0, 1,
  0, 0, 0,
  0, 0, 0
), byrow = TRUE, ncol = 3)
C <- matrix(c(
  0, 0, 0,
  1, 0, 0,
  0, 0, 0
), byrow = TRUE, ncol = 3)
matrixList <- list(A, B, C)
cumulativeSumMatrices(matrixList)

Description

dag_check verifies whether a directed adjacency matrix represents a directed acyclic graph (DAG) and, when it does not, removes a set of arcs that breaks every cycle. dag_sort returns the node names in topological order (sources first, sinks last).

Usage

dag_check(A)

dag_sort(A)

Arguments

Details

In the convention used throughout the main path functions, A[i,j] > 0 means that knowledge flows from paper i to paper j, that is, paper j cites paper i (Liu and Lu, 2012; Kuan, 2020). Sources (in-degree zero) are the papers that cite nobody in the corpus; sinks (out-degree zero) are the papers that nobody in the corpus cites. A matrix in which A[i,j] > 0 means that i cites j should be transposed first with t(A).

The order is obtained with the algorithm of Kahn (1962): the sources are placed first, their arcs are removed, and the nodes left without incoming arcs are placed next, until no node remains. When some nodes are never left without incoming arcs, they lie on a cycle.

A citation network can have cycles, for instance when two papers published at the same time cite each other, and a paper citing itself is a cycle of length one. dag_check removes the self-citations and then the arcs that point backwards in the ordering of Eades, Lin and Smyth (1993). Arcs that can be restored without closing a cycle are restored, so that no removed arc is unnecessary. The set is small but is not guaranteed to be the smallest possible, which is a hard problem. Liu, Lu and Ho (2019) discuss alternatives that keep all the citations, such as merging the papers of a cycle into one node, which should be applied before this function when the removed arcs matter.

Value

dag_check returns a named list:

is_dag

Logical; TRUE if the original matrix was already acyclic.

n_removed

Integer number of arcs removed, self-citations included.

A

Adjacency matrix with the removed arcs set to zero.

dag_sort returns a character vector of node names in topological order.

Author(s)

Alejandro Espinosa-Rada

References

Eades, P., Lin, X. and Smyth, W.F. (1993). A fast and effective heuristic for the feedback arc set problem. Information Processing Letters. 47(6): 319-323. doi:10.1016/0020-0190(93)90079-O.

Hummon, N.P. and Doreian, P. (1989). Connectivity in a citation network: The development of DNA theory. Social Networks. 11(1): 39-63. doi:10.1016/0378-8733(89)90017-8.

Kahn, A.B. (1962). Topological sorting of large networks. Communications of the ACM. 5(11): 558-562. doi:10.1145/368996.369025.

Liu, J.S., Lu, L.Y.Y. and Ho, M.H.C. (2019). A few notes on main path analysis. Scientometrics. 119(1): 379-391. doi:10.1007/s11192-019-03034-x.

Examples

# P1 is cited by P2 and P3, which are both cited by P4
A <- matrix(c(
  0, 1, 1, 0,
  0, 0, 0, 1,
  0, 0, 0, 1,
  0, 0, 0, 0
), byrow = TRUE, nrow = 4)
rownames(A) <- c("P1", "P2", "P3", "P4")
colnames(A) <- c("P1", "P2", "P3", "P4")

dag_check(A)

# P4 is also cited by P1, which closes a cycle
A_cycle <- A
A_cycle["P4", "P1"] <- 1
dag_check(A_cycle)
dag_sort(A)

Description

Neighbourhood-inclusion preorders for directed networks (Marmulla and Brandes, 2026), which extend the vicinal preorder of the undirected case to the criteria that different families of centrality indices preserve.

Usage

dir_inclusion(
  A,
  type = c("radial_out", "radial_in", "hierarchical_down", "hierarchical_up", "medial"),
  strength = c("strong", "weak"),
  direction = c("dominated", "dominates")
)

Arguments

Details

Let $N^+(i)$ be the nodes that $i$ sends ties to, $N^-(i)$ the ones that send ties to $i$, and $N[i]$ the same set including $i$. The strong relations use the open neighbourhoods and the weak ones the closed neighbourhoods:

radial_out: $N^+(i) \subseteq N^+(j)$, preserved by the indices that measure how far a node reaches, such as out-degree and closeness.

radial_in: $N^-(i) \subseteq N^-(j)$, the same for the ties received.

hierarchical_down: $N^+(i) \subseteq N^+(j)$ and $N^-(i) \supseteq N^-(j)$, so $j$ sends more and receives less than $i$.

hierarchical_up: $N^-(i) \subseteq N^-(j)$ and $N^+(i) \supseteq N^+(j)$, so $j$ receives more and sends less, which is the criterion of the indices of status.

medial: $N^+(i) \subseteq N^+[j]$ and $N^-(i) \subseteq N^-[j]$, with two extra conditions when $i$ and $j$ are adjacent, so that the advantage that the dominated node has from that tie is compensated. It is the criterion preserved by betweenness.

Value

This function returns a binary matrix P of the criterion asked for in type, where P[u, v] = 1 when the neighbourhoods of u are included in those of v, in the direction asked for.

Author(s)

Alejandro Espinosa-Rada

References

Marmulla, G. and Brandes, U. (2026). Centrality in directed networks. Social Networks, 86, 23–34. doi:10.1016/j.socnet.2026.01.001

Schoch, D. and Brandes, U. (2016). Re-conceptualizing centrality in social networks. European Journal of Applied Mathematics, 27(6), 971–985. doi:10.1017/S0956792516000401

Examples

A <- matrix(c(
  0, 1, 1, 0, 0,
  0, 0, 1, 0, 0,
  0, 0, 0, 1, 1,
  0, 0, 0, 0, 1,
  0, 0, 0, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

dir_inclusion(A, type = "radial_out")
dir_inclusion(A, type = "medial")

Description

This function calculate some geographical distances considering a list of places specifying their latitud and longitud. The function currently works for degree decimal or radians formats.

Usage

dist_geographic(
  latitude,
  longitud,
  method = c("spherical", "harvesine", "manhattan", "minkowski"),
  places = NULL,
  dd_to_radians = FALSE,
  p = NULL
)

Arguments

Value

This function return a distance matrix.

Source

Adapted from Mario Pineda-Krch (Great-circle distance calculations in R)

Examples

set.seed(1234)
x <- cbind(latitud = rnorm(5, -90), longitud = rnorm(5, 45))
dist_geographic(x[, 1], x[, 2], method = "harvesine")

Description

In the literature of social network, Euclidean distance (Burt, 1976) or correlations (Wasserman and Faust, 1994) were considered as measures of structural equivalence.

Usage

dist_sim_matrix(
  A,
  method = c("euclidean", "hamming", "jaccard"),
  bipartite = FALSE
)

Arguments

Value

This function returns a distance matrix between nodes of the same matrix.

Author(s)

Alejandro Espinosa-Rada

References

Burt, Ronald S. (1976) Positions in networks. Social Forces, 55(1): 93-122.

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 0, 0, 1,
  0, 0, 0, 1, 1,
  0, 1, 0, 0, 1,
  0, 0, 1, 1, 0,
  0, 1, 0, 0, 0
), nrow = 5, ncol = 5, byrow = TRUE)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)
dist_sim_matrix(A, method = "jaccard")

A <- matrix(c(
  0, 0, 3, 0, 5,
  0, 0, 2, 0, 4,
  5, 4, 0, 4, 0,
  0, 3, 0, 1, 0,
  0, 0, 0, 0, 2
), nrow = 5, ncol = 5, byrow = TRUE)
dist_sim_matrix(A, method = "euclidean")

# Several relations are compared at the same time, stacking the rows and the
# columns of every matrix
B <- matrix(c(
  0, 1, 0, 0, 1,
  1, 0, 0, 0, 1,
  0, 0, 0, 1, 0,
  0, 0, 1, 0, 0,
  1, 1, 0, 0, 0
), nrow = 5, ncol = 5, byrow = TRUE)
dist_sim_matrix(list(A, B), method = "euclidean")

Description

Distances between nodes using breadth-first search (BFS) or Dijkstra's algorithm to find shortest path distances.

Usage

bfs_ugraph(A, from = NULL)

count_geodesics(A)

short_path(A, from = NULL, to = NULL)

wlocal_distances(A, select = c("all", "in", "out"), from, to, path = c())

wall_distances(A, select = c("all", "in", "out"))

Arguments

Value

This function returns the distances o shortest path distance between two nodes for unweighted graph (bfs_ugraph, count_geodesics and short_path respectively) and weighted graphs (wlocal_distances or wall_distances)

Author(s)

Alejandro Espinosa-Rada

References

Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik. 1: 269–271.

Examples

A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  0, 0, 0, 1, 1, 0,
  0, 0, 0, 0, 1, 0,
  0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 1,
  0, 0, 0, 0, 0, 0
), byrow = TRUE, nrow = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- letters[1:ncol(A)]

bfs_ugraph(A, from = "a")


A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  0, 0, 0, 1, 1, 0,
  0, 0, 0, 0, 1, 0,
  0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 1,
  0, 0, 0, 0, 0, 0
), byrow = TRUE, nrow = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- letters[1:ncol(A)]

count_geodesics(A)


A <- matrix(c(
  0, 1, 1, 0, 0, 0,
  0, 0, 0, 1, 1, 0,
  0, 0, 0, 0, 1, 0,
  0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 1,
  0, 0, 0, 0, 0, 0
), byrow = TRUE, nrow = 6)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- letters[1:ncol(A)]

short_path(A, from = "a", to = "d")


A <- matrix(
  c(
    0, 3, 3, 10, 15, 0, 0, 0,
    1, 0, 5, 2, 7, 0, 0, 0,
    3, 5, 0, 0, 0, 0, 0, 0,
    10, 2, 0, 0, 2, 7, 12, 0,
    11, 3, 0, 3, 0, 11, 2, 0,
    0, 0, 0, 7, 11, 0, 3, 2,
    0, 0, 0, 12, 2, 3, 0, 2,
    0, 0, 0, 0, 0, 2, 2, 0
  ),
  byrow = TRUE, ncol = 8, nrow = 8
)
rownames(A) <- c("a", "b", "s", "c", "d", "e", "f", "z")
colnames(A) <- rownames(A)
wlocal_distances(A, from = "a", to = "d")


A <- matrix(
  c(
    0, 3, 3, 10, 15, 0, 0, 0,
    1, 0, 5, 2, 7, 0, 0, 0,
    3, 5, 0, 0, 0, 0, 0, 0,
    10, 2, 0, 0, 2, 7, 12, 0,
    11, 3, 0, 3, 0, 11, 2, 0,
    0, 0, 0, 7, 11, 0, 3, 2,
    0, 0, 0, 12, 2, 3, 0, 2,
    0, 0, 0, 0, 0, 2, 2, 0
  ),
  byrow = TRUE, ncol = 8, nrow = 8
)
rownames(A) <- c("a", "b", "s", "c", "d", "e", "f", "z")
colnames(A) <- rownames(A)
wall_distances(A, select = "in")

Description

Layers, status and transitive reduction of a strict dominance relation.

Usage

dominance_layers(D, reduction = FALSE, direction = c("dominated", "dominates"))

Arguments

Details

The first layer contains the nodes that are not dominated by any other node (maximal elements). These nodes are removed, and the procedure is repeated until all nodes are assigned. Each layer is an antichain, i.e. the nodes within a layer do not dominate each other.

The maximal elements are of two kinds (Espinosa-Rada, 2026): dominant nodes dominate at least one other node, while independent nodes neither dominate nor are dominated. The remaining nodes are dominated. The net dominance is the number of nodes dominated minus the number of nodes dominating.

The transitive reduction removes the tie $u \to v$ when there is another node $k$ such that $u \to k \to v$, which is the usual representation of a hierarchy (Hasse diagram).

Value

This function returns the layers, the layer, status and net dominance of each node and, if requested, the transitive reduction (with the same direction as D).

Author(s)

Alejandro Espinosa-Rada

References

Brandes, U. (2016). Network positions. Methodological Innovations, 9, 1–19. doi:10.1177/2059799116630650

Espinosa-Rada, A. (2026). Network positions within scholars and intellectual networks. Journal of Informetrics, 20, 101854. doi:10.1016/j.joi.2026.101854

Examples

X <- matrix(c(
  1, 1, 1, 0,
  1, 1, 0, 0,
  1, 0, 0, 0,
  0, 1, 0, 0,
  0, 0, 1, 1
), byrow = TRUE, ncol = 4)
rownames(X) <- c("a1", "a2", "a3", "a4", "a5")
colnames(X) <- c("w1", "w2", "w3", "w4")

D <- pareto_dominance(list(set_inclusion(X)))
dominance_layers(D, reduction = TRUE)

Description

Comparable and incomparable pairs of a dominance relation.

Usage

dominance_pairs(P, direction = c("dominated", "dominates"))

Arguments

Details

Two nodes are comparable if one is dominated by the other. Centrality indices resolve the incomparable pairs with a total ranking, which is not required by the structure of the network.

Value

This function returns the number and proportion of comparable pairs, and a table of the unordered pairs, where u < v means that u is dominated by v.

Author(s)

Alejandro Espinosa-Rada

References

Schoch, D. and Brandes, U. (2016). Re-conceptualizing centrality in social networks. European Journal of Applied Mathematics, 27(6), 971–985. doi:10.1017/S0956792516000401

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

dominance_pairs(neigh_inclusion(A))

Description

The ranks that a node can take in the rankings that are consistent with a dominance relation (Schoch and Brandes, 2016).

Usage

dominance_ranks(P, direction = c("dominated", "dominates"))

Arguments

Details

A dominance relation only orders some pairs of nodes. Any centrality index that preserves it gives a complete ranking, but different indices give different rankings. The interval of a node contains every rank it can take in such a ranking: it cannot be ranked below the nodes it dominates, nor above the nodes that dominate it. A node with a wide interval is one whose position depends on the index that is chosen, and a node with an interval of a single value has the same rank under every index that preserves the relation.

The rank one is the lowest.

Value

This function returns the minimum and the maximum rank of every node, and the width of the interval.

Author(s)

Alejandro Espinosa-Rada

References

Schoch, D. and Brandes, U. (2016). Re-conceptualizing centrality in social networks. European Journal of Applied Mathematics, 27(6), 971–985. doi:10.1017/S0956792516000401

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

dominance_ranks(neigh_inclusion(A))

Description

This function explores dyads and triads (Simmel, 1950), building from the 'forbidden triad' (Granovetter, 1973). First, the minimum structure is an isolated node, then dyads. Afterwards, different combinations of 'forbidden triads' are explored.

Usage

dyad_triad_table(A, adjacency_list = FALSE, min = NULL, max = NULL)

Arguments

Details

For each node, every pair of its neighbours forms a triad with the node at its centre. The triad is a forbidden triad (type 201) when the two neighbours are not tied, which Granovetter (1973) argued is unlikely when both ties are strong, and it is closed (type 300) when they are. A node with a single neighbour is listed with its dyad (type 102) and an isolated node alone (type 003). The underlying graph of the network is used.

The same triad receives the same number in triad for every node that lists it: a closed triad is listed by its three nodes, and a forbidden triad only by its centre.

Value

This function returns a data frame with the triads of each node: the node, the number of the triad, its members and its type.

If adjacency_list = TRUE it also return the adjacency list of the triads per node.

Author(s)

Alejandro Espinosa-Rada

References

Granovetter, M.S. (1973). The Strength of Weak Ties. American Journal of Sociology. 78 (6): 1360–80. doi:10.1086/225469.

Simmel, G. (1950). Individual and Society. In K. H. Wolff (Ed.), The Sociology of George Simmel. New York: Free Press.

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 1, 0, 0,
  1, 1, 0, 0, 0,
  1, 0, 0, 0, 1,
  0, 0, 0, 1, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- letters[1:ncol(A)]

dyad_triad_table(A)

# Nodes at the centre of at least two forbidden triads
dyad_triad_table(A, adjacency_list = TRUE, min = 2)

Description

Dyad census

Usage

dyadic_census(G, directed = TRUE, loops = FALSE)

Arguments

Value

This function return the counts of the dyad census.

Author(s)

Alejandro Espinosa-Rada

References

Wasserman, S. and Faust, K. (1994). Social network analysis: Methods and applications. Cambridge University Press.

Examples

data(krackhardt_friends)
dyadic_census(krackhardt_friends)

data(FIFAin)
dyadic_census(FIFAin[[1]], directed = FALSE)

Description

Everett and Borgatti specification of the constraint measure for binary, directed and valued matrices

Usage

eb_constraint(A, ego = NULL, digraph = FALSE, weighted = FALSE)

Arguments

Details

The constraint of Burt (1992) is computed in the ego network, from the proportion $p_{ij}$ of the ties of each node $i$ that go to $j$, and is split in the three terms of Everett and Borgatti (2020: Eq. 2):

$\sum_j p_{ij}^2 + 2 \sum_j p_{ij} \sum_q p_{iq} p_{qj} + \sum_j \left(\sum_q p_{iq} p_{qj}\right)^2$