This post introduces the R package signnet, available on CRAN, which implements several methods to analyze signed networks.

Signed networks include two types of relations, positive and negative ones (say friends and enemies). They are fundamentally different from traditional networks and require a very different set of tools to analyse them. This post covers everything from data structures to methods and should give a comprehensive overview of what is possible to do with the package.

# install.packages("signnet")
library(igraph)
library(signnet)

Data structures

The foundation of signnet is provided by igraph. All functions in the package assume that an igraph object is a signed network if it has an edge attribute “sign” with values 1 (positive) or -1 (negative).

g <- graph.full(5,directed = FALSE,loops = FALSE)
E(g)$sign <- sample(c(-1,1), ecount(g), replace = TRUE)
g
## IGRAPH 6a43e14 U--- 5 10 -- Full graph
## + attr: name (g/c), loops (g/l), sign (e/n)
## + edges from 6a43e14:
##  [1] 1--2 1--3 1--4 1--5 2--3 2--4 2--5 3--4 3--5 4--5

All methods (should) throw an error if the sign attribute is missing or contains other values than -1 and 1.

Matrices associated with a signed network follow the igraph naming scheme. The signed adjacency matrix can be obtained with as_adj_signed().

as_adj_signed(g)
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0    1    1   -1   -1
## [2,]    1    0    1    1   -1
## [3,]    1    1    0   -1   -1
## [4,]   -1    1   -1    0    1
## [5,]   -1   -1   -1    1    0

The signed Laplacian matrix is obtained by laplacian_matrix_signed().

laplacian_matrix_signed(g)
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    4   -1   -1    1    1
## [2,]   -1    4   -1   -1    1
## [3,]   -1   -1    4    1    1
## [4,]    1   -1    1    4   -1
## [5,]    1    1    1   -1    4

Included datasets

The package includes two well known datasets.

The “tribes” dataset is a signed social network of tribes of the Gahuku–Gama alliance structure of the Eastern Central Highlands of New Guinea. The network contains sixteen tribes connected by friendship (“rova”) and enmity (“hina”).

The “cowList” dataset contains a list of 52 signed networks of inter-state relations over time (1946-1999). Two countries are connected by a positive tie if they form an alliance or have a peace treaty. A negative tie exists between countries who are at war or in other kinds of conflicts. The dataset is derived from the correlates of war.

Structural Balance

The principles underlying structural balance are based on a theory in social psychology dating back to the work of Heider in the 1940s, which was generalized and extended to graphs by Cartwright and Harary in the 1950s. In its simplest form, it is defined via triangles. A triangle is balanced if all ties are positive (“the friend of a friend is a friend”) or only one tie is positive (“the enemy of my enemy is my friend”). The remaining configurations are said to be unbalanced.

A network is balanced if i.a., it can be partitioned into two vertex subsets, such that intra-group edges are all positive and inter-group edges are all negative.

A (random) balanced network can be obtained with the function sample_islands_signed() which is pretty much the same as sample_islands() from the igraph package, except that inter-group edges are negative ties.

g <- sample_islands_signed(islands.n = 2,islands.size = 10,
                           islands.pin = 0.8,n.inter = 5)

Increasing islands.n leads to “clusterable” networks as defined by Davis.

The function ggsigned() can be used to get a rudimentary visualization of signed networks. Note that this requires the package ggraph to be installed.

ggsigned(g)

A balanced network only contains balanced triangles. This can be verified with count_signed_triangles().

count_signed_triangles(g)
## +++ ++- +-- --- 
## 137   0   4   0

Note the absence of ++- and --- triangles.

To list all triangles use signed_triangles().

head(signed_triangles(g))
##      V1 V2 V3 P
## [1,] 85  6 77 1
## [2,] 85 36 80 1
## [3,] 85 78 64 1
## [4,] 43 37 48 3
## [5,] 43 42 70 3
## [6,] 43 39 60 3

The column P indicated the number of positive ties in the triangle. A value of 3 indicates that the triangle is “+++”.

Balancedness measures

Determining if a network is balanced or not is easy, but measuring a degree of balancedness (i.e. how close is a network to be balanced?) is not. The package, so far, implements three methods to calculate balance scores. All are defined such that a value of one indicates perfect balance and zero perfect unbalance. Though for intermediate networks, results may vary significantly. Check the paper by Samin Aref (and his other work) for more details.

balance_score(g, method = "triangles")
## [1] 1
balance_score(g, method = "walk")
## [1] 1
balance_score(g, method = "frustration")
## [1] 1

“triangles” returns the fraction of balanced triangles.

“walk” is based on eigenvalues of the signed and underlying unsigned network. Check the paper by Estrada for details.

“frustration” assumes that the network can be partitioned into two groups, where intra group edges are positive and inter group edges are negative. The index is defined as the sum of intra group negative and inter group positive edges. Note that the problem is NP complete and only an upper bound is returned (based on simulated annealing). Exact methods can be found in the work of Aref.

There disagreement for non-balanced networks can be seen with the included “tribes” dataset.

data("tribes")
balance_score(tribes, method = "triangles")
## [1] 0.8676471
balance_score(tribes, method = "walk")
## [1] 0.3575761
balance_score(tribes, method = "frustration")
## [1] 0.7586207

Blockmodeling

In signed blockmodeling, the goal is to determine k blocks of nodes such that all intra-block edges are positive and inter-block edges are negative. In the example below, we construct a network with a perfect block structure with sample_islands_signed(). The network consists of 10 blocks with 10 vertices each, where each block has a density of 1 (of positive edges). The function signed_blockmodel() is used to construct the blockmodel. The parameter k is the number of desired blocks. alpha is a trade-off parameter. The function minimizes \(f(C)=\alpha N+(1-\alpha)P\), where \(N\) is the total number of negative ties within blocks,\(P\) be the total number of positive ties between blocks and \(C\) a blockmodel assignment.

g <- sample_islands_signed(10,10,1,20)
clu <- signed_blockmodel(g,k = 10,alpha = 0.5)

The function returns a list with two entries. The block membership of nodes and the value of \(f(C)\).

table(clu$membership)
## 
##  1  2  3  4  5  6  7  8  9 10 
## 10 10 10 10 10 10 10 10 10 10
clu$criterion
## [1] 0

The function ggblock() can be used to plot the outcome of the blockmodel (ggplot2 is required).

ggblock(g,clu$membership,show_blocks = TRUE)

If the parameter annealing is set to TRUE, simulated annealing is used in the optimization step. This generally leads to better results but longer runtimes.

data("tribes")
set.seed(44) #for reproducibility

signed_blockmodel(tribes,k = 3,alpha=0.5,annealing = TRUE)
## $membership
##  [1] 1 1 3 3 2 3 3 3 2 2 3 3 2 2 1 1
## 
## $criterion
## [1] 2
signed_blockmodel(tribes,k = 3,alpha=0.5,annealing = FALSE)
## $membership
##  [1] 1 1 2 3 2 2 2 2 3 3 2 2 3 3 1 1
## 
## $criterion
## [1] 5

Generalized Blockmodeling

The function signed_blockmodel() is only able to provide a blockmodel where the diagonal blocks are positive and off-diagonal blocks are negative. The function signed_blockmodel_general() can be used to specify different block structures. In the below example, we construct a network that contains three blocks. Two have positive and one has negative intra-group ties. The inter-group edges are negative between group one and two, and one and three. Between group two and three, all edges are positive.

g1 <- g2 <- g3 <- graph.full(5)

V(g1)$name <- as.character(1:5)
V(g2)$name <- as.character(6:10)
V(g3)$name <- as.character(11:15)

g <- Reduce("%u%",list(g1,g2,g3))
E(g)$sign <- 1
E(g)$sign[1:10] <- -1
g <- add.edges(g,c(rbind(1:5,6:10)),attr = list(sign=-1))
g <- add.edges(g,c(rbind(1:5,11:15)),attr = list(sign=-1))
g <- add.edges(g,c(rbind(11:15,6:10)),attr = list(sign=1))

The parameter blockmat is used to specify the desired block structure.

set.seed(424) #for reproducibility
blockmat <- matrix(c(1,-1,-1,-1,1,1,-1,1,-1),3,3,byrow = TRUE)
blockmat
##      [,1] [,2] [,3]
## [1,]    1   -1   -1
## [2,]   -1    1    1
## [3,]   -1    1   -1
general <- signed_blockmodel_general(g,blockmat,alpha = 0.5)
traditional <- signed_blockmodel(g,k = 3,alpha = 0.5,annealing = TRUE)

c(general$criterion,traditional$criterion)
## [1] 0 6

Centrality indices

There exist dozens of indices for networks with positive ties, but for signed networks they are rather scarce. The package implements three indices so far. Versions of degree and eigenvector centrality, and PN centrality by Everett & Borgatti.

Degree centrality can be calculated in four different ways with degree_signed(), specified by the type parameter:

  • type="pos" count only positive neighbors
  • type="neg" count only negative neighbors
  • type="ratio" positive neighbors/(positive neighbors+negative neighbors)
  • type="net" positive neighbors-negative neighbors

The mode parameter can be used to get “in” and “out” versions for directed networks.

The PN index is very similar to Katz status and Hubbell’s measure for networks with only positive ties. The technical details can be found in the paper by Everett & Borgatti.

Eigenvector centrality is defined as the eigenvector associated with the dominant eigenvalue of the signed adjacency matrix.

The below example illustrates all indices with a network where signed degree can not distinguish vertices.

A <- matrix(c(0,  1,  0,  1,  0,  0,  0, -1, -1,  0,  
               1,  0,  1, -1,  1, -1, -1,  0,  0,  0,  
               0,  1,  0,  1, -1,  0,  0,  0, -1,  0,  
               1, -1,  1,  0,  1, -1, -1,  0,  0,  0,  
               0,  1, -1,  1,  0,  1,  0, -1,  0, -1,  
               0, -1,  0, -1,  1,  0,  1,  0,  1, -1,  
               0, -1,  0, -1,  0,  1,  0,  1, -1,  1,  
              -1,  0,  0,  0, -1,  0,  1,  0,  1,  0,  
              -1,  0, -1,  0,  0,  1, -1,  1,  0,  1,  
               0,  0,  0,  0, -1, -1,  1,  0,  1,  0),10,10)

g <- graph_from_adjacency_matrix(A,"undirected",weighted = "sign")
ggsigned(g)

degree_signed(g,type="ratio")
##  [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
eigen_centrality_signed(g)
##  [1] -0.62214960  1.00000000 -0.74518850  1.00000000 -0.89990041
##  [6]  0.64289592  0.35828159 -0.37471921 -0.28087411 -0.07834568
pn_index(g)
##  [1] 0.9009747 0.8613482 0.9076997 0.8613482 0.8410658 0.8496558 0.8617321
##  [8] 0.9015909 0.8509848 0.9072930

Note that PN centrality and eigenvector centrality differ significantly for this network.

cor(eigen_centrality_signed(g),pn_index(g),method = "kendall")
## [1] -0.2

A note on eigenvector centrality

The adjacency matrix of a signed network may not have a dominant eigenvalue. This means it is not clear which eigenvector should be used. In addition it is possible for the adjacency matrix to have repeated eigenvalues and hence multiple linearly independent eigenvectors. In this case certain centralities can be arbitrarily assigned. The eigen_centrality_signed() function returns an error if this is the case.

A <- matrix(c( 0,  1,  1, -1,  0,  0, -1,  0,  0, 
               1,  0,  1,  0, -1,  0,  0, -1,  0, 
               1,  1,  0,  0,  0, -1,  0,  0, -1, 
              -1,  0,  0,  0,  1,  1, -1,  0,  0, 
               0, -1,  0,  1,  0,  1,  0, -1,  0, 
               0,  0, -1,  1,  1,  0,  0,  0, -1, 
              -1,  0,  0, -1,  0,  0,  0,  1,  1, 
               0, -1,  0,  0, -1,  0,  1,  0,  1, 
               0,  0, -1,  0,  0, -1,  1,  1,  0), 9, 9)

round(eigen(A)$values,6)
## [1]  3  3  0  0  0  0  0 -3 -3
g <- graph_from_adjacency_matrix(A,"undirected",weighted = "sign")
eigen_centrality_signed(g)
## Error in eigen_centrality_signed(g): no dominant eigenvalue exists

Signed two-mode networks

Blockmodeling

Most research for signed two-mode networks concerns the use of blockmodeling. The package does not include any functionality for blockmodeling signed two-mode networks yet. This will be included in a future version.

Projections

A common analytic tool for two-mode networks is to project the network onto on relevant mode. This is easily done using the adjacency matrix \(A\). \(AA^T\) yields the row projection and \(A^TA\) the column projection. The resulting networks will thus be weighted. Several methods exist to turn a weighted projection into an unweighted network where only the most significant edges are included. A number of these methods are implemented in the backbone package.

Projecting signed networks, however, is not as straightforward. Consider the following simple example.

el <- matrix(c(1,"a",1,"b",1,"c",2,"a",2,"b"),ncol = 2,byrow = TRUE)
g <- graph_from_edgelist(el,directed = FALSE)
E(g)$sign <- c(1,1,-1,1,-1)
V(g)$type <- c(FALSE,TRUE,TRUE,TRUE,FALSE)

(An igraph two-mode network requires a logical vertex attribute type)

If we use the regular projection rules we obtain

A <- as_incidence_signed(g)
R <- A%*%t(A)
C <- t(A)%*%A
R
##   1 2
## 1 3 0
## 2 0 2
C
##    a  b  c
## a  2  0 -1
## b  0  2 -1
## c -1 -1  1

The row projection suggests that there is no relation between 1 and 2, when in fact there is a negative path (via b) and a positive path (via a) between them. The same holds for the column projection and the nodes a and b.

The paper of Schoch introduces two projection methods that circumvent this “nullification”. The package implements the duplication approach since it plays well with existing binarization tools. The first step is to turn the signed two-mode network into an unsigned one. This is done by duplicating all vertices of the primary mode (i.e. the one to project on). For example, vertex a turns into two vertices “a-pos” and “a-neg”. The vertices of the secondary mode connect to these new vertices depending on the sign of edge. For instance, 1 has a positive edge to a and thus 1 connects to a-pos.

This can be done for the whole network with the function as_unsigned_2mode() by specifying the primary mode (either TRUE or FALSE).

gu <- as_unsigned_2mode(g,primary = TRUE)
gu
## IGRAPH 0c1ac62 UN-B 8 5 -- 
## + attr: name (v/c), type (v/l)
## + edges from 0c1ac62 (vertex names):
## [1] a-pos--1 b-pos--1 c-neg--1 a-pos--2 b-neg--2

Now, any binarization toll (e.g. from the backbone package) can be applied since the network is an unsigned two-mode network. For illustration, we include all edges with a weight greater one (the “universal” approach) since it can be done without the backbone package.

pu <- bipartite_projection(gu,which = "true")
pu <- delete_edge_attr(pu,"weight")
pu
## IGRAPH f4bf1d9 UN-- 6 4 -- 
## + attr: name (v/c)
## + edges from f4bf1d9 (vertex names):
## [1] a-pos--b-pos a-pos--c-neg a-pos--b-neg b-pos--c-neg

After binarization, the network is turned back to an unsigned network using a contraction rule. The contraction rule works as follows:

If there is an edge (a-pos,b-pos) or (a-neg,b-neg) in the projection then there is a positive edge (a,b) in the signed projection.

If there is an edge (a-pos,b-neg) or (a-neg,b-pos) in the projection then there is a negative edge (a,b) in the signed projection.

If there is an edge (a-pos,b-pos) and (a-neg,b-pos) (or, e.g., (a-neg,b-neg) and (a-pos,b-neg)) in the projection then there is an ambivalent edge (a,b) in the signed projection.

This is done with the function as_signed_proj().

ps <- as_signed_proj(pu)
as_data_frame(ps,"edges")
##   from to type
## 1    a  b    A
## 2    a  c    N
## 3    b  c    N

The projection of a signed two-mode network thus may contain three types of edges (positive (“P”), negative (“N”) or ambivalent (“A”)). The concept of ambivalent ties comes from work by Abelson & Rosenberg and Cartwright & Harary.

More technical details can be found in the original paper by Schoch.

Representing networks with ambivalent ties

Ambivalent ties add a new level of complexity for analytic tasks (especially involving matrices) since it is not clear which value to assign to them. Intuitively they should be “somewhere” between a positive and a negative tie but zero is already taken for the null tie.

We can construct a kind of adjacency matrix with the character values, but we can’t really work with characters analytically.

as_adj(ps,type = "both", attr = "type", sparse = FALSE)
##   a   b   c  
## a ""  "A" "N"
## b "A" ""  "N"
## c "N" "N" ""

This is where complex matrices come in. Instead of thinking about edge values being only in one dimension, we can add a second one for negative ties. That is, a positive tie would be coded as \((1,0)\) and a negative one as \((0,1)\). It is much easier in this case to include ambivalent ties by assigning \((0.5,0.5)\) to them.

Tuples like these can also be written as a complex number, i.e. \((1,0)\) turns into \(1+0i\), \((0,1)\) into \(0+1i\), and \((0.5,0.5)\) into \(0.5+0.5i\). Complex numbers may be scary to some, but they have a kind of intuitive interpretation here. The real part is the positive value of an edge and the imaginary part is the negative part. So we could actually also have something like \(0.3+0.7i\) which is an edge that is 30% positive and 70% negative. For now, though, the three values from above suffice.

The function as_adj_complex() can be used to return the complex adjacency matrix of a signed network with ambivalent ties.

as_adj_complex(ps,attr = "type")
##          a        b    c
## a 0.0+0.0i 0.5+0.5i 0+1i
## b 0.5-0.5i 0.0+0.0i 0+1i
## c 0.0-1.0i 0.0-1.0i 0+0i

When there is a complex adjacency matrix, then there is also a complex Laplacian matrix. This matrix can be obtained with laplacian_matrix_complex().

laplacian_matrix_complex(ps, attr = "type")
##                a              b    c
## a  1.707107+0.0i -0.500000-0.5i 0-1i
## b -0.500000+0.5i  1.707107+0.0i 0-1i
## c  0.000000+1.0i  0.000000+1.0i 2+0i

Functions supporting ambivalent ties

So far, only the triangle routines support networks with ambivalent ties.

g <- graph.full(5)
E(g)$type <- c(rep("P",3),rep("N",3),rep("A",4))

count_complex_triangles(g,attr = "type")
## PPP PPN PNN NNN PPA PNA NNA PAA NAA AAA 
##   0   2   0   0   1   3   1   0   2   1

References

Abelson, Robert P., and Milton J. Rosenberg. 1958. “Symbolic Psycho-Logic: A Model of Attitudinal Cognition.” Behavioral Science 3 (1): 1–13.

Aref, Samin, and Mark C. Wilson. 2018. “Measuring Partial Balance in Signed Networks.” Journal of Complex Networks 6 (4): 566–95.

Bonacich, Phillip, and Paulette Lloyd. 2004. “Calculating Status with Negative Relations.” Social Networks 26 (4): 331–38.

Cartwright, Dorwin, and Frank Harary. 1956. “Structural Balance: A Generalization of Heider’s Theory.” Psychological Review 63 (5): 277.

Cartwright, Dorwin, and Frank Harary. 1970. “Ambivalence and Indifference in Generalizations of Structural Balance.” Behavioral Science 15 (6).

Davis, James A. 1967. “Clustering and Structural Balance in Graphs.” Human Relations 20 (2): 181–87.

Doreian, Patrick, and Andrej Mrvar. 1996. “A Partitioning Approach to Structural Balance.” Social Networks 18 (2): 149–68.

Doreian, Patrick, and Andrej Mrvar. 2009. “Partitioning Signed Social Networks.” Social Networks 31 (1): 1–11.

Doreian, Patrick, Paulette Lloyd, and Andrej Mrvar. 2013. “Partitioning Large Signed Two-Mode Networks: Problems and Prospects.” Social Networks, Special Issue on Advances in Two-mode Social Networks, 35 (2): 178–203.

Doreian, Patrick, and Andrej Mrvar. 2015. “Structural Balance and Signed International Relations.” Journal of Social Structure 16: 1.

Estrada, Ernesto. 2019. “Rethinking Structural Balance in Signed Social Networks.” Discrete Applied Mathematics

Everett, Martin G., and Stephen P. Borgatti. 2014. “Networks Containing Negative Ties.” Social Networks 38: 111–20.

Heider, Fritz. 1946. “Attitudes and Cognitive Organization.” The Journal of Psychology 21 (1): 107–12.

Schoch, David. 2020. “Projecting Signed Two-Mode Networks” Mathematical Sociology, forthcoming