Beyond ‘Standard’ Networks

Social Network Analysis

Termeh Shafie

Ego networks

Egocentric network data typically include at least three types of data

• Ego-level attribute data

• Alter-level attribute data

• Alter-alter tie data

Ego networks in

In worksheet 5a we will use egor package for

• data import
• data manipulation
• network measures
• visualization

of egocentric networks

Two-mode networks

A two-mode network is a network that consists of two disjoint sets of nodes (like people and events)

Common examples include:

• Affiliation networks (Membership in institutions)
• Voting/Sponsorship networks (politicians and bills)
• Citation network (authors and papers)
• Co-Authorship networks (authors and papers)

Toy example

data("southern_women")

Analyzing two-mode networks

The adjacency matrix is called incidence matrix

A <- as_incidence_matrix(southern_women)
A[1:8, ]
#>           6/27 3/2 4/12 9/26 2/25 5/19 3/15 9/16 4/8 6/10 2/23 4/7 11/21 8/3
#> EVELYN       1   1    1    1    1    1    0    1   1    0    0   0     0   0
#> LAURA        1   1    1    0    1    1    1    1   0    0    0   0     0   0
#> THERESA      0   1    1    1    1    1    1    1   1    0    0   0     0   0
#> BRENDA       1   0    1    1    1    1    1    1   0    0    0   0     0   0
#> CHARLOTTE    0   0    1    1    1    0    1    0   0    0    0   0     0   0
#> FRANCES      0   0    1    0    1    1    0    1   0    0    0   0     0   0
#> ELEANOR      0   0    0    0    1    1    1    1   0    0    0   0     0   0
#> PEARL        0   0    0    0    0    1    0    1   1    0    0   0     0   0

tnet and bipartite offer some methods to analyse two mode networks directly, by adapting tools for standard networks.

Projecting two-mode networks

Projecting two-mode networks

# Rows: People (P1, P2, P3), Columns: Events (E1, E2)
incidence_matrix <- matrix(c(
  1, 0,  # P1 attended E1
  1, 1,  # P2 attended E1 and E2
  0, 1   # P3 attended E2
), nrow = 3, byrow = TRUE)
rownames(incidence_matrix) <- c("P1", "P2", "P3")
colnames(incidence_matrix) <- c("E1", "E2")

incidence_matrix
#>    E1 E2
#> P1  1  0
#> P2  1  1
#> P3  0  1

Projecting two-mode networks

# People-to-people projection: A %*% t(A)
people_projection <- incidence_matrix %*% t(incidence_matrix)
people_projection
#>    P1 P2 P3
#> P1  1  1  0
#> P2  1  2  1
#> P3  0  1  1

Projecting two-mode networks

# Event-to-event projection: t(A) %*% A
event_projection <- t(incidence_matrix) %*% incidence_matrix
event_projection
#>    E1 E2
#> E1  2  1
#> E2  1  2

Can you do it by hand?

A =
1 0
1 1
0 1

Aᵗ =
1 1 0
0 1 1

Multiply A by Aᵗ: People × People Multiplying row by column:

• P1-P1: (1×1) + (0×0) = 1
• P1-P2: (1×1) + (0×1) = 1
• P1-P3: (1×0) + (0×1) = 0
• P2-P1: (1×1) + (1×0) = 1
• P2-P2: (1×1) + (1×1) = 2
• P2-P3: (1×0) + (1×1) = 1
• P3-P1: (0×1) + (1×0) = 0
• P3-P2: (0×1) + (1×1) = 1
• P3-P3: (0×0) + (1×1) = 1

1 1 0
1 2 1
0 1 1

Multiply Aᵗ by A: Events × Events

• E1-E1: (1×1) + (1×1) + (0×0) = 2
  
• E1-E2: (1×0) + (1×1) + (0×1) = 1
  
• E2-E1: (0×1) + (1×1) + (1×0) = 1
  
• E2-E2: (0×0) + (1×1) + (1×1) = 2
  

2 1
1 2

Projecting two-mode networks

A <- as_incidence_matrix(southern_women)
A[1:8, ]
#>           6/27 3/2 4/12 9/26 2/25 5/19 3/15 9/16 4/8 6/10 2/23 4/7 11/21 8/3
#> EVELYN       1   1    1    1    1    1    0    1   1    0    0   0     0   0
#> LAURA        1   1    1    0    1    1    1    1   0    0    0   0     0   0
#> THERESA      0   1    1    1    1    1    1    1   1    0    0   0     0   0
#> BRENDA       1   0    1    1    1    1    1    1   0    0    0   0     0   0
#> CHARLOTTE    0   0    1    1    1    0    1    0   0    0    0   0     0   0
#> FRANCES      0   0    1    0    1    1    0    1   0    0    0   0     0   0
#> ELEANOR      0   0    0    0    1    1    1    1   0    0    0   0     0   0
#> PEARL        0   0    0    0    0    1    0    1   1    0    0   0     0   0

B <- A%*%t(A)
B[1:5,1:5]
#>           EVELYN LAURA THERESA BRENDA CHARLOTTE
#> EVELYN         8     6       7      6         3
#> LAURA          6     7       6      6         3
#> THERESA        7     6       8      6         4
#> BRENDA         6     6       6      7         4
#> CHARLOTTE      3     3       4      4         4

Projecting two-mode networks

Filtering projections

naïve
delete all edge with a weight less than x

advanced
statistical tools using null models: backbone
Introduction to the package

“Backbone” null models

Compare an edge’s observed weight in the projection to the distribution of weights expected in a projection obtained from a random bipartite network where both the row vertex degrees and column vertex degrees are …

Fixed degree sequence model
… exactly fixed at their original values

Stochastic degree sequence model
… approximately fixed at their original values

“Backbone” example

library(backbone)
sw_bb <- sdsm(southern_women,alpha = 0.4)

Application of projections

Co-voting behavior of US senators

Link to paper

Signed networks

Signed networks include two types of relations:
positive (“friends”) and negative (“foes”)

typical research questions involve (implemented in signnet):

• structural balance
• blockmodeling
• (centrality)

Structural balance theory

Beyond triangles
A network is balanced if it can be partitioned into two groups such that all intra group edges are positive and all inter group edges are negiative

Extended form of balance (Davis 1960s)
A network is balanced if it can be partitioned into k groups …

Toy example

library(signnet)
data("tribes")

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”).

ggsigned(tribes)

ggsigned(tribes,weights = T)

Measuring structural balance

• triangles: Fraction of balanced triangles.
• walks: fraction of signed to unsigned walks
• frustration: optimal partition such that the sum of intra group negative and inter group positive edges is minimized

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

“triangles” returns the fraction of balanced triangles.

“walk” is based on eigenvalues of the signed and underlying unsigned network.

“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.

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

set.seed(141)
bl <- signed_blockmodel(tribes,3)
bl
#> $membership
#>  [1] 3 3 1 1 2 1 1 1 2 2 1 1 2 2 3 3
#> 
#> $criterion
#> [1] 2

ggblock(tribes,blocks = bl$membership,show_blocks = TRUE,show_labels = TRUE)

Blockmodeling

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 diagonal block structure is not always the most optimal representation of the data

Generalized blockmodeling

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.

The function signed_blockmodel_general() allows to specify arbitrary block structures.

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

Generalized blockmodeling

Other functions

Most functions us the igraph name + _signed:

• as_adj_signed()
• as_incidence_signed()
• laplacian_matrix_signed()
• triad_census_signed()
• degree_signed()
• eigen_centrality_signed()
• sample_islands_signed()

All signed triads