Getting started with netmem

Alejandro Espinosa-Rada

netmem (Network Measures using Matrices) computes social network measures with matrix algebra. A network is a matrix, and every function takes and returns matrices, vectors or data frames, so the results can be combined with the rest of R without converting between classes of objects. The package imports only Matrix and stats.

This vignette shows the standard tools of social network analysis with a single network. The vignette What netmem adds covers the measures that are not available elsewhere, and the vignette Multilayer networks covers two-mode, multilevel and multiplex networks.

install.packages("netmem")

library(netmem)

---

The data

The Campnet data were collected among the 18 people of a three-week course, 14 participants and 4 instructors (Borgatti et al., 2018). At the end of the second week, each person ranked the others by how much they had interacted with them. campnet$network[i, j] is one when j is among the three people with whom i interacted most.

data(campnet)
A <- campnet$network
gender <- campnet$attributes$gender # 1 = woman, 2 = man
role <- campnet$attributes$role # 1 = participant, 2 = instructor

matrix_report(A)
#> The matrix A might have the following characteristics:
#> --> The vectors of the matrix are `numeric`
#> --> Matrix is asymmetric (network is directed)
#> --> The matrix is square, 18 by 18
#>      nodes arcs
#> [1,]    18   54

Some measures are defined for undirected networks. The underlying graph keeps a tie between two people when either of them chose the other:

U <- pmax(A, t(A))

---

Describing the network

Density, reciprocity and transitivity:

gen_density(A)
#> [1] 0.1764706
recip_coef(A)
#>    Mutual 
#> 0.8954248
trans_coef(A)
#> [1] 0.483871
geo_summary(A)
#> $diameter
#> [1] 7
#> 
#> $average_distance
#> [1] 2.873786
#> 
#> $prop_reachable
#> [1] 0.6732026

Each person chose three others, so the density is fixed by the design. What the choices reveal is their arrangement: most of them are reciprocated. The dyad and triad censuses show this arrangement in detail. triad_uman() compares each type of triad with its expectation given the number of mutual, asymmetric and null dyads (Holland and Leinhardt, 1976):

dyadic_census(A)
#>      Mutual Asymmetrics       Nulls 
#>          19          16         118
triad_uman(A)
#>    label OBS     EXP    VAR   STD
#> 1    003 345 372.138 40.876 6.393
#> 2    012 177 153.988 57.856 7.606
#> 3    102 223 182.861 70.421 8.392
#> 4   021D   3   4.936  4.501 2.122
#> 5   021U   4   4.936  4.501 2.122
#> 6   021C   9   9.871  8.134 2.852
#> 7   111D  21  25.007 20.673 4.547
#> 8   111U   6  25.007 20.673 4.547
#> 9   030T   0   0.586  0.556 0.746
#> 10  030C   0   0.195  0.192 0.438
#> 11   201  10  28.132 17.272 4.156
#> 12  120D   5   0.795  0.776 0.881
#> 13  120U   1   0.795  0.776 0.881
#> 14  120C   0   1.589  1.516 1.231
#> 15   210   8   3.815  3.440 1.855
#> 16   300   4   1.351  1.235 1.111

The complete triads (300) and the triads with two mutual ties (210) are more frequent than expected, and the open triads with two mutual ties (201) less frequent: reciprocated ties tend to close.

A network can be moved between a matrix and an edge list:

E <- matrix_to_edgelist(A, digraph = TRUE)
head(E)
#>      [,1]     [,2]   
#> [1,] "HOLLY"  "PAM"  
#> [2,] "HOLLY"  "PAT"  
#> [3,] "HOLLY"  "DON"  
#> [4,] "BRAZEY" "LEE"  
#> [5,] "BRAZEY" "STEVE"
#> [6,] "BRAZEY" "BERT"
identical(edgelist_to_matrix(E, label = rownames(A)), A)
#> [1] TRUE

---

Centrality

The functions for centrality return a named vector, so several indices can be gathered in a data frame. The closeness is harmonic because not every person can be reached from every other:

centrality <- data.frame(
  indegree = gen_degree(A, type = "in"),
  closeness = closeness_centrality(A, type = "in", harmonic = TRUE),
  betweenness = betweenness_centrality(A),
  eigenvector = eigenvector_centrality(A)$vector,
  pagerank = page_rank_centrality(A)
)
round(centrality, 2)
#>         indegree closeness betweenness eigenvector pagerank
#> HOLLY          4      8.73       78.33         0.4     0.07
#> BRAZEY         1      2.92        0.00         0.0     0.02
#> CAROL          2      6.82        1.33         0.5     0.05
#> PAM            5      9.15       32.50         1.0     0.10
#> PAT            4      8.65       39.50         0.8     0.09
#> JENNIE         3      7.15        6.33         0.8     0.08
#> PAULINE        4      7.82       12.50         0.7     0.07
#> ANN            2      6.65        0.50         0.6     0.06
#> MICHAEL        4      8.15       58.83         0.1     0.05
#> BILL           0      0.00        0.00         0.0     0.01
#> LEE            3      4.33        5.00         0.0     0.06
#> DON            4      8.07       16.33         0.2     0.06
#> JOHN           0      0.00        0.00         0.0     0.01
#> HARRY          3      7.05        2.33         0.1     0.04
#> GERY           2      3.67       54.67         0.0     0.03
#> STEVE          5      5.50       16.83         0.0     0.08
#> BERT           4      5.00       13.67         0.0     0.07
#> RUSS           4      5.00       47.33         0.0     0.06

In a directed network, the eigenvector centrality is zero for the people who are not reached by the chains of choices that start in the group with the largest eigenvalue (Bonacich and Lloyd, 2001), here eight people, among them the four instructors. PageRank avoids this with its damping factor, which lets every person receive a small share of the status.

The centralization of Freeman (1978) compares the network with a star of the same size:

centrality_centralization(A, measure = "degree", digraph = TRUE, type = "in")$centralization
#> [1] 0.1323529
centrality_centralization(A, measure = "betweenness", digraph = TRUE)$centralization
#> [1] 0.4429066

---

Cohesive subgroups and communities

components_id(A, mode = "weak")$size
#> components
#>  1 
#> 18
k_core(U)
#>   HOLLY  BRAZEY   CAROL     PAM     PAT  JENNIE PAULINE     ANN MICHAEL    BILL 
#>       3       3       3       3       3       3       3       3       3       3 
#>     LEE     DON    JOHN   HARRY    GERY   STEVE    BERT    RUSS 
#>       3       3       3       3       3       3       3       3
clique_max(U, min = 3)
#> [[1]]
#> [1] "HOLLY"   "MICHAEL" "DON"     "HARRY"  
#> 
#> [[2]]
#> [1] "BRAZEY" "LEE"    "STEVE"  "BERT"  
#> 
#> [[3]]
#> [1] "MICHAEL" "BILL"    "DON"     "HARRY"  
#> 
#> [[4]]
#> [1] "CAROL"   "PAM"     "PAULINE"
#> 
#> [[5]]
#> [1] "CAROL"   "PAT"     "PAULINE"
#> 
#> [[6]]
#> [1] "PAM"    "JENNIE" "ANN"   
#> 
#> [[7]]
#> [1] "PAM"     "PAULINE" "ANN"    
#> 
#> [[8]]
#> [1] "JOHN" "GERY" "RUSS"
#> 
#> [[9]]
#> [1] "GERY"  "STEVE" "RUSS" 
#> 
#> [[10]]
#> [1] "STEVE" "BERT"  "RUSS"

The Leiden algorithm (Traag et al., 2019) finds three communities. One has six of the eight women, another has Holly with four men who were participants, and the third gathers the four instructors with three participants:

set.seed(18)
communities <- leiden(U)
communities$modularity
#> [1] 0.5497959
table(community = communities$partition, gender = gender)
#>          gender
#> community 1 2
#>         1 6 0
#>         2 1 6
#>         3 1 4
table(community = communities$partition, role = role)
#>          role
#> community 1 2
#>         1 6 0
#>         2 3 4
#>         3 5 0

---

Homophily and positions

The mixing matrix counts the choices between the categories, and the E-I index of Krackhardt and Stern (1988) summarises them, from -1 (every tie within the categories) to 1 (every tie between them):

mix_matrix(A, gender)
#>     To
#> From  1  2
#>    1 20  4
#>    2  5 25
ei_index(A, att = gender)
#> [1] -0.6666667
block_density(A, gender)
#> $densities
#>           1         2
#> 1 0.3571429 0.0500000
#> 2 0.0625000 0.2777778
#> 
#> $image
#>   1 2
#> 1 1 0
#> 2 0 1
#> 
#> $density
#> [1] 0.1764706

The core-periphery model of Borgatti and Everett (2000):

set.seed(18)
core_periphery(U)[c("core", "periphery")]
#> $core
#> [1] "PAM"     "PAT"     "PAULINE" "MICHAEL" "DON"     "HARRY"   "GERY"   
#> [8] "STEVE"   "BERT"   
#> 
#> $periphery
#> [1] "HOLLY"  "BRAZEY" "CAROL"  "JENNIE" "ANN"    "BILL"   "LEE"    "JOHN"  
#> [9] "RUSS"

---

Structural holes

Effective size, efficiency and constraint of Burt (1992) for every person. The ties are used in both directions, as in Burt (1992):

round(structural_holes(A), 2)
#>         alters effective_size efficiency constraint
#> HOLLY        5           3.86       0.77       0.44
#> BRAZEY       3           1.00       0.33       1.02
#> CAROL        3           2.00       0.67       0.80
#> PAM          5           3.88       0.78       0.52
#> PAT          4           3.57       0.89       0.39
#> JENNIE       3           2.33       0.78       0.61
#> PAULINE      5           3.86       0.77       0.54
#> ANN          3           1.60       0.53       0.93
#> MICHAEL      5           3.07       0.61       0.60
#> BILL         3           1.00       0.33       1.08
#> LEE          3           1.67       0.56       0.82
#> DON          4           2.14       0.54       0.70
#> JOHN         3           2.33       0.78       0.73
#> HARRY        4           1.75       0.44       0.79
#> GERY         4           2.90       0.73       0.65
#> STEVE        5           3.06       0.61       0.61
#> BERT         4           2.21       0.55       0.72
#> RUSS         4           2.79       0.70       0.63

The constraint of an ego can be split into its three terms and normalized between the minimum and the maximum that an ego with the same number of alters can have (Everett and Borgatti, 2020):

eb_constraint(A, ego = "HOLLY", digraph = TRUE)
#> $results
#>       term1 term2 term3 constraint normalization
#> HOLLY 0.224 0.152  0.06      0.436          0.38
#> 
#> $maximum
#> HOLLY 
#> 0.823

---

Statistical tests

Is the network more transitive than a random network with the same number of ties? The conditional uniform graph test compares the observed value with the values of random networks:

set.seed(18)
transitivity <- cug_test(A, trans_coef, cmode = "edges", reps = 500)
transitivity[c("observed", "mean", "p_greater")]
#> $observed
#> [1] 0.483871
#> 
#> $mean
#> [1] 0.1715542
#> 
#> $p_greater
#> [1] 0

The quadratic assignment procedure (Krackhardt, 1988) keeps the structure of the networks and permutes the labels of the nodes. Do people who share a gender choose each other?

same_gender <- outer(gender, gender, "==") * 1
dimnames(same_gender) <- dimnames(A)

set.seed(18)
homophily <- qap_cor(A, same_gender, reps = 500)
homophily[c("correlation", "p_greater")]
#> $correlation
#> [1] 0.3301313
#> 
#> $p_greater
#> [1] 0

---

How the results are checked

Each function is compared with another implementation (igraph, sna, netseg, netrankr, signnet) or with the tables of the publication that defines it. The comparisons are kept in the folder dev/validation of the GitHub repository.

---

References

Bonacich, P. and Lloyd, P. (2001). Eigenvector-like measures of centrality for asymmetric relations. Social Networks, 23(3), 191–201. https://doi.org/10.1016/S0378-8733(01)00038-7

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

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

Burt, R. S. (1992). Structural Holes: The Social Structure of Competition. Harvard University Press.

Everett, M. G. and Borgatti, S. P. (2020). Unpacking Burt’s constraint measure. Social Networks, 62, 50–57. https://doi.org/10.1016/j.socnet.2020.02.001

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

Holland, P. W. and Leinhardt, S. (1976). Local structure in social networks. Sociological Methodology, 7, 1–45. https://doi.org/10.2307/270703

Krackhardt, D. (1988). Predicting with networks: Nonparametric multiple regression analysis of dyadic data. Social Networks, 10(4), 359–381. https://doi.org/10.1016/0378-8733(88)90004-4

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

Traag, V. A., Waltman, L. and van Eck, N. J. (2019). From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9, 5233. https://doi.org/10.1038/s41598-019-41695-z