Short Assignment 2

Describing Network Data

I am working with a dataset accessed from the network package - Florentine Families. This dataset is already in a format suitable for network analysis (an adjancecy matrix) and ready to work with. The matrix shows the marriage links of different families in Florence. A 1 indicates the presence of a marriage link, while a 0 indicates the absence of a marriage link.

library(network)
library(igraph)
library(statnet)
data("flo")
network_adjacencey<- flo
network_adjacencey 

             Acciaiuoli Albizzi Barbadori Bischeri Castellani Ginori
Acciaiuoli            0       0         0        0          0      0
Albizzi               0       0         0        0          0      1
Barbadori             0       0         0        0          1      0
Bischeri              0       0         0        0          0      0
Castellani            0       0         1        0          0      0
Ginori                0       1         0        0          0      0
Guadagni              0       1         0        1          0      0
Lamberteschi          0       0         0        0          0      0
Medici                1       1         1        0          0      0
Pazzi                 0       0         0        0          0      0
Peruzzi               0       0         0        1          1      0
Pucci                 0       0         0        0          0      0
Ridolfi               0       0         0        0          0      0
Salviati              0       0         0        0          0      0
Strozzi               0       0         0        1          1      0
Tornabuoni            0       0         0        0          0      0
             Guadagni Lamberteschi Medici Pazzi Peruzzi Pucci Ridolfi
Acciaiuoli          0            0      1     0       0     0       0
Albizzi             1            0      1     0       0     0       0
Barbadori           0            0      1     0       0     0       0
Bischeri            1            0      0     0       1     0       0
Castellani          0            0      0     0       1     0       0
Ginori              0            0      0     0       0     0       0
Guadagni            0            1      0     0       0     0       0
Lamberteschi        1            0      0     0       0     0       0
Medici              0            0      0     0       0     0       1
Pazzi               0            0      0     0       0     0       0
Peruzzi             0            0      0     0       0     0       0
Pucci               0            0      0     0       0     0       0
Ridolfi             0            0      1     0       0     0       0
Salviati            0            0      1     1       0     0       0
Strozzi             0            0      0     0       1     0       1
Tornabuoni          1            0      1     0       0     0       1
             Salviati Strozzi Tornabuoni
Acciaiuoli          0       0          0
Albizzi             0       0          0
Barbadori           0       0          0
Bischeri            0       1          0
Castellani          0       1          0
Ginori              0       0          0
Guadagni            0       0          1
Lamberteschi        0       0          0
Medici              1       0          1
Pazzi               1       0          0
Peruzzi             0       1          0
Pucci               0       0          0
Ridolfi             0       1          1
Salviati            0       0          0
Strozzi             0       0          0
Tornabuoni          0       0          0

Following the script, I’ve created both a statnet and igraph network object from the dataset (adjancecy matrix).

network_statnet <- network(network_adjacencey, direct = FALSE)

network_igraph <- graph_from_adjacency_matrix(network_adjacencey, mode = "upper", weighted = NULL)

List of the objects available:

ls()

[1] "flo"                "network_adjacencey" "network_igraph"    
[4] "network_statnet"   

We already know that the Florentine Families dataset is in the format of a matrix. In terms of network size, the network has 16 vertices or nodes (16 families) connected by 20 edges (representing ties of marriage, in this case).

print(network_statnet)

 Network attributes:
  vertices = 16 
  directed = FALSE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  total edges= 20 
    missing edges= 0 
    non-missing edges= 20 

 Vertex attribute names: 
    vertex.names 

No edge attributes

When it comes to network features, we learn from running the codes below that the network is unweighted/binary (with 0 and 1 inidicating the absence and presence of a marriage tie, respectively), undirected (meaning that the relationship between nodes is inherently symmetric, as marriage relationships are), and single/not bipartite.

is_weighted(network_igraph)

[1] FALSE

is_directed(network_igraph)

[1] FALSE

is_bipartite(network_igraph)

[1] FALSE

Network attributes:

network::list.vertex.attributes(network_statnet)

[1] "na"           "vertex.names"

network::list.edge.attributes(network_statnet)

[1] "na"

Network structure comments

sna::dyad.census(network_statnet)

     Mut Asym Null
[1,]  20    0  100

Since the ties are undirected, the dyad census command confirms the expected - that there are no assymetric edges. In ohter words, that all edges are reciprocal/mutual.

sna::triad.census(network_statnet, mode = "graph")

       0   1  2 3
[1,] 324 195 38 3

sum(sna::triad.census(network_statnet, mode = "graph"))

[1] 560

The statnet command allowed me to indicate that the netwrok is undirected with the option mode=“graph”. There are 4 undirected triads, for a total number of 560.

How transitive are the relationships? What proportion of the connected triads are complete?

Local clustering (below)

transitivity(network_igraph, type = "average")

[1] 0.2181818

Global clustering (below)

transitivity(network_igraph, type = "global")

[1] 0.1914894

Local clustering is higher than global clustering, but overall low proportion of connected triads that are complete, I think. But I am having difficulties understanding/explaining what this means in this marriage dataset.

gtrans(network_statnet)

[1] 0.1914894

Average path length in whole network:

average.path.length(network_igraph, directed = F)

[1] 2.485714

names(igraph::components(network_igraph))

[1] "membership" "csize"      "no"        

igraph::components(network_igraph)$no

[1] 2

The network has 2 components…

igraph::components(network_igraph)$csize

[1] 15  1

… 1 of the components is larger, with 15 members and the other has one single member.

isolates(network_statnet)

[1] 12

There are 12 isolates in the network - 12 nodes that have no link to the rest of the network.

as.vector(network_statnet%v%'vertex.names')[c(isolates(network_statnet))]

[1] "Pucci"

When we retrieve the names of the isolates, they all refer to the Pucci family, which is not linked by marriage to any of the other Florentine families.

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