First: copy over the import script provided:
#################################################################################################
"Florentine Families.R" has imported a network of weddings among powerful Florentine families.
This is a small, unweighted, undirected network stored as an adjacency matrix with no additional node or edge attributes. For more information use "?flo".
The import script has created three objects that represent the network:
-network_adjacency (an adjacency matrix containing edges and node names)
-network_igraph (an igraph object)
-network_statnet (a network object compatable with statnet packages like sna & ergm)
Each object name starts, quite generically, with "network_" and ends with the type of object it is. Note that the names are generic so that they are compatable with other scripts you will use in this course. Feel free to rename the objects for your purposes.
################################################################################################A brief description of the network:
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[1] FALSE[1] "name"character(0)Nodes are the names of 16 prominent families in Florentine during the earily 1400s. Ties are marriage connections between two families. This is an unweighted (binary), undirected graph.
First, we’ll take a brief look at the structure of this network.
[1] "membership" "csize" "no" [1] 2[1] 15 1 Mut Asym Null
[1,] 20 0 100 003 012 102 021D 021U 021C 111D 111U 030T 030C 201 120D 120U
[1,] 324 0 195 0 0 0 0 0 0 0 38 0 0
120C 210 300
[1,] 0 0 3[1] 0.1914894From here, we see there are two components in this graph: one consisting of fifteen of the elements, and one consisting of one (one isolate). This is an undirected graph so there are no asymmetrical dyads. Our global transitivity statistic is approximately 0.191.
This plot is a visual representation of our two-component network, with 15 nodes conneted and 1 isolate (the Pucci family).
Now we’ll run some measures of centrality.
name degree indegree outdegree eigen Bonacich
1 Acciaiuoli 2 1 1 0.13215429 -0.3796632
2 Albizzi 6 3 3 0.24395611 -2.0881476
3 Barbadori 4 2 2 0.21170525 -1.5186528
4 Bischeri 6 3 3 0.28280009 0.0000000
5 Castellani 6 3 3 0.25902617 -1.3288212
6 Ginori 2 1 1 0.07492271 -1.8983160
7 Guadagni 8 4 4 0.28911560 -0.1898316
8 Lamberteschi 2 1 1 0.08879189 0.0000000
9 Medici 12 6 6 0.43030809 -0.5694948
10 Pazzi 2 1 1 0.04481344 0.0000000
11 Peruzzi 6 3 3 0.27573037 -0.5694948
12 Pucci 0 0 0 0.00000000 0.0000000
13 Ridolfi 6 3 3 0.34155264 1.3288212
14 Salviati 4 2 2 0.14591720 -0.1898316
15 Strozzi 8 4 4 0.35598045 0.1898316
16 Tornabuoni 6 3 3 0.32584230 1.1389896
rc eigen.rc dc eigen.dc
1 0.1666667 0.02202572 0.8333333 0.11012858
2 0.2727273 0.06653348 0.7272727 0.17742262
3 0.2222222 0.04704561 0.7777778 0.16465964
4 0.2727273 0.07712730 0.7272727 0.20567279
5 0.3333333 0.08634206 0.6666667 0.17268411
6 0.3333333 0.02497424 0.6666667 0.04994847
7 0.4000000 0.11564624 0.6000000 0.17346936
8 0.2500000 0.02219797 0.7500000 0.06659392
9 0.4285714 0.18441775 0.5714286 0.24589034
10 0.5000000 0.02240672 0.5000000 0.02240672
11 0.3000000 0.08271911 0.7000000 0.19301126
12 0.0000000 0.00000000 1.0000000 0.00000000
13 0.2307692 0.07881984 0.7692308 0.26273280
14 0.2857143 0.04169063 0.7142857 0.10422657
15 0.3333333 0.11866015 0.6666667 0.23732030
16 0.2307692 0.07519438 0.7692308 0.25064792The question is, now that we’ve calculated all of these scores, what do we do with them?
If we simply sort by degree:
name degree indegree outdegree eigen Bonacich
1 Medici 12 6 6 0.43030809 -0.5694948
2 Guadagni 8 4 4 0.28911560 -0.1898316
3 Strozzi 8 4 4 0.35598045 0.1898316
4 Albizzi 6 3 3 0.24395611 -2.0881476
5 Bischeri 6 3 3 0.28280009 0.0000000
6 Castellani 6 3 3 0.25902617 -1.3288212
7 Peruzzi 6 3 3 0.27573037 -0.5694948
8 Ridolfi 6 3 3 0.34155264 1.3288212
9 Tornabuoni 6 3 3 0.32584230 1.1389896
10 Barbadori 4 2 2 0.21170525 -1.5186528
11 Salviati 4 2 2 0.14591720 -0.1898316
12 Acciaiuoli 2 1 1 0.13215429 -0.3796632
13 Ginori 2 1 1 0.07492271 -1.8983160
14 Lamberteschi 2 1 1 0.08879189 0.0000000
15 Pazzi 2 1 1 0.04481344 0.0000000
16 Pucci 0 0 0 0.00000000 0.0000000
rc eigen.rc dc eigen.dc
1 0.4285714 0.18441775 0.5714286 0.24589034
2 0.4000000 0.11564624 0.6000000 0.17346936
3 0.3333333 0.11866015 0.6666667 0.23732030
4 0.2727273 0.06653348 0.7272727 0.17742262
5 0.2727273 0.07712730 0.7272727 0.20567279
6 0.3333333 0.08634206 0.6666667 0.17268411
7 0.3000000 0.08271911 0.7000000 0.19301126
8 0.2307692 0.07881984 0.7692308 0.26273280
9 0.2307692 0.07519438 0.7692308 0.25064792
10 0.2222222 0.04704561 0.7777778 0.16465964
11 0.2857143 0.04169063 0.7142857 0.10422657
12 0.1666667 0.02202572 0.8333333 0.11012858
13 0.3333333 0.02497424 0.6666667 0.04994847
14 0.2500000 0.02219797 0.7500000 0.06659392
15 0.5000000 0.02240672 0.5000000 0.02240672
16 0.0000000 0.00000000 1.0000000 0.00000000We find, unsurprisingly, that the Medici family has the highest overall degree measure (12) as well as both the highest in- and out-degree measures, which suggest to me that the Medici family is the most highly connected of the Florentine Families. They also have the highest eigenvalue measure, which suggests again that they are most centrally connected and that they are connected to the most centrally connected other families.
Of note is the Bonacich measure:
name degree indegree outdegree eigen Bonacich
1 Ridolfi 6 3 3 0.34155264 1.3288212
2 Tornabuoni 6 3 3 0.32584230 1.1389896
3 Strozzi 8 4 4 0.35598045 0.1898316
4 Bischeri 6 3 3 0.28280009 0.0000000
5 Lamberteschi 2 1 1 0.08879189 0.0000000
6 Pazzi 2 1 1 0.04481344 0.0000000
7 Pucci 0 0 0 0.00000000 0.0000000
8 Guadagni 8 4 4 0.28911560 -0.1898316
9 Salviati 4 2 2 0.14591720 -0.1898316
10 Acciaiuoli 2 1 1 0.13215429 -0.3796632
11 Medici 12 6 6 0.43030809 -0.5694948
12 Peruzzi 6 3 3 0.27573037 -0.5694948
13 Castellani 6 3 3 0.25902617 -1.3288212
14 Barbadori 4 2 2 0.21170525 -1.5186528
15 Ginori 2 1 1 0.07492271 -1.8983160
16 Albizzi 6 3 3 0.24395611 -2.0881476
rc eigen.rc dc eigen.dc
1 0.2307692 0.07881984 0.7692308 0.26273280
2 0.2307692 0.07519438 0.7692308 0.25064792
3 0.3333333 0.11866015 0.6666667 0.23732030
4 0.2727273 0.07712730 0.7272727 0.20567279
5 0.2500000 0.02219797 0.7500000 0.06659392
6 0.5000000 0.02240672 0.5000000 0.02240672
7 0.0000000 0.00000000 1.0000000 0.00000000
8 0.4000000 0.11564624 0.6000000 0.17346936
9 0.2857143 0.04169063 0.7142857 0.10422657
10 0.1666667 0.02202572 0.8333333 0.11012858
11 0.4285714 0.18441775 0.5714286 0.24589034
12 0.3000000 0.08271911 0.7000000 0.19301126
13 0.3333333 0.08634206 0.6666667 0.17268411
14 0.2222222 0.04704561 0.7777778 0.16465964
15 0.3333333 0.02497424 0.6666667 0.04994847
16 0.2727273 0.06653348 0.7272727 0.17742262This seems to be capturing something different about network position than the eigen score, and on this measure, the Medici family is at number 11. Per the igraph documentation:
“…positive values imply that vertices become more powerful as their alters become more powerful (as occurs in cooperative relations), while negative values imply that vertices become more powerful only as their alters become weaker (as occurs in competitive or antagonistic relations).”
Only three families have positive Bonacich scores (I am not sure how to interpret a Bonacich score of 0?).
The last measure I’ll look at is the reflected and derived centrality scores. Note: a higher score of eigen reflected centrality indicates the node may occupy a hub position.
name degree indegree outdegree eigen Bonacich
1 Medici 12 6 6 0.43030809 -0.5694948
2 Strozzi 8 4 4 0.35598045 0.1898316
3 Guadagni 8 4 4 0.28911560 -0.1898316
4 Castellani 6 3 3 0.25902617 -1.3288212
5 Peruzzi 6 3 3 0.27573037 -0.5694948
6 Ridolfi 6 3 3 0.34155264 1.3288212
7 Bischeri 6 3 3 0.28280009 0.0000000
8 Tornabuoni 6 3 3 0.32584230 1.1389896
9 Albizzi 6 3 3 0.24395611 -2.0881476
10 Barbadori 4 2 2 0.21170525 -1.5186528
11 Salviati 4 2 2 0.14591720 -0.1898316
12 Ginori 2 1 1 0.07492271 -1.8983160
13 Pazzi 2 1 1 0.04481344 0.0000000
14 Lamberteschi 2 1 1 0.08879189 0.0000000
15 Acciaiuoli 2 1 1 0.13215429 -0.3796632
16 Pucci 0 0 0 0.00000000 0.0000000
rc eigen.rc dc eigen.dc
1 0.4285714 0.18441775 0.5714286 0.24589034
2 0.3333333 0.11866015 0.6666667 0.23732030
3 0.4000000 0.11564624 0.6000000 0.17346936
4 0.3333333 0.08634206 0.6666667 0.17268411
5 0.3000000 0.08271911 0.7000000 0.19301126
6 0.2307692 0.07881984 0.7692308 0.26273280
7 0.2727273 0.07712730 0.7272727 0.20567279
8 0.2307692 0.07519438 0.7692308 0.25064792
9 0.2727273 0.06653348 0.7272727 0.17742262
10 0.2222222 0.04704561 0.7777778 0.16465964
11 0.2857143 0.04169063 0.7142857 0.10422657
12 0.3333333 0.02497424 0.6666667 0.04994847
13 0.5000000 0.02240672 0.5000000 0.02240672
14 0.2500000 0.02219797 0.7500000 0.06659392
15 0.1666667 0.02202572 0.8333333 0.11012858
16 0.0000000 0.00000000 1.0000000 0.00000000By this metric, the Medici family is a hub, as well as the Strozzi and Guadagni families. If we look at our plot again, this makes sense.
Recall that nodes with higher eigen derived centrality scores may occupy a bridge position.
name degree indegree outdegree eigen Bonacich
1 Ridolfi 6 3 3 0.34155264 1.3288212
2 Tornabuoni 6 3 3 0.32584230 1.1389896
3 Medici 12 6 6 0.43030809 -0.5694948
4 Strozzi 8 4 4 0.35598045 0.1898316
5 Bischeri 6 3 3 0.28280009 0.0000000
6 Peruzzi 6 3 3 0.27573037 -0.5694948
7 Albizzi 6 3 3 0.24395611 -2.0881476
8 Guadagni 8 4 4 0.28911560 -0.1898316
9 Castellani 6 3 3 0.25902617 -1.3288212
10 Barbadori 4 2 2 0.21170525 -1.5186528
11 Acciaiuoli 2 1 1 0.13215429 -0.3796632
12 Salviati 4 2 2 0.14591720 -0.1898316
13 Lamberteschi 2 1 1 0.08879189 0.0000000
14 Ginori 2 1 1 0.07492271 -1.8983160
15 Pazzi 2 1 1 0.04481344 0.0000000
16 Pucci 0 0 0 0.00000000 0.0000000
rc eigen.rc dc eigen.dc
1 0.2307692 0.07881984 0.7692308 0.26273280
2 0.2307692 0.07519438 0.7692308 0.25064792
3 0.4285714 0.18441775 0.5714286 0.24589034
4 0.3333333 0.11866015 0.6666667 0.23732030
5 0.2727273 0.07712730 0.7272727 0.20567279
6 0.3000000 0.08271911 0.7000000 0.19301126
7 0.2727273 0.06653348 0.7272727 0.17742262
8 0.4000000 0.11564624 0.6000000 0.17346936
9 0.3333333 0.08634206 0.6666667 0.17268411
10 0.2222222 0.04704561 0.7777778 0.16465964
11 0.1666667 0.02202572 0.8333333 0.11012858
12 0.2857143 0.04169063 0.7142857 0.10422657
13 0.2500000 0.02219797 0.7500000 0.06659392
14 0.3333333 0.02497424 0.6666667 0.04994847
15 0.5000000 0.02240672 0.5000000 0.02240672
16 0.0000000 0.00000000 1.0000000 0.00000000By this measure, the Ridolfi and Tornabuoni occupy bridge positions. Of note is that the Medici family is also high on this list, which perhaps speaks to the centrality of that family, regardless of measure.
