Author: JJ Merelo
Date: 2024-04-10

# Computing the betweenness of the Florentine social network using DNSL method

#### JJ Merelo

#### 2024-04-10

## Introduction

Using data from the well known dataset of relations between Florentine families Padgett and Ansell (1993), we will, in this vignette, correct the betweenness centrality of the resulting social network. The new dataset, which is included in this package, adds a self-loop that connects the Medici family with itself; since it was a big family, it did have some marriages, as reflected by the Medici family tree published in the Wikipedia. We will use the `dupNodes` package (Merelo and Molinari 2024) to compute the betweenness centrality of the doges social network, which includes self-loops (some doges married to members of the same extended family).

## Set up

    library(dupNodes)
    data("florentine.sn")

This loads this library, as well as loads the `igraph` object `florentine.sn`, containing the data we’re interested in.

    library(igraph)
    print(V(florentine.sn))
    #> + 15/15 vertices, named, from f163606:
    #>  [1] ACCIAIUOL ALBIZZI   BARBADORI BISCHERI  CASTELLAN GUADAGNI  MEDICI
    #>  [8] PAZZI     PERUZZI   RIDOLFI   GINORI    LAMBERTES STROZZI   TORNABUON
    #> [15] SALVIATI
    print(E(florentine.sn))
    #> + 36/36 edges from f163606 (vertex names):
    #>  [1] ACCIAIUOL--MEDICI    ALBIZZI  --GINORI    ALBIZZI  --GUADAGNI
    #>  [4] ALBIZZI  --MEDICI    BARBADORI--CASTELLAN BARBADORI--MEDICI
    #>  [7] BISCHERI --GUADAGNI  BISCHERI --PERUZZI   BISCHERI --STROZZI
    #> [10] CASTELLAN--PERUZZI   CASTELLAN--STROZZI   GUADAGNI --LAMBERTES
    #> [13] GUADAGNI --TORNABUON MEDICI   --RIDOLFI   MEDICI   --SALVIATI
    #> [16] MEDICI   --TORNABUON PAZZI    --SALVIATI  PERUZZI  --STROZZI
    #> [19] RIDOLFI  --STROZZI   RIDOLFI  --TORNABUON BARBADORI--CASTELLAN
    #> [22] BARBADORI--GINORI    BARBADORI--MEDICI    BARBADORI--PERUZZI
    #> [25] BISCHERI --GUADAGNI  BISCHERI --LAMBERTES BISCHERI --PERUZZI
    #> [28] CASTELLAN--LAMBERTES CASTELLAN--PERUZZI   MEDICI   --GINORI
    #> + ... omitted several edges
    plot(florentine.sn)

    print(E(florentine.sn)[which_loop(florentine.sn)])
    #> + 1/36 edge from f163606 (vertex names):
    #> [1] MEDICI--MEDICI

We have a single self-loop, but in such a small network, it will prove enough to change the values of betweenness centrality. But first we have to convert this graph to another duplicating nodes, to then proceed and compute betweenness centrality.

    dn.florentine.sn <- dup.nodes.from.graph(florentine.sn)
    V(florentine.sn)$DNSLbetweenness <- betweenness.with.duplicated.nodes(dn.florentine.sn)

Which, shown sorted in a table, are:

    DNSLBetweenness <- data.frame( family=V(florentine.sn)$name, betweenness=V(florentine.sn)$DNSLbetweenness)
    knitr::kable(DNSLBetweenness[order(DNSLBetweenness$betweenness, decreasing=TRUE),])

| family | betweenness
--- | --- | ---
7 | MEDICI | 22.713740
3 | BARBADORI | 20.681499
6 | GUADAGNI | 12.444657
14 | TORNABUON | 8.566008
10 | RIDOLFI | 7.473844
9 | PERUZZI | 6.679051
2 | ALBIZZI | 5.526482
5 | CASTELLAN | 4.507246
13 | STROZZI | 4.340510
4 | BISCHERI | 3.504216
12 | LAMBERTES | 2.347951
11 | GINORI | 0.501054
1 | ACCIAIUOL | 0.000000
8 | PAZZI | 0.000000
15 | SALVIATI | 0.000000

## Conclusions

Intra-family links have its importance in the status achieved by a family; not only supports its resilience, but also explains the position they have achieved. `dupNodes` enables computation of betweenness with self-loops in a principled way, giving a more accurate picture of the status of families in its social network. This status can then be correlated successfully to wealth or position, but this is an exercise left to the reader.

## References

Merelo, J. J., and M. C. Molinari. 2024. “Intra-Family Links in the Analysis of Marital Networks.” Journal of Computational Social Science. https://doi.org/https://doi.org/10.1007/s42001-023-00245-4.
Padgett, John F, and Christopher K Ansell. 1993. “Robust Action and the Rise of the Medici, 1400-1434.” American Journal of Sociology 98 (6): 1259–319.
Wang, Dan. 2022. “Padgett-Florence-Families_Multiplex_Social (1).zip,” November. https://doi.org/10.6084/m9.figshare.21545577.v1.