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

## Introduction

Using data from dogesr (Merelo-Guervós 2022), we will, in this vignette, correct the betweenness centrality of the doges social network, which was computed using the igraph package, and therefore does not take into account self-loops. We will use the dupNodes package (Merelo and Molinari 2024) to compute the betweenness centrality of the doges social network, which is a network with self-loops.

## Set up

library(dogesr)
library(dupNodes)
data("doges")

This loads the two involved libraries, as well as loads the data frame data.doges that contains data on all doges, including who they married to. Let us see how many self-loops are there

family.marriages <- data.doges[ data.doges$Family.doge == data.doges$Family.dogaressa,]

So we have 7 self-loops, including 5 different families. It’s not a lot, but quite enough to make a difference in the status of certain families as expressed by betweenness centrality. These are the families with self-loops:

knitr::kable(table(family.marriages$Family.doge)) Var1 Freq Candiano 2 Corner 2 Faliero 1 Michiel 1 Selvo 1 Let’s compute betweenness centrality, with and without self-loops, and see the difference, after extracting only those doges that actually married library(igraph) married.doges <- data.doges[ data.doges$Family.dogaressa != '',]
original.betweenness <- betweenness(
graph_from_data_frame(
data.frame(married.doges$Family.doge, married.doges$Family.dogaressa), directed=FALSE
)
)
dnsl.betweenness <- DNSL.betweenness(
married.doges,
first.node="Family.doge",
second.node="Family.dogaressa")

Which, shown sorted in a table, are:

knitr::kable(head(sort(original.betweenness, decreasing=TRUE), n=10))
x
Dandolo 446.0000
Contarini 315.6667
Priuli 249.3333
Morosini 243.3333
Loredan 242.1667
Malipiero 205.0000
Mocenigo 193.5000
Orseolo 171.0000
Barbarigo 136.0000
knitr::kable(head(sort(dnsl.betweenness, n=10, decreasing=TRUE), n=10))
x
Dandolo 520.6667
Contarini 375.2667
Priuli 305.6667
Loredan 266.7500
Malipiero 264.0000
Morosini 263.5000
Mocenigo 239.5833
Orseolo 229.0000
Barbarigo 148.0000

There is some change, due to the fact that there were very few central families, and some of the most prominent had self-loops. The Candiano family was very peripheral, and the change is not enough to make it scale the top 10; the Mocenigo and Gradenigo families have been the one that have changed the most, possibly due to the connection to these families with self-loops. We can see the graph with duplicated nodes here

dup.graph <- dup.nodes.from.data.frame(
data.frame(V1=married.doges$Family.doge, V2=married.doges$Family.dogaressa)
)
components <- igraph::components(dup.graph, mode="weak")
biggest_cluster_id <- which.max(components$csize) vert_ids <- V(dup.graph)[components$membership == biggest_cluster_id]

doges.sn.connected <- igraph::induced_subgraph(dup.graph, vert_ids)
plot(doges.sn.connected,vertex.label.cex=0.9,vertex.size=5)

## 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. This is why it is important to include this new method to compute betweenness centrality, taking into account not only the existence of self-loops, but also its value. This dupNodes package, which is available in CRAN, can be used to correct betweenness centrality in any eligible network, as well as to represent the position in the network of these self-loops-converted-into-duplicated nodes so that it explains whatever change in the value of centrality or in ranking.

## 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.
Merelo-Guervós, J. J. 2022. “What Is a Good Doge? Analyzing the Patrician Social Network of the Republic of Venice.” arXiv. https://doi.org/10.48550/ARXIV.2209.07334.