# 12 Centrality

In this tutorial, we look at measures of network centrality, which we use to identify structurally important actors. We also discuss possible ideas for identifying important edges.

Centrality originally referred to how central actors are in a network’s structure. It has become abstracted as a term from its topological origins and now refers very generally to how important actors are to a network. Topological centrality has a clear definition, but many operationalizations. Network “importance” on the other hand has many definitions and many operationalizations. We will explore the possible meanings and operationalizations of centrality here. There are four well-known centrality measures: degree, betweenness, closeness and eigenvector - each with its own strengths and weaknesses. The main point we want to make is that the analytical usefulness of each depends heavily on the context of the network, the type of relation being analyzed and the underlying network morphology. We don’t want to leave you with the impression that one is better than another - only that one might serve your research goals better than another.

Every node-level measure has its graph-level analogue. Centralization measures the extent to which the ties of a given network are concentrated on a single actor or group of actors. We can also look at the degree distribution. It is a simple histogram of degree, which tells you whether the network is highly unequal or not.

As always, we need to load igraph. We will load in tidyverse too in case we need to do some data munging or plotting.

library(igraph)
library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──
## ✓ tibble  3.1.3     ✓ dplyr   1.0.7
## ✓ tidyr   1.1.3     ✓ stringr 1.4.0
## ✓ readr   2.0.0     ✓ forcats 0.5.1
## ✓ purrr   0.3.4
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::as_data_frame() masks tibble::as_data_frame(), igraph::as_data_frame()
## x purrr::compose()       masks igraph::compose()
## x tidyr::crossing()      masks igraph::crossing()
## x dplyr::filter()        masks stats::filter()
## x dplyr::groups()        masks igraph::groups()
## x dplyr::lag()           masks stats::lag()
## x purrr::simplify()      masks igraph::simplify()
library(reshape2)

We will use John Padgett’s Florentine Families dataset. It is part of a famous historical datset about the relationships of prominent Florentine families in 15th century Italy. The historical puzzle is how the Medici, an upstart family, managed to accumulate political power during this period. Padgett’s goal was to explain their rise.

He looked at many relations. On the github, we have access to marriage, credit, and business partnership ties, but we will focus on marriage for now. Marriage was a tool in diplomacy, central to political alliances. A tie is drawn between families if the daughter of one family was sent to marry the son of another.

Ron Breiger, who analyzed these data in a famous paper on local role analysis, has argued these edges should be directed, tracing where daughters were sent or where finances flowed.

As always, we first load and prepare the dataset. We will do so directly from GitHub again.

# prepare the marriage adjacency matrix
florentine_edj <- florentine_edj[,2:3]

# prepare the attributes file

# graph the marriage network
marriageNet <- graph.edgelist(as.matrix(florentine_edj), directed = T)
V(marriageNet)\$Wealth <- florentine_attributes\$Gwealth[match(V(marriageNet)\$name, florentine_attributes\$Family)]

# Gross wealth (Florins), for 87 (92) families
# simple mean imputation of wealth (alternatively, we might think that those with NA were too poor to show up in historical records?)
V(marriageNet)\$Wealth <- ifelse(is.na(V(marriageNet)\$Wealth), mean(V(marriageNet)\$Wealth, na.rm = T), V(marriageNet)\$Wealth)

# Number of Priors, The Priorate (or city council), first created in 1282, was Florence's governing body. Count of how many seats a family had on that city council from 1282-1344
# measure of the aggregate political influence of the family over a long period of time
V(marriageNet)\$Priorates <- florentine_attributes\$Npriors[match(V(marriageNet)\$name, florentine_attributes\$Family)]

Let’s see how it looks.

plot(marriageNet, vertex.size = 8, vertex.label.cex = .4, vertex.label.color = "black", vertex.color = "tomato", edge.arrow.size = 0.4)

Based on this plot, which family do you expect is most central?

### 12.0.2 Degree Centrality

The simplest measure of centrality is degree centrality. It counts how many edges each node has - the most degree central actor is the one with the most ties.

Note: In a directed network, you will need to specify if in or out ties should be counted. These will be referred to as in or out degree respectively. If both are counted, then it is just called degree

Degree centrality is calculated using the degree function in R. It returns how many edges each node has.

degree(marriageNet)
##    Acciaiuoli  Guicciardini        Medici       Adimari     Arrigucci
##             4            14            40             6             2
##     Barbadori       Strozzi       Albizzi      Altoviti    Della_Casa
##            14            50            28            12             4
##         Corsi     Davanzati   Frescobaldi        Ginori      Guadagni
##             2             6            12             8            20
##      Guasconi         Nerli   Del_Palagio   Panciatichi       Scolari
##            24             2             4            14             6
##  Aldobrandini    Alessandri       Tanagli    Bencivenni Gianfigliazzi
##             4             2             4             2            20
##         Spini  Dall'Antella      Martelli    Rondinelli   Ardinghelli
##             8             6             4            10            10
##       Peruzzi         Rossi       Arrighi  Baldovinetti     Ciampegli
##            30             4             2             8             2
##       Manelli      Carducci    Castellani      Ricasoli         Bardi
##             4             4            14            20            14
##       Bucelli      Serragli     Da_Uzzano    Baroncelli         Raugi
##             4             4             4             2             2
##       Baronci     Manovelli       Bartoli      Solosmei     Bartolini
##             2             4             2             2             2
##   Belfradelli    Del_Benino        Donati       Benizzi      Bischeri
##             2             2             4             2             8
##     Brancacci       Capponi       Nasi(?)     Sacchetti       Vettori
##             4             8             2             6             2
##         Doffi          Pepi    Cavalcanti          Ciai    Corbinelli
##             2             2             6             2             2
##          Lapi     Orlandini    Dietisalvi        Valori     Federighi
##             2             4             2             8             2
##  Dello_Scarfa    Fioravanti      Salviati        Giugni    Pandolfini
##             2             4             6             6             4
##  Lamberteschi    Tornabuoni       Mancini  Di_Ser_Segna      Macinghi
##             2             8             2             2             2
##          Masi        Pecori         Pitti    Popoleschi     Portinari
##             2             2             4             4             2
##       Ridolfi    Serristori    Vecchietti    Minerbetti         Pucci
##             8             2             2             2             2
##    Da_Panzano       Parenti         Pazzi      Rucellai    Scambrilla
##             4             2             4             2             2
##      Soderini
##             2

Who is the most degree central?

We can assign the output to a variable in the network and size the nodes according to degree.

V(marriageNet)\$degree <- degree(marriageNet) # assignment

plot(marriageNet, vertex.label.cex = .6, vertex.label.color = "black", vertex.size = V(marriageNet)\$degree, vertex.label.cex = .2) # sized by degree

The problem is that the degree values are a little small to plot well. We can use a scalar to increase the value of the degree but maintain the ratio.

plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$degree*3)

### 12.0.3 Betweenness Centrality

Betweenness centrality captures which nodes are important in the flow of the network. It makes use of the shortest paths in the network. A path is a series of adjacent nodes. For any two nodes we can find the shortest path between them, that is, the path with the least amount of total steps (or edges). If a node C is on a shortest path between A and B, then it means C is important to the efficient flow of goods between A and B. Without C, flows would have to take a longer route to get from A to B.

Thus, betweenness effectively counts how many shortest paths each node is on. The higher a node’s betweenness, the more important they are for the efficient flow of goods in a network.

In igraph, betweenness() computes betweenness in the network

betweenness(marriageNet, directed = FALSE)
##    Acciaiuoli  Guicciardini        Medici       Adimari     Arrigucci
##      0.000000    327.850305   1029.609288     93.009524      0.000000
##     Barbadori       Strozzi       Albizzi      Altoviti    Della_Casa
##    162.697344   1369.979110    856.436111    125.620147     11.577398
##         Corsi     Davanzati   Frescobaldi        Ginori      Guadagni
##      0.000000    260.000000    145.532681    180.571429    277.059921
##      Guasconi         Nerli   Del_Palagio   Panciatichi       Scolari
##    583.679251      0.000000      0.000000    167.994891     19.801010
##  Aldobrandini    Alessandri       Tanagli    Bencivenni Gianfigliazzi
##     23.351190      0.000000     88.000000      0.000000    187.915043
##         Spini  Dall'Antella      Martelli    Rondinelli   Ardinghelli
##     89.500000    174.000000     88.000000     43.186597     58.278211
##       Peruzzi         Rossi       Arrighi  Baldovinetti     Ciampegli
##    604.369691      0.000000      0.000000     95.613889      0.000000
##       Manelli      Carducci    Castellani      Ricasoli         Bardi
##      0.000000      5.009524    194.199423    205.097092    280.248232
##       Bucelli      Serragli     Da_Uzzano    Baroncelli         Raugi
##      1.066667     81.000000      3.666667      0.000000      0.000000
##       Baronci     Manovelli       Bartoli      Solosmei     Bartolini
##      0.000000     88.000000      0.000000      0.000000      0.000000
##   Belfradelli    Del_Benino        Donati       Benizzi      Bischeri
##      0.000000      0.000000     88.000000      0.000000     63.995238
##     Brancacci       Capponi       Nasi(?)     Sacchetti       Vettori
##      0.000000    177.000000      0.000000    197.651515      0.000000
##         Doffi          Pepi    Cavalcanti          Ciai    Corbinelli
##      0.000000      0.000000    125.467749      0.000000      0.000000
##          Lapi     Orlandini    Dietisalvi        Valori     Federighi
##      0.000000     88.000000      0.000000    202.285859      0.000000
##  Dello_Scarfa    Fioravanti      Salviati        Giugni    Pandolfini
##      0.000000      5.951190     35.571429     96.664141      0.000000
##  Lamberteschi    Tornabuoni       Mancini  Di_Ser_Segna      Macinghi
##      0.000000     23.831746      0.000000      0.000000      0.000000
##          Masi        Pecori         Pitti    Popoleschi     Portinari
##      0.000000      0.000000     19.610606      0.000000      0.000000
##       Ridolfi    Serristori    Vecchietti    Minerbetti         Pucci
##    213.727670      0.000000      0.000000      0.000000      0.000000
##    Da_Panzano       Parenti         Pazzi      Rucellai    Scambrilla
##     16.961111      0.000000      4.361111      0.000000      0.000000
##      Soderini
##      0.000000

We can again assign the output of betweenness() to a variable in the network and size the nodes according to it.

V(marriageNet)\$betweenness <- betweenness(marriageNet, directed = F) # assignment

plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$betweenness) # sized by betweenness

Betweenness centrality can be very large. It is often helpful to normalize it by dividing by the maximum and multiplying by some scalar when plotting.

plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$betweenness/max(V(marriageNet)\$betweenness) * 20)

### 12.0.4 Closeness Centrality

With closeness centrality we again make use of the shortest paths between nodes. We measure the distance between two nodes as the length of the shortest path between them. Farness, for a given node, is the average distance from that node to all other nodes. Closeness is then the reciprocal of farness (1/farness).

closeness(marriageNet)
## Warning in closeness(marriageNet): At centrality.c:2784 :closeness centrality is
## not well-defined for disconnected graphs
##    Acciaiuoli  Guicciardini        Medici       Adimari     Arrigucci
##  0.0011723329  0.0012468828  0.0012836970  0.0011961722  0.0010822511
##     Barbadori       Strozzi       Albizzi      Altoviti    Della_Casa
##  0.0012210012  0.0012970169  0.0012658228  0.0012165450  0.0011792453
##         Corsi     Davanzati   Frescobaldi        Ginori      Guadagni
##  0.0011389522  0.0011467890  0.0012180268  0.0011737089  0.0012254902
##      Guasconi         Nerli   Del_Palagio   Panciatichi       Scolari
##  0.0012903226  0.0011389522  0.0011834320  0.0012562814  0.0011834320
##  Aldobrandini    Alessandri       Tanagli    Bencivenni Gianfigliazzi
##  0.0011834320  0.0010141988  0.0011135857  0.0010989011  0.0012690355
##         Spini  Dall'Antella      Martelli    Rondinelli   Ardinghelli
##  0.0011792453  0.0011709602  0.0010638298  0.0012195122  0.0012330456
##       Peruzzi         Rossi       Arrighi  Baldovinetti     Ciampegli
##  0.0012722646  0.0011534025  0.0011061947  0.0012091898  0.0010928962
##       Manelli      Carducci    Castellani      Ricasoli         Bardi
##  0.0011668611  0.0011933174  0.0012254902  0.0012269939  0.0012531328
##       Bucelli      Serragli     Da_Uzzano    Baroncelli         Raugi
##  0.0011627907  0.0011389522  0.0011737089  0.0001108033  0.0001108033
##       Baronci     Manovelli       Bartoli      Solosmei     Bartolini
##  0.0010416667  0.0011467890  0.0001108033  0.0001108033  0.0009727626
##   Belfradelli    Del_Benino        Donati       Benizzi      Bischeri
##  0.0011001100  0.0009469697  0.0010330579  0.0011441648  0.0012360939
##     Brancacci       Capponi       Nasi(?)     Sacchetti       Vettori
##  0.0012048193  0.0010976948  0.0010010010  0.0011806375  0.0010010010
##         Doffi          Pepi    Cavalcanti          Ciai    Corbinelli
##  0.0011061947  0.0011061947  0.0012285012  0.0010638298  0.0011534025
##          Lapi     Orlandini    Dietisalvi        Valori     Federighi
##  0.0010416667  0.0010438413  0.0010638298  0.0011337868  0.0001108033
##  Dello_Scarfa    Fioravanti      Salviati        Giugni    Pandolfini
##  0.0001108033  0.0011682243  0.0011723329  0.0011641444  0.0010718114
##  Lamberteschi    Tornabuoni       Mancini  Di_Ser_Segna      Macinghi
##  0.0011061947  0.0011890606  0.0011587486  0.0011587486  0.0011641444
##          Masi        Pecori         Pitti    Popoleschi     Portinari
##  0.0011441648  0.0011534025  0.0012195122  0.0011587486  0.0011534025
##       Ridolfi    Serristori    Vecchietti    Minerbetti         Pucci
##  0.0012315271  0.0011534025  0.0011534025  0.0011641444  0.0009560229
##    Da_Panzano       Parenti         Pazzi      Rucellai    Scambrilla
##  0.0011389522  0.0011641444  0.0011337868  0.0011641444  0.0010683761
##      Soderini
##  0.0011641444

We assign it to a node variable and plot the network, adjusting node size by closeness.

V(marriageNet)\$closeness <- closeness(marriageNet)
## Warning in closeness(marriageNet): At centrality.c:2784 :closeness centrality is
## not well-defined for disconnected graphs
plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$closeness/max(V(marriageNet)\$closeness) * 20)

## 12.1 Eigenvector Centrality

Degree centrality only takes into account the number of edges for each node, but it leaves out information about ego’s alters.

However, we might think that power comes from being tied to powerful people. If A and B have the same degree centrality, but A is tied to all high degree people and B is tied to all low degree people, then intuitively we want to see A with a higher score than B.

Eigenvector centrality takes into account alters’ power. It is calculated a little bit differently in igraph. It produces a list object and we need to extract only the vector of centrality values.

evcent(marriageNet)\$vector
##    Acciaiuoli  Guicciardini        Medici       Adimari     Arrigucci
##  1.399699e-01  3.830752e-01  6.707160e-01  1.831812e-01  2.433106e-02
##     Barbadori       Strozzi       Albizzi      Altoviti    Della_Casa
##  3.547849e-01  1.000000e+00  5.323660e-01  3.372469e-01  1.414314e-01
##         Corsi     Davanzati   Frescobaldi        Ginori      Guadagni
##  7.071156e-02  7.332195e-02  3.436670e-01  1.057862e-01  3.460880e-01
##      Guasconi         Nerli   Del_Palagio   Panciatichi       Scolari
##  6.811084e-01  7.071156e-02  1.383317e-01  5.090917e-01  1.367356e-01
##  Aldobrandini    Alessandri       Tanagli    Bencivenni Gianfigliazzi
##  1.509870e-01  4.503879e-03  3.390835e-02  4.479485e-02  7.162939e-01
##         Spini  Dall'Antella      Martelli    Rondinelli   Ardinghelli
##  2.096823e-01  1.893404e-01  2.560082e-02  3.998861e-01  4.000982e-01
##       Peruzzi         Rossi       Arrighi  Baldovinetti     Ciampegli
##  7.915554e-01  1.582815e-01  4.596917e-02  2.497482e-01  3.317283e-02
##       Manelli      Carducci    Castellani      Ricasoli         Bardi
##  1.659979e-01  1.799494e-01  4.238792e-01  5.324287e-01  4.080706e-01
##       Bucelli      Serragli     Da_Uzzano    Baroncelli         Raugi
##  1.218222e-01  5.798025e-02  1.593404e-01  1.498662e-17  9.937879e-18
##       Baronci     Manovelli       Bartoli      Solosmei     Bartolini
##  1.421582e-02  1.070266e-01  1.321330e-17  2.188422e-17  3.400431e-03
##   Belfradelli    Del_Benino        Donati       Benizzi      Bischeri
##  4.564759e-02  1.231075e-03  9.268390e-03  1.051384e-01  3.348147e-01
##     Brancacci       Capponi       Nasi(?)     Sacchetti       Vettori
##  2.232934e-01  2.844521e-02  3.778238e-03  1.486187e-01  3.778238e-03
##         Doffi          Pepi    Cavalcanti          Ciai    Corbinelli
##  5.630179e-02  5.630179e-02  2.359640e-01  1.405106e-02  8.908790e-02
##          Lapi     Orlandini    Dietisalvi        Valori     Federighi
##  9.738994e-03  9.913900e-03  1.405106e-02  6.854784e-02  1.354681e-17
##  Dello_Scarfa    Fioravanti      Salviati        Giugni    Pandolfini
##  1.851919e-17  1.350571e-01  1.467173e-01  1.011873e-01  2.254509e-02
##  Lamberteschi    Tornabuoni       Mancini  Di_Ser_Segna      Macinghi
##  4.596917e-02  1.834366e-01  9.046827e-02  9.046827e-02  1.328251e-01
##          Masi        Pecori         Pitti    Popoleschi     Portinari
##  1.051384e-01  8.908790e-02  2.219130e-01  1.134529e-01  8.908790e-02
##       Ridolfi    Serristori    Vecchietti    Minerbetti         Pucci
##  2.507818e-01  8.908790e-02  8.908790e-02  1.328251e-01  1.316815e-03
##    Da_Panzano       Parenti         Pazzi      Rucellai    Scambrilla
##  9.046017e-02  1.328251e-01  9.020762e-02  1.328251e-01  2.785107e-02
##      Soderini
##  1.328251e-01

Then we can assign that vector to our network and plot it.

V(marriageNet)\$eigenvector <- evcent(marriageNet)\$vector

plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$eigenvector/max(V(marriageNet)\$eigenvector) * 20)

## 12.2 Bonacich Centrality

Perhaps marrying your daughters off to weaker families is a good way to ensure their loyalty? We could evaluate this using bonacich centrality. From igraph: “Interpretively, the Bonacich power measure corresponds to the notion that the power of a vertex is recursively defined by the sum of the power of its alters. The nature of the recursion involved is then controlled by the power exponent: 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).”

V(marriageNet)\$bonacich <- power_centrality(marriageNet, exponent = -2, rescale = T)
V(marriageNet)\$bonacich <- ifelse(V(marriageNet)\$bonacich < 0, 0, V(marriageNet)\$bonacich)

plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$bonacich/max(V(marriageNet)\$bonacich) * 20)

## 12.3 Page Rank

Here is Google’s page rank measure. It uses random walks to identify individuals who are commonly encountered along such walks. Those individuals are viewed as central.

V(marriageNet)\$page_rank <- page_rank(marriageNet, directed = TRUE)\$vector

plot(marriageNet,
vertex.label.cex = .6,
vertex.label.color = "black",
vertex.size = V(marriageNet)\$page_rank/max(V(marriageNet)\$page_rank) * 20)

### 12.3.1 Measure Correlations

Most of these measures are highly correlated, meaning they don’t necessarily capture unique aspects of pwoer. However, the amount of correlation depends on the network structure. Let’s see how the correlations between centrality measures looks in the Florentine Family network. cor.test(x,y) performs a simple correlation test between two vectors.

# extract all the vertex attributes
all_atts <- lapply(list.vertex.attributes(marriageNet),function(x) get.vertex.attribute(marriageNet,x))
# bind them into a matrix
all_atts <- do.call("cbind", all_atts)
# add column nams
colnames(all_atts) <- list.vertex.attributes(marriageNet)
# drop the family variable
all_atts <- data.frame(all_atts[,2:ncol(all_atts)])
# convert all to numeric
all_atts <- sapply(all_atts, as.numeric)
# produce a correlation matrix
cormat <- cor(all_atts)
# melt it using reshape to function melt() to prepare it for ggplot which requires long form data
melted_cormat <- melt(cormat)
ggplot(data = melted_cormat, aes(x=Var1, y=Var2, fill=value)) +
geom_tile() +
scale_fill_distiller(palette = "Spectral", direction=-2) +
xlab("") +
ylab("")

What do we learn?

### 12.3.2 Centralization and Degree Distributions

To understand the measures further, we can look at their distributions. This will tell us roughly how many nodes have centralities of a given value.

# fitting a degree distribution on the log-log scale
alter_hist = table(degree(marriageNet))
vals = as.numeric(names(alter_hist))
vals = vals[2:length(vals)]
alter_hist = alter_hist[2:length(alter_hist)]
df = data.frame(Vals = log(vals), Hist = log(as.numeric(alter_hist)), stringsAsFactors = F)

# plot log-log degree distribution
plot(Hist ~ Vals, data = df)

# regression line
abline(lm(Hist ~ Vals, data = df))

Do their marriage partners have more marriage partners than they do?

# degrees of your friends
neighbor_degrees <- knn(marriageNet)\$knn
degrees <- degree(marriageNet)

mean(neighbor_degrees, na.rm = T)
## [1] 18.72979
mean(degrees)
## [1] 6.541667
# plot neighbor degrees vs. ego degress
hist(neighbor_degrees)

hist(degrees)

We can see that most nodes in the marriage network have low betweenness centrality, and only one node has more than 40 betweenness. Degree distributions tend to be right-skewed; that is, only a few nodes in most networks have most of the ties. Evenly distributed degree is much rarer.

Finally centralization measures the extent to which a network is centered around a single node. The closer a network gets to looking like a star, the higher the centralization score will be.

degcent <- centralization.degree(marriageNet)\$centralization
centralization.betweenness(marriageNet)\$centralization
## [1] 0.2881759
centralization.evcent(marriageNet)\$centralization
## [1] 0.8503616
centralization.closeness(marriageNet)\$centralization
## Warning in centralization.closeness(marriageNet): At
## centrality.c:2784 :closeness centrality is not well-defined for disconnected
## graphs
## [1] 0.02030026

Can we compare our centralization scores against some baseline? Here is an example with Barabasi-Albert model, which simulates a network in which there is preferential attachment with respect to degree, the amount of which is controlled by the power parameter.

N <- vcount(marriageNet)
degcent <- centralization.degree(marriageNet)\$centralization

centralizations = c()
powers <- seq(from = 0, to = 3, by = 0.1)
for(e in powers){
net <- barabasi.game(N, directed = F, power=e)
centralizations <- c(centralizations, centralization.degree(net)\$centralization)
}

power_df <- data.frame(Centralization = centralizations, Power = powers)

ggplot(power_df, aes(x = Power, y = Centralization)) +
geom_point() +
geom_hline(yintercept = degcent, linetype="dashed", color = "red") +
theme_bw()

## Reach N What proportion of nodes can any node reach at N steps?

reach_n =function(x, n = 2){
r=vector(length=vcount(x))
for (i in 1:vcount(x)){
neighb =neighborhood(x, n, nodes=i)
ni=unlist(neighb)
l=length(ni)
r[i]=(l)/vcount(x)
}
return(r)
}

two_reach = reach_n(marriageNet, 2)

plot(marriageNet, vertex.size = two_reach * 10, vertex.label.cex = .4, vertex.label.color = "black", vertex.color = "tomato")

three_reach = reach_n(marriageNet, 3)

plot(marriageNet, vertex.size = three_reach * 10, vertex.label.cex = .4, vertex.label.color = "black", vertex.color = "tomato")

four_reach = reach_n(marriageNet, 4)

plot(marriageNet, vertex.size = four_reach * 10, vertex.label.cex = .4, vertex.label.color = "black", vertex.color = "tomato")

five_reach = reach_n(marriageNet, 5)

plot(marriageNet, vertex.size = five_reach * 10, vertex.label.cex = .4, vertex.label.color = "black", vertex.color = "tomato")

## 12.4 Distance weighted reach

distance_weighted_reach=function(x){
distances=shortest.paths(x) #create matrix of geodesic distances
diag(distances)=1 # replace the diagonal with 1s
weights=1/distances # take the reciprocal of distances
return(apply(weights,1,sum)) # sum for each node (row)
}

dw_reach = distance_weighted_reach(marriageNet)
dw_reach = dw_reach/max(dw_reach)

plot(marriageNet, vertex.size = dw_reach * 10, vertex.label.cex = .4, vertex.label.color = "black", vertex.color = "tomato")