avaialable from (https://rpubs.com/staszkiewicz/cw6Net)

Introduction

A network (graph) consists of a vertex (node) and an edge. The way many packages available for network analysis, we will use the `igraph’

#rm(list = ls()) # Gdyby była potrzeba wyczyszczenia środowiska
#install.packages("igraph") # jeśli po raz pierwszy instalujemy bibliotekę

library(igraph) # ładowanie pakietu

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

Igraph overlays its names in the workspace, so I rather recommend its use alone (without overloading the namespace libraries)

As usual, we will use primary data from the article Audit fee and communication sentiment. Economic Research-Ekonomska Istraživanja. https://doi.org/10.1080/1331677X.2021.1985567 Staszkiewicz and Karkowska (2021). The data, as usual, are available in the Essential (“Public Materials”) file named Bank.cvs. Please download it to your computer and upload it to R. Please note that the data may be used after class only for non-commercial purposes with attribution of the source.

# z bazowych funkcji systemu wczytamy klasyczny plik z csv ale w taki sposób
#   że wybierzemy z okienka umiejscowienie pliku na włanym komuterze 
# dlaego zagnieżdzamy polecenie "file.choose()"

# bank <-  read.csv(file.choose()) # not acitve for this part

Basics

Igraph allows you to visualize a network (graph). Let’s name vertices 1,2,3 and their relationships that 1 connects to 2, further 2 connects to 3, while 3 connects to 1. Now let’s define such a network as an object S1 and see its structure:

S1<-graph( edges=c(1,2, 2,3, 3, 1), n=3, directed=F ) 
class(S1)

## [1] "igraph"

S1

## IGRAPH ba5b536 U--- 3 3 -- 
## + edges from ba5b536:
## [1] 1--2 2--3 1--3

S1 is an “igraph” object, which is a list. Described by the following characteristics: 4 letters:

D or U, a directed or undirected network
N when, the mesh is named (vertices have the name attribute)
W when the network is weighted (edges have the weight attribute)
B for a bipartite (two-mode) graph (where vertices have the type attribute)

Two numbers (3 3) the number of vertices and edges respectively.
And a letter designation of the features (attributes) of the vertices and edges:

(g/c) - network-level character attribute
(v/c) - vertex - level character attribute
(e/n) - edge - level numeric attribute

in our case we are dealing with an undirected network, because when creating the object we chose the option: directed=F and so let’s visualize our network with the plot function

plot(S1)

Example (fictitious data) introduction

ABB Group in 2018 was audited by KPMG in 2019 and 2016 and 2015 by Deloitte, in 2020 and 2017 by PWC and in 2021 by EY. Average annual audit firms hire 223, 198, 204, 187 auditors data for KPMG, Deloitte, PWC and EY respectively.

Let us create a network showing the direction of ABB flow between audit firms. Our nodes are the audit firms (W), while the edges are the change of audit firm in a given year. Since it matters whether ABB moves from EY to KPMG or vice versa in a given year, so our network will be directed.

W<-c("KPMG","Deloitte","PWC","EY") # zdefinowaliśmy wierzchołki

# Definiujemy krawędzie odpowiadające transfer firm audytorskich w poszczególnych latach

#    2015->2016; 2016->2017;    2017->2018    2018->2019  2019->2020  2020->2021
K<-c("KPMG","KPMG", "KPMG","Deloitte", "Deloitte","KPMG", "KPMG","Deloitte", "Deloitte","PWC", "PWC","EY")

Let’s visualize the network:

S2 <- S1<-graph( edges=K, n=4, directed=T )

## Warning in graph(edges = K, n = 4, directed = T): 'n' is ignored for edge list
## with vertex names

plot(S2)

Note that when we define vertices with proper names we do not need to specify them with parameter n. In 2015 and 2016 ABB used the services of KPMG so we have a cycled (loop) edge to KPMG, while in subsequent years there were exchanges between KPMG and Deloitte. We don’t always need such detailed information and sometimes we may want to omit unnecessary repeats or zcycled edges between our vertices, hence we can simplify the graph as follows by removing repeats and loops.

#upraszczamy sieć S2 poprze usunięcie powtórzeń i pętli
S2b <- simplify(S2, remove.multiple = T, remove.loops = T)
# zobrazujmy uproszczoną sieć
plot(S2b)

So far, we have operated on nodes and edges, but for both nodes and edges we can create attributes (features). Let’s first see what the current strycture of the S2 network is and then assign attributes to it. We can check the vertices of V(object_name),

V(S2)

## + 4/4 vertices, named, from 752f6e4:
## [1] KPMG     Deloitte PWC      EY

edges E(object_name).

E(S2)

## + 6/6 edges from 752f6e4 (vertex names):
## [1] KPMG    ->KPMG     KPMG    ->Deloitte Deloitte->KPMG     KPMG    ->Deloitte
## [5] Deloitte->PWC      PWC     ->EY

The relationship between vertices and edges can also be shown in matrix form (generally sparse), as follows:

S2[]

## 4 x 4 sparse Matrix of class "dgCMatrix"
##          KPMG Deloitte PWC EY
## KPMG        1        2   .  .
## Deloitte    1        .   1  .
## PWC         .        .   .  1
## EY          .        .   .  .

So far, we have not depicted on the graph the information when the transfer of the audit firm occurs. Since this is an edge feature, let us build such a feature by using the function V and adding the years of transfer.

lata<- c("2015->2016","2016->2017","2017->2018","2018->2019","2019->2020","2020->2021")
E(S2)$lata<-lata

edge_attr(S2)

## $lata
## [1] "2015->2016" "2016->2017" "2017->2018" "2018->2019" "2019->2020"
## [6] "2020->2021"

Let’s visualize

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 0)

Let’s zoom in this visualization on the size of the default font in the graph edge.label.cex=0.4 and show the returns of edges edge.arrow.size = 0.05

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 1,edge.label.cex=0.4)

So far in our gaf we have extracted information about audit firms, dynamics of change (edge characteristics), what we have not illustrated is employment in audit firms. Employment is not a characteristic of an edge but of a vertex. Now we have a vertex described with the name of the audit firm, we will use two features of the node i.e. the color of the circle and the color of the circle field to show the employment data. Let’s start with the color of the circle field: Let us assign a color to the employment size. In view of the above: 1. will assign an attribute to the ‘staff’ hanger describing the employment multiplicity

V(S2)$staff<-c(223,198,204,187) # przypisujemy atrybuty
vertex_attr(S2) # zobaczmy przypisanie

## $name
## [1] "KPMG"     "Deloitte" "PWC"      "EY"      
## 
## $staff
## [1] 223 198 204 187

Let’s visualize the network, with the circles colored according to the number of auditors

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 0.05,edge.label.cex=0.4,vertex.color=colors(V(S2)$staff) )

## Warning in if (distinct) c[!duplicated(t(col2rgb(c)))] else c: the condition has
## length > 1 and only the first element will be used

We can also assign individual colors to specific audit firms e.g.:

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 0.05,edge.label.cex=0.4,vertex.color=c("blue","yellow","green","red") )

And by framing the oval, divide into those sub-entities that have more than 200 auditors

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 0.05,edge.label.cex=0.4,vertex.color=c("brown","yellow","green","red"),
vertex.frame.color=c( "black", "cyan")[1+(V(S2)$staff<200)])

Let’s combine area color scaling and circle borders into one

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 0.05,edge.label.cex=0.4,vertex.color=colors(V(S2)$staff),vertex.frame.color=c( "black", "cyan")[1+(V(S2)$staff<200)] )

## Warning in if (distinct) c[!duplicated(t(col2rgb(c)))] else c: the condition has
## length > 1 and only the first element will be used

Let us complement our network with vertex graphics. We would often like to depict vertices graphically, e.g. with logos of auditing companies. To do so, we need to 1) build a database of graphics 2) translate the graphics into their digital representation (raster) 3) create a graphic attribute in the igraphic object. So.

let’s build a connection between companies and their logos

I’m building a container for images

rasters <- as.list(c(imgType1='',imgType2='',imgType3='',imgType4=''))

rasters is a list containing four image objects. In the folder “path” folder I put 4 graphics logs of each submot, to convert them to graphics objects we need to run library library(jpeg) library or install it.

rasterize the images

library(jpeg)
rasters$imgType1 <- readJPEG("path/KPMG.jpg",native=TRUE)
rasters$imgType2 <- readJPEG("path/PWC.jpg",native=TRUE)
rasters$imgType3 <- readJPEG("path/D.jpg",native=TRUE)
rasters$imgType4 <- readJPEG("path/EY.jpg",native=TRUE)

linking vertices to images

We are building a data frame in which we have vertex names and corresponding rasterized image types:

lkp_mat <- data.frame(from=c('KPMG','PWC','Deloitte','EY'),type=c('imgType1','imgType2','imgType3','imgType4'))

and then associate the image types with the vertices in our igraph object, using the vertex name and the vertex ids in both objects, i.e. To do this, we will use the vertex name ‘name’ and the vertex ids in both objects, i.e. lkp_mat and S2 as follows:

for(i in V(S2)$name){
  imgtype <- lkp_mat$type[lkp_mat["from"]==i]
  V(S2)[name==i]$raster <- rasters[imgtype]
}

What we are left with is a visualization add 1) vertex.shape=“raster” - to insert images 2)vertex.label="" to not insert text description

and here is the effect

plot(S2, edge.label=E(S2)$lata,edge.arrow.size = 0.05,edge.label.cex=0.4,vertex.color=colors(V(S2)$staff),
     vertex.frame.color=c( "black", "cyan")[1+(V(S2)$staff<200)], 
     vertex.shape="raster",vertex.size=20, vertex.size2=20,vertex.label="")

## Warning in if (distinct) c[!duplicated(t(col2rgb(c)))] else c: the condition has
## length > 1 and only the first element will be used

## Generating Subgraphs (Subnetworks)

Often we don’t need the whole network, but just a part of it, so we can build the network based on the characteristics of hangers or edges. Let us analyze only two hangers, namely KPMG and Deloitte.

sel<-c("KPMG","Deloitte") #generujemy listę wierzchołków

S3<- induced.subgraph(graph=S2,vids=sel) #bubujemy subsieć
 
plot(S3, main="ABB zmiany firm audytorskich") #wizualizujemy

#Parameters of the plot function

vertices vertex.color color of the vertex vertex.frame.color color of vertex’s border vertex.shape One of “none”, “circle”, “square”, “csquare”, “rectangle” “crectangle”, “vrectangle”, “pie”, “raster”, or “sphere” vertex.size Size of vertex (default is 15) vertex.size2 Secondary vertex size (for example for a rectangle) vertex.label A string of characters used as vertex name vertex.label.family Font (“Times”, “Helvetica”, etc.) vertex.label.font Type of font: 1 plain, 2 bold, 3, italic, 4 bold and italic , 5 simile vertex.label.cex Font size (multiplication factor, device-dependent) vertex.label.dist distance label to vertex vertex.label.degree position of label in relation to vertex, where 0 right, “pi” left, “pi/2” below and “-pi/2” above

Edges. edge.colour color of edge edge.width width of edge, default =1, edge.arrow.size size of the arrow, default = 1 edge.arrow.width width of the arrow, default =1 edge.lty line type 0 for “none”, 1 for “continuous”, 2 for “dashed”, 3 for “dotted”, 4 for “dotted and dashed”, 5 for “long dashed”, 6 for “double dashed” edge.label String of characters used for the carver label edge.label.family Font type (“Times”, “Helvetica”, etc.) edge.label.font Font: 1 plain, 2 bold, 3, italic, 4 bold italic, 5 symbol edge.label.cex size of the font in the label edge.curved edge curvature, in range 0-1 (FALSE to 0, TRUE to 0.5) arrow.mode A vector indicating whether there should be an arrow, possible stay: 0 no arrow, 1 back, 2 forward, 3 both Other. margin Empty space margins around the plot, vector with length 4 frame if TRUE, the graph will be framed main if set, the gaf title will be displayed sub subtitle of graph

Tasks to work on independently

Task

Marco Van Fun Marco Van Fun is the director of a cooperative bank in Gliwice, and today he has just received the monthly report on the realisation of the loan portfolio. Among the top 10 exposures is a loan granted to the Agricultural Advisory Centre (ODR) in Dobrzeniu Wielkie, for PLN 70 million. The loan was secured with ODR’s shares in RodzinaNaSwoim Sp. z o.o. at the time of granting the loan, i.e. 12 months ago, RodzinaNaSwoim was a capital group structured as follows:

Grandma sp. z.o.o. Wnuczek sp. z o.o. Uncle s.a. Cousin s.r.o The returns vectors indicate control, while percentages indicate the level of control and influence. The shareholders of the family on its own apart from Wnuczek sp. o.o. (40%) were also individuals, grandmother (10)%, mother (20%), father (15%), child (10%), granddaughter (5%). The report indicates the following transactions with related parties for the last 12 months.

Page nal.		Transaction value	Hourly value
RnS	Mom	100	90
Mummy	RnS	300	350
	Mum	Mum	200
RnS	Grandson	500	550
Uncle	RnS	111	100
Uncle	Dad	120	220
Uncle	Grandma	100	100
Grandma	RnS	500	350
Grandma	Mommy	702	560
Ta-Ta	RnS	200	210
In the case of the RnS	Dad	300	100

In the reports submitted in the credit application, there was one transaction with related parties Mama sp. z o.o. sold to RodzinieNaSwoim, products for the amount of 100 million with a fair value of 130 million million.

** You are required to:**

Show graphically transactions with related parties in net net

Tell Mark what measure of hedge risk he should use.

Hint Discuss the idea of the article: Staszkiewicz, P. (2011). Risk structures. A conceptual sketch. (K. Jajuga & W. Ronka-Chmielowiec, Eds.). Financial investments and insurance - global trends vs. Poland, 183, 378–384.

To read:

Interacting network drawing in 3d interactive

A well written guide to igraph here.

How to polyline graphics to vertices image link

How to visualize a network segment. subnet generation

References

Staszkiewicz, Piotr, and Renata Karkowska. 2021. “Audit Fee and Banks’ Communication Sentiment.” Economic Research-Ekonomska Istraživanja, October, 1–21. https://doi.org/10.1080/1331677x.2021.1985567.

EX. 6 Network analysis - intro

Piotr Staszkiewicz

23 01 2022