# Fuzzy Social Network Analysis: Theory and Application in a University
Department’s Collaboration Network

Annamaria Porreca  Fabrizio Maturo  Viviana Ventre

###### Abstract

Social network analysis (SNA) helps us understand the relationships and
interactions between individuals, groups, organisations, or other social
entities. In SNA, ties are generally binary or weighted based on their
strength. Nonetheless, when actors are individuals, the relationships between
actors are often imprecise and identifying them with simple scalars leads to
information loss. Social relationships are often vague in real life. Despite
many classical social network techniques contemplate the use of weighted
links, these approaches do not align with the original philosophy of fuzzy
logic, which instead aims to preserve the vagueness inherent in human language
and real life. Dealing with imprecise ties and introducing fuzziness in the
definition of relationships requires an extension of social network analysis
to fuzzy numbers instead of crisp values. The mathematical formalisation for
this generalisation needs to extend classical centrality indices and
operations to fuzzy numbers. For this reason, this paper proposes a
generalisation of the so-called Fuzzy Social Network Analysis (FSNA) to the
context of imprecise relationships among actors. The article shows the theory
and application of real data collected through a fascinating mouse tracking
technique to study the fuzzy relationships in a collaboration network among
the members of a University department.

###### keywords:

Fuzzy Social Network , Fuzzy Centrality Indices , Vague Relationships ,
Uncertainty , Fuzzy Numbers , University Department Collaboration Network.

###### MSC:

62A86 , 94D05 , 91D30

[inst1]organization=Department of Medical, Oral and Biotechnologies Science,
University G. d’Annunzio of Chieti-Pescara, addressline=Via dei Vestini, 31,
city=Chieti, postcode=66100, country=Italy

[inst2]organization=Faculty of Technological and Innovation Sciences,
Universitas Mercatorum, addressline=Piazza Mattei, 10, city=Rome,
postcode=00186, country=Italy

[inst3]organization=Department of Mathematics and Physics, University of
Campania Luigi Vanvitelli, addressline=Viale Abramo Lincoln, 5, city=Caserta,
postcode=81100, country=Italy

##  1 Introduction

Social network analysis (SNA) helps us understand the relationships and
interactions between individuals, groups, organisations, or other social
entities [18, 31, 29, 40, 26]. Each type of social structure can be
represented with a graph, in which the social entities are represented as
nodes or vertices in a network, and their relationships are represented as
edges or ties. SNA examines relationships between entities and how
information, resources, or influence flows through them. It identifies
patterns, structures, and dynamics of social systems. Ties in a network are
generally binary based on the presence or absence of the link or weighted
based on their strength. Each connection can be identified by a number in
$[0,1]$ or $\\{0,1\\}$, respectively. Nonetheless, when the actors of a social
network are individuals, the relationships are very complex and measuring them
using a single number leads to a loss of information. In other words,
relationships are inherently imprecise in real life, and thus, social network
analysis should consider vagueness. For example, let’s consider a social
network like Facebook. The nodes may represent people registered on the
network, and the links can be the connections between pairs of individuals.
Typically, in a social network of this type, these connections are based on
the presence or absence of ”friendship”. We must consider whether being
friends with someone on Facebook should have the same significance for all
connections. It’s evident that not all friendships are of equal weight and
intensity; the level of interaction determines the closeness of a relationship
rather than merely being connected online. However, this issue applies to
social networks and real-life relationships because connections are fuzzy and
real-life phenomena are not black or white but often grey [45, 47, 46]. What
does it mean to be a friend? Is it being a close friend, an acquaintance, or
just a casual acquaintance? What does being close collaborators or barely
knowing each other in a university department mean?

One way to address this issue is using a fuzzy network analysis approach.
Despite this, fuzzy network analysis has mainly been studied by introducing
weighted ties within the network [see, e.g., 20, 8]. This approach of weighing
the links between nodes differs from the original philosophy of fuzzy logic,
which instead aims to preserve the vagueness inherent in human language.
Assigning a link a value in $[0,1]$ or $\\{0,1\\}$ can sometimes be a
simplification that sacrifices information for the sake of convenience.
Translating vague concepts (such as close collaborator, occasional
collaborator, real friend, and similar) necessarily results in a loss of
information regarding the inherently imprecise nature of these expressions.
Hence, considering a link as ”fuzzy” merely because it is represented by a
number in the interval $[0,1]$ does not align with the basic principles of
fuzzy logic, which begins and thrives where opposites coexist. In other words,
fuzzy logic deals with situations where A and non-A coexist simultaneously
with a certain degree of truth [45, 47, 46, 41]. Dealing with imprecise ties
and introducing fuzziness in the definition of relationships needs an
extension of classical social network analysis to fuzzy numbers instead of
crisp values. The mathematical formalisation of this generalisation requires
rewriting the classical centrality indices into fuzzy centrality indices
involving fuzzy relationships. In other words, this research’s primary
motivation and basic idea is that relationships between nodes should be
characterised by functions rather than scalars to preserve the information
about the vagueness of ties. The great advantage of this approach is the
ability to create social networks and centrality indices that preserve
information about the vagueness and imprecision of ties because they are
inherent in human nature, human language, and interpersonal relationships. In
statistics, it is known that valuable information is contained within
variability. Similarly, in social networks, true information lies in
understanding the vagueness of relationships.

The article shows the theory of Fuzzy Centrality Measures (FCM) in Social
Network Analysis and their application to real data. The dataset adopted in
this study is collected through a fascinating technique that includes mouse
tracking to capture the vagueness of the relationships among the members of a
University department. The first part of the work introduces the preliminaries
necessary for formally understanding the proposed indices. Subsequently, based
on the idea mentioned above, new centrality indices are proposed: fuzzy degree
centrality, fuzzy out-degree centrality index, total fuzzy degree centrality
index, fuzzy betweenness centrality, fuzzy closeness centrality, and fuzzy
out-closeness centrality. The application Section proposes a fascinating
investigation based on a University department collaboration network. The
paper ends with a discussion and conclusions.

##  2 A brief introduction to Graph Theory (GT)

A graph $G=(V,E)$ is an abstract object formed by a set $V$ of vertices
(nodes) and a set $E$ of edges (links) that join pairs of vertices. The vertex
set and edge set of a graph $G$ are denoted by $V(G)$ and $E(G)$,
respectively. The cardinality of $V$ is usually denoted by $n$, and the
cardinality of $E$ by $m$. If an edge joins two vertices, they are adjacent,
and we call them neighbors. Graphs can be undirected or directed. In
undirected graphs, the order of the end vertices of an edge does not indicate
a specific direction between them; the relationship is symmetric. In directed
graphs, each directed edge (arc) has an origin (tail) and a destination
(head). An edge with origin $u\in V$ and destination $v\in V$ is represented
by an ordered pair $(u,v)$. In shorthand notation, an edge $\\{u,v\\}$ or
$(u,v)$ can also be denoted by $uv$. In a directed graph, $uv$ is short for
$(u,v)$, while in an undirected graph, $uv$ and $vu$ are the same and both
stand for $\\{u,v\\}$. Edge weights are numerical values associated with edges
or vertices in a graph $G=(V,E)$. Here, we focus on edge weights. These
weights are often represented by a function $w:E\to\mathbb{R}$, assigning each
edge $e\in E$ a weight $w(e)$. Depending on the application, edge weights can
represent properties such as cost, capacity, strength of interaction, or
similarity.

A walk from $v_{0}$ to $v_{k}$ in a graph $G=(V,E)$ is an alternating sequence
$v_{0},e_{1},v_{1},e_{2},v_{2},$
$\ldots,v_{k-1},e_{k},v_{k}$ of vertices and edges, where
$e_{i}=\\{v_{i-1},v_{i}\\}$ in the undirected case and $e_{i}=(v_{i-1},v_{i})$
in the directed case. The length of the walk is defined as the number of edges
on the walk. The walk is called a path if $e_{i}\neq e_{j}$ for $i\neq j$, and
a path is a simple path if $v_{i}\neq v_{j}$ for $i\neq j$. A path with
$v_{0}=v_{k}$ is a cycle. A cycle is a simple cycle if $v_{i}\neq v_{j}$ for
$0\leq i<j\leq(k-1)$. For a path $p$ in a graph $G=(V,E)$ with edge weights
$w$, the weight of the path, denoted by $w(p)$, is defined as the sum of the
weights of the edges on $p$. A path from $u$ to $v$ in $G$ is the shortest
path (with respect to $w$) if its weight is the smallest possible among all
paths from $u$ to $v$. The length of the shortest path from $u$ to $v$, also
called the shortest-path distance between $u$ and $v$, is denoted by $d(u,v)$.

The single-source shortest paths problem is defined as follows: Given a graph
$G=(V,E)$ with edge weights $w:E\to\mathbb{R}$ and a vertex $v\in V$ (the
source), compute the shortest paths from $v$ to all other vertices in the
graph. The problem is only well-defined if the graph does not contain a
negative weight cycle. If the edge weights are non-negative, the shortest
paths problem can be solved in time $O(m+n\log n)$ using an efficient
implementation of Dijkstra’s algorithm [15]. If the edge weights are
arbitrary, the Bellman-Ford algorithm [4] uses time $O(mn)$ to detect a cycle
of negative length or, if no such cycle exists, solve the problem. For the
case of unit edge weights, the Bellman-Ford algorithm solves the problem in
linear time $O(n+m)$. In the all-pairs shortest paths problem, one is given a
graph $G=(V,E)$ with edge weights $w:E\to\mathbb{R}$ and wants to compute the
shortest-path distances for all pairs of nodes. Provided that $G$ does not
contain a cycle of negative length, this problem can be solved by the Floyd-
Warshall algorithm [17] in time $O(n^{3})$, or by running the shortest paths’
problem computations in time $O(nm+n^{2}\log n)$.

Adjacency and incidence matrices are tools used in graph theory to represent
and analyze the relationships between vertices and arcs of a graph. The
adjacency matrix (or sociomatrix) of a directed graph $G=(V,E)$ with vertices
$V=\\{v_{1},\ldots,v_{n}\\}$ is an $n\times n$ matrix $A_{G}$ where
$a_{i,j}=1$ if $(v_{i},v_{j})\in E$, and 0 otherwise. For undirected graphs,
this matrix is symmetric. In weighted graphs, entries $a_{i,j}$ can represent
weights $w(v_{i},v_{j})$ instead of binary adjacency. In a simple directed
graph, the incidence matrix $B_{G}$ with $n$ rows and $m$ columns captures
relationships between nodes and arcs. Entries $b_{i,j}$ indicate whether node
$v_{i}$ is the origin or destination of arc $e_{j}$. A simple graph is a graph
without loops (self-edges) and multiple arcs between any pair of vertices. For
a simple direct graph, we can assume by convention $b_{i,i}=0$ $\forall i\in
I_{n}$, for $i\neq j$.

##  3 Overview of the Main Classical Centrality Measures

When examining relationships within a group, our initial inquiry often
revolves around identifying influential individuals within the network. To
address such queries effectively, it is essential to establish measures that
quantify centrality. Many vertex centrality indices were introduced in the
1950s, such as the Bavelas index [2, 3], degree centrality [21], and a first
feedback centrality introduced by Seeley [36]. Before providing a formal
definition of centrality indices, it is necessary to introduce the concept of
isomorphism.

###### Definition 1

Two directed graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ are
isomorphic (denoted as $G_{1}\simeq G_{2}$) if there exists a bijection
$\varrho:V_{1}\rightarrow V_{2}$ such that for all $u,v\in V_{1}$:

$(u,v)\in E_{1}\Longleftrightarrow(\varrho(u),\varrho(v))\in E_{2}.$

Such a bijection is called an isomorphism.

An isomorphism that maps a graph onto itself is called an automorphism.
Usually, we consider two graphs to be the same if they are isomorphic.
Isomorphism and automorphism for a graph with symmetric relations are defined
analogously.

Formally [24], a centrality index is a structural index which is defined as
follows:

######  Definition 2 (Structural Index)

Let $G=(V,E)$ be a weighted, directed, or undirected multigraph, and let $V$
represent the set of vertices or edges of $G$, respectively. A real-valued
function $s$ is called a structural index if and only if the following
condition holds: for all $v\in V$, for any graph $H=(V^{\prime},E^{\prime})$
isomorphic to $G$, there exists a bijection $\varrho:V\rightarrow V^{\prime}$
such that $s_{G}(v)=s_{H}(\varrho(v))$, where $s_{G}(v)$ denotes the value of
$s(v)$ in graph $G$.

A centrality index $C$ is required to be a structural index and thus induces
at least a semi-order on the set of vertices or edges, respectively. According
to this order, $v\in V$ is considered at least as central as $u\in V$ with
respect to a given centrality $C$ if $C(v)\geq C(u)$. Centrality indices are
structural measures used to quantify the importance of a node $v\in V_{G}$ in
a graph $G$. For instance, $\sigma^{G}:V_{G}\rightarrow\Re_{0}^{+}$ assigns
higher values to nodes that are more interconnected. In cases where two nodes
$v$ and $u$ exhibit identical network structures, the centrality index should
yield the same value $\sigma^{G}(v)=\sigma^{G}(u)$ for both nodes [28, 35].
Nodes $v$ and $u$ are considered identically structurally integrated into the
network if there exists an automorphism $\eta:V_{G}\rightarrow V_{G}$ with
$u=\eta(v)$. Centrality represents the extent to which an actor occupies a
central position within a network. Empirical research has shown that occupying
a central position in a network of relationships often correlates with access
to resources and influence [14, 32, 33].

####  3.0.1 Degree

The number of edges connected to a vertex is referred to as its “degree” in
graph theory. The degree of a node $v_{i}$ is commonly denoted as $d_{i}$. In
the context of directed graphs, nodes have both in-degrees (edges pointing
towards the node) and out-degrees (edges pointing away from the node), denoted
as $d_{i}^{in}$ and $d_{i}^{out}$, respectively. For example, in social
platforms like Facebook, the degree of a user corresponds to the number of
friends [48].

###### Definition 3

In any undirected graph $G(V,E)$, the sum of all node degrees equals twice the
number of edges:

$\sum_{i}d_{i}=2|E|$ (1)

This equation holds because each edge contributes to the degree sum of exactly
two nodes.

In directed graphs, the sum of in-degrees equals the sum of out-degrees:

$\sum_{i}d_{i}^{in}=\sum_{j}d_{j}^{out}$ (2)

This equality arises because each directed edge contributes exactly one unit
to the in-degree of its destination node and one unit to the out-degree of its
origin node.

####  3.0.2 Degree centrality

In real-world interactions, individuals with many connections are often
perceived as important. Degree centrality quantifies this idea into a
measurable index. The degree centrality index ranks nodes higher based on the
number of connections they have.

For an undirected graph, the degree centrality $C_{d}$ of a node $v_{i}$ is
given by:

$C_{d}(v_{i})=d_{i}$ (3)

where $d_{i}$ represents the degree (number of adjacent edges) of node
$v_{i}$.

In directed graphs, we distinguish between in-degree centrality, out-degree
centrality, and total degree centrality, which combines in and out degrees:

$C_{d}(v_{i})=d_{i}^{in}\quad\text{(In-degree centrality)}$ (4)
$C_{d}(v_{i})=d_{i}^{out}\quad\text{(Out-degree centrality)}$ (5)
$C_{d}(v_{i})=d_{i}^{in}+d_{i}^{out}\quad\text{(Total degree centrality)}$ (6)

In-degree centrality indicates how popular a node is within the network. Out-
degree centrality measures how gregarious a node is.

Comparing degree centrality values across different networks directly is not
straightforward due to variations in network size. To standardize comparison,
degree centrality can be normalized as:

$C_{d}^{*}(v_{i})=\frac{d_{i}}{n-1}$ (7)

where $n$ is the number of vertices in the network.

####  3.0.3 Closeness centrality

Closeness centrality measures the average distance from a vertex to all other
vertices in the network. Unlike other network metrics, closeness centrality
offers a distinct perspective on each individual’s network position, capturing
how quickly each vertex can reach all others [19]. The central idea is that
nodes with higher closeness centrality are more central, allowing them to
reach other nodes more efficiently [48]. Mathematically, closeness centrality
$C_{c}(v_{i})$ for vertex $v_{i}$ is defined as:

$C_{c}(v_{i})=\frac{1}{\bar{l}_{D_{j}}}$ (8)

where

$\bar{l}_{D_{j}}=\frac{1}{n-1}\sum_{v_{j}\neq v_{i}}l_{i,j}$ (9)

represents the average shortest path length from node $v_{i}$ to all other
nodes in the network. Nodes with smaller average shortest path lengths
$l_{i,j}$ exhibit higher closeness centrality.

####  3.0.4 Betweenness centrality

Another perspective on centrality considers how crucial nodes are in
connecting other nodes. For a node $v_{i}$, one approach is to quantify the
number of shortest paths between other nodes that pass through $v_{i}$:

$C_{b}(v_{i})=\sum_{\begin{subarray}{c}s\neq v_{i}\neq
t\end{subarray}}\frac{\sigma_{st}(v_{i})}{\sigma_{st}}$ (10)

Here, $\sigma_{st}$ denotes the total number of shortest paths from node $s$
to $t$ (also referred to as information pathways), and $\sigma_{st}(v_{i})$
represents the number of those paths that pass through $v_{i}$. Essentially,
this metric evaluates how central $v_{i}$ is in facilitating connections
between any pair of nodes $s$ and $t$, known as betweenness centrality.

To standardize betweenness centrality across networks, normalization is
necessary. This involves calculating its maximum possible value:

$C_{b}^{*}(v_{i})=\frac{C_{b}(v_{i})}{\max_{s\neq t}C_{b}(v_{i})}$

The maximum value occurs when $v_{i}$ lies on all shortest paths from $s$ to
$t$ for every pair $(s,t)$, i.e., $\forall(s,t),s\neq t\neq
v_{i},\frac{\sigma_{st}(v_{i})}{\sigma_{st}}=1$.

##  4 A brief introduction to Fuzzy Numbers

The limit of bivalent logic is that it addresses the real problems as if they
were white or black [25]. Understanding the critical point at which a thing
becomes a non-thing undermines traditional logic, which is based on black and
white, all or nothing, true or false; according to Aristotelian logic, an
element belongs to a set or does not belong at all. Aristotelian thought is
essentially based on the principles of identity, non-contradiction, and the
excluded middle. Since ancient times, Mathematicians and Philosophers have
tried to find a rational explanation for paradoxes to save the dichotomy
assumption and avoid dealing with the grey that exists in all things. The
first to use the term fuzzy to describe logic based on uncertainty and
vagueness was Lotfi Zadeh. When Zadeh published his first article on “Fuzzy
Sets”, he introduced the concept of “membership value” which, in contrast to
the bivalent theory, considers the possibility of intermediate values of
truth; in particular, the degree of membership can take any value in the range
$[0,1]$[45]. Fuzzy logic contemplates the grey and starts where the
contradictions inherent in things begin, where A and non-A melt to a certain
extent. Thus, fuzzy logic essentially deals with uncertainty due to vagueness.

In classical set theory, a set comprises all the elements of the discourse
universe that satisfy a given membership function. For a traditional set,
which we define as crisp, to distinguish it from a fuzzy set, the membership
function is Boolean; it associates to each element $x$ of the universe a value
of “true” or “false” depending on whether or not $x$ belongs to the set.
Formally, a crisp set has the following characteristic function:

$\mu_{A}(x)=\begin{cases}1&\text{if x $\in$ A}\\\ 0&\text{if x $\notin$
A}\end{cases}$ (11)

The theory of fuzzy sets posits that an element can partially belong to a set,
determined by a membership function that assigns real values in the interval
$[0,1]$.

A fuzzy set _A_ is defined by a set of ordered pairs and a binary relation as
follows:

$\emph{A}=\\{(x,\mu_{A}(x))\mid x\in A,\mu_{A}(x)\in[0,1]\\}$ (12)

where $\mu_{A}(x)$ is a function, known as the membership function, that
specifies the degree to which any element $x$ in $A$ belongs to the fuzzy set
_A_. This implies a gradual transition from membership to non-membership [47].

Formally, a fuzzy set _A_ is defined by its characteristic function:

$\mu_{A}:X\rightarrow[0,1]$ (13)

being $X$ the universe of definition.

###  4.1 Definition of a fuzzy number

A fuzzy number $\mu(x)$ is a special case of a fuzzy set; in fact it can be
defined as a fuzzy set, defined on real number, with a normal and convex
membership function, such that there exists at least one point where the
membership function takes the value “one”; it is a very useful tool for
working with imprecise numerical quantities [45]. A fuzzy number is a function
having as domain the set of real numbers and with values in $[0,1]$:

$\mu:\mathbb{R}\rightarrow[0,1]$ (14)

such that the following characteristics apply:

  1. 1.

Bounded support: there are two real numbers a and b, with $a\leq b$, called
the endpoints of $\mu$, such that:

$\begin{cases}\mu(x)=0&\text{for}\qquad x\not\in[a,b]\\\
\mu(x)>0&\text{for}\qquad x\in(a,b);\end{cases}$ (15)

  2. 2.

Normality: there are two real numbers $c$ and $d$, with $a\leq c\leq d\leq b$
such that:

$\mu(x)=1\qquad\text{if and only if}\qquad x\in[c,d].$ (16)

  3. 3.

Convexity: $\mu(x)$ is a function increasing in the interval $[a,c]$ and
decreasing in the interval $[d,b]$;

  4. 4.

Compactness: for every $\alpha\in(0,1)$, the set
{$x\in\mathbb{R}:\mu(x)=\alpha$} is a closed interval.

The set of the real numbers $x$ such that $\mu(x)>0$ is said the support of
the fuzzy number, and the interval $[c,d]$ is the core or central part. The
intervals $[a,c)$ and $(d,b]$ are, respectively, the left part and the right
part. The real numbers ${\mu}_{L}=c-a$, ${\mu}_{C}=d-c$, and ${\mu}_{R}=b-d$
are the left, middle, and right spreads, respectively. Their sum
${\mu}_{T}=b-a$ is the total spread of the fuzzy number [23]. The choice of
the type of fuzzy numbers is delicate as it influences the characteristics of
the processes of fuzzification and defuzzification of the outputs and the
inputs [47]. The fuzzification allows to calculate the degree of membership of
each numerical value assumed by one input variable to each fuzzy set defined
for it. The defuzzification, instead, starting from the result obtained in the
fuzzy inference process, calculates a real value for the output variable. The
main membership functions representing fuzzy variables are triangular (TFN),
trapezoidal, and bell-shaped. In this work, we concentrate on TFNs. A
triangular fuzzy number $A$ is defined by the following membership function:

$\mu_{A}(x)=\begin{cases}\frac{x-a_{l}}{a_{m}-a_{l}}&\text{for}\quad a_{l}\leq
x\leq a_{m}\\\ \frac{x-a_{r}}{a_{m}-a_{r}}&\text{for}\quad a_{m}\leq x\leq
a_{r}\\\ \ 0&\text{otherwise}.\end{cases}$ (17)

where $[a_{l},a_{r}]$ is the support and $a_{m}$ is the core. As illustrated
in Fig.1 $a_{l}$ and $a_{r}$ are respectively the left and right endpoints,
while $a_{m}$ is the point where the membership function is equal to one.
However, a TFN is often indicated using a simpler notation as follows:
$A=(a_{l},a_{m},a_{r})$.

Figure 1: Triangular fuzzy number.

###  4.2 Ranking fuzzy numbers

Given a universe set $X$, a fuzzy set $A_{i}$ on $X$ is defined as
$\\{(x,\mu_{A_{i}}(x))\mid x\in X\\}$, where $\mu_{A_{i}}(x)$ denotes the
membership degree of $x$ in $A_{i}$ [9]. The support of a fuzzy number is
defined as:

$A_{i}=\operatorname{cl}\\{x\in X\mid\mu_{A_{i}}(x)>0\\},$

where $\operatorname{cl}$ represents the topological closure operator. Let
$\mathcal{F}$ denote the set of all fuzzy numbers. Since a fuzzy number
$A_{i}$ is fully characterized by its membership function values, denote this
function by $A_{i}(x)$.

The primary ranking methods can be categorized into two types [9]. The first
type maps fuzzy numbers directly onto the real line using transformation
functions $M:\mathcal{F}\rightarrow\mathbb{R}$. These functions associate each
fuzzy number with a real number and use the ordering $\geq$ on the real line.
For most methods, a higher associated value indicates a higher rank:

$M(A_{i})\geq M(A_{j})\Rightarrow A_{i}\succeq_{M}A_{j},$

where $\succeq_{M}$ denotes the dominance relation induced by $M$.

The second type of ranking method generates fuzzy binary relations, defined as
functions $M:\mathcal{F}\times\mathcal{F}\rightarrow[0,1]$, where
$M(A_{i},A_{j})\in[0,1]$ measures the degree to which $A_{i}$ is greater than
$A_{j}$. Accordingly, fuzzy numbers are ranked based on:

$M(A_{i},A_{j})\geq M(A_{j},A_{i})\Rightarrow A_{i}\succeq_{M}A_{j}.$

From the fuzzy binary relation among fuzzy numbers, certain procedures derive
an ordering relation [38, 39]. One widely used ranking method is the Center of
Gravity (CoG) of a fuzzy number, introduced by [30]:

$\operatorname{CoG}(A)=\frac{\int_{-\infty}^{\infty}xA(x)\,\mathrm{d}x}{\int_{-\infty}^{\infty}A(x)\,\mathrm{d}x}.$
(18)

There exist numerous other proposals in the literature for ranking fuzzy
numbers [see, e.g., 42, 7, 16, 11, 12, 22, 1, 27]; however, ranking methods
are secondary to our study focus and serve a functional role in our
application. Hereafter, we employ the Center of Gravity, though our approach
may extend to other strategies.

###  4.3 A brief introduction to Fuzzy Graphs (FGs)

Graphs are fundamental mathematical structures commonly used to model networks
and relationships [5, 6]. Rosenfeld [34] introduced the concept of a fuzzy
graph, applying fuzzy set theory to graphs. According to Rosenfeld [34], a
fuzzy graph is defined as a graph composed of vertices and edges, where the
edges have membership functions defined as follows:

######  Definition 4 (Rosenfeld [34])

A fuzzy graph $G=(\sigma,\mu)$ consists of functions
$\sigma:S\rightarrow[0,1]$ and $\mu:S\times S\rightarrow[0,1]$, where for all
$x,y\in S$, $\mu(x,y)\leq\min\\{\sigma(x),\sigma(y)\\}$.

This definition specifies that in a fuzzy graph, both vertices and edges have
membership values in the interval $[0,1]$. Yeh and Bang [44] proposed an
alternative formulation of fuzzy graphs tailored for clustering analysis:

######  Definition 5 (Yeh and Bang [44])

A fuzzy graph $G=(V,R)$ consists of a set of vertices $V$ and a fuzzy relation
$R$ on $V$, where the edges between vertices have a membership function
$\mu_{R}:V\times V\rightarrow[0,1]$.

Unlike Definition 4, Yeh and Bang’s formulation allows for both fuzzy vertices
and fuzzy edges. They designed this version of fuzzy graphs to better suit
clustering applications, providing a more flexible representation.

Since their inception, fuzzy graphs by Rosenfeld [34] and Yeh and Bang [44]
have been extensively studied for various applications. Blue et al. [5, 6]
categorized fuzzy graphs into different types, including:

  1. 1.

Fuzzy sets of crisp graphs:

$G=\\{(G_{1},\mu(G_{1})),(G_{2},\mu(G_{2})),\ldots,(G_{n},\mu(G_{n}))\\}$

where each crisp graph $G_{i}$ is associated with a membership function
$\mu(G_{i})$.

  2. 2.

Crisp vertices and edges with fuzzy connectivity:

$G=\\{V,E\\},\quad V=\\{v_{1},v_{2},\ldots,v_{n}\\},\quad
E=\\{(h_{i},\sigma_{i}),(t_{i},\tau_{i})\mid i=1,2,\ldots,m\\}$

The edges’ connectivity is fuzzy, where heads $h_{i}$ and tails $t_{i}$ have
membership values in $[0,1]$, while vertices and edges themselves are crisp.

  3. 3.

Fuzzy vertices and crisp edges:

$G=\\{V,E\\},\quad
V=\\{(v_{1},\mu(v_{1})),(v_{2},\mu(v_{2})),\ldots,(v_{n},\mu(v_{n}))\\},\quad
E=\\{e_{1},e_{2},\ldots,e_{n}\\}$

The graph $G$ consists of fuzzy vertices with membership functions and crisp
edges.

  4. 4.

Crisp graph with fuzzy weight:

$G=\\{V,E\\},\quad V=\\{v_{1},v_{2},\ldots,v_{n}\\},\quad
E=\\{e_{1},e_{2},\ldots,e_{n}\\}$

where edges have fuzzy weights defined by:

$w_{i}=\\{(w_{i,1},\mu(w_{i,1})),(w_{i,2},\mu(w_{i,2})),\ldots\\}$

These classifications reflect the varying degrees of fuzziness applied to
graphs, ranging from fully crisp vertices and edges to combinations of fuzzy
vertices or edges with crisp counterparts.

###  4.4 Fuzzy operations

A fuzzy operation of $n$ variables is a function,

$f:(v_{1},v_{2},\ldots,v_{n})\in[0,1]^{n}\rightarrow[0,1]$

The most commonly used binary fuzzy operations are the “fuzzy intersections”
or “t-norms” and the “fuzzy union” or “t-conorms”.

####  4.4.1 The “fuzzy intersection” or “t-norms”

###### Definition 6

Let $f:[0,1]\times[0,1]\rightarrow[0,1]$, it is a “fuzzy intersection” or
“t-norm” if the following properties hold:

  * 1.

(Commutative) $f(a,b)=f(b,a)$;

  * 2.

(Associative) $f(f(a,b),c)=f(a,f(b,c))$;

  * 3.

(Monotonicity) $a\leq a^{\prime}\Rightarrow f(a,b)\leq f(a^{\prime},b);\quad
b\leq b^{\prime}\Rightarrow f(a,b)\leq f(a,b^{\prime})$;

  * 4.

(Unity as a neutral element) $f(a,1)=a;\quad f(1,b)=b$.

We then write $afb$ to indicate $f(a,b)$. Some examples of fuzzy intersections
are:

  * 1.

Standard intersection $\cap$: $a\cap b=\min(a,b)$;

  * 2.

Algebraic product $\odot$: $a\odot b=ab$;

  * 3.

Limited difference $\odot_{d}$: $a\odot_{d}b=\max(0,a+b-1)$;

  * 4.

Drastic intersection $\Delta$: $a\Delta b=0$ if $(a<1$ and $b<1)$, $a\Delta
b=\min(a,b)$ otherwise.

The following theorems hold:

###### Theorem 1

For each $a,b\in[0,1]$, $a\Delta b\leq a\odot b\leq a\cap b$.

###### Theorem 2

If $\otimes$ is any fuzzy intersection $\implies$ $a\Delta b\leq a\odot b\leq
a\cap b$.

###### Theorem 3

The standard intersection is the only fuzzy intersection $\times$ which
satisfies the following property:

  * 1.

(Idempotence) $\forall a\in[0,1],a\odot a=a$.

####  4.4.2 The “fuzzy union” or “t-conorms”

###### Definition 7

Let $f:[0,1]\times[0,1]\rightarrow[0,1]$, it is a “fuzzy union” or “t-conorm”
if the following properties hold:

  * 1.

(Commutative) $f(a,b)=f(b,a)$;

  * 2.

(Associative) $f(f(a,b),c)=f(a,f(b,c))$;

  * 3.

(Monotonicity) $a\leq a^{\prime}\Rightarrow f(a,b)\leq f(a^{\prime},b);\quad
b\leq b^{\prime}\Rightarrow f(a,b)\leq f(a,b^{\prime})$;

  * 4.

(Zero as a neutral element) $f(a,0)=a;\quad f(0,b)=b$.

We then write $afb$ to indicate $f(a,b)$. Some examples of fuzzy unions are:

  * 1.

Standard union $\cup$: $a\cup b=\max(a,b)$;

  * 2.

Algebraic sum $\oplus_{s}$: $a\oplus_{s}b=a+b-ab$;

  * 3.

Limited sum $\oplus_{L}$: $a\oplus_{L}b=\min(1,a+b)$;

  * 4.

Drastic union $\Gamma$: $a\Gamma b=1$ if $(a>0$ and $b>0)$, $a\Gamma
b=\max(a,b)$ otherwise.

The following theorems hold:

###### Theorem 4

For each $a,b\in[0,1]$, $a\cup b\leq a\oplus_{s}b\leq a\oplus_{L}b\leq a\Gamma
b$.

###### Theorem 5

If $\oplus$ is any fuzzy union $\implies$ $a\cup b\leq a\oplus b\leq a\Gamma
b$.

###### Theorem 6

The standard union is the only fuzzy union $\oplus$ which satisfies the
following property:

  * 1.

(Idempotence) $\forall a\in[0,1],a\oplus a=a$.

###### Definition 8

A fuzzy intersection $\otimes$ and a fuzzy union $\oplus$ are said to be
associated if it holds: $a\otimes b+a\oplus b=a+b$ (Grassmann relation).

###### Theorem 7

The following fuzzy intersection and fuzzy union pairs are associated with
each other:

  * 1.

Standard Intersection and Standard Union;

  * 2.

Algebraic product and Algebraic sum;

  * 3.

Limited Difference and Limited Sum;

  * 4.

Drastic intersection and Drastic union.

####  4.4.3 Fuzzy n-air operations

Let $h:(v_{1},v_{2},\ldots,v_{n})\in[0,1]^{n}\rightarrow[0,1]$ be an $n$-ary
fuzzy operation. In order for it to represent an average of the values
$v_{1},v_{2},\ldots,v_{n}$, it is necessary to require some axioms:

  * 1.

(M1) (Idempotence)
$(v_{1}=v_{2}=\ldots=v_{n}=k\in[0,1])\rightarrow
h(v_{1},v_{2},\ldots,v_{n})=k$;

  * 2.

(M2) (Growth)
$(u_{i}\leq v_{i},i=1,2,\ldots,n)\rightarrow h(u_{1},u_{2},\ldots,u_{n})\leq
h(v_{1},v_{2},\ldots,v_{n})$;

  * 3.

(M3) (Commutativity)
If $(u_{1},u_{2},\ldots,u_{n})$ is a permutation of
$(v_{1},v_{2},\ldots,v_{n})$ then
$h(u_{1},u_{2},\ldots,u_{n})=h(v_{1},v_{2},\ldots,v_{n})$;

  * 4.

(M4) (Continuity)
$h$ is a continuous function on $[0,1]^{n}$.

###### Definition 9

Let $w=(w_{1},w_{2},\ldots,w_{n})$ be a vector formed by elements belonging to
the interval $[0,1]$ and such that $w_{1}+w_{2}+\ldots+w_{n}=1$. For each
tuple $(a_{1},a_{2},\ldots,a_{n})$ of elements of $[0,1]$, let
$(b_{1},b_{2},\ldots,b_{n})$ be the permutation of
$(a_{1},a_{2},\ldots,a_{n})$ satisfying the condition $b_{1}\geq
b_{2}\geq\ldots\geq b_{n}$. The ordered weighted average of
$a_{1},a_{2},\ldots,a_{n}$, weighted with the weight vector $w$ (OWA = Ordered
Weighted Average), is the number:

$m_{w}(a_{1},a_{2},\ldots,a_{n})=b_{1}w_{1}+b_{2}w_{2}+\ldots+b_{n}w_{n}$

The following theorems hold:

###### Theorem 8

$m_{w}(a_{1},a_{2},\ldots,a_{n})\in[0,1]$.

###### Theorem 9

For any weight vector $w=(w_{1},w_{2},\ldots,w_{n})$ that is non-negative and
such that $w_{1}+w_{2}+\ldots+w_{n}=1$, the operation
$h:(a_{1},a_{2},\ldots,a_{n})\in[0,1]^{n}\rightarrow
m_{w}(a_{1},a_{2},\ldots,a_{n})\in[0,1]$ satisfies properties (M1), (M2),
(M3), and (M4).

###### Theorem 10

Let $w=(w_{1},w_{2},\ldots,w_{n})$ and
$w^{\prime}=(w_{1},w_{2},\ldots,w_{i}+h,\ldots,w_{j}-h,\ldots,w_{n})$ with
$h>0$, $i<j$, two non-negative weight vectors such that
$w_{1}+w_{2}+\ldots+w_{n}=1$. Then for all
$(a_{1},a_{2},\ldots,a_{n})\in[0,1]^{n}$, $m_{w}(a_{1},a_{2},\ldots,a_{n})\leq
m_{w^{\prime}}(a_{1},a_{2},\ldots,a_{n})$.

###### Theorem 11

Let $w_{\text{max}}$ be the weight vector with the first component equal to 1
and all others zero, and let $w_{\text{min}}$ be the weight vector with the
last component equal to 1 and all others zero. For any weight vector
$w=(w_{1},w_{2},\ldots,w_{n})$ that is non-negative and such that
$w_{1}+w_{2}+\ldots+w_{n}=1$, it holds that for all
$(a_{1},a_{2},\ldots,a_{n})\in[0,1]^{n}$,

$m_{w_{\text{min}}}(a_{1},a_{2},\ldots,a_{n})\leq
m_{w}(a_{1},a_{2},\ldots,a_{n})\leq
m_{w_{\text{max}}}(a_{1},a_{2},\ldots,a_{n}).$

###### Definition 10

Let $\diamondsuit$ be a binary operation on $[0,1]$ which satisfies the
commutative, associative, and monotonic properties. The $n$-ary operation
generated by $\diamondsuit$ is the function:

$\diamondsuit_{n}:(v_{1},v_{2},\ldots,v_{n})\in[0,1]^{n}\rightarrow
v_{1}\diamondsuit v_{2}\diamondsuit\ldots\diamondsuit v_{n}\in[0,1].$

We observe that based on the commutative and associative properties,
$v_{1}\diamondsuit v_{2}\diamondsuit\ldots\diamondsuit v_{n}$ does not depend
on the order in which the operations are carried out or on any permutations of
the elements. Additionally, properties (M2) and (M3) apply. If $\diamondsuit$
is also idempotent (M1) and continuous (M4), then the $n$-ary operation
generated by $\diamondsuit$ is a weighted ordered average (OWA). This occurs
if there exists a weight vector $w=(w_{1},w_{2},\ldots,w_{n})$ such that:

$v_{1}\diamondsuit v_{2}\diamondsuit\ldots\diamondsuit
v_{n}=m_{w}(v_{1},v_{2},\ldots,v_{n}).$

###### Theorem 12

If $\diamondsuit$ is the standard intersection $\cap$, then
$\diamondsuit_{n}=v_{1}\cap v_{2}\cap\ldots\cap v_{n}$ is a weighted ordered
average associated with the weight $w_{\text{min}}=(0,0,\ldots,1)$, i.e.,
$m_{w_{\text{min}}}(v_{1},v_{2},\ldots,v_{n})$.

###### Theorem 13

If $\diamondsuit$ is the standard union $\cup$, then
$\diamondsuit_{n}=v_{1}\cup v_{2}\cup\ldots\cup v_{n}$ is a weighted ordered
average associated with the weight $w_{\text{max}}=(1,0,\ldots,0)$, i.e.,
$m_{w_{\text{max}}}(v_{1},v_{2},\ldots,v_{n})$.

An example of a weighted ordered average not generated by a binary operation
is the simple arithmetic average where
$m_{w}(v_{1},v_{2},\ldots,v_{n})=\frac{v_{1}+v_{2}+\ldots+v_{n}}{n}$, and
therefore $w=\left(\frac{1}{n},\frac{1}{n},\ldots,\frac{1}{n}\right)$.

In fact, if $\diamondsuit$ is the binary operation that associates the mean
$\frac{a+b}{2}$ to each ordered pair $(a,b)$ of elements of $[0,1]$, then the
commutative, monotonic, continuous, and idempotent properties hold, but the
associative property does not.

###  4.5 Ordering and metric space structures of the OWA operators

####  4.5.1 Sorting structure

Let $W$ be the set of vectors $w=(w_{1},w_{2},\ldots,w_{n})$ formed by $n$
elements belonging to the interval $[0,1]$ such that
$w_{1}+w_{2}+\ldots+w_{n}=1$. In $W$, an order $\leq$ can be defined for each
pair
$w=(w_{1},w_{2},\ldots,w_{n}),w^{\prime}=(w^{\prime}_{1},w^{\prime}_{2},\ldots,w^{\prime}_{n})$
as follows:

$w\leq w^{\prime}\Leftrightarrow(\forall i\in\\{1,2,\ldots,n-1\\},w_{i}\leq
w^{\prime}_{i})$ (19)

###### Theorem 14

The pair $(W,\leq)$ forms a lattice with at least $w_{\min}=(0,0,\ldots,1)$
and maximum $w_{\max}=(1,0,\ldots,0)$.

####  4.5.2 Structure of the metric space

Let $W$ be the set of vectors $w=(w_{1},w_{2},\ldots,w_{n})$ formed by $n$
elements belonging to the interval $[0,1]$ such that
$w_{1}+w_{2}+\ldots+w_{n}=1$. Let $k=(k_{1},k_{2},\ldots,k_{n})$ be a vector
satisfying $k_{1}>k_{2}>\ldots>k_{n}$, with $k_{1}=1$ and $k_{n}=0$. In $W$, a
distance (or metric) $d$, associated with $k$, can be defined for each
$w,w^{\prime}\in W$ as:

$d(w,w^{\prime})=k_{1}|w_{1}-w^{\prime}_{1}|+k_{2}|w_{2}-w^{\prime}_{2}|+\ldots+k_{n}|w_{n}-w^{\prime}_{n}|$
(20)

Some authors [43, 8] have proposed the expression

$k_{i}=\frac{n_{i}}{n-1}$ (21)

Thus, (20) can be rewritten as:

$d(w,w^{\prime})=\frac{\left[\sum_{i=1}^{n-1}(n-i)|w_{i}-w^{\prime}_{i}|\right]}{n-1}$
(22)

###### Theorem 15

Let $d$ be the distance defined by (20). The pair $(W,d)$ forms a metric
space. Furthermore,

  * 1.

(D1) $d(w,w^{\prime})\leq 1$;

  * 2.

(D2) $d(w,w^{\prime})=1\iff\\{w,w^{\prime}\\}=\\{w_{\min},w_{\max}\\}$;

  * 3.

(D3) For all $w\in W$, $d(w,w_{\min})+d(w,w_{\max})=1$.

According to Theorem 15, we can define a ”degree of proximity” $v(w,w^{*})$ of
a vector $w$ from a fixed vector $w^{*}$ by setting $v(w,w^{*})=1-d(w,w^{*})$.
In particular, we can consider:

  * 1.

The proximity $v(w,w_{\max})$ of a vector $w=(w_{1},w_{2},\ldots,w_{n})$ from
the vector $w_{\max}$, called orness (unionity). If we assume (21), it results
in:

$\text{orness}(w)=v(w,w_{\max})=\left[\sum_{i=1}^{n-1}\frac{(n-i)w_{i}}{n-1}\right]$
(23)

With simple calculations, we find:

$\text{orness}(w_{\max})=1,\quad\text{orness}(w_{\min})=0,\quad\text{orness}(w_{\text{med}})=0.5$

where $w_{\text{med}}$ is the vector having all components equal to $1/n$.

  * 2.

The proximity $v(w,w_{\min})$ of a vector $w=(w_{1},w_{2},\ldots,w_{n})$ from
the vector $w_{\min}$, called andness (intersection). From (D3), it follows
that:

$\text{andness}(w)=1-\text{orness}(w)$ (24)

If we assume (21), then it results in:

$\text{andness}(w)=1-\left[\sum_{i=1}^{n-1}\frac{(n-i)w_{i}}{n-1}\right]$ (25)

  * 3.

The mediality of a vector of weights $w$, defined as $v(w,w_{\text{med}})$,
where $w_{\text{med}}$ is the vector having all components equal to $1/n$.

####  4.5.3 FOWA: The Fuzzy Ordered Weighted Averaging Operator

Recently, Chen and Chen [13] formulated the FOWA operator to deal with
information expressed by fuzzy numbers.

###### Definition 11

[37] The fuzzy ordered weighted averaging operator, denoted by FOWA, is the
mapping
$\text{FOWA}:\widetilde{\mathscr{P}}(\mathbb{R}^{+})^{n}\rightarrow\widetilde{\mathscr{P}}(\mathbb{R}^{+})$,
which has an associated weighting vector $(w_{1},\ldots,w_{n})$, such that

$\text{FOWA}(\tilde{x}_{1},\ldots,\tilde{x}_{n})=\bigoplus_{i\in[n]}(w_{i}\odot\tilde{x}_{(i)})$

where the operations $\oplus$ and $\odot$ are a t-conorm and a t-norm
respectively, and $(\cdot)$ denotes a permutation on $[n]$ such that
$\tilde{x}_{(1)}\leq\ldots\leq\tilde{x}_{(n)}$.

It is evident that the FOWA operator extends the OWA operator. Particularly,
for a weight vector $w$, we have:

  * 1.

If $\mathbf{w}=(1,0,\ldots,0)$, then FOWA $=\tilde{x}_{(1)}$ (fuzzy minimum).

  * 2.

If $\mathbf{w}=(0,\ldots,0,1)$, then FOWA $=\tilde{x}_{(n)}$ (fuzzy maximum).

  * 3.

If $w_{i}=\frac{1}{n}$ for all $i$, then FOWA $=$ FAA (fuzzy arithmetic
averaging).

If FOWA concerns triangular fuzzy numbers, it is simply referred to as FTOWA.

##  5 Fuzzy Graphs: Generalization of Weighted Graphs

In this work, we propose the use of Fuzzy Graphs (FG) as a generalization of
Weighted Graphs (WG). A WG is one in which edges are associated with weights.
A WG can be represented as $G(V,E,W)$, where $W$ represents the weights
associated with each edge ($|W|=|E|$). For an adjacency matrix representation,
instead of $0$ and $1$, we can use the weight associated with the edge. This
saves space by combining $E$ and $W$ into one adjacency matrix $A$, assuming
an edge exists between $v_{i}$ and $v_{j}$ if and only if $W_{i,j}\neq 0$.

In the literature, there are different definitions and proposals for the
concept of fuzzy graph [see e.g. 49]. Here, we focus on graphs where only the
edges are fuzzy. Thus, FGs are considered as a generalization of WG, where
edges are associated with fuzzy numbers.

###### Definition 12

An undirected fuzzy social network is defined as a fuzzy relational structure
$\widetilde{G}_{un}=(V,E,\widetilde{A}_{un})$, where
$V=\\{v_{1},v_{2},\ldots,v_{n}\\}$ is a non-empty set of actors or nodes, and

$\widetilde{A}_{un}=\left(\begin{array}[]{ccc}\widetilde{A_{11}}&\cdots&\widetilde{A_{1n}}\\\
\vdots&\ddots&\vdots\\\
\widetilde{A_{n1}}&\cdots&\widetilde{A_{nn}}\end{array}\right)$

is an undirected fuzzy relation on $V$.

Many fuzzy relations are directional. A fuzzy relation is directional if the
ties are oriented from one actor to another. Thus, Hu et al. [20] defined the
directed fuzzy social network (DFSN) as follows:

###### Definition 13

A directed fuzzy social network is defined as a fuzzy relational structure
$\widetilde{G}_{dir}=(V,E,\widetilde{A}_{dir})$, where
$V=\\{v_{1},v_{2},\ldots,v_{n}\\}$ is a non-empty set of actors or nodes, and
$\widetilde{A}_{dir}=\left(\begin{array}[]{ccc}\widetilde{A_{11}}&\cdots&\widetilde{A_{1n}}\\\
\vdots&\ddots&\vdots\\\
\widetilde{A_{n1}}&\cdots&\widetilde{A_{nn}}\end{array}\right)$ is an
undirected fuzzy relation on $V$.

Fuzzy Social Network (FSN) includes Undirected Fuzzy Social Network (UFSN) and
Directed Fuzzy Social Network (DFSN). The significant difference between DFSN
and UFSN is that a directed fuzzy relation is considered. According to
Definition 12, $\widetilde{A}_{ij}$ is equal to $\widetilde{A_{ji}}$ in an
undirected fuzzy social network. However, $\widetilde{A}_{ij}$ is not always
equal to $\widetilde{A}_{ji}$ in a directed fuzzy social network. In
Definition 13, $\widetilde{A}_{dir}$ is called the directed fuzzy adjacency
matrix of $\widetilde{G}_{dir}$:

$\mu_{\widetilde{A}_{dir}}=\left(\begin{array}[]{ccc}\mu\left(\widetilde{A_{11}}\right)&\cdots&\mu\left(\widetilde{A_{1n}}\right)\\\
\vdots&\ddots&\vdots\\\
\mu\left(\widetilde{A_{n1}}\right)&\cdots&\mu\left(\widetilde{A_{nn}}\right)\end{array}\right)$

where, in this case, we have an asymmetrical membership function. The related
concepts of UFSNs can also be used in DFSNs, but the direction of the fuzzy
relation between actors must be considered.

###### Definition 14

Assume that $v_{0}A_{1}v_{1}A_{2}v_{2}\ldots A_{k}v_{k}$ is a path from
$v_{0}$ to $v_{k}$ in $\widetilde{G}_{dir}$, then
$\widetilde{s}_{d}(\widetilde{\omega})=\bigwedge_{i=1}^{k}\mu\left(A_{i}\right)$
is called the directed fuzzy intensity of path $\widetilde{\omega}$.

In Definition 2.3, $\mu(e_{i})$ is a membership function.

###### Definition 15

If $\bar{\omega}_{k}$ are $n$ paths from $u$ to $v$ in $\widetilde{G}_{dir}$,
then
$\widetilde{s}_{d}(u,v)=\bigvee_{k=1}^{n}\widetilde{s}_{d}(\widetilde{\omega}_{k})$
is called the directed fuzzy connected intensity from $u$ to $v$.

Here, $k=1,2,\ldots,n$. If there is no path from $u$ to $v$, then
$\widetilde{s}_{d}(u,v)=0$. If $u=v$, then $\widetilde{s}_{d}(u,v)=1$. In
UFSN, $\bar{s}(u,v)$ always equals $\widetilde{s}(v,u)$. However,
$\widetilde{s}_{d}(u,v)$ is not always equal to $\widetilde{s}_{d}(v,u)$ in
DFSN.

###### Definition 16

Assume $G_{dn}=(V,E_{dn})$ is a DFSN,

$\widetilde{C}_{d}=\left(\begin{array}[]{ccc}\widetilde{s}\left(v_{1},v_{1}\right)&\cdots&\widetilde{s}\left(v_{1},v_{n}\right)\\\
\vdots&\ddots&\vdots\\\
\widetilde{s}\left(v_{n},v_{1}\right)&\cdots&\widetilde{s}\left(v_{n},v_{n}\right)\end{array}\right)$

then $\widetilde{C}_{d}$ is called the directed fuzzy connected intensity
matrix of $\widetilde{G}_{dn}$.

The directed fuzzy connected intensity matrix is a crucial concept in DFSN.
With it, we can determine the directed fuzzy connected intensity between any
two actors in DFSN, indicating the relationship (direct or indirect) between
them.

Hereafter, we will focus only on triangular fuzzy numbers using a simple
notation like the following:

$\tilde{A}=(a_{l},a_{m},a_{r}).$

An example of FG can be done to represent relations in a social network like
Facebook or in a University Department, where the nodes are the department
members, and the edges are the relations between them assessed according to a
questionnaire. If we collect information about the relationship between pairs
of Professors and researchers according to how much they feel they have
relations with another member (for example, not at all, a little, a lot,
etc.), summarizing this information with a scalar, as we do in the WG, would
lead to a loss of information. Indeed, when dealing with this type of opinion,
mainly when it is turned off through a human language attribute, the data
always has a degree of inaccuracy. In such a context, it is natural to
consider that the FG is direct because the opinion of member $A$ about his
relationship with member $B$ may differ from that of member B about his
relationship with member $A$. For this reason, we consider only the case of
direct graphs. We will have the symmetrical FG be only a particular case of
the asymmetrical one if all couples give the same opinion. This latter case
would be improbable since we deal with personal opinions.

Assuming that through a fuzzy inference procedure, we have translated the
qualitative answers of the department members into fuzzy numbers, we could
have a situation like the one illustrated in Figure 2.

Figure 2: Example of fuzzy social network representing relationship in a
University Department. The edges are represented via triangular fuzzy numbers.
For simplicity, we do not plot the fuzzy number when there is a null
relationship.

##  6 Fuzzy Centrality Indexes in a Fuzzy Social Network

Centrality indices quantify the importance of a node in a given network, which
is often identified with the importance of the corresponding entity in the
complex system modelled by the network. In the following sections, we extend
the classical centrality indexes to the case of a network where the edges are
represented by triangular fuzzy numbers. In other words, we define fuzzy
degree centrality, fuzzy betweenness centrality, fuzzy closeness centrality,
and fuzzy eigenvector centrality.

###  6.1 Fuzzy degree centrality

In real-world interactions, we often consider people with many connections to
be important. Degree centrality transfers the same idea into a measure. The
degree centrality measure ranks nodes with more connections higher in terms of
centrality. In the classical framework, the degree centrality $C_{d}(v_{i})$,
in directed graphs can be computed in different ways according to the number
of in-connections and out-connections, leading to the in-degree index
$d_{i}^{in}$ and the out-degree index $d_{i}^{out}$, respectively, or the
combination of both. However, in the context of (directed) fuzzy social
networks, the edges are represented by fuzzy numbers, and thus they are not
scalars. Therefore, in the next Sections, we extend the concepts of in-degree
centrality and out-degree centrality to the case of fuzzy numbers.

The first step is introducing the concepts of fuzzy triangular ordered
weighted arithmetic operator [37]. The Fuzzy Triangular Ordered Weighted
Arithmetic Operator (FTOWA) can be defined as follows.

###### Definition 17

The fuzzy triangular ordered weighted arithmetic operator, denoted by FTOWA,
is the mapping FTOWA:
$\widetilde{\mathscr{P}}([0,1])^{n}\rightarrow\widetilde{\mathscr{P}}([0,1])$,
such that

$\operatorname{FTOWA}\left(\tilde{x}_{1},\ldots,\tilde{x}_{n}\right)=\bigoplus_{i\in[n]}\left(w_{i}\odot\tilde{x}_{(i)}\right)$

where $\oplus$ is a t-conorm, $\odot$ is a t-norm,
$\left(w_{1},\ldots,w_{n}\right)$ is a weighting vector associated with
$\oplus$, and $(\cdot)$ is a permutation on $[n]$ such that
$\tilde{x}_{(1)}\leq\cdots\leq\tilde{x}_{(n)}$

####  6.1.1 Fuzzy in-degree centrality index

When using in-degrees, fuzzy degree centrality measures how popular a node is
and its value shows prominence or prestige.

###### Definition 18

Assume that $\widetilde{G}_{dir}=\left(V,E,\widetilde{A}_{dir}\right)$ is a
DFSN, $\widetilde{D}_{in}\left(v_{i}\right)$ is the sum of the fuzzy relations
that are adjacent to $v_{i}$, then $\widetilde{D}_{in}\left(v_{i}\right)$ is
called fuzzy in-degree centrality of $v_{i}$.

For a given node $i$, let $\widetilde{O}_{j}^{in}$ be the set of fuzzy numbers
$\widetilde{A}_{ji}$ ranked in decreasing order, indicating the relationship
between each node $j$ and the node $i$ (directed), the formula of fuzzy in-
degree centrality of $v_{i}$ is given by

$\widetilde{D}_{in}\left(v_{i}\right)=\bigoplus_{j\in[n],j\neq
i}\left(w_{j}\odot\tilde{O}_{j}^{in}\right).$ (26)

To rank the fuzzy numbers, we can use one of the approaches presented in
Section 2.12. In the following, we will focus on the centre of gravity
approach to understand the ranking of fuzzy numbers.

It follows that we can obtain the following extreme cases:

  * 1.

$w_{j}=\\{1,0,...,0\\}$ $\implies\widetilde{D}_{in}$ is determined by the
greatest $\widetilde{A}_{ji}$;

  * 2.

$w_{j}=\\{0,0,...,1\\}$ $\implies\widetilde{D}_{in}$ is determined by the
lowest $\widetilde{A}_{ji}$;

  * 3.

$w_{j}=\\{1/n,1/n,...,1/n\\}$ $\implies\widetilde{D}_{in}$ is determined by
all the $\widetilde{A}_{ji}$ with the same weights. Therefore, we give the
same importance to low and high fuzzy relationships in computing fuzzy in-
degree centrality.

The index just illustrated in Equation 26 does not allow us to compare the
centrality values between different networks. To overcome this drawback, we
can normalize the index so that it becomes a pure number between 0 and 1. The
reasoning behind the normalization is that we calculate the maximum that the
in-degree centrality index can take. The maximum of this index would, of
course, be obtained if an actor is perfectly connected with all the other
nodes. For this aim, the edge represented by the fuzzy number should have the
characteristic of possessing a core with the maximum possible value on the
support of the fuzzy number (with zero fuzziness). If the latter happens for
all the actor’s edges, then we can compute the maximum of Equation 26 as
follows.

The easiest way to find this relative index is to normalize the fuzzy numbers.
In fact, unlike the classical case in which the number of edges is necessary
to normalize the in-degree centrality, in this case, all the nodes are
connected and therefore, what matters is the intensity of the edges. Once the
fuzzy numbers have been normalized, we already have a relative index as
follows:

$\widetilde{D}^{in*}\left(v_{i}\right)=\bigoplus_{j\in[n],j\neq
i}\left(w_{j}\odot\widetilde{O}_{j}^{in*}\right).$ (27)

where $\widetilde{O}_{j}^{in*}$ is the order set of fuzzy numbers with
normalized support in $[0,1]$.

####  6.1.2 Fuzzy out-degree centrality index

When using out-degrees, it measures the gregariousness of a node.

###### Definition 19

Assume that $\widetilde{G}_{dir}=\left(V,E,\widetilde{A}_{dir}\right)$ is a
DFSN, $\widetilde{D}^{out}\left(v_{i}\right)$ is the sum of the fuzzy
relations that are adjacent from $v_{i}$, then
$\widetilde{O}_{j}^{out}\left(v_{i}\right)$ is called fuzzy out-degree
centrality of $v_{i}$

For a given node $i$, let $\widetilde{O}_{j}^{out}$ be the set of fuzzy
numbers $\widetilde{A}_{ij}$ ranked in decreasing order, indicating the
relationship between each node $i$ and the node $j$ (directed), the formula of
fuzzy out-degree centrality of $v_{i}$ is given by

$\widetilde{D}^{out}\left(v_{i}\right)=\bigoplus_{j\in[n],j\neq
i}\left(w_{j}\odot\tilde{O}_{j}^{out}\right).$ (28)

According to the vector $w_{j}$, we can obtain the following cases:

  * 1.

$w_{j}=\\{1,0,...,0\\}$ $\implies\widetilde{D}_{out}$ is determined by the
greatest $\tilde{A}_{ij}$;

  * 2.

$w_{j}=\\{0,0,...,1\\}$ $\implies\widetilde{D}_{out}$ is determined by the
lowest $\tilde{A}_{ij}$;

  * 3.

$w_{j}=\\{1/n,1/n,...,1/n\\}$ $\implies\widetilde{D}_{out}$ is determined by
all the $\tilde{A}_{ij}$ with the same weights. Therefore, we give the same
importance to low and high fuzzy relationships in computing fuzzy out-degree
centrality.

Naturally, a possible solution can also be to adopt a vector of decreasing
weights so that the importance of fuzzy numbers decreases as the relationship
is less important.

As for the in-degree centrality index, also in this case, we can define a
relative out-degree centrality index as follows:

$\widetilde{D}^{out*}\left(v_{i}\right)=\bigoplus_{j\in[n],j\neq
i}\left(w_{j}\odot\tilde{O}_{j}^{out*}\right).$ (29)

####  6.1.3 Total Fuzzy degree centrality index

In DFSN applications, the previous degrees can be of great interest. The fuzzy
out-degree centralities are measures of expansiveness and the fuzzy in-degree
centralities are measures of receptivity, or popularity.

However, we can consider a formula for the total degree centrality of $v_{i}$
given by:

$\widetilde{D}\left(v_{i}\right)=\widetilde{D}^{out}\left(v_{i}\right)\oplus\widetilde{D}^{in}\left(v_{i}\right)$
(30)

###  6.2 Fuzzy betweenness centrality

Another way of looking at centrality is by considering how important nodes are
in connecting other nodes. In the classical setting, for a node $v_{i}$, we
can compute the relative frequency of the shortest paths to connect other
nodes that pass through $v_{i}$.

In the classic case, the search for the shortest path consists of finding the
path from one node to another node that is as fast as possible, crossing the
least number of intermediate nodes. In our case, however, the nodes of the
social network are all connected to each other with a different degree of
truth; consequently, the shortest path, in a case where all the fuzzy edges
would be equal, would always be the directed path from one node to another
node. But usually, in reality, the connections are not all the same.

At this point in our setting, we are forced to reinterpret the concept of the
shortest path as follows: In a context of fuzzy social network analysis, the
importance of a node in connecting the other nodes must be evaluated based on
the quality of the existing fuzzy edges: one path must be considered
privileged over another if it is more effective in connecting two elements.

In our context, a path can be considered more effective than another path if
it is able to connect two nodes, ensuring a higher connection strength.
Clearly, it may happen that a longer path in terms of intermediate nodes to
cross is better in terms of the strength of the edges and, therefore, more
effective in connecting two elements $i$ to $j$. Therefore, in a network where
everyone is (“more” or “less”) connected, an indirect path to go from node $i$
to $j$ could be better than the directed path from $i$ to $j$ (for example $i$
and $j$ know each other little directly but are well-connected through another
subject).

###### Definition 20

Given two different paths $P_{i\rightarrow j}^{(1)}$ and $P_{i\rightarrow
j}^{(2)}$, of possible different lengths $k={1,2,...,K}$ and $t={1,2,...T}$,
respectively, we say that $P_{i\rightarrow j}^{(1)}$ is better than
$P_{i\rightarrow j}^{(2)}$ when:

$P_{i\rightarrow j}^{(1)}\succ P_{i\rightarrow
j}^{(2)}\iff\bigoplus_{k\in[1,K]}\left(w_{k}\odot\tilde{N}_{k}^{(1)}\right)>\bigoplus_{t\in[1,T]}\left(w_{t}\odot\tilde{N}_{t}^{(2)}\right)$
(31)

where $\widetilde{N}_{k}^{(1)}$ is the set of $K$ fuzzy edges in the first
path between $i$ and $j$ with normalized support in $[0,1]$ and ranked in
decreasing order, $\widetilde{N}_{t}^{(2)}$ is the set of fuzzy numbers in the
second path between $i$ and $j$ also with normalized support in $[0,1]$ and
ranked in decreasing order, and $w_{k}$ and $w_{t}$ is a vector of weights. We
define $P_{i\rightarrow j}^{(1)}$ as the best path from $i$ to $j$ when for
any other path $P_{i\rightarrow j}^{(2)}$ from $i$ to $j$ we have that
$P_{i\rightarrow j}^{(1)}\succ P_{i\rightarrow j}^{(2)}$. Equation 31 allows
us to consider different possibilities for selecting the so-called best path,
e.g. average fuzzy number, minimum, maximum, etc. From now on, we will use the
minimum by introducing $w_{k}=\\{0,0,...,1\\}$ and $w_{t}=\\{0,0,...,1\\}$ as
the vectors of weights. In other words, when looking at two possible
alternative paths to reach $j$ starting from the node $i$, we examine all the
fuzzy edges in those paths, and we select the best one with the higher minimum
fuzzy number (edge) in the path.

An example of this reasoning is illustrated in Figure 3. The best path for
going from Dr. Perry to Dr. Porreca is via Prof. Smith. In fact, if we
consider the direct connection, we have a very weak relationship, while if we
connect the two nodes indirectly, we have two very strong relationships. In
other words, if the direct relationship is weak, it is better to take more
steps characterized by strong relationships. In the case of the example shown,
the same applies if Dr. Porreca wants to get in touch with Perry (reverse
path). This graphic example helps to better understand that in our context
where the links are blurred, the concept of the shortest path is replaced by
the concept of the best and most effective path to connect two distinct nodes.
So we can say that the concept of betweenness centrality we have in the
classic social network, that is, that an actor is highly central when he is in
the middle of many shorter paths to go from a generic $j$ node to a generic
$i$ node, in this context, it is replaced by the following concept: a node $z$
has high centrality if, considering all the best paths to go from a generic
node $j$ to a generic node $i$, $z$ is frequently in the middle of these
paths.

Figure 3: Example of fuzzy social network where the directed path between two
nodes is worse than an indirected path in terms of fuzzy numbers edges. In
other words, if the direct relationship is weak it is better to take more
steps characterized by strong relationships.

Having identified the concept of the best path, we can therefore formalize the
betweenness centrality of node $i$ as the ratio between all the best paths to
join two nodes $s$ and $t$ via $i$ (obviously they are indirect paths), and
all the best paths between all the pairs of nodes $s$ and $t$ (direct or
indirect paths via $i$ or not). Formally, we can write as follows:

$B\left(v_{i}\right)=\sum_{s\neq t\neq v_{i}}\frac{P_{st}(v_{i})}{P_{st}}$
(32)

where $P_{st}(v_{i})$ indicate the number of “best” paths between $s$ and $t$
that passes thought $i$, and $P_{st}$ the total number of “best” paths between
generic $s$ and $t$.

The search of the so-called “best” paths is done according to the minimum
fuzzy number edge occurring in the path between $s$ and $t$, even if it is
indirect. Clearly, to compare the multiple different paths that occur in a
social network, we have to rank them according to the mentioned criteria to
order fuzzy numbers, e.g. the center of gravity approach.

###  6.3 Fuzzy closeness centrality

In classical closeness centrality, the intuition is that the more central
nodes are, the more quickly they can reach other nodes. Formally, these nodes
should have a smaller average shortest path length to other nodes. The smaller
the average shortest path length, the higher the centrality for the node.

As we have seen for fuzzy betweenness centrality, also in this case we refer
to the concept of “shortest” path as “best” path, i.e. to reach $j$ starting
from $i$, one path could better than another one despite the number of
intermediate nodes is higher when the relationship between them is strong.

####  6.3.1 Fuzzy in-closeness centrality

Let the average (fuzzy number) of the best paths from node $i$ to all the
nodes $j$ be represented by:

$\widetilde{M}_{v_{i},s}=\bigoplus_{v_{j}\neq v_{i}\in
V}\left(w_{j}\odot\widetilde{A}_{ij}\right)$ (33)

where $\widetilde{A}_{ij}$ is the set of the best fuzzy edges to connect $i$
and $j$ ranked in decreasing order, s indicates the maximum number of steps
considered evaluating the best path, and $w_{j}$ a vector of weights of the
FOWA operator. Clearly, being a weighted average of fuzzy numbers,
$\widetilde{M}_{v_{i},s}$ is also a fuzzy number.

Hence, we can define the fuzzy out-closeness centrality index as follows:

$\widetilde{C}^{in}_{v_{i},s}=\widetilde{M}_{v_{i},s}$ (34)

Thus, differently from Equation 32 that is a relative frequency,
$\widetilde{C}_{v_{i},s}$ is a centrality index that incorporates the
fuzziness of the edges and thus remains fuzzy.

Another possible approach to define fuzzy in-closeness centrality, in a way
that is more close to the classical index, would be to consider:

$\widetilde{C^{(2)}}^{in}_{v_{i},s}=\frac{1}{\widetilde{M}_{v_{i},s}}.$ (35)

However, in the following we refer to Equation 34 because our basic idea is to
look for an immediate representation of the in-closeness concept rather than
to identically reproduce the classical indexes using fuzzy numbers. In fact,
Equation 34 is quite interpretable, because provides a measure of how a node
is central in considering the fuzzy incoming edges. However, Equation 34,
differently from 26 takes into account the concept of the best path and
maximum number of steps to reach another node.

####  6.3.2 Fuzzy out-closeness centrality

Following the same reasoning, we can define the fuzzy in-closeness centrality
index of the node $v_{i}$ as follows:

$\widetilde{C}^{out}_{v_{i},s}=\widetilde{Z}_{v_{i},s}$ (36)

where $\widetilde{Z}_{v_{i},s}$ is the average (fuzzy number) of the best
paths from nodes $j$ to $j$ in decreasing order. Thus,

$\widetilde{Z}_{v_{i},s}=\bigoplus_{v_{j}\neq v_{i}\in
V}\left(w_{j}\odot\widetilde{A}_{ji}\right)$ (37)

is computed by applying the FOWA operator to fuzzy numbers representing the
edges from the generic nodes $j$ and the node $i$, which we consider
centrality.

As before, the vector of weights can be of different types. From now on, we
will compute the average fuzzy number $\widetilde{M}_{v_{i},s}$ or
$\widetilde{Z}_{v_{i},s}$ using $w=\\{1/n,...,1/n\\}$. However, we stress that
to select the best paths $P_{i\rightarrow j}$ or $P_{j\rightarrow i}$ will
always consider $w=\\{0,...,1\\}$. Naturally, we could also consider different
combinations of the weights according to the different meanings we aim to give
to our application. In other words, we apply the FOWA operator twice when
computing betweenness centrality indexes.

As for the fuzzy in-closeness centrality index, also in this case we can
consider an alternative measure of the fuzzy out-closeness centrality index as
follows:

$\widetilde{C^{(2)}}^{out}_{v_{i},s}=\frac{1}{\widetilde{Z}_{v_{i},s}}$ (38)

In this case, we get an index more similar to the classical non-fuzzy in-
closeness centrality index. The interpretation is contrary to 36; the lower
the index, the higher the fuzzy in-closeness centrality index. Therefore, in
the following, we focus on Equation 36 because the interpretation is quite
immediate; indeed, the higher the index in Equation 36, the higher the
strength of the fuzzy outcoming edges of a node (given the number of steps to
consider the best path), the higher the fuzzy out-closeness centrality of that
node.

####  6.3.3 Total fuzzy closeness centrality

As we have seen for fuzzy degree centrality, we can also sum up the two
different components of closeness centrality in this case. Therefore, we can
define the total fuzzy closeness centrality index as follows:

$\widetilde{C}^{tot}_{v_{i},s}=_{s}\tilde{C}^{in}_{v_{i},s}+_{s}\tilde{C}^{out}_{v_{i},s}$
(39)

##  7 An application of fuzzy social network analysis to collaboration
networks in University Departments

Collaboration is a complex and multifaceted concept that cannot be evaluated
using a single metric. Attempting to measure it directly takes a lot of work.
One possible approach is to use proxies, such as counting the number of shared
papers or similar information, but this method still needs to fully capture
the essence of collaboration. Scientific collaboration is a broad concept
involving exchanging knowledge and ideas between researchers. It includes
sharing research procedures and ideas, which ultimately lead to the production
of scientific knowledge. However, this definition alone is not enough to help
us empirically measure a collaboration network of this type.

For this reason, this study starts from the basic idea of collecting data
using the intuition of previous research in a different context. Notably,
Calcagnì and Lombardi [10] proposed a new method to collect qualitative data
using fuzzy set theory: the so-called Dynamic Fuzzy Rating Tracker (DYFRAT).
The authors developed an applet to collect directly triangular fuzzy numbers
data without the fuzzy inference process [10]. Their work used mouse movements
to model human rating evaluations from a fuzzy-set perspective, capturing the
imprecision in point answers. In particular, DYFRAT captures the fuzziness of
human ratings by modelling some real-time biometric events that occur during
the cognitive rating process the noisy and dynamic x-y trajectory is initially
mapped into a set of polar objects (distances and angles). Next, a linear
histogram model of angles of movements is built, which stores the most
important spatial features of the computer mouse trajectory, such as location,
directions, and amplitudes. Finally, the histogram model is used to determine
a quantification of spatial events involved in the original movement path.
DYFRACT is not available, and thus, in this paper, following the intuition by
[10], we built an interactive fuzzy questionnaire to collect fuzzy numbers
data to quantify and measure the collaboration relations between professors in
the Economic Studies Department of the University “G. d’Annunzio” in Italy.

The software implemented to collect fuzzy data is based on the LAMP platform,
i.e., Linux (operating system) + Apache (webserver) + MySql (database
management system) + PHP (server-side programming language). For the
management and use of the graphical interface was used Bootstrap (free
framework initially created by Twitter to standardize graphical components,
based in turn on HTML \+ CSS \+ JS) with its many modules, while also
retrieving the position of the mouse, were used: JS (JavaScript, object-
oriented programming language and client-side events) + jQuery (development
library for client-side web applications based on JS).

Figure 4 shows an example of a single questionnaire answer inserted on a
semicircumference. Each user is called to answer the same question with
respect to her/his collaboration relay with all the other members of the
department. This pseudo-circular scale is justified by its ability to provide
greater degrees of freedom for mouse movement recordings than traditional
linear or arc-type scales. All the information related to the questions is
saved directly on Database. At the same time, the users’ answers are stored on
DB in JSON format (JS-based information interchange language).

Figure 4: Fuzzy questionnaire to collect mouse movements of respondents.

The distribution of mouse movements within the semicircumference, through the
tracking procedure, allowed us to grasp the uncertainty in the answer before
clicking on the final answer. This allowed us to build triangular
distributions and reconstruct a triangular fuzzy number with its vagueness,
support, core and membership function. In other words, we were able to
establish, for each professor, a relationship with all the other professors
based on their subjective view of the relationship with their colleagues. This
approach assumes the possibility that the relationship is asymmetric. Prof A
might overestimate a collaboration with Prof. B, who, on the other hand,
believes that the collaboration is of minor or significant importance. Each
professor answered the question for every possible department colleague,
leading us to the creation of the entire departmental network.

Figure 5 and Table 1 present the results of the fuzzy in-degree centrality
indexes of the actors involved in the Department of Economic Studies of the
University of Chieti-Pescara, Italy. The most central professors are 29, 44,
and 21. Clearly, for privacy reasons, we do not disclose the names of the
department members. The results can be evaluated both graphically and by
observing the centre of gravity. Naturally, we could also consider other
ranking methods among those available in the literature. Figure 6 and Table 2
represent the results of the fuzzy out-degree centrality indexes of the actors
involved in the Department of Economic Studies of the University of Chieti-
Pescara, Italy. The most central professors are 7, 38, and 35. It’s
interesting to note how we get very different values if comparing in- and out-
degree centrality indexes.

Figure 5: Fuzzy in-degree centrality indexed of the actors in the collaborative network of the Department of Economic Studies of the University of Chieti-Pescara, Italy.  Prof | Left | Core | Right | CoG
---|---|---|---|---
29 | 0.09 | 0.11 | 0.13 | 0.11
44 | 0.08 | 0.10 | 0.12 | 0.10
21 | 0.07 | 0.10 | 0.11 | 0.09
26 | 0.07 | 0.08 | 0.10 | 0.08
28 | 0.06 | 0.08 | 0.09 | 0.08
37 | 0.06 | 0.08 | 0.09 | 0.08
38 | 0.07 | 0.08 | 0.08 | 0.08
45 | 0.06 | 0.08 | 0.09 | 0.08
46 | 0.07 | 0.08 | 0.09 | 0.08
47 | 0.06 | 0.08 | 0.10 | 0.08
48 | 0.07 | 0.08 | 0.09 | 0.08
17 | 0.05 | 0.07 | 0.09 | 0.07
24 | 0.05 | 0.07 | 0.08 | 0.07
30 | 0.06 | 0.08 | 0.08 | 0.07
2 | 0.04 | 0.06 | 0.07 | 0.06
3 | 0.04 | 0.06 | 0.07 | 0.06
16 | 0.05 | 0.06 | 0.08 | 0.06
20 | 0.06 | 0.06 | 0.07 | 0.06
4 | 0.03 | 0.05 | 0.06 | 0.05
7 | 0.04 | 0.06 | 0.06 | 0.05
Table 1: First twenty Professors according to normalized Fuzzy in-degree centrality indexes in the collaborative network of the Department of Economic Studies of the University of Chieti-Pescara, Italy. Figure 6: Fuzzy out-degree centrality indexed of the actors in the collaborative network of the Department of Economic Studies of the University of Chieti-Pescara, Italy.  Prof | Left | Core | Right | CoG
---|---|---|---|---
7 | 0.36 | 0.42 | 0.47 | 0.42
38 | 0.31 | 0.41 | 0.48 | 0.40
35 | 0.20 | 0.33 | 0.42 | 0.31
9 | 0.15 | 0.24 | 0.28 | 0.22
11 | 0.16 | 0.18 | 0.27 | 0.21
27 | 0.15 | 0.19 | 0.21 | 0.18
32 | 0.10 | 0.10 | 0.12 | 0.11
46 | 0.10 | 0.11 | 0.12 | 0.11
21 | 0.05 | 0.08 | 0.09 | 0.07
29 | 0.04 | 0.06 | 0.06 | 0.06
43 | 0.05 | 0.05 | 0.08 | 0.06
44 | 0.05 | 0.05 | 0.08 | 0.06
53 | 0.04 | 0.06 | 0.07 | 0.06
22 | 0.04 | 0.05 | 0.05 | 0.05
8 | 0.03 | 0.05 | 0.05 | 0.04
12 | 0.03 | 0.03 | 0.04 | 0.04
18 | 0.03 | 0.04 | 0.05 | 0.04
26 | 0.03 | 0.04 | 0.04 | 0.04
17 | 0.03 | 0.03 | 0.03 | 0.03
37 | 0.02 | 0.03 | 0.03 | 0.03
Table 2: First twenty Professors according to normalized Fuzzy out-degree
centrality indexes in the collaborative network of the Department of Economic
Studies of the University of Chieti-Pescara, Italy.

Figure 7 presents the findings of the fuzzy betweenness centrality indexes of
the nodes of the Department of Economic Studies of the University of Chieti-
Pescara, Italy. The most fundamental professors are 38, 35, and 7. In other
words, professor 38 is often on the best path for many couple of professors
because, for them, it’s better to reach other colleagues passing via professor
38.

Figure 7: Fuzzy betweenness centrality indexed of the actors in the
collaborative network of the Department of Economic Studies of the University
of Chieti-Pescara, Italy.

Figure 8, Table 3, Figure 9, Table 4, illustrate the results of the fuzzy in-
closeness and out-closeness centrality indexes, respectively. The most
important nodes are 7, 35, 38, and 29, 44, 21, respectively. In summary,
professor 29 is the most powerful and approachable.

Figure 8: Fuzzy in-closeness centrality indexed of the actors in the collaborative network of the Department of Economic Studies of the University of Chieti-Pescara, Italy.  Prof | Left | Core | Right | CoG
---|---|---|---|---
7 | 0.0115 | 0.0131 | 0.0143 | 0.0130
35 | 0.0094 | 0.0128 | 0.0160 | 0.0127
38 | 0.0093 | 0.0131 | 0.0149 | 0.0124
22 | 0.0104 | 0.0113 | 0.0120 | 0.0112
46 | 0.0099 | 0.0110 | 0.0125 | 0.0111
32 | 0.0101 | 0.0108 | 0.0119 | 0.0109
27 | 0.0080 | 0.0110 | 0.0124 | 0.0105
11 | 0.0073 | 0.0081 | 0.0114 | 0.0089
9 | 0.0067 | 0.0087 | 0.0102 | 0.0085
17 | 0.0056 | 0.0067 | 0.0076 | 0.0066
12 | 0.0034 | 0.0042 | 0.0053 | 0.0043
21 | 0.0023 | 0.0032 | 0.0040 | 0.0032
37 | 0.0022 | 0.0024 | 0.0027 | 0.0024
8 | 0.0017 | 0.0024 | 0.0026 | 0.0022
29 | 0.0016 | 0.0023 | 0.0025 | 0.0022
44 | 0.0017 | 0.0019 | 0.0027 | 0.0021
53 | 0.0016 | 0.0021 | 0.0025 | 0.0021
49 | 0.0013 | 0.0017 | 0.0018 | 0.0016
26 | 0.0013 | 0.0015 | 0.0019 | 0.0015
10 | 0.0012 | 0.0013 | 0.0014 | 0.0013
Table 3: First twenty Professors according to normalized Fuzzy in-closeness centrality indexes in the collaborative network of the Department of Economic Studies of the University of Chieti-Pescara, Italy. Figure 9: Fuzzy out-closeness centrality indexed of the actors in the collaborative network of the Department of Economic Studies of the University of Chieti-Pescara, Italy.  Prof | Left | Core | Right | CoG
---|---|---|---|---
29 | 0.0039 | 0.0045 | 0.0052 | 0.0045
44 | 0.0037 | 0.0045 | 0.0053 | 0.0045
21 | 0.0033 | 0.0039 | 0.0044 | 0.0039
28 | 0.0033 | 0.0039 | 0.0044 | 0.0039
19 | 0.0029 | 0.0035 | 0.0050 | 0.0038
41 | 0.0033 | 0.0037 | 0.0041 | 0.0037
45 | 0.0028 | 0.0035 | 0.0040 | 0.0034
23 | 0.0028 | 0.0034 | 0.0040 | 0.0034
26 | 0.0029 | 0.0033 | 0.0038 | 0.0033
36 | 0.0024 | 0.0035 | 0.0038 | 0.0033
30 | 0.0026 | 0.0033 | 0.0036 | 0.0032
48 | 0.0028 | 0.0032 | 0.0035 | 0.0032
17 | 0.0024 | 0.0030 | 0.0040 | 0.0031
47 | 0.0025 | 0.0031 | 0.0035 | 0.0031
20 | 0.0027 | 0.0028 | 0.0034 | 0.0030
16 | 0.0024 | 0.0028 | 0.0031 | 0.0028
49 | 0.0024 | 0.0028 | 0.0031 | 0.0027
8 | 0.0022 | 0.0029 | 0.0032 | 0.0027
35 | 0.0023 | 0.0027 | 0.0032 | 0.0027
39 | 0.0023 | 0.0027 | 0.0032 | 0.0027
Table 4: First twenty Professors according to normalized Fuzzy out-closeness
centrality indexes in the collaborative network of the Department of Economic
Studies of the University of Chieti-Pescara, Italy.

##  8 Discussion e conclusions

In the literature on social network analysis, the links are binary, i.e.
present or absent. A possible extension of this approach, widely used in the
literature, considers the so-called weighted networks. In the latter, the link
between the different nodes of a network is generally expressed with a number
in the range $[0,1]$. This approach of considering connections with a weight
does not align with fuzzy logic’s philosophy. Fuzzy logic embraces the
coexistence of opposites and, for this reason, typically conveys information
about vagueness. Attempting to translate an inherently imprecise term (for
example, being very good friends) into a single scalar (a number between zero
and one) is a strategy that sacrifices information for simplicity.

This work considers the possibility that a vague measure can express the link
between different actors in a social network. This approximate measure could
be due to an attribute of human language that is often inaccurate by nature;
for example, the inaccuracy inherent in concepts such as ”being friends,”
”being little friends,” ”being very close friends,” ”being close
collaborators,” or ”having a poor collaboration.” In all these cases, we
should have imprecise edges to preserve information about the vagueness of
these links. In other cases, as in the application proposed in this work, we
can also experience another kind of uncertainty due to indecision in choosing
the final qualitative answer to a questionnaire, for example, by checking the
mouse’s movements before providing the definitive answer.

The translation of these sources of vagueness has been addressed using
triangular fuzzy numbers. This approach extends classic social networks to
scenarios where actors are fully connected through nuanced ties, represented
by functions. Consequently, this paper introduces a novel method for analyzing
social networks, where the connections between actors are expressed through
fuzzy numbers. In other words, the links between nodes are represented by
functions rather than scalars, as in traditional social network theory.

In light of this new context, this study presented an extension of the
traditional centrality indices to the case where the network consists of
shaded edges. This research introduces several new measures that consider the
imprecision of the links in calculating the centrality measures and, in
particular, often provides an original interpretation of these new measures.
In the application part of this work, we presented a case of a collaboration
network within a university department. The goal was to measure individual
collaboration through questionnaires to understand how much each pair of
professors or researchers collaborates. The basic idea of this approach is to
try to understand the links between department members that go beyond simple
indicators such as the number of publications in common or other parameters
that can be measured through scalar measures. Indeed, in a department or even
in another type of network, there are many cases in which the friendship or,
more generally, the collaboration between two members goes beyond the number
of tangible products resulting from this collaboration. Strong cooperation
between individuals can also exist simply because they have projects in
common, exchange ideas, exchange didactic material, go out together outside
the working environment, are relatives, and help each other with research
questions even if they are not published together, have interests in common
outside the university for example for professional activities, or different
types of interests in common. In all these cases, considering objective
parameters such as the number of joint publications would be highly limiting,
and the information would need more effective collaboration between subjects.
In any case, since many of these concepts we have listed are by their nature
vague, the relationship between individuals and, therefore also, their
centrality in a more complex network must be measured through suitable methods
that consider the imprecision of human language as well as the uncertainty
connected with their way of seeing things. For the latter reason, the social
network we propose in this research is rarely symmetrical. In fact, in most
cases, an individual’s opinion regarding his relationship with another
individual is rarely the same in the opposite direction. For this reason, we
have focused on the case of direct social networks in this study. Even if
highly improbable, we could have as an extreme case that every link between
two nodes is equal in both directions; naturally, as the size of the network
or the number of nodes involved increases, the probability that this extreme
case occurs tends to zero.

An exciting research component is the possibility of choosing a system of
different weights to calculate the proposed centrality indices. Indeed, in
some specific contexts and depending on the research objective, it could be
interesting to consider other systems of weights to give different importance
to solid bonds rather than weak bonds or vice versa. In the theoretical part
of the research, we introduced generalized measures to allow the calculation
of different centrality measures based on different possible systems of
weights. In the application part, there were no particular contextual
requirements to think of weights other than those all equal to each other, so
we focused on a system of equal weights.

In attributing weights to calculate the centrality indices, choosing the
method to create a ranking of fuzzy numbers also assumes great importance.
Different methods of sorting fuzzy numbers could lead to different results.
This work focused on the centre of gravity method, but our approach can be
extended to other procedures. The research focused on triangular fuzzy
numbers, but of course, a possible exciting extension would be to consider
different membership functions to represent the links existing in a social
network. Other interesting future developments could be creating an R package
to implement all the functions we have made in this research and developing
other indices proposed in the literature on classical social network analysis.

Our approach’s main advantage is to analyse networks of actors in which the
relationships are not precise because they are defined by heterogeneous
components or simply because they are collected through questionnaires whose
possible answers are imprecise or simply attributes of human language. The
possible applications include analysing the network of friendships on social
networks such as Facebook, Instagram, YouTube, etc. Limiting oneself to simple
friendships or followed contacts could be more informative. Actually, being
friends with a person on Facebook can have a very nuanced meaning, depending
on many other aspects that could be considered. Therefore, considering a
relationship between two individuals based on a binary outcome or a simple
weight between zero and one would be a complete loss of information because it
does not consider the variability or, even better, the so-called fuzziness
inherent in the relationship. In this research, we presented an application of
the data collected within university departments, but this approach can be
extended to multiple real contexts. For example, we could consider workplaces
or all networks of individuals involved in sports or government teams.
Consequently, the practical applications are numerous and of considerable
interest.

## Competing interests

No competing interest is declared.

## Acknowledgments

The authors thank Dr. Simone Di Nardo for creating the data collection
platform and professors Marta Di Nicola and Francesca Scozzari for supporting
Dr. Annamaria Porreca in her doctoral path, during which she collected data
and worked on the project. We also thank the members of the Department of
Economics of the University of Pescara-Chieti for responding to the
questionnaire and allowing the study to be carried out.

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