Content uploaded by Andrea Fronzetti Colladon
Author content
All content in this area was uploaded by Andrea Fronzetti Colladon on Nov 29, 2024
Content may be subject to copyright.
Why distinctiveness centrality is distinctive
Andrea Fronzetti Colladon
a,b,*,1
, Maurizio Naldi
a,c
a
Department of Civil, Computer Science and Aeronautical Technologies Engineering, Via della Vasca Navale 79, Rome 00146, Italy
b
Department of Engineering, University of Perugia, Via G. Duranti 93, Perugia 06125, Italy
c
Department of Law, Economics, Politics, and Modern Languages, LUMSA University, Rome 00192, Italy
ARTICLE INFO
Keywords:
Distinctiveness centrality
Social network analysis
Beta centrality
Gamma centrality
ABSTRACT
This paper responds to a commentary by Neal (2024) regarding the Distinctiveness centrality metrics introduced
by Fronzetti Colladon and Naldi (2020). Distinctiveness centrality offers a novel reinterpretation of degree
centrality, particularly emphasizing the signicance of direct connections to loosely connected peers within
(social) networks. This response paper presents a more comprehensive analysis of the correlation between
Distinctiveness and the Beta and Gamma measures. All ve Distinctiveness measures are considered, as well as a
more meaningful range of the
α
parameter and different network topologies, distinguishing between weighted
and unweighted networks. Findings indicate signicant variability in correlations, supporting the viability of
Distinctiveness as alternative or complementary metrics within social network analysis. Moreover, the paper
presents computational complexity analysis and simplied R code for practical implementation. Encouraging
initial ndings suggest potential applications in diverse domains, inviting further exploration and comparative
analyses.
1. Introduction
In 2020, our publication introduced ‘Distinctiveness Centrality’(DC)
as a suite of 5 new measures of centrality in social networks (Fronzetti
Colladon and Naldi, 2020), offering a reinterpretation of degree cen-
trality (Freeman, 1979) tailored to pinpoint the role played by direct
node connections, particularly when these occur with loosely connected
neighbors. Though loosely connected, these nodes are not necessarily
pertinent to the network periphery; for example, they may serve as
crucial links between a network’s core and periphery despite their few
connections. Distinctiveness, as a metric, scrutinizes the dening char-
acteristics of a node’s direct connections.
The initial motivation and value of introducing Distinctiveness cen-
trality stems from transferring the Term Frequency –Inverse Document
Frequency (TF-IDF) concept to networks. In text mining, TF-IDF is a
widely used weighting system (Ramos, 2003), for example, to determine
keywords based on their frequency within a document relative to their
frequency across the corpus, thus offsetting common terms (i.e.,
assigning a zero score to terms that appear in every document or giving
low scores to terms that occur in a large number of documents).
Distinctiveness, particularly in its rst two formulations, represents an
effort to enrich studies exploring word networks’properties and
analytical methodologies. Much relevant research, indeed, converges at
the intersection of text mining and social network analysis (Fronzetti
Colladon et al., 2020).
This paper responds to Neal’s (2024) comment, aiming to enrich the
ongoing debate surrounding our metrics, shed light on their properties,
and stimulate further scientic debate. We extend our gratitude to Neal
for dedicating time to analyzing two of our proposed metrics and pub-
lishing a commentary on them. Additionally, we appreciate the author’s
mention of the Gamma (Neal, 2011) and Beta (Bonacich, 1987) cen-
trality metrics, which bear similarities to Distinctiveness under certain
circumstances.
In the following, we briey summarize the main points of Neal’s
(2024) critique for the reader’s convenience. Neal argued that the
Distinctiveness metrics are redundant, suggesting they are minor vari-
ations of Beta and Gamma centralities (Bonacich, 1987; Neal, 2011). To
demonstrate this similarity, the author conducted a correlation analysis,
albeit considering only two out of the ve Distinctiveness measures.
Moreover, the values of the alpha parameter used in the Distinctiveness
formulas were constrained and less signicant than those envisioned
during the parameter’s conceptualization.
* Corresponding author at: Department of Civil, Computer Science and Aeronautical Technologies Engineering, Via della Vasca Navale 79, Rome 00146, Italy.
E-mail addresses: andrea.fronzetticolladon@uniroma3.it (A. Fronzetti Colladon), m.naldi@lumsa.it (M. Naldi).
1
ORCID: 0000–0002-5348–9722
Contents lists available at ScienceDirect
Social Networks
journal homepage: www.elsevier.com/locate/socnet
https://doi.org/10.1016/j.socnet.2024.11.001
Received 4 August 2024; Received in revised form 14 October 2024; Accepted 6 November 2024
Social Networks 81 (2025) 1–16
0378-8733/© 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ).
Given the results of this analysis, the author advised against using
Distinctiveness centrality, recommending Gamma and Beta centralities
instead. The justication was that these two metrics are already well-
established in the literature and do not require a specialized software
package for computation –as they “can be written elegantly in a matrix
form that expresses the complete vector of centrality scores as a function
of an adjacency matrix”(Neal, 2024, p. 7).
In Section 3, we thoroughly examine the correlations between the
rankings generated by the ve Distinctiveness metrics and those by Beta
and Gamma centralities, surpassing the mere consideration of only the
rst two Distinctiveness measures –D1 and D2 (formulas provided in
Section 2). In addition, we carefully explore a larger range of
α
param-
eter values within the Distinctiveness formulas. We further extend the
analysis to scrutinize both weighted and unweighted networks charac-
terized by Small-World and Scale-Free topologies. Our results show that
while Distinctiveness may bear similarities to Beta and Gamma cen-
tralities in certain congurations, it yields distinct rankings. Therefore,
we respectfully disagree with Neal’s (2024) position advising against the
use of Distinctiveness centrality. Despite their potential similarities with
other measures, we maintain that our metrics operate on distinct prin-
ciples and conceptualization, with Distinctiveness serving as a potential
source of inspiration for scholars investigating word networks or other
domains. Indeed, the signicance of Distinctiveness has been illustrated
in various contexts. For example, it has proven valuable for analyzing
semantic networks, as also demonstrated by its integration into the Se-
mantic Brand Score (Fronzetti Colladon, 2018) as a substitute for degree
centrality (e.g., Santomauro et al., 2021;Vestrelli et al., 2024). Addi-
tionally, it was used in the mapping of technological interdependence –
proving valuable in the analysis of the structure of the innovators’
network (Fronzetti Colladon et al., 2025)–and the analysis of urban
networks (Fronzetti Colladon et al., 2024)–providing valuable insights
for the pricing strategies of gas stations –, among other applications (e.
g., Silva, 2021;Yudhoatmojo, 2024).
In Section 2, we address Neal’s comment regarding the elegance of
code, which we believe is a less compelling argument. We also discuss
the potential benets of utilizing a software package designed to manage
graph objects rather than relying on potentially unwieldy adjacency
matrices. Furthermore, we demonstrate that all Distinctiveness metrics
can be efciently expressed in just a few lines of R code, similar to how
the author presented Beta and Gamma centralities, using adjacency
matrices as input. We also explore the asymptotic complexity of each
metric, emphasizing that this aspect is more pertinent to their practical
application than the current structure of the R code.
Lastly, we agree with Neal (2024) on the perspective that Distinc-
tiveness measures, contingent upon the selection of their
α
parameter,
can be construed as metrics of power
2
–potentially reecting a social
actor’s ability to exert inuence or control over, or exploit,
poorly-connected neighbors, with power coming from “being connected
to those who are powerless”(Bonacich, 1987, p. 1171).
This research demonstrates the absence of inherent limitations in the
application of Distinctiveness centrality and encourages its continued
exploration and utilization within academic research. We elaborate on
our perspectives further in the remaining sections of the paper.
2. R code and computational complexity
Neal contends that opting for Beta or Gamma centralities over
Distinctiveness centrality presents certain advantages as they “can be
written elegantly in a matrix form that expresses the complete vector of
centrality scores as a function of an adjacency matrix”(Neal, 2024, p. 7).
However, we argue that the availability of R and Python packages
tailored for Distinctiveness calculations adds signicant value. These
packages streamline the process by incorporating essential functions,
such as data validation checks, and accommodate the computation of
ve distinct metrics rather than a single one. Moreover, they include
variants of Distinctiveness metrics tailored for directed networks. In
other words, in developing our Python and R packages, our primary
emphasis was ensuring robustness by validating input data and deliv-
ering accurate results. We designed the packages to seamlessly integrate
with graph objects from libraries like igraph (https://igraph.org) or
networkx (https://networkx.org/), thus enhancing their versatility and
compatibility. While there is potential for further optimization to
enhance efciency, it is important to note that any such renements do
not impede the usability or effectiveness of our metrics.
Additionally, the memory-intensive nature of handling large adja-
cency matrices during computation may pose resource constraints. In
contrast, utilizing graph-type objects provided by libraries such as
igraph or networkx could offer more efcient storage and operations for
large-scale graphs. Therefore, there are potential efciency gains from
leveraging these well-known and widely-used packages that should be
considered and explored further.
3
In Table 1, we show the R code proposed by Neal (2024) to calculate
Beta and Gamma centralities that we consider for our analysis, together
with the mathematical formulas for these two metrics, to support the
reader, although their detailed discussion is deferred to previous papers
(Bonacich, 1987; Neal, 2011, 2024).
In the above formulas, Ais an adjacency matrix, 1 is a column vector
of 1 s, and γis the tuning parameter of Gamma centrality, which is used
to assign higher scores to nodes that are connected to well-connected
neighbors (if γ>0) or to nodes connected to poorly-connected neigh-
bors (if γ<0). Iis an identity matrix, inv is the matrix inverse function,
and βis the tuning parameter of Beta centrality, which controls whether
connections to well-connected or poorly-connected nodes lead to higher
Beta centrality scores, similar to the tuning parameter of Gamma
centrality.
Should we choose to reimagine the code for computing all the ve
Distinctiveness centrality metrics in R, we could achieve this with the
following functions, which maintain an “elegant”structure akin to the
one proposed for Beta and Gamma centralities by Neal (2024). We are
presenting this additional R code solely to address the author’s critique
of our use of a software package. However, the following code (as well as
the code for Gamma centrality) may still encounter problems, e.g., when
dealing with networks that include isolates. Therefore, we keep rec-
ommending using the distinctiveness package, which offers more robust
code to prevent errors and handle such cases effectively. Table 2 pre-
sents the ve Distinctiveness centrality formulas for undirected net-
works (Fronzetti Colladon and Naldi, 2020) and a new R code for their
computation that takes an adjacency matrix as input, considering net-
works without isolates (for which Distinctiveness would be zero). The
proposed functions essentially serve as an analog to utilizing the
Table 1
Beta and Gamma centralities. Formulas and R code.
Metric Formula R Code
Beta BC =inv(I−βA)A1 (1) B <- 0 #Set value of beta
I<- diag(nrow(A))
O<- matrix(1, nrow =nrow(A))
beta <- solve(I - (B * A)) %*% A %*% O
Gamma GC =A(A1)γ(2) G <- 0 #Set value of gamma
O<- matrix(1, nrow =nrow(A))
gamma <- A %*% ((A %*% O)^G)
2
For the sake of simplicity, we will continue to use the term "centrality"
throughout the remainder of the paper.
3
It is worth noting that Beta Centrality is available in the igraph package
through the ‘power_centrality()’function. In contrast, to the best of our
knowledge, Gamma Centrality has not yet been implemented in any package.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
2
distinctiveness package in R (https://github.com/iandreafc/distinct
iveness-R), skipping normalization. However, a notable distinction lies
in the input structure: while the package operates on igraph class graph
objects, our suggested reformulation accepts an adjacency matrix as
input. It is worth noting the equivalence of D1 and D2 measures on
unweighted networks, allowing for seamless interchangeability between
the two on such networks.
In the above formulas, nrepresents the number of network nodes, gj
denotes the degree of node j,wij signies the weight of the edge con-
necting nodes iand j, and I(wij >0)stands as a function that equals 1 if
the weight of the edge linking iand jexceeds zero, and 0 otherwise (this
function indicates the presence or absence of an edge between iand j, as
we only consider positively weighted networks). Lastly,
α
serves as the
tuning parameter employed for the Distinctiveness centrality metrics,
with higher values of
α
indicating a higher penalization of connections
toward well-connected neighbors.
It is worth noting that D5 and Gamma centralities produce the same
scores when the adjacency matrix is binary (i.e., the network is un-
weighted) and
α
= − γ(see our discussion in Section 3).
2.1. Computational complexity
So far, we have reformulated the R code to compute the ve
Distinctiveness metrics. However, as previously mentioned, we believe
that computational time and efciency in the use of computer memory
are the most critical factors to consider when conducting an analysis –
for example, avoiding large adjacency matrices and using more efcient
graph objects. The elegance or brevity of the code is less important,
especially when software packages that allow for one-line computation
of these metrics are available. Accordingly, we provide an analysis of the
asymptotic complexity of all the metrics discussed so far, including
Distinctiveness, Beta, and Gamma centrality.
4
It is to be noted that each metric requires the computation of the
node degree. This is explicitly considered in all the Distinctiveness
metrics, while it is embodied in the A1 term in the Beta and Gamma
centralities. When the degrees for all nodes are computed beforehand,
they can be subsequently retrieved from a look-up table. The compu-
tational complexity of node degree computation is O(n2), as it involves
summing all the nterms in each row of the weight matrix for all the n
rows.
5
Since this task has to be carried out for all the centrality metrics
we are considering, we can view it as a lower bound for the computa-
tional complexity of these metrics. In the following, we assume that the
computation of the degree has been carried out beforehand and will
evaluate the computational complexity of the other operations. If that
subsequent complexity should result lower than O(n2), the overall
complexity will anyway be O(n2).
First, we introduce some notations that we will use for all the met-
rics. We indicate the number of nodes in the network as n, while kis the
number of digits of the numbers for all the quantities involved (for
simplicity, we do not make differences in the numerical resolution
among, e.g., node degrees,
α
values, weights, etc.). Several metrics
require the computation of a logarithm function. For this elementary
function, we assume that the computational complexity is O(M(k)logk)
following Brent (1976), where the Arithmetic-Geometric mean method
is employed (Brent and Zimmermann, 2010), and M(k)is the compu-
tational effort for multiplication. The same can be said for exponentia-
tion. As to the multiplication computational effort, it depends on the
algorithm to be used, going from O(klogk)with the Harvey-Hoeven al-
gorithm (Harvey and Van Der Hoeven, 2021) to O(n1.585)with Kar-
atsuba’s algorithm. Please refer to Bernstein (2001) for a survey of
multiplication algorithms. For the time being, since multiplications
occur in all metrics, we will not specify their computational effort. In the
following, we consider the computational complexity of computing each
metric for all the nodes in the network, and also that n>k.
D1. We have to raise the mode degrees of all nodes to the power of
α
,
carry out ndivisions (for which we can assume the same computational
effort as for multiplications), nlogarithm computations, n2products by
the wij coefcients and nsums. The overall complexity is then
C(D1) = nO(M(k)log k) + nO(M(k)) + nO(M(k)log k) + n2O(M(k)) + nO(k)
=n2O(M(k))
(8)
D2. The difference with respect to D1 is that we do not have to
multiply each logarithmic term by the arc weights. The computational
complexity is then
Table 2
Distinctiveness Centrality. Formulas and R code.
Metric Formula R Code
D1 D1(i) = n
j=1
j∕=i
wijlog10
n−1
g
α
j
(3) alpha <- 1 #Set value of alpha
N<- nrow(adj_matrix)
degrees <- colSums(adj_matrix !=0)
d1 <- adj_matrix %*% log10((N - 1) / degrees^alpha)
D2 D2(i) = n
j=1
j∕=i
log10
n−1
g
α
j
I(wij>0)
(4) alpha <- 1 #Set value of alpha
N<- nrow(adj_matrix)
degrees <- colSums(adj_matrix !=0)
d2 <- (adj_matrix !=0) %*% log10((N - 1) / degrees^alpha)
D3
D3(i) = n
j=1
j∕=i
wijlog10
n
k,l=1
k∕=l
wkl
2
n
k=1
k∕=j
w
α
jk
−w
α
ij +1
(5) alpha <- 1 #Set value of alpha
numerator <- sum(adjacency_matrix[upper.tri(adjacency_matrix)])
denominator <- rowSums(adjacency_matrix^alpha) - adjacency_matrix^alpha +1
d3 <- colSums(adjacency_matrix * log10(numerator / denominator))
D4 D4(i) = n
j=1
j∕=i
wij
w
α
ij
n
k=1
k∕=j
w
α
jk
(6) alpha <- 1 #Set value of alpha
numerator <- adjacency_matrix * (adjacency_matrix^alpha)
denominator <- rowSums(adjacency_matrix^alpha)
d4 <- colSums(numerator / denominator)
D5 D5(i) = n
j=1
j∕=i
1
g
α
j
I(wij>0)
(7) alpha <- 1 #Set value of alpha
degrees <- colSums(adj_matrix !=0)
d5 <- (adj_matrix !=0) %*% (1 / degrees^alpha)
4
As already discussed in past research, the computational complexity of
Gamma centrality is lower than that of Beta centrality (Neal, 2013).
5
The complexity of degree computation arises from using the adjacency
matrix as the sole representation of the graph. If an edge list were also avail-
able, the computation would require O(m)operations.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
3
C(D2) = nO(M(k)logk) + nO(M(k)) + nO(M(k)logk) + nO(k)
=nO(M(k)logk)
(9)
However, we must consider the computation of node degrees so that
the overall
complexity, as far as the dependence on the network size is con-
cerned, is again O(n2).
D3. Here, we have to raise the weights wij to the power of
α
and also
sum them. The computational effort is
C(D3) = n2O(M(k)logk) + nO(k) + nO(k) + nO(k)
+n(M(k)) + n2O(M(k)logk) + n2O(M(k))
+nO(k) = n2O(M(k)logk)
(10)
D4. The major effort here is to raise the weights wij to the power of
α
and also sum them. The computational effort is
C(D4) = n2O(M(k)logk) + nO(k) + n2O(M(k))
+n2O(M(k)) + nO(k) = n2O(M(k)logk)(11)
D5. The major effort here is to raise the node degrees to the power of
α
, compute their reciprocals, and sum them. The computational effort is
C(D5) = nO(M(k)logk) + nO(M(k)) + nO(k) = nO(M(k)logk)
(12)
Again, we must consider the lower bound represented by the
computation of the node degrees. As to the dependence on the network
size, the computational complexity is then O(n2).
2.1.1. Beta centrality
Assuming that the product A1, leading to the node degrees, has
already been carried out, this metric involves the inversion of a matrix,
which is probably the most relevant operation, and a multiplication. If
we consider the inversion to be carried out through Gauss-Jordan
elimination, its computational complexity is O(n3). Other algorithms
achieve a slightly lower exponent of n. The computational effort is
C(BC) = n2O(M(k)) + nO(k) + nO(M(k)) + n2O(k)
+O(n3) = O(n3)
(13)
2.1.2. Gamma centrality
This metric involves again the computation of A1 and the exponen-
tiation of the resulting vector plus a matrix-by-vector multiplication.
The computational effort is
C(GC) = n2O(M(k)) + nO(k) + nO(M(k)logk) + n2O(M(k))
+nO(k) = n2O(M(k))
(14)
A brief comparison of the metrics shows that Beta centrality is the
computationally most expensive one, growing with n3. A lower cost is
required for metrics D1, D3, D4, and Gamma, whose cost grows with the
square of the number of nodes. The computational complexity of metrics
D2 and D5 would grow linearly with the number of nodes if the node
degrees were available beforehand. If that is not the case, their
computational complexity grows again as O(n2).
3. Beta, gamma, and distinctiveness: a new comparison
Neal (2024) examined only two of the ve Distinctiveness centrality
metrics, providing a limited perspective compared to a comprehensive
analysis of the entire set. Specically, their paper outlines the formula
for D1, which aligns with D2 in unweighted networks, and compares
them with Beta and Gamma centralities. In the ensuing discussion, we
present a thorough examination encompassing all ve metrics,
distinguishing between weighted and unweighted networks. Notably,
we employ D2, D3, and D5 for unweighted networks, as they are tailored
for such contexts, while using D1, D3, and D4 for weighted networks.
Furthermore, our analysis includes correlation values between
Gamma and Beta centralities, which were not elucidated in Neal’s paper.
To ensure consistency across different metrics, we adhere to formulas
4 and 5 presented in Neal’s (2024) paper, which they suggested to
harmonize the parameters of the different metrics. However, since our
emphasis lies on Distinctiveness centrality, we illustrate the variations in
correlations as the parameter
α
changes. Consequently, we establish the
parameter γfor Gamma centrality as γ= −
α
, while dening the
parameter βfor Beta centrality as β=2
e
α
−1×1
λ1, where
λ1represents the largest eigenvalue of the adjacency matrix.
While Neal (2024) only provides a short explanation of these re-
lationships, we notice that they may derive from the metric D5 when we
have a non-weighted network. In fact, the product A1 appearing in the
Gamma centrality is the vector of degree values
A1=
w11 ⋯w1n
⋮ ⋱ ⋮
wn1⋯wnn
1
⋮
1
=
n
j=1w1j
⋮
n
j=1wnj
=
g1
⋮
gn
(15)
so that the Gamma centrality can be written as
GC =A(A1)γ=
w11 ⋯w1n
⋮ ⋱ ⋮
wn1⋯wnn
gγ
1
⋮
gγ
n
=
n
j=1w1jgγ
j
⋮
n
j=1wnjgγ
j
(16)
which is exactly the D5 metric when wij ∈ {0,1}so that wij =I(wij >0)
and γ= −
α
.
However, this relationship is valid for D5 only. As can be seen from
the formulas for the other metrics, a similar equivalence does not apply
to D1 through D4. Hence, any conclusion based on the proposed
harmonization formulas to make the metrics comparable should be
considered with caution. Having said that, in order to allow for a direct
comparison with the analyses carried out by Neal (2024), we adopt
those formulas throughout this paper.
In formulating Distinctiveness centrality (Fronzetti Colladon and
Naldi, 2020), the parameter
α
was conceived to offer exibility, allowing
it to deviate from its standard value of 1. This deviation enables the
penalization of connections to highly connected nodes to a greater
extent. Thus, we suggested that
α
should be greater than or equal to one.
We highlighted the case
α
>1 because the contributions of the log-
arithmic term to D1, for example, may be negative in that case, while
they are all positive when 0 <
α
<1. What happens when the contri-
butions are all positive is, however, that the nodes with a higher degree
contribute less to the nal score than the nodes with a lower degree. If
we consider two nodes kand m, with degrees gk>gm, we see that their
contributions to D1 are such that logn−1
g
α
k<logn−1
g
α
m, regardless of the
value of
α
>0. Hence, the general principle that nodes with a higher
degree are penalized in that metric is maintained even if 0 <
α
<1.
In addition, we can notice that the rst derivative of D1 is always
negative and does not depend on
α
, so there is no discontinuity when
α
crosses the border value 1:
∂
D1
∂α
= − 1
ln10
n
j=1
j∕=i
wijlngj<0 (17)
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
4
Fig. 1. Spearman’s correlations –Scale-Free Networks.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
5
Fig. 2. Spearman’s correlations –Small-World Networks.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
6
Neal (2024) also wonders why we should choose larger values of
α
.
Well, larger values of
α
, being used in the exponent of the degree of node
jin the formula for D1, amplify the differences in the contributions
brought by high-degree and low-degree nodes, respectively. However,
as shown by formula 17, D1 overall value decreases linearly when
α
grows.
Consequently, we revised the range of variation for
α
when
comparing the metrics, selecting it from 0.5 to 3 –though higher values
could have been considered, we constrain it to 3 for conciseness. Unlike
Neal (2024), we do not normalize the metrics as it is irrelevant to the
computation of correlations.
Figs. 1 and 2 illustrate the outcomes of our analysis, carried out using
Spearman’s correlation. In all our analyses, the correlation scores would
be lower if Kendall’s
τ
were used instead of Spearman’s
ρ
; e.g., for
allowing comparison with the results presented by Schoch et al. (2017).
Kendall’s correlation scores can be easily obtained from the code we
have shared alongside this paper.
We computed the average correlations among the different metrics
on randomly generated Small-World and Scale-Free networks using the
igraph package in R. For each conguration (Scale-Free vs. Small-World
and Weighted vs. Unweighted), we generated 200 random networks
each containing 1000 nodes.
6
The code required to replicate these
ndings and the other analyses presented in this paper is available at htt
ps://github.com/iandreafc/distinctiveness_comparisons.
In the gures, we have arranged separate plots for the correlations of
Distinctiveness with Gamma and Beta centralities to enhance read-
ability. Fig. 1 illustrates the average Spearman’s correlations observed
for Scale-Free networks, while Fig. 2 focuses on Small-World networks.
Across all scenarios, we notice variability in the correlations among the
different metrics, giving evidence to each measure’s capacity to assign
distinct importance scores to network nodes.
The vertical line denoting an
α
value of 1 serves as a reference point
for meaningful consideration of the parameter. As
α
increases, we
observe a decline in correlation values, highlighting an increasing
divergence between metrics and the adaptability inherent in Distinc-
tiveness centrality to accommodate varying interpretations with
changes in the
α
parameter. As expected, in unweighted networks, we
observe a maximal correlation between Gamma and D5 –which produce
the same scores for γ= −
α
. Additionally, correlations between D2 and
Beta, D2 and Gamma and D3 and Gamma, and sometimes D3 and Beta,
remain relatively high. However, these values may decrease when
extending the analysis to higher alpha values.
7
Importantly, this is not
always the case, and results can vary depending on network topology.
Our aim is not to generalize these results but rather to demonstrate the
variability in correlation values across metrics based on the chosen
alpha parameter, network topologies, and arc weights –see the Ap-
pendix for another illustrative analysis conducted on random graphs
generated using the Erdos-Renyi model (Erdos and Renyi, 1959).
Therefore, it is challenging to designate one metric as the substitute for
another. Each metric could showcase an optimal use case, and we
advocate for future research in this direction. An exception lies in the
equal scores produced by D5 and Gamma centrality in unweighted
networks. Conversely, other Distinctiveness measures diverge from
Gamma and Beta centralities, particularly with increasing alpha values.
The ability to easily adjust the alpha parameter, coupled with the fact
that it encompasses a set of ve metrics, endows Distinctiveness with
unique exibility, rendering it potentially valuable for diverse
applications.
Lastly, the gures reveal a degree of correlation between Gamma and
Beta centralities, which is not unexpectedly high in many cases (see also
the Appendix). In general, signicant and high correlations can be
observed between many measures of network centrality (or power),
such as degree, betweenness, eigenvector, and closeness centrality
(Schoch et al., 2017). However, this does not support choosing one or
two metrics and disregarding the others. While often correlated, these
diverse metrics succeed in capturing distinct facets of social actors’
positions. Moreover, they may demonstrate varying degrees of explan-
atory power concerning external variables (i.e., centrality effects), such
as employee performance levels within a company (Wen et al., 2020).
4. Discussion and conclusion
In this paper, we presented a more comprehensive examination of
Distinctiveness centrality in comparison with Beta and Gamma cen-
tralities than the one offered by Neal (2024). This involved exploring
more appropriate values for the
α
parameter, incorporating all ve
Distinctiveness metrics, distinguishing between weighted and un-
weighted networks, and including the correlation between Beta cen-
trality and Gamma centrality.
Our ndings indicate that, except for the correlation between D5 and
Gamma centrality in unweighted networks, the remaining correlations
demonstrate signicant variability and tend to decrease as the alpha
parameter increases. In particular, the lowest correlations (as alpha in-
creases), between Distinctiveness and Gamma and Beta centralities, vary
by network topology, and our experiments show the following results. In
unweighted scale-free networks, D3 and Beta, as well as D5 and Beta,
exhibit the lowest correlations, while in weighted scale-free networks, it
is D4 and Beta, D4 and Gamma, and D1 and Gamma. D5 and Beta have
the lowest correlations for unweighted small-world networks, while in
weighted versions, it is D1 and Gamma, along with D4 and Gamma. In
unweighted Erdos-Renyi graphs (see the Appendix), the lowest corre-
lations are observed between D2 and Gamma, and D2 and Beta. In their
weighted counterparts, these are D1 and Gamma, D1 and Beta, D3 and
Gamma, and D3 and Beta. This variability, which could be further
investigated and justied in future research, demonstrates that
Distinctiveness can produce scores that differ from those generated by
Beta and Gamma centralities, positioning it as an alternative or com-
plementary set of metrics. This addresses Neal’s (2024) rst critique, in
which they questioned the novelty of our metrics.
Additionally, in Section 2, we presented the formulas accompanied
by simplied R code and an analysis of the computational complexity of
all the metrics discussed in this paper. In particular, we assessed their
asymptotic computational complexities and found that Beta centrality
exhibits the highest complexity at O(n
3
). The complexity of Gamma
centrality grows with the square of the number of nodes, similar to all
Distinctiveness centrality metrics when node degrees are not pre-
computed. If node degrees are available beforehand, the computational
complexity of metrics D2 and D5 instead grows linearly with the number
of nodes.
All these ndings support the viability of Distinctiveness as either an
alternative or complementary set of metrics to Beta, Gamma, and other
conventional measures of centrality (or power) within social networks.
Distinctiveness has an additional advantage in that it comprises a set of
ve metrics, all based on a shared conceptualization. This is signicant
per se, as it connects to the study of words and networks, setting it apart
from the conceptualizations of Beta and Gamma centralities. Further-
more, having ve metrics increases application exibility beyond what a
single metric can offer.
There is also a potential limitation worth considering when using
6
We used the R igraph package for generating the random networks with the
following functions sample_pa(n =1000, m=2, directed =FALSE) and sample_
smallworld(dim =1, size =1000, nei =2, p=0.05). For the weighted versions,
we attributed random weights in the range from 1 to 20 to each edge. In the
shared code, we set random seeds to restrict our analysis to networks without
isolates due to limitations in the Gamma code, which generates NaNs when
isolates are present. Alternatively, one could use a function –such as this one g_
no_isolates <- delete_vertices(g, V(g)[degree(g) == 0]) –to remove isolates.
7
The code we share can be tested with various parameters of the network
generation functions. Experimentation can include adjusting sizes and
exploring different alpha values.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
7
Spearman correlation. Take, for example, two hypothetical centrality
measures applied to a network of three nodes, A, B, and C. Let us say
Measure 1 assigns scores {A: 100, B: 99, C: 98}, while Measure 2 assigns
scores {A: 100, B: 9, C: 1} on the same measurement scale. Spearman’s
correlation would yield a result of one, indicating perfect alignment in
node rankings. However, this assessment overlooks the variability of
scores. While a comprehensive discussion falls outside the scope of this
paper, we incorporate a comparison of the score distributions generated
by the various metrics in the Appendix and assess their distance using
the Ruzicka index (Cha, 2007). It might be valuable for future research
to delve more into these nuances, exploring additional methods for
comparing scores derived from Distinctiveness and other centrality
measures.
Similarly, future research should consider the inherent limitations of
using correlation analysis to assess the redundancy of new centrality
metrics. Correlations among centrality metrics, which are often high,
may not necessarily reect formal or conceptual similarities, as they can
be confounded by underlying network structures (Schoch et al., 2017).
Therefore, not only correlations may be questioned as the most appro-
priate method for assessing the redundancy of centrality metrics, but
this adds to the potential concerns raised in Section 3 regarding the
proposed harmonization formulas for the
α
,β, and γparameters, which
are used to compare the metrics.
Despite being a relatively recent introduction, Distinctiveness cen-
trality metrics already demonstrate promise across various domains. For
example, they have shown potential in analyzing semantic networks,
urban networks, and technological interdependence relationships be-
tween sectors (Fronzetti Colladon et al., 2024, 2025; Vestrelli et al.,
2024).
In terms of future research directions, it would be important to
conduct comparative analyses of the explanatory capabilities of
Distinctiveness metrics, Beta, Gamma, and other traditional centrality
measures in diverse contexts. Such investigations could provide insights
into certain observed phenomena and contribute to their understanding
through empirical examination.
Certainly, the encouraging initial ndings of the aforementioned
studies should catalyze scholars to delve deeper into potential and
possible applications of Distinctiveness rather than constraining its
exploration.
Furthermore, the conceptualization of Distinctiveness holds promise
as a source of inspiration for researchers interested in the study of se-
mantic networks. By bringing the logic of TF-IDF transformation to
networks, Distinctiveness centrality introduces an inherently innovative
approach that has the potential to inspire researchers engaged in studies
at the intersection of text mining and network analysis.
While not the primary focus of this paper, it is worth noting that
Distinctiveness has also been adapted for computation on directed net-
works, thereby extending the logic of metrics such as in-degree and out-
degree. Future research could explore these extensions and broaden the
scope of comparisons discussed herein.
In conclusion, we appreciate Neal’s (2024) dedication and interest in
analyzing two out of ve Distinctiveness metrics. In general, we believe
that discussions such as the one sparked by our 2020 paper are pivotal
for the advancement of science, and we are grateful for that.
Nevertheless, we respectfully disagree with the authors’suggestion to
forgo the utilization of Distinctiveness centrality in research. Their
correlation analysis is somewhat limited, as it focuses solely on D1 and
D2 and does not encompass
α
values greater than 1, which are integral to
the original logic of the metrics. Moreover, the arguments regarding the
elegance of Beta and Gamma centrality formulations with respect to
Distinctiveness appear weak. In Section 2, we have presented a simpli-
ed R code for computing the ve metrics and an analysis of their
asymptotic complexity –an aspect we believe holds more relevance for
researchers than considerations of programming code structure. As
previously mentioned in Section 2, utilizing a package enabling metric
calculation directly from an igraph object, as opposed to an adjacency
matrix, offers the additional advantage of conserving computer memory
resources. This approach allows for calculations even on sizable net-
works, facilitating analysis on standard commercial PCs –a task prob-
ably unfeasible if working with large adjacency matrices.
The exploration we undertook unveils new avenues of research
regarding the potential of Distinctiveness. It also highlights how Neal’s
(2024) judgment may have been rendered without considering more
comprehensive comparisons and broader perspectives.
Accordingly, we maintain that researchers should be afforded the
freedom to conduct their own evaluations and select the metrics that
best suit their research needs, whether it be Beta, Gamma, Distinctive-
ness centrality, or alternative metrics.
Declaration of Generative AI and AI-assisted technologies in the
writing process
While preparing this work, we used Grammarly and ChatGPT solely
to rene the language. After using these tools, we reviewed and edited
the content as needed. We take full responsibility for the content of the
publication.
CRediT authorship contribution statement
Andrea Fronzetti Colladon: Writing –review &editing, Writing –
original draft, Visualization, Validation, Software, Methodology, Fund-
ing acquisition, Formal analysis, Conceptualization. Maurizio Naldi:
Writing –review &editing, Writing –original draft, Methodology,
Formal analysis, Conceptualization.
Acknowledgments
We are very grateful to the two anonymous reviewers for their
constructive feedback.
We also wish to thank Zachary Neal for sparking this scientic debate
and offering valuable suggestions during the revision of this manuscript.
This work was partially supported by the University of Perugia
through the program Fondo Ricerca di Ateneo 2022, Proj. “Argo-
mentazione Astratta, Text Mining e Network Analysis per il Supporto
alle Decisioni (RATIONALISTS)”. The funders had no role in study
design, data analysis, decision to publish, or preparation of the
manuscript.
Appendix
Fig. A1 illustrates a correlation analysis akin to the one outlined in Section 3. Here, we employed the sample_gnp(100, 0.1) function from the
igraph package to generate 100 random networks, each with a size of 100 (to provide an example on smaller networks), using the Erdos-Renyi model
(Erdos and Renyi, 1959). For the weighted versions, we attributed random weights in the range from 1 to 80 to each edge (to provide an example with
greater variability in arc weights). As we can observe from the gure, some correlations drop fast and even become negative for
α
values bigger than
two.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
8
Figure A1. Spearman’s correlations –Erdos-Renyi Networks.
In the subsequent gures, we present an analysis aimed at comparing the distributions of scores generated by the Distinctiveness metrics and Beta
and Gamma centralities. This comparison extends beyond Spearman’s correlations, focusing on assessing the distances between scores and their
densities. As mentioned in Section 4, this analysis represents only a partial exploration and invites further investigation in future research endeavors.
We used a random Scale-Free network comprising 1000 nodes to conduct the analysis.
8
Random weights ranging from 1 to 80 were assigned to the
edges to calculate the weighted version of the metrics. Each of the following gures presents density plots of the scores obtained from the metrics
alongside a heatmap illustrating the values of Ruzicka’s index (Cha, 2007) for comparisons between each pair of distributions.
Figs. A2,A3, and A4 show density plots and Ruzicka indices on the unweighted network for alpha parameter values of 1, 2, and 3 (with
8
The network was generated using the R code "sample_pa(n =1000, m=2, directed =FALSE)". The same analysis could be replicated on Small-World or Erdos-
Renyi networks, by simply adapting the R code we made available.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
9
corresponding adjustments of values for gamma and beta parameters, as discussed in Section 3). In this context, we explore Distinctiveness metrics D2,
D3, and D5, tailored for unweighted networks, along with Gamma and Beta centralities.
Figure A2. Comparing distributions. Unweighted scale-free network, alpha =1.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
10
Figure A3. Comparing distributions. Unweighted scale-free network, alpha =2.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
11
Figure A4. Comparing distributions. Unweighted scale-free network, alpha =3.
Similarly, Figs. A5,A6, and A7 present the same analysis but with random weights assigned to the edges. In this scenario, we examine Distinc-
tiveness metrics D1, D3, and D4, specically designed for weighted networks, in addition to Gamma and Beta centralities.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
12
Figure A5. Comparing distributions. Weighted scale-free network, alpha =1.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
13
Figure A6. Comparing distributions. Weighted scale-free network, alpha =2.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
14
Figure A7. Comparing distributions. Weighted scale-free network, alpha =3.
Our ndings highlight the variability inherent in the proposed metrics, aligning with the outcomes derived from Spearman’s correlations. Once
again, we nd perfect equivalence between D5 and Gamma centrality in unweighted networks when
α
= − γ. Moreover, as alpha values increase, we
note a decline in the average of the Ruzicka indices, indicating a widening gap among the metrics under examination. While this trend holds true for
most cases, there are exceptions. For example, the distributions of Beta and D2 centrality scores on the unweighted network reduce their distance as
alpha increases, as do Beta and D1 scores in the weighted network when alpha increases from 2 to 3.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
15
References
Bernstein, D., 2001. Multidigit multiplication for mathematicians. Adv. Appl. Math. 12,
1–19.
Bonacich, P., 1987. Power and centrality: a family of measures. Am. J. Sociol. 92 (5),
1170–1182.
Brent, R.P., 1976. Multiple-precision zero-nding methods and the complexity of
elementary function evaluation. Analytic Computational Complexity. Elsevier,
pp. 151–176. https://doi.org/10.1016/B978-0-12-697560-4.50014-9.
Brent, R.P., Zimmermann, P., 2010. Modern Computer Arithmetic, 1st ed.). Cambridge
University Press. https://doi.org/10.1017/CBO9780511921698.
Cha, S.-H., 2007. Comprehensive survey on distance/similarity measures between
probability density functions | BibSonomy. Int. J. Math. Models Methods Appl. Sci. 1
(4), 300–307.
Erdos, P., Renyi, A., 1959. On random graphs. Publ. Math. 6, 290–297.
Freeman, L.C., 1979. Centrality in social networks conceptual clarication. Soc. Netw. 1
(3), 215–239. https://doi.org/10.1016/0378-8733(78)90021-7.
Fronzetti Colladon, A., Naldi, M., 2020. Distinctiveness centrality in social networks.
PLOS ONE 15 (5), e0233276. https://doi.org/10.1371/journal.pone.0233276.
Fronzetti Colladon, A., 2018. The Semantic Brand Score. J. Bus. Res. 88, 150–160.
https://doi.org/10.1016/j.jbusres.2018.03.026.
Fronzetti Colladon, A., Gloor, P., Iezzi, D.F., 2020. Editorial introduction: The power of
words and networks. Int. J. Inf. Manag. 51, 102031. https://doi.org/10.1016/j.
ijinfomgt.2019.10.016.
Fronzetti Colladon, A., Verdoliva, G., Segneri, L., Vitali, A.G., 2024. Analyzing gasoline
prices in ve Italian cities: Insights from social network analysis. Cities 150, 105075.
https://doi.org/10.1016/j.cities.2024.105075.
Fronzetti Colladon, A., Guardabascio, B., Venturini, F., 2025. A new mapping of
technological interdependence. Res. Policy 54 (1), 105126. https://doi.org/
10.1016/j.respol.2024.105126.
Harvey, D., Van Der Hoeven, J., 2021. Integer multiplication in time O(nlog n). Ann.
Math. 193 (2). https://doi.org/10.4007/annals.2021.193.2.4.
Neal, Z., 2011. Differentiating Centrality and Power in the World City Network. Urban
Stud. 48 (13), 2733–2748. https://doi.org/10.1177/0042098010388954.
Neal, Z., 2013. A computationally efcient approximation of beta centrality. Connections
33 (1), 11–17.
Neal, Z., 2024. Is ‘distinctiveness centrality’actually distinctive? A comment on Fronzetti
Colladon and Naldi (2020). PLOS ONE 19 (4), e0298666. https://doi.org/10.1371/
journal.pone.0298666.
Ramos, J. (2003). Using TF-IDF to Determine Word Relevance in Document Queries.
Proceedings of the First Instructional Conference on Machine Learning, 1–4. https://doi.
org/10.1.1.121.1424.
Santomauro, G., Alderuccio, D., Ambrosino, F., Migliori, S., 2021. Ranking
Cryptocurrencies by Brand Importance: A Social Media Analysis in ENEAGRID. In:
Bitetta, V., Bordino, I., Ferretti, A., Gullo, F., Ponti, G., Severini, L. (Eds.), Mining
Data for Financial Applications, MIDAS 2020. Springer, pp. 92–100. https://doi.org/
10.1007/978-3-030-66981-2_8.
Schoch, D., Valente, T.W., Brandes, U., 2017. Correlations among centrality indices and a
class of uniquely ranked graphs. Soc. Netw. 50, 46–54. https://doi.org/10.1016/j.
socnet.2017.03.010.
Silva, P.F. (2021). An Exploration of FPGAs as Accelerators for Graph Analysis via High-
Level Synthesis [University of Porto]. 〈https://repositorio-aberto.up.pt/bitstream/1
0216/137155/2/511069.pdf〉.
Vestrelli, R., Fronzetti Colladon, A., Pisello, A.L., 2024. When attention to climate change
matters: the impact of climate risk disclosure on rm market value. Energy Policy
185, 113938. https://doi.org/10.1016/j.enpol.2023.113938.
Wen, Q., Gloor, P.A., Fronzetti Colladon, A., Tickoo, P., Joshi, T., 2020. Finding top
performers through email patterns analysis. J. Inf. Sci. 46 (4), 508–527. https://doi.
org/10.1177/0165551519849519.
Yudhoatmojo, S.B. (2024). Quantitative Insights on the Use and Mis-use of Social Media in
Scientic, Extremist, and Political Contexts [State University of New York at
Binghamton]. 〈https://www.proquest.com/openview/b5ef808c0302a6995f4869ec
83be2c65/1?pq-origsite=gscholar&cbl=18750&diss=y〉.
A. Fronzetti Colladon and M. Naldi Social Networks 81 (2025) 1–16
16