# Dynamics of senses of new physics discourse: co-keywords analysis

Yurij L. Katchanov<EMAIL_ADDRESS>Institute for Statistical
Studies and Economics of Knowledge, National Research University Higher School
of Economics 20 Myasnitskaya Ulitsa, Moscow 101000, Russian Federation  Yulia
V. Markova<EMAIL_ADDRESS>American Association for the Advancement
of Science 1200 New York Ave NW, 20005, Washington, DC, USA

###### Abstract

The paper presents a longitudinal analysis of the evolution of new physics
keywords co-occurrence patterns. For that, we explore the documents indexed in
the INSPIRE database from 1989 to 2018. Our purpose is to quantify the
knowledge structure of the fast-growing subfield of new physics. The
development of a novel approach to keywords co-occurrence analysis is the main
point of the paper. In contrast to traditional co-keyword network analysis, we
investigate structures that unite physics concepts in different documents and
bind different documents with the same physics concepts. We consider the
structures that reveal relationships among concepts as topological and call
them “physics senses”. Based on the notion of trajectory mutual information,
the paper offers clustering of physics senses, determines their period of
life, and constructs a classification of senses’ “authority”.

###### keywords:

Co-occurrence analysis , Keywords , New physics , INSPIRE

††journal: Journal of Informetrics

##  1 Introduction

Throughout the modern history of physics, high energy physics (or HEP for
convenience) has been and remains one of the main driving forces of conceptual
and methodological innovations. It is also a front-runner in the development
of scientific tools for publication and communication. E.g., the
INSPIRE111https://inspirehep.net database is the leading publication platform
indexing HEP publications (Hecker, 2017). All these factors make HEP an
interesting subject of study inside the science of science approach (Fortunato
et al., 2018) (hereafter, abbreviated as SciSci). One of the principal
objectives of SciSci is to comprehend the emergence of new knowledge, and HEP
can provide a first-choice object of research in this regard.

The somewhat bureaucratic term of “the Standard Model” traditionally defines
the monumental achievement of the twentieth-century HEP—the product of efforts
by thousands of researchers spanning over 50 years (Langacker, 2017). The
Standard Model constitutes a contemporary view of particle physics that
combines theoretical and experimental results of research into the properties
and interactions of particles. With no deviations from the Standard Model
discovered to date, it remains on the cutting edge of our experimentally
proven understanding of the fundamental laws of the microworld. However, there
has to be a new physics beyond the Standard Model—something that researchers
have been working to uncover for decades now. This smart term is used to
describe the fundamental layer of the physics reality at the core of the
structure of the universe, destined to replace the Standard Model as the
leading theory.

New physics is not a specific theory, but rather a collective term for
everything that is at variance with the Standard Model (Ghosh et al., 2020).
Judging by abundant circumstantial evidence, new physics does exist but
remains hard to define. For that reason, a multitude of various methods have
been applied in search of its manifestations. Spanning decades, these
concentrated and diverse experimental and theoretical quests for new physics
make it into a unique testing ground for the development of new methods of the
quantitative study of science.

This paper presents a longitudinal analysis of the evolution of new physics
keywords co-occurrence patterns based on the documents indexed in the INSPIRE
database from 1989 to 2018. We consider keywords co-occurrence structure as an
operationalization of the new physics knowledge structure. These patterns
reveal the relationships between new physics knowledge entities.

Our purpose is to quantify the knowledge structure of the fast-growing new
physics’ subfield. The issue addressed in the paper is the development of a
novel approach to keywords co-occurrence analysis.

The remainder of this paper is organized as follows. Section 2 refers to
related studies on a given topic. Section 3 presents the methodological
framework. Section 4 explains the data used in our analyses. Section 5 reports
our main results and generalizes the findings of the paper. Finally, in
Section 6, we draw the conclusions.

##  2 Background

Quantitative research on physics, physicists and the discourse of physics has
by now become a well-established tradition (Sinatra et al., 2015; Colavizza
and Franceschet, 2016; Battiston et al., 2019; Sun and Latora, 2020). It
allows for a shift from descriptive analysis and general arguments to more
keen and detailed research on the evolution of physics as a scientific field
(Liu et al., 2019). The availability of extensive publication databases and
advanced data processing tools open the door to new interdisciplinary research
opportunities. Take, e.g., the study of the evolution of research interests or
physics subject areas (Jia et al., 2017; Aleta et al., 2019; Zeng et al.,
2019; Coccia, 2020), physics research institutions (Katchanov et al., 2016),
careers pathways (Deville et al., 2014; Petersen, 2018; Xing et al., 2019), or
strategies and talent of physicists (Pluchino et al., 2019; Tripodi et al.,
2020). Due to its nature, physics has its own information environment, and
researchers have access not only to general commercial databases like Web of
Science or Scopus but also to specific physics publications repositories like
arXiv, INSPIRE, or APS. Thanks to the openness of these specialized data
sources, researchers may use the full range of quantitative methods to study
physics.

Quantitative analysis of physics concepts provides a good opportunity to
explore the relations between external classifications of publications and the
outcomes of community detection algorithms (Palchykov et al., 2016). The
authors demonstrated that classifications made by authors had some
discrepancies from automatically retrieved communities. These differences may
carry important information about relations between physics fields that the
external observer does not realize. M. Krenn and A. Zeilinger used a large
corpus of publications covering about a hundred years to train a neural
network and try to predict future trends in physics (Krenn and Zeilinger,
2020). A large body of literature opens the way to interesting studies of
cognitive aspects of science. Thus, S. Milojević compared the growth of
scientific literature and the lexical diversity of research articles titles.
She found that periods of fast cognitive growth did not coincide with the rise
of publication growth (Milojević, 2015).

Recent advances in text analysis methods have facilitated detailed
quantitative research on physics concepts. Thus, M. Herrera, using
publications and PACS categories, built physics fields and traced their
evolution for a period of $20$ years (Herrera et al., 2010). M. Fontana et al.
(Fontana et al., 2020) analyzed more than $230$ thousand articles published in
$8$ journals of the American Physical Society and $2.4$ million citations to
measure novelty and interdisciplinarity. R.K. Pan et al. studied the evolution
of interdisciplinary research in physics (Pan et al., 2012). They constructed
a network of publications and showed that interactions between physics fields
were increasing, and the area of interdisciplinary research was growing. Using
publications data spanning more than a century, M. Perc explored patterns and
trends of physics terminology. The analysis revealed that the evolution of
scientific paradigms in physics was governed by principles of self-
organization (Perc, 2013).

Machine learning methods gained popularity in SciSci in recent years (Xu et
al., 2020a). Thus, K. McKeown et al. (McKeown et al., 2016) presented a system
that predicts the impact of a physics concept; M. Chinazzi et al. (Chinazzi et
al., 2019) mapped research space in physics using the American Physical
Society publications database; A. Palmucci et al. (Palmucci et al., 2020)
suggested a representation for the relative movements of physics’ domains in a
multi-dimensional space based on the PACS codes.

Network analysis has a crucial part to play in the SciSci (Zeng et al., 2017)
these days. Given its universal scope, network analysis allows the study of
various aspects of physics as a scientific field. In recent years, a wealth of
research has been produced on co-authorship networks in physics (Heiberger and
Wieczorek, 2016; Silva et al., 2019), co-citation networks (Zhang et al.,
2013; Renoust et al., 2017; de Arruda et al., 2018), and keyword co-occurrence
networks (Palchykov et al., 2021; You et al., 2021; Karimi et al., 2021). In
the past two decades, the application of co-keyword analysis, first used by
Callon et al. (Callon et al., 1983), was both far-ranging and far-reaching,
covering relevant aspects of research topic networks (Cobo et al., 2011; Yi
and Choi, 2012; Leydesdorff and Nerghes, 2017; Radhakrishnan et al., 2017; Wen
et al., 2017; Behrouzi et al., 2020; Wang et al., 2021).

When studying co-authorship networks, co-citation or co-keyword networks,
graph theory is often perceived as a mathematical framework underlying the
SciSci (Mingers and Leydesdorff, 2015). At the same time, quantitative science
studies are evolving, and so is their mathematical apparatus. In a number of
papers, a depiction produced with customary network tools is supplemented with
that created with tools from applied topology. E.g., (Salnikov et al., 2018;
Christianson et al., 2020) use a simplicial complex approach to word co-
occurrences in order to explore the conceptual landscape of mathematical
publications. A. Patania et al. (Patania et al., 2017) develop simplicial
descriptions of publications and persistent homology to track the growth and
development of collaborations based on co-authorship data.

##  3 Methods

###  3.1 Problem formulation

We apply the co-word analysis technique in order to map the structural
dynamics of the “new physics” subfield. This analysis has proven effective for
the study of relationships and interrelationships among scientific concepts
(van Raan, 2019). The basic principle of co-word analysis is to characterize
the underlying texts through the use of co-occurrence networks consisting of
nodes that represent the keywords and edges between them (Callon et al.,
1991).

Physics knowledge is a complex system, at its basic level consisting of
physics concepts and their interactions. From the SciSci point of view, the
complexity of the physics knowledge system largely results from the variety of
interactions between concepts. The network approach focuses on the analysis of
pairwise relationships between concepts (terms, keywords), invariably reducing
multidimensional structures and losing information. Nevertheless, the recent
decades saw a great number of intellectual structures described successfully
as networks where co-occuring pairs of concepts are connected by links (Li et
al., 2016; Wen et al., 2017; Radhakrishnan et al., 2017; Lozano et al., 2019).
These achievements are prompting the quantitative science studies to develop
new methods of analysis (Cheng et al., 2020; Joslyn et al., 2021).

The scientometrics community is in agreement that system of well-separated and
well-connected clusters representing keywords in a co-occurrence network can
ensure an adequate description of a research discourse (van Eck and Waltman,
2014; Hosseini et al., 2021). When such network representation of scientific
discourse is assumed, a strong assumption is made: the overall interplay among
the concepts is suggested to be completely described by pairwise co-occurrence
data. In reality, however, the constraints in a co-occurrence network may
prove to be fairly weak, and so the identification of the true structure of
research themes may require that more complex interactions are taken into
account that help reveal more interplay among physics concepts. The problem of
the context in which a keyword co-occurrence network is generated was first
described in (Bornmann et al., 2018) and was studied in more detail in (Cheng
et al., 2020). However, the proposed heuristic solutions concern the use a
citation network and are beyond the scope of keyword co-occurrence analysis.

Research communication has a distinguishing feature, which is that all
components of the system influence each other in a direct or indirect way.
Each scientific publication features several concepts simultaneously, and one
should not assume _a priori_ that their interactions can be explained in terms
of dyads (see details in (Battiston et al., 2020)).

###  3.2 Physics sense: definition

The sense of a physics concept cannot be fully explained by its formal
definition. Its understanding is greatly improved by a series of examples
illustrating its use in documents. For working physicists, these examples
represent both motivation and the sense of a concept. It is for this reason
that exploring the co-occurrence of physics concepts can prove to be a
significant element in the study of the cognitive structure of physics. The
sense helps to reveal the essence of a physics concept in the research
context. A physics sense determines the place of the concept in scientific
communication. The sense introduces the “whole—part” relationship, making the
concept essential as part of a whole which is the system of a physics
discourse.

Specification of documents that mention a physics concept is imperative for
the understanding of its physics sense. We consider a physics sense in the
framework of an “information field”. The concept of the field emerges
naturally as one analyzes the forming of a physics sense within a system of
relationships between physics concepts. Information field is a holistic
structure that describes both direct and indirect mutual influences of physics
concepts in documents. In this paper, the intuitive viewpoint is that an
information field that determines a physics sense is a topological space of
physics concepts and their properties.

Following the generally accepted definition (see, e.g., (Klement, 2001;
Deleuze, 1993)), one understands physics sense as a proposition including a
set of physics concepts. The intellectual value of a concept is the result of
its position in a complex network of semantic relations. The study of physics
discourse in its entirety involves a structural comparison of the concepts.
Our methodology supposes that to study physics concepts, one needs to
construct statistical relations between them. This type of relations we call
“sense”.

Figure 1: Construction of physics sense. Part A shows how the documents are
related together in the orbit $O_{A}$ of the concept
$x_{{\alpha}0}\equiv{c_{1}}$ (marked with a dark circle). Part B depicts the
support $X_{A}$ of a sense $A$. Part C displays the sense $A$.

For our further reasoning we will need auxiliary concepts. Let us refer to the
set of all documents that mention the physics concept $x_{{\alpha}0}$ as the
orbit $O_{A}$ of that concept. We will understand the spectrum $S_{A}$ of
physics concept $x_{{\alpha}0}$ as a set of physics concepts
$x_{{\alpha}0},x_{{\alpha}1},\ldots,x_{{\alpha}k}$ corresponding to the orbit
$O_{A}$. (Clearly, $x_{{\beta}0}\in S_{A}\Leftrightarrow x_{{\alpha}0}\in
S_{B}$, etc.) Let $X_{A}$ (with or without indices) be a sample of $M$ physics
concepts that occur in the orbit $O_{A}$ with frequencies
$m_{{\alpha}0},m_{{\alpha}1},\ldots,m_{{\alpha}k}$, such that
$\sum_{\alpha}m_{\alpha}={M}$.

Let $\sigma_{A}$ be a configuration of the set
$X_{A}=\\{x_{\alpha}\\}_{\alpha\in A}$. The configuration $\sigma_{A}$ can be
represented in the form

$(\forall{x_{\alpha}}\in S_{A}\subset
O_{A})\colon\sigma_{A}=\begin{pmatrix}x_{{\alpha}0}&x_{{\alpha}1}&\ldots&x_{{\alpha}k}\\\
m_{{\alpha}0}&m_{{\alpha}1}&\ldots&m_{{\alpha}k}\end{pmatrix}{.}$ (1)

The set $X_{A}$ will be called hereafter the support of a physics sense $A$
(see Fig. 1). A topological structure on the support $X_{A}$ is a collection
$\tau_{A}$ of subsets of $X_{A}$ which includes the empty set $\emptyset$ and
the whole support $X_{A}$ and which is such that: a) the intersection of any
number of elements of $\tau_{A}$ belongs to $\tau_{A}$ and b) the union of any
set of elements of $\tau_{A}$ belongs to $\tau_{A}$ (Singh, 2019). In our
case, the collection of all subsets of $X_{A}$ is a topology on $X_{A}$, and
$\tau_{A}$ is called the discrete topology.

One can think of $S_{A}$ as having been obtained by “identifying” each pair of
equivalent elements of the support $X_{A}$. Hence, the surjective map
$\pi\colon X_{A}\rightarrow{S_{A}}$ carries each point of the support $X_{A}$
to the equivalence class $\omega$ containing it. In the quotient topology
induced by $\pi$, the set $\Omega$ such that $\Omega=\cup_{\alpha\in
X}\omega_{\alpha}$ is called a quotient space of $X_{A}$ (Singh, 2019). For
further details, we refer to A.

In this paper, a _physics sense_ $A$ of concept $x_{{\alpha}0}$ is a finite
topological space $(X_{A},\tau_{A})$—a support $X_{A}$ with a discrete
topological structure $\tau_{A}$. By putting it this way, we symbolically
equate the space $(X_{A},\tau_{A})$ with its focal point—the concept
$x_{{\alpha}0}$. The main point here is to realize that the topological
structure $\tau_{A}$ is the elementary property of physics sense as a whole.
It means this structure refers to the physics concepts collection $X_{A}$, but
not to its parts or connections between these parts.

Knowing the configuration $\sigma_{A}$ Eq.(1) of the physics sense $A$ enables
us to define a state function $h(A)$
$=-\sum{p(\omega)}\ln{p(\omega)}=\sum_{\alpha}p(\alpha)\ln{p(\alpha)}$ (see
Eq. (11))—the topological entropy, where $p(\alpha)$ is the probability of the
realization of the physics concept $x_{\alpha}$ in the orbit $O_{A}$. The
interpretation of entropy $h(A)$ is that it is a measure of the number of
arrangements the physics concepts could be in.

###  3.3 Basic quantities

####  3.3.1 Dimension

The quantity of dimension $\dim(A)$ Eq. (8) is the number of the physics
concepts in the spectrum $S_{A}$. It is possible to interpret a dimension as a
number of degrees of freedom of the information field of a physics sense. A
more straightforward interpretation of a dimension is that $\dim$ constitutes
the number of information channels that connect a single sense with others.

####  3.3.2 Trajectory mutual information

Thus, we attempted to describe the sense of a physics concept through a
configuration of the information field where the concept functions in the
discourse. In order to characterize the thematic structure of physics
discourse itself, we need to examine the configuration of the field induced by
all relevant physics senses. Since we have conceptualized a physics sense as a
topological space of concepts within a single orbit, it is only natural to
consider the interaction of physics senses as a topological space constructed
on the intersection of respective orbits. If the orbits of physics concepts
intersect, it indicates that the concepts share a common context and we can
evaluate its significance using the concept of topological entropy.

Let us consider physics senses $A$ and $B$, i.e., topological spaces
$(X_{A},\tau_{A})$ and $(X_{B},\tau_{B})$ with orbits $O_{A}$ and $O_{B}$ and
spectra $S_{A}$ and $S_{B}$ so that $x_{\alpha 0}\in S_{B}$ (and,
respectively, $x_{\beta 0}\in S_{A}$). On the intersection of orbits $O_{A}$
and $O_{B}$ as well as spectra $S_{A}$ and $S_{B}$, a topological space
$(X_{AB},\tau_{AB})$ is defined that describes the interplay between the
physics senses $A$ and $B$. One can easily define the support
$X_{AB}=X_{A}\cap X_{B}$

$\left(O_{A}\cap{O_{B}}\right)\left(\forall{x_{\alpha\beta}}\in{S_{A}\cap{S_{B}}}\right)\colon
X_{AB}=\left\\{x_{\alpha\beta}\right\\}_{\alpha\beta\in{AB}},$ (2)

the discrete topology $\tau_{AB}$ on $X_{AB}$, and the quotient space
$(\Omega_{AB},\varpi_{AB})$ as shown in A.

We can relate the trajectory of the physics sense
$A^{T}=\left\\{{A^{1},A^{2},\ldots{,}A^{l}}\right\\}$ in time $T=\\{{t}\\}$
with the trajectories of all other physics senses $\\{{B^{t}_{\gamma}}\\}$
from the totality $\Gamma\setminus\left\\{{A^{t}}\right\\}$. Our goal here is
to characterize information transmission between senses’ trajectories via
their path mutual information

$\mu\left(A^{T}\right)=\sum_{t\in
T}\sum_{\gamma=1}^{\dim(A)}p_{(A^{t},B^{t}_{\gamma})}(\alpha,\beta)\ln\frac{p_{(A^{t},B^{t}_{\gamma})}(\alpha,\beta)}{p_{A^{t}}(\alpha)p_{B^{t}_{\gamma}}(\beta)},$
(3)

where $p_{(A,B)}(\cdot)$ is the joint probability mass function of $A$ and
$B$, $p_{A}(\cdot)$ and $p_{B}(\cdot)$ are the marginal probability mass
function of $A$ and $B$, respectively. The quantity $\mu\left(A^{T}\right)$
will be called the trajectory mutual information (TMI, from now on) of physics
sense $A$.

To derive a distribution of probabilities $\mu\left(A^{T}\right)$, we must
characterize the principle whereby out of the set of all possible quotient
spaces $\\{{\left(\Omega_{A},\varpi_{A}\right)}\\}$ (see A), we select those
that are realized. The comparison of quotient topological spaces we have drawn
here allows for formulation of an extreme principle for the changeability of a
physics sense: from the given state $\left(\Omega_{A},\varpi_{A}\right)$ the
sense tends towards a state $\left(\Omega_{B},\varpi_{B}\right)$ with the
maximum value of ${\mathsf{P}}\\{{B}\\}$ (Eq. (10)). We now admit that the
actual trajectory of the physics sense represents a most probable trajectory.
This is equivalent to the transition to a physics sense with the maximum value
of the TMI. It is easy to derive the Fréchet extreme distribution for TMI.

The challenge facing us at this point is to approximate the probability
distribution function of $\mu\left(A^{T}\right)$. The quantity of TMI
represents a sum total of the information activity of a physics sense,
understood as a sum of its own activity and that of the senses whose orbits
intersect its own. We are dealing here with values of the sums of the form

$Z_{\gamma}=\sum_{\gamma=1}^{\dim(A)}\zeta_{\gamma}=\sum_{\gamma=1}^{\dim(A)}\sum_{t\in
T}p_{(A^{t},B^{t}_{\gamma})}(\alpha,\beta)\ln\frac{p_{(A^{t},B^{t}_{\gamma})}(\alpha,\beta)}{p_{A^{t}}(\alpha)p_{B^{t}_{\gamma}}(\beta)},$

such that $\mu\left(A^{T}\right)=Z_{\gamma}$. Since we postulated that the
true trajectory of the physics sense $A$ is corresponding to the maximum of
$\mu\left(A^{T}\right)$, we must consider the limit behavior of extremes

$(d\rightarrow\infty)\left(\dim(A)\rightarrow\infty\right)\colon\widehat{Z}_{d\gamma}=\max_{1\leq\delta\leq
d}\sum_{\gamma}\zeta_{\delta\gamma}.$ (4)

Let the variables $(\delta,\gamma\geq 1)\colon\zeta_{\delta\gamma}$ be
mutually independent and have a common distribution $F$. We assume that $F$
belongs to the class of subexponential distributions (Foss et al., 2013): the
convolution tail $\overline{F{\ast}F}(u)$ is equivalent to $2\overline{F}(u)$
as $u\to\infty$, where $\overline{F}(u)=1-F(u)$. Assume also that $F$ belongs
to the class of distributions with regularly varying tails, i.e.,
$\overline{F}(u)\sim u^{-a}L(u)$, $u\to\infty$, where $a>0$ and $L(u)$ is a
slowly varying function (Bingham et al., 1987). This is not necessarily true,
but it represents a reasonably good approximation in a large number of
subsystems of science (Zeng et al., 2017). Under the conditions considered
here, can be obtained that the family of extremes similar to
$\widehat{Z}_{d\gamma}$ has well-known Fréchet extreme distribution
$\mathsf{P}{\\{Z^{\prime}\leq z\\}}\rightarrow\exp(-z^{-a})$, $0<a<2$
(Lebedev, 2005). It means, among other things, that bigger TMI values are
normally a result of individual senses having higher information activity
rather than a combined activity of a large number of senses.

###  3.4 Assistant quantities

####  3.4.1 Similarity

To derive similarity between $(X_{A},\tau_{A})$ and $(X_{B},\tau_{B})$, one
should keep in mind that the value Eq. (10) characterizes the probability of a
particular state of a physics sense. It follows that the similarity between
physics senses $A$ and $B$ will be given by

$s_{AB}=\ln\frac{p_{(A,B)}(\alpha,\beta)}{p_{A}(\alpha)p_{B}(\beta)}.$

This expression is nothing but the point-wise mutual information.

####  3.4.2 Normalized entropy

To suppress the dependence of the entropy $h(A)$ on $k$ in Eq. (11) and
keeping in the mind Eq. (8), we consider the normalized entropy $h_{n}(A)$
defined as followed:

$h_{n}(A)=\frac{h(A)}{\ln{\dim(A)}}.$ (5)

####  3.4.3 Complexity

Following (López-Ruiz et al., 1995; Martin et al., 2006; Rosso et al., 2007),
we computed the values of the so-called statistical complexity

$c(A)=h_{n}(A)d(A),$ (6)

where the “disequilibrium” $d(A)$, which measures the deviation of the
probability distribution $\mathsf{P}\\{{A}\\}$ from uniform probability
distribution, is defined as

$d(A)=\sum_{\alpha=1}^{\dim(A)}\left(p(\alpha)-\frac{1}{\dim(A)}\right)^{2}.$

By construction, the quantity $d(A)$ vanishes for probability distributions
$\mathsf{P}\\{{A}\\}$ that correspond to absolute order and maximal
randomness.

####  3.4.4 Transitivity

The fraction of pairs of physics senses that are themselves intersected
defines the sense’s transitivity $tr(A)$ (Newman, 2018):

$tr(A)=\frac{\\#\>\mathsf{pairs\>of\>senses\>whose\>orbits\>intersect\>with\>}O_{A}}{\\#\>\mathsf{pairs\>of\>senses}}.$
(7)

The transitivity is an important quantity, which informs on the density of a
sense’s information field.

####  3.4.5 Citation rate

To construct the citation rate of the concepts, we used a three-year citation
window and fractional counting. It means the citation rate of the concept $x$
of document $y$ is a share of all other concepts of this document. Thus the
total concept citation rate is the sum of all its fractional citations in all
documents where it is used.

##  4 Data

The publication data for this study comes from the INSPIRE database. INSPIRE
is a leading information system in HEP (Xu et al., 2020b) that provides open
access to more than $1$ million scientific publications. It continues the
SPIRES physics literature information system. The INSPIRE database is the
result of close collaboration between such institutes as CERN, DESY, Fermilab,
and SLAC. It brings together information from arXiv.org, NASA–ADS, PDG, and
other HEP publishers. In addition to documents, INSPIRE integrates information
about authors, institutes, journals, conferences, and jobs in the field of
HEP. This complex integrated system provides opportunities to analyze
information processes in HEP from different points of view. The INSPIRE system
is of interest not only to physicists but also to researchers in quantitative
science studies (Zingg et al., 2020). Researchers who have worked with INSPIRE
datasets pointed out that the data is well-curated and easily accessible (Jang
and Ko, 2019; Strumia and Torre, 2019).

Another significant aspect of the INSPIRE information system is the keywords
that accompany documents. The keywords are automatically extracted with the
help of regular expressions from full-text papers by the BibClassify
extraction algorithm222For details,
http://cds.cern.ch/help/hacking/bibclassify-extraction-algorithm, last
accessed on 08 December 2020.. This algorithm draws upon a controlled
vocabulary resulting from the HEP Index (HEPI). The HEPI thesaurus is
maintained by the German High Energy Physics Laboratory DESY since 1963. It is
prepared by the subject specialists and updated occasionally. The HEPI
thesaurus is actively used by the principal HEP institutes and information
databases. The BibClassify algorithm extracts two types of keywords: single
keywords and composite keywords. Composite keywords are combined with several
single keywords. If single keywords are placed next to each other, they form a
composite keyword. The words of composite keywords are separated by a colon
e.g. “Higgs particle: mass” or “CP: violation”. The final keyword list
consists of top $20$ first best single and composite keywords. These keywords
serve as an information base for the study.

Our dataset counts documents with the term “new physics” in titles, abstracts,
or keyword lists published in 1989—2018333Data was retrieved from the INSPIRE
database in October 2019. Processing of the XML dump, data preparation, and
statistical analysis were performed with the R programming language. Citation
data was downloaded from the INSPIRE database in August 2021.. To make the
sample more stable and consistent, we excluded some irrelevant documents and
concepts. First of all, we removed the documents featuring only one physics
concept or no concepts at all. Secondly, we removed keywords having errors due
to the indexing procedure and keywords, denoting document types like “book”,
“thesis”, “report”, and so on. After these procedures, the final dataset
counted $19{,}830$ documents and $8{,}822$ physics concepts. The collection of
documents thusly obtained was subjected to further statistical analysis.

##  5 Results and discussion

###  5.1 Statistical features of new physics discourse

Fig. 2A illustrates the number of new physics unique concepts indexed in
INSPIRE in the years 1989—2018. It can be seen that the figures gradually
increase from between 1989 and 1995. Then they show some fluctuation to
proceed with rapid growth after 2006. In 2006—2015, one can note a quick rise
of numbers and even a spike in 2013 and 2015. In 2015, the number of concepts
reached its maximum value of $4259$ items, followed by the curve dropping
sharply to $3557$ concepts in 2018. For the whole 30–year period, the number
of unique concepts rises from $373$ to $4259$ items, and the curve fits the
exponential regression line (cf. (Milojević, 2015)).

Figure 2: Features of new physics discourse.
(A) Dynamics of new physics concepts in 1989—2018. The equation of the
regression curve: $\\#\>\mathsf{concepts}=312.612\cdot\exp(0.087t)$
$\left(R^{2}=0.954,p=0.000\right)$. Dashed vertical lines indicate the
generations of physics senses (see 5.3 “Generations of physics senses”).
(B) The approximation of the probability density function of the dimension: a
Fréchet distribution with parameters $\alpha=1.1978,\>\beta=34.672$
(Kolmogorov–Smirnov statistic $0.0137$.)
(C) The approximation of the probability density function of the citation: a
Pareto Type II distribution with parameters $\alpha=0.7986,\>\beta=3.0318$
(Kolmogorov–Smirnov statistic $0.0145$.)
(D) The approximation of the probability density function of the TMI: a
Fréchet distribution with parameters $\alpha=0.8947,\>\beta=382.26$
(Kolmogorov–Smirnov statistic $0.0128$.)

Fig. 2B, Fig. 2C, and Fig. 2D show approximations of the probability density
functions of dimension, citation, and TMI, respectively. One can spot at a
glance the qualitative difference of the dimension and the TMI probability
distributions from that of citation. Indeed, the probability density functions
in Fig. 2B and Fig. 2D are of Fréchet type, whereas Fig. 2C shows a power-law
distribution function. It seems only natural that the citation distribution
can be fitted by a Pareto distribution (Zeng et al., 2017; Fortunato et al.,
2018). The Fréchet distribution observed for dimension is untypical of
scientometric quantities of this kind (cf. (Börner and Milojević, 2019)). A
quantity homologous to dimension, the “node degree” is normally distributed
according to a power law (Zeng et al., 2017; Palchykov et al., 2021). We have,
in turn, predicted the Fréchet distribution for TMI.

Despite the obvious differences between the dimension and the TMI probability
distributions, on the one hand, and that of citation, on the other, they share
an internal similarity. The fact is, the Pareto distribution belongs to the
maximum domain of attraction of the Fréchet distribution (Foss et al., 2013).
The asymptotic behavior of citation distribution is similar to that which
defines the distribution of dimension and TMI. In particular, larger values of
citation, dimension, and TMI are achieved not through a larger number of
physics senses but due to individual physics senses with higher activity.

The Fréchet extreme distribution is a distribution of maximums; it expresses
relatively enduring structures of the dimension and the TMI. These structures
are characterized via the probability of extreme events (exceedances of a high
level). Both the dimension Eq. (8) and the TMI Eq. (3) are high outliers by
nature, so it is only logical that they are described by the Fréchet extreme
distribution. This distribution is a pattern or arrangement of physics senses;
we can surmise, therefore, that it is the trends defined by record values that
form the basis of a nascent new physics discourse.

The citation structure of physics senses is, in turn, described by the Pareto
distribution. This distribution signifies the fact that very few physics
senses possess an extremely high citation rate, while very many demonstrate an
extremely low one.

Table 1: The Top–30 physics senses of the PageRank value PageRank | Physics sense | $\dim$
---|---|---
1 | numerical calculations | 5762
2 | supersymmetry | 3962
3 | cp: violation | 3523
4 | new physics: search for | 3040
5 | p p: scattering | 3007
6 | higgs particle: mass | 2506
7 | electroweak interaction | 2398
8 | numerical calculations: monte carlo | 2351
9 | neutral current: flavor changing | 2324
10 | minimal supersymmetric standard model | 2287
11 | atlas | 2238
12 | electron positron: annihilation | 2119
13 | cms | 2138
14 | quantum chromodynamics | 2052
15 | dark matter | 2099
16 | effective lagrangian | 2013
17 | effective field theory | 2099
18 | neutrino: mass | 2055
19 | p p: colliding beams | 1902
20 | neutrino: oscillation | 1823
21 | higgs particle: coupling | 1781
22 | lepton: flavor: violation | 1725
23 | lhc-b | 1644
24 | coupling: yukawa | 1784
25 | numerical calculations: interpretation of experiments | 1512
26 | angular distribution: asymmetry | 1600
27 | electroweak interaction: symmetry breaking | 1643
28 | interpretation of experiments | 1504
29 | batavia tevatron coll | 1608
30 | top: pair production | 1592
Figure 3: Scatterplot of PageRank against the TMI. Figure 4: Scatterplot of
$\\#\>\mathsf{citations}$ against the TMI. The equation of the regression
curve of $\\#\>\mathsf{citations}$ is the following:
$\\#\>\mathsf{citations}=1{.}547\cdot{10}^{-8}\cdot\mu^{2}+0{.}11\cdot\mu-3{.}278\>\left(R^{2}=0.903,\>p=0.000\right)$.

###  5.2 PageRank and TMI

In order to study the dynamics of the “importance” and “authority” of physics
senses in the years 1989–2018, the PageRank value of physics senses was
calculated with the use of the $\mathsf{igraph}$ software package. The Top–30
physics senses of the PageRank value are listed for illustration and
comparison, as shown in Table 1.

Fig. 3 shows PageRank as a linear function of the TMI. The relationship
between the two quantities is described by the regression equation
$\mathsf{PageRank}=b\cdot\mu\left(A^{T}\right)+a$, where
$b=2.187\cdot{10^{-8}}$, $a=5.157\cdot{10^{-5}}$ ($R^{2}=0.958$, $p=0.000$).
The importance of physics senses is directly proportional to their measure of
the mutual dependence: the stronger the sense is connected with others, the
higher its authority. This means that the authority of a sense is determined
by its interconnections with a totality of physics senses. This
interrelationship between PageRank and the TMI illustrates the structure of
the informational process of new physics.

Table 2: The characteristics of generations Generation | Years | Total number | Number of | Percent of
---|---|---|---|---
|  | of senses | new senses | new senses
I | 1989—1994 | 1363 | 1363 | 15.4
II | 1995—2007 | 4071 | 3030 | 34.3
III | 2008—2012 | 4952 | 2274 | 25.8
IV | 2013—2018 | 7183 | 2156 | 24.4

###  5.3 Generations of physics senses

“Generations” offer a convenient method of classifying physics senses. We
chose the doubling of the number of senses in 1995, 2008, and 2013 as the
criterion for identifying generations. Following this way, we got four
generations (see Table 2 and Fig. 2A). A physics sense belongs to a particular
generation if its first occurrence happens in the period corresponding to that
generation. Thus, all senses that occur in the years 1989–1994 are generation
I senses. Generation II senses are those that occurred for the first time in
documents published in 1995–2007, etc. Table 2 shows that the total number of
senses increases from generation to generation, although new senses become
less common over time. Of the total dataset, $15.4\%$ are generation I,
$34.3\%$ generation II, $25.8\%$ generation III, and $24.4\%$ generation IV
senses. Analysis of the generations helped to identify the “core” of physics
senses—senses that occur in the discourse of new physics throughout all four
generations. Such senses amount to $7.3\%$ of the total dataset. They comprise
the most frequently occurring physics senses with connections to many other
senses. Their average $\dim$ is $471$. For reference, an average $\dim$ for
senses that occur for the duration of three generations is $204$, whereas
senses occurring for the period of one or two generations have an average
$\dim$ of $34$ and $74$, respectively.

Figure 5: Scatterplot of the dimension against the transitivity. The equation
of the regression curve of $\dim$ is the following:
$\dim=150{.}418\cdot{tr^{-1}}-215{.}254\>\left(R^{2}=0.895,\>p=0.000\right)$.

###  5.4 TMI, dimension, number of citations, and number of documents

Fig. 4 through Fig. 7 show interconnections between quantities essential for
understanding the structure of physics senses.

  * 1.

In Fig. 4, we see TMI as a quadratic function of the number of citations. This
is the way in which the Fréchet extreme distribution manifests itself:
maximums of different values belong to the same physics senses. Fig. 4
indicates that, for the orbit of the sense, its measure of intersection with
other orbits directly depends on the citation rate of the sense. This is how
the difference between the Fréchet distribution of the TMI and the Pareto
distribution of the citation expresses itself: the quadratic dependence stems
from the fact that the citation is distributed more unevenly than the TMI.

  * 2.

Fig. 5 shows a dimension as an inverse function of transitivity. This is in
line with results already published in the literature (Newman, 2018). This can
be easily interpreted: the higher the value of the dimension, the bigger the
number of pairs of physics senses whose orbits intersect with the orbit of the
given sense and, therefore, the lower the transitivity value (see Eq. (7)).
Since physics senses with a high dimension value are few, and those with a low
one are many (the fact that determines the Fréchet extreme distribution of the
dimension), the chart in Fig. 5 resembles a hyperbola.

  * 3.

A TMI—$\dim$ linear graph in double logarithmic coordinates is shown in Fig.
6; and Fig. 7 has a TMI—$\\#$ documents linear graph in double logarithmic
coordinates. It is worth emphasizing that the straight line in double
logarithmic coordinates indicates a power function. The level of TMI changes
in proportion to the dimension to the power of $1.327$ and the number of
documents to the power of $1.059$. This fact indicates heavy tails of TMI
distributions. Moreover, the power functions TMI($\dim$) and TMI($\\#$
documents) are stable under changes in the scales of TMI, $\dim$, and number
of documents.

  * 4.

Fig. 6 shows the influence of the topology of physics senses on their
informational value in the new physics subfield. It appears that knowledge
structure is a function of the dimension. To put it simply, the principal
topological characteristic of a physics sense determines its position in the
information structure of new physics. Notably, the TMI grows faster than the
linear function of the dimension. It follows that physics senses with high TMI
values play a disproportionately more significant role, lending a hierarchical
character to the knowledge structure of the new physics subfield.

  * 5.

Importantly, extreme values in all four charts belong to the same physics
senses as listed in Table 1. These Top–30 physics senses determine, to a
certain extent, the properties of the scholarly discourse of new physics.

Figure 6: Scatterplot of the TMI ($\ln$) against the dimension ($\ln$). The
equation of the regression curve of $\ln\mu$ is the following:
$\ln\mu=1{.}327\cdot\ln\dim+1{.}245\>\left(R^{2}=0.981,\>p=0.000\right)$.
Figure 7: Scatterplot of the TMI ($\ln$) against $\ln\\#\>\mathsf{documents}$.
The equation of the regression curve of $\ln\mu$ is the following:
$\ln\mu=1{.}059\cdot\ln\\#\>\mathsf{documents}+4{.}615\>\left(R^{2}=0.936,\>p=0.000\right)$.
Figure 8: Space of physics sense clusters.
A. The clusters of physics senses in the TMI—the dimension plane.
B. The clusters of physics senses in the $\\#\>\mathsf{citations}$—the
$\\#\>\mathsf{documents}$ plane. Figure 9: Scatterplot of the statistical
complexity $c(A)$ against the normalized entropy $h_{n}(A)$. Figure 10: The
generations of physics senses in the statistical complexity $c(A)$—the
normalized entropy $h_{n}(A)$ plane.

###  5.5 Clusters of physics senses

We calculated the values of $s_{AB}$ for all pairs of physics senses and
carried out the procedure of sense clustering with the help of the function
$\mathsf{cluster.walktrap}$444See $\mathsf{igraph}$ software package.. This
function uses a random walk algorithm to find dense subgraphs in complex
networks (see details in (Masuda et al., 2017)). The algorithm assumes that
short random walks are in the same community. Based on this, the algorithm
tries to find objects that form subgraphs within one network.

Obtained were $12$ clusters Fig. 8. We interpret the clusters on the basis of
average values calculated for physics senses that form the clusters. Analysis
of the behavior of the senses with regard to TMI and $\dim$ allows us to
characterize their information interaction with other senses (see Fig. 8A).
The closer senses get in terms of their information interaction, the bigger
their chance of ending up in the same cluster and sharing other similarities.
Fig. 8B shows the interdependence between the citation rate of the senses in a
cluster and the number of documents where they occur. The figure shows that
the first three clusters are comprised of senses with the highest citation
rate and the most frequently occurring in the studied documents. The same
three clusters contain senses with the most authority. It is these clusters
that boast a concentration of the PageRank’s Top–30 senses Table 1. Of the
latter, the majority can be found in the second cluster: it includes ten of
the Top–30 senses. For reference, clusters from 8 to 12 do not feature a
single sense with a high PageRank value. Clusters 1 through 3 comprise senses
with the highest TMI and $\dim$ values.

As becomes clear from Fig. 8A and Fig. 8B, the set of clusters splits into
three groups. The first group contains basic physics senses. Due to their high
$\dim$ and TMI values, they consolidate the scholarly discourse about new
physics. The first group of clusters forms a key part of the HEP subfield that
relates to new physics. The second can be interpreted as the mainstream of the
discourse; it consists of the main physics senses. Main discourse events can
occur within the group’s perimeter. Finally, the third group of clusters
appears largely unstructured and chaotic. The most likely reason is that it is
formed by physics senses that are either peripheral or emerging or declining
(cf. (Hosseini et al., 2021)).

###  5.6 Complexity and entropy

The statistical complexity $c(A)$ and the normalized entropy $h_{n}(A)$ allow
us to study the structural properties of a physics sense. The quantity $c(A)$
indicates the “state of disorder” such that senses with extremely ordered or
disordered structures will receive lower scores, while senses with non-trivial
structures—higher ones. At the same time, $h_{n}(A)$ measures the degree of
diversity of the support $X(A)$ of a sense and stochasticity. Thus, the
statistical complexity $c(A)$—the normalized entropy $h_{n}(A)$ plane (see
Fig. 9) partially reflects the relationship of the notions of
disorder/complexity versus order/simplicity with regard to physics senses.
Although these quantities are loosely connected statistically (the Pearson
correlation coefficient is equal to $0.557$ between $c(A)$ and $h_{n}(A)$),
their relationship partially encodes structural features of each physics
sense. For example, smaller values of $h_{n}(A)$ and larger values of $c(A)$
correspond to senses that consist of regular sets of recurring ordinal
concepts. By contrast, senses that feature a significant diversity of concepts
grouped into stochastic sets will generate larger values of $h_{n}(A)$ and
smaller values of $c(A)$.

$75\%$ of physics senses have the value $c(A)\leq 0.045$ and $h_{n}(A)\leq
6.67\cdot 10^{-5}$, i.e., they occur in that domain of the statistical
complexity $c(A)$—the normalized entropy $h_{n}(A)$ plane that can be
identified through the paired notion of order/simplicity. As illustrated in
Fig. 9, the Top–30 physics senses possess the highest values of normalized
entropy: from $0.0159$ for “numerical calculations” to $0,0019$ for
“interpretation of experiments”. However, the values of statistical complexity
observed for the leaders of the rating are far from unprecedented: they range
from $0.2348$ (“neutrino oscillation”) to $0.0988$ (“quantum chromodynamics”).
We may argue that physics senses in the Top–30 rated positions are
characterized by the highest level of disorder and a fairly high complexity
level. It is these structural properties that, on the qualitative level,
explain the position of the central senses in the scholarly discourse of new
physics.

We studied the evolution of physics senses in the statistical complexity
$c(A)$—normalized entropy $h_{n}(A)$ plane. For each physics sense, we
calculated the values of $c(A)$ and $h_{n}(A)$ by years and then averaged the
results by generation. The results are presented for each generation in Fig.
10. We can see that each successive generation demonstrates lower values of
statistical complexity and normalized entropy than their predecessors. This
observation conforms to the overall picture as shown in Fig. 9. As we know,
the vast majority of physics senses occur in the domain of low values of
$c(A)$ and $h_{n}(A)$. At the same time, it follows from Fig. 2A that the
number of physics senses was growing exponentially in time. Therefore, physics
senses with lower values of $c(A)$ and $h_{n}(A)$ belong mainly to younger
generations. In other words, generation I constitutes the semantic core of the
scholarly discourse of new physics.

##  6 Conclusion

Our paper contributes to the literature by addressing the question of
quantitative research on the new physics knowledge structure.

  1. 1.

We develop a new method of keyword co-occurrence analysis that unites physics
concepts of different documents and binds different documents with the same
physics concepts. It is a more general approach than the network one since it
offers a natural framework for quantifying knowledge structure of the new
physics subfield. We consider topological structures that reveal relationships
among concepts and call them physics senses. These senses represent complex
systems of direct and indirect interactions between physics concepts.

  2. 2.

The proposed phenomenological model has the advantage of providing a relevant
quantitative evaluation of the distribution of physics senses without recourse
to a relatively limited scientometric framework. In case of the new physics’
subfield, the Fréchet extreme distribution indicates a high level of
competition between physics concepts.

  3. 3.

Physics senses form certain network clusters. The position of physics sense in
the new physics’ subfield is completely conditioned by the value of dimension
and the belonging to a generation. Our findings indicate that the research
interests of physicists are focused on a limited number of physics senses
which, due to their higher dimension, act as a kind of glue that holds
together a heterogeneous discourse of new physics.

However, some limitations need to be considered. First of all, our results are
limited to the content of the INSPIRE database. However, given its state as
one of the most comprehensive databases in high energy physics, we believe it
provides the most possibly complete picture of new physics discourse. A second
potential drawback of our study is that it depends on the BibClassify
algorithm that extracts keywords from documents. Nevertheless, BibClassify
draws upon the HEP Index maintained by the physics community. This index
manifests the state of physics discourse as the physicists see it. Therefore,
despite the above shortcomings, we believe our study gives a relevant view of
the historical dynamics of new physics discourse.

That various potential studies can be conducted in the future using the
proposed methodology. It would be interesting to carry out a comparison of
several HEP fields: for example, new physics vs. the standard model. Also,
assessment and improvement of the suggested methodology would benefit from the
involvement of new data sources and other text analysis tools.

##  Appendix A Framework

Recall that a partition (or a cluster expansion) of $X$ is a closed disjoint
cover of $X$, and each partition of a set $X$ determines the equivalence
relation on physics concepts: Two elements of $X$ are said to be equivalent if
they belong to the same cluster of the partition. In what follows, we shall
denote by $\Omega$ a fixed cluster expansion of the finite $T_{1}$ topological
space $(X,\tau)$. There is defined a canonical projection $\pi\colon
X\rightarrow\Omega$ of the discrete topological space $(X,\tau)$ onto the set
$\Omega$, i.e. the mapping $\pi$ assign to the physics concept $x\in X$ the
equivalence class $[x]\in\Omega$. We can now assign to $\Omega$ a quotient
topology on $\Omega$ by calling a set $\omega\subset\Omega$ to open if and
only if its preimage $\pi^{-1}(\omega)$ under the canonical projection is open
in $X$ (Singh, 2019). The collection of these sets $\\{{\omega}\\}$ is a
topological structure $\varpi$ in the quotient set $\Omega$, and
$\Omega=\cup_{\alpha\in X}\omega_{\alpha}$,
$\omega_{\alpha}\cap\omega_{\beta}=\varnothing$ when $\alpha\notin\beta$. We
stress that by construction, each cluster $\omega_{\alpha}$ of the cluster
expansion $\Omega$ corresponds uniquely to one physics concept $x_{\alpha}$
according to the configuration $\sigma$ (see Eq (1)), and each cluster
$\omega_{\alpha}$ is indiscrete subspace of $(X,\tau)$.

Here $|\cdot|\colon X\rightarrow\mathbb{N}$ denotes the counting measure, so
$|{X}|=M$ is the number of physics concepts in the support $X$,
$|\omega_{\alpha}|=m_{\alpha}$ is the number of elements in the cluster
$\omega_{\alpha}$, and $\sum_{\alpha}m_{\alpha}={M}$. We can introduce the
corresponding non-strict partial order on $(\Omega,\varpi)$:
${\omega_{1}}\preceq{\omega_{2}}\preceq{\ldots}\preceq{\omega_{k}}$. Here, the
non-strict partial order $\preceq$ on the set $\\{m_{\alpha}\\}$ gives the
non-strict partial order $\preceq$ on $(\Omega,\varpi)$ — on the maximal
$T_{0}$-quotient of $(X,\tau)$. Hence, a finite topological space $(X,\tau)$
is fully described by a non-strict partially ordered set
$\left(\Omega,\preceq\right)$ whose clusters
$\omega_{1},\omega_{2},\ldots,\omega_{k}$ are equipped with multiplicities
$m_{1},m_{2},\ldots,m_{k}$.

The nonempty set
$C_{\Omega}(\omega_{\alpha})=\\{\omega_{\alpha}\in\Omega\colon\omega_{\alpha}\preceq\omega_{\beta}\\}$
is said to be a cone of $\omega_{\alpha}$ over $\left(\Omega,\preceq\right)$.
We can look at this in another way: If $\omega_{\beta}\in
C_{\Omega}(\omega_{\alpha})$, then $\omega_{\alpha}$ precedes or equal to
$\omega_{\beta}$, i.e. $\omega_{\alpha}\preceq\omega_{\beta}$. For elements
$\omega_{\alpha}$, $\omega_{\beta}$ of $\left(\Omega,\preceq\right)$,
$C_{\Omega}(\omega_{\alpha})=C_{\Omega}(\omega_{\beta})$ implies
$\omega_{\alpha}=\omega_{\beta}$, and $C_{\Omega}(\omega_{1})\subset
C_{\Omega}(\omega_{2})\subset\cdots\subset C_{\Omega}(\omega_{k})$ implies
$\omega_{1}\preceq\omega_{2}\preceq{\ldots}\preceq\omega_{k}$.

The order topologies on the quotient space $(\Omega,\varpi)$ are in bijective
correspondence with the relation $\preceq$. The collection of all cones
$C_{\Omega}(\omega_{\alpha})$ forms a minimal base for the topology $\varpi$
(Singh, 2019). Further, the dimension $\dim(A)$ is the maximum of all non-
negative integers $k$ such that exists a chain
$\omega_{1}\preceq\omega_{2}\preceq{\ldots}\preceq\omega_{k}$ in $\Omega$:

$(\omega_{\alpha}\in\Omega)(k\in\mathbb{N})\colon\dim(A)=\max\left\\{{k}\colon\exists\>{\omega_{1}\preceq\omega_{2}\preceq{\ldots}\preceq\omega_{k}}\right\\}.$
(8)

Since our key intuition is to treat the order topology $\varpi$ as an internal
property of $\Omega$, our next task is to find an easy way of computing some
topological quantity that characterizes $(\Omega,\varpi)$. We can describe the
difference between the physics sense $A$ and physics sense $B$ by mapping
quotient space $\left(\Omega_{A},\varpi_{A}\right)$ into quotient space
$\left(\Omega_{B},\varpi_{B}\right)$. The topological structures on
$\Omega_{A}$ and $\Omega_{B}$ are completely identical if there exists a
homeomorphism of $\left(\Omega_{A},\varpi_{A}\right)$ onto
$\left(\Omega_{B},\varpi_{B}\right)$ (in topological terms, these spaces are
homeomorphic:
$\left(\Omega_{A},\varpi_{A}\right)\cong\left(\Omega_{B},\varpi_{B}\right)$).
Let us remember that a homeomorphism is an invertible continuous map
$\varphi\colon\left(\Omega_{A},\varpi_{A}\right)\to\left(\Omega_{B},\varpi_{B}\right)$
such that its inverse is continuous (Singh, 2019). Physics senses $A$ and $B$
are homeomorphic if and only if there exists a continuous bijective map
between $\left(\Omega_{A},\varpi_{A}\right)$ and
$\left(\Omega_{B},\varpi_{B}\right)$ that preserves the multiplicities
$m_{1},m_{2},\ldots,m_{k}$.

To give a precise meaning to the statements about distinctions or similarities
of physics senses, we need to investigate the mappings of
$\varphi\colon\left(\Omega_{A},\varpi_{A}\right)\to\left(\Omega_{B},\varpi_{B}\right)$.
It is clear that, if $\varphi$ is the homeomorphism, physics senses $A$ and
$B$ are equivalent from topological point of view. Let
$\nu=\left|\\{\varphi\\}\right|$ be the cardinality of the set of
homeomorphisms. Suppose the equality $\nu_{A}=\nu_{B}$ holds, then
$\left(\Omega_{A},\varpi_{A}\right)\cong\left(\Omega_{B},\varpi_{B}\right)$.
In this case, $\varphi$ is the a one-to-one correspondence between each class
$\omega_{\alpha}\in\Omega$ and itself, and we get

$\left(\sum_{\alpha=1}^{\dim(A)}m_{\alpha}=M\right)\colon\nu\left\\{\left(\Omega_{A},\varpi_{A}\right)\right\\}=\prod_{\alpha=1}^{\dim(A)}m_{\alpha}^{m_{\alpha}}.$
(9)

Let ${\mathsf{P}}\left\\{{A}\right\\}$ be the ratio of the number of
homeomorphisms $\nu_{A}$ to the number of all possible mappings from $\Omega$
to $\Omega$

${\mathsf{P}}\left\\{A\right\\}=\frac{\nu_{A}}{M^{M}}.$ (10)

Thus, the number ${\mathsf{P}}\left\\{A\right\\}$ is the probability of a
partition $\Omega_{A}$ of a support $X_{A}$ with a fixed topological structure
$\varpi_{A}$. It is obvious that ${\mathsf{P}}\left\\{A\right\\}$ is
proportional to $M$. We now introduce a normalized logarithmic representation
of ${\mathsf{P}}\left\\{A\right\\}$

$h(A)=-\frac{1}{M}\ln{\frac{\nu_{A}}{M^{M}}}=-\left(\sum_{\alpha=1}^{\dim(A)}\frac{m_{\alpha}}{M}\ln\frac{m_{\alpha}}{M}\right).$
(11)

It is readily seen that the right-hand side $\eqref{eq:g3}$ is nothing other
than the entropy, as defined by Shannon.

## Funding

The article was prepared in the framework of a research grant funded by the
Ministry of Science and Higher Education of the Russian Federation (grant ID:
075-15-2020-928).

## References

  * Aleta et al. (2019) Aleta, A., Meloni, S., Perra, N., Moreno, Y., 2019\.  Explore with caution: Mapping the evolution of scientific interest in physics.  EPJ Data Science 8.  doi:10.1140/epjds/s13688-019-0205-9.
  * Battiston et al. (2020) Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.G., Petri, G., 2020\.  Networks beyond pairwise interactions: Structure and dynamics.  Physics Reports 874, 1–92.  doi:10.1016/j.physrep.2020.05.004.
  * Battiston et al. (2019) Battiston, F., Musciotto, F., Wang, D., Barabási, A.L., Szell, M., Sinatra, R., 2019\.  Taking census of physics.  Nature Review Physics 1, 89–97.  doi:10.1038/s42254-018-0005-3.
  * Behrouzi et al. (2020) Behrouzi, S., Sarmoor, Z.S., Khosrow, H., Kavousi, K., 2020\.  Predicting scientific research trends based on link prediction in keyword networks.  Journal of Informetrics 14, 101079\.  doi:10.1016/j.joi.2020.101079.
  * Bingham et al. (1987) Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987.  Regular Variation.  Cambridge University Press, Cambridge.  doi:10.1017/CBO9780511721434.
  * Börner and Milojević (2019) Börner, K., Milojević, S., 2019\.  Springer Handbook of Science and Technology Indicators. Springer International Publishing, Cham. chapter Science Forecasts: Modeling and Communicating Developments in Science, Technology, and Innovation.  pp. 145–157.  doi:10.1007/978-3-030-02511-3_6.
  * Bornmann et al. (2018) Bornmann, L., Haunschild, R., Hug, S.E., 2018.  Visualizing the context of citations referencing papers published by Eugene Garfield: a new type of keyword co-occurrence analysis.  Scientometrics 114, 427–437.  doi:10.1007/s11192-017-2591-8.
  * Callon et al. (1991) Callon, M., Courtial, J., Laville, F., 1991.  Co-word analysis as a tool for describing the network of interactions between basic and technological research: The case of polymer chemsitry.  Scientometrics 22, 155–205.  doi:10.1007/BF02019280.
  * Callon et al. (1983) Callon, M., Courtial, J.P., Turner, W.A., Bauin, S., 1983\.  From translations to problematic networks: An introduction to co-word analysis.  Social Science Information 22, 191–235.  doi:10.1177/053901883022002003.
  * Cheng et al. (2020) Cheng, Q., Wang, J., Lu, W., Huang, Y., Bu, Y., 2020\.  Keyword-citation-keyword network: a new perspective of discipline knowledge structure analysis.  Scientometrics 124, 1923–1943.  doi:10.1007/s11192-020-03576-5.
  * Chinazzi et al. (2019) Chinazzi, M., Gonçalves, B., Zhang, Q., Vespignani, A., 2019\.  Mapping the physics research space: A machine learning approach.  EPJ Data Science 8.  doi:10.1140/epjds/s13688-019-0210-z.
  * Christianson et al. (2020) Christianson, N.H., Sizemore, B.A., Bassett, D.S., 2020.  Architecture and evolution of semantic networks in mathematics texts.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, 20190741\.  doi:10.1098/rspa.2019.0741.
  * Cobo et al. (2011) Cobo, M., López-Herrera, A., Herrera-Viedma, E., Herrera, F., 2011\.  An approach for detecting, quantifying, and visualizing the evolution of a research field: A practical application to the Fuzzy Sets Theory field.  Journal of Informetrics 5, 146–166.  doi:10.1016/j.joi.2010.10.002.
  * Coccia (2020) Coccia, M., 2020.  The evolution of scientific disciplines in applied sciences: dynamics and empirical properties of experimental physics.  Scientometrics 124, 451–487.  doi:10.1007/s11192-020-03464-y.
  * Colavizza and Franceschet (2016) Colavizza, G., Franceschet, M., 2016\.  Clustering citation histories in the Physical Review.  Journal of Informetrics 10, 1037–1051.  doi:10.1016/j.joi.2016.07.009.
  * de Arruda et al. (2018) de Arruda, H.F., Comin, C.H., Costa, L.d.F., 2018.  How integrated are theoretical and applied physics?  Scientometrics 116, 1113–1121.  doi:10.1007/s11192-018-2777-8.
  * Deleuze (1993) Deleuze, G., 1993.  Logic of Sense.  Columbia University Press, New York, NY.  URL: http://cup.columbia.edu/book/the-logic-of-sense/9780231059831.
  * Deville et al. (2014) Deville, P., Wang, D., Sinatra, R., Song, C., Blondel, V.D., Barabási, A.L., 2014\.  Career on the move: Geography, stratification, and scientific impact.  Scientific Reports 4, 4770\.  doi:10.1038/srep04770.
  * Fontana et al. (2020) Fontana, M., Iori, M., Montobbio, F., Sinatra, R., 2020\.  New and atypical combinations: An assessment of novelty and interdisciplinarity.  Research Policy 49, 104063\.  doi:10.1016/j.respol.2020.104063.
  * Fortunato et al. (2018) Fortunato, S., Bergstrom, C.T., Börner, K., Evans, J.A., Helbing, D., Milojević, S., Petersen, A.M., Radicchi, F., Sinatra, R., Uzzi, B., Vespignani, A., Waltman, L., Wang, D., Barabási, A.L., 2018\.  Science of science.  Science 359, eaao0185.  doi:10.1126/science.aao0185.
  * Foss et al. (2013) Foss, S., Korshunov, D., Zachary, S., 2013.  An Introduction to Heavy–Tailed and Subexponential Distributions.  Springer–Verlag, New York, NY; Heidelberg; Dordrecht; London.  doi:10.1007/978-1-4614-7101-1.
  * Ghosh et al. (2020) Ghosh, S.S., Hebbeker, T., Meyer, A., Pook, T., 2020\.  General Model Independent Searches for Physics Beyond the Standard Model.  Springer International Publishing, Cham.  doi:10.1007/978-3-030-53783-8.
  * Hecker (2017) Hecker, B.L., 2017.  Four decades of open science.  Nature Physics 13, 523–525.  doi:10.1038/nphys4160.
  * Heiberger and Wieczorek (2016) Heiberger, R.H., Wieczorek, O.J., 2016\.  Choosing collaboration partners. How scientific success in physics depends on network positions.  arXiv [physics.soc-ph] , 1608.03251URL: http://arxiv.org/abs/1608.03251.
  * Herrera et al. (2010) Herrera, M., Roberts, D.C., Gulbahce, N., 2010.  Mapping the evolution of scientific fields.  PLoS ONE 5, e10355.  doi:10.1371/journal.pone.0010355.
  * Hosseini et al. (2021) Hosseini, S., Baziyad, H., Norouzi, R., Jabbedari Khiabani, S., Gidòfalvi, G., Albadvi, A., Alimohammadi, A., Seyedabrishami, S., 2021\.  Mapping the intellectual structure of GIS-T field (2008–2019): a dynamic co-word analysis.  Scientometrics 126, 2667–2688.  doi:10.1007/s11192-020-03840-8.
  * Jang and Ko (2019) Jang, Y.S., Ko, Y.J., 2019.  How latecomers catch up to leaders in high-energy physics as Big Science: transition from national system to international collaboration.  Scientometrics 119, 437–480.  doi:10.1007/s11192-019-03030-1.
  * Jia et al. (2017) Jia, T., Wang, D., Szymanski, B.K., 2017.  Quantifying patterns of research-interest evolution.  Nature Human Behaviour 1, 0078\.  doi:10.1038/s41562-017-0078.
  * Joslyn et al. (2021) Joslyn, C.A., Aksoy, S.G., Callahan, T.J., Hunter, L.E., Jefferson, B., Praggastis, B., Purvine, E., Tripodi, I.J., 2021\.  Hypernetwork science: From multidimensional networks to computational topology, in: Braha, D., de Aguiar, M.A.M., Gershenson, C., Morales, A.J., Kaufman, L., Naumova, E.N., Minai, A.A., Bar-Yam, Y. (Eds.), Unifying Themes in Complex Systems X. ICCS 2020. Springer, Cham. Springer Proceedings in Complexity, pp. 377–392.  doi:10.1007/978-3-030-67318-5_25.
  * Karimi et al. (2021) Karimi, F., Green, D., Matous, P., Varvarigos, M., Khalilpour, K.R., 2021.  Network of networks: A bibliometric analysis.  Physica D: Nonlinear Phenomena 421, 132889.  doi:10.1016/j.physd.2021.132889.
  * Katchanov et al. (2016) Katchanov, Y.L., Markova, Y.V., Shmatko, N.A., 2016.  How physics works: scientific capital in the space of physics institutions.  Scientometrics 108, 875–893.  doi:10.1007/s11192-016-2005-3.
  * Klement (2001) Klement, K.C., 2001.  Frege and the Logic of Sense and Reference.  Routledge, New York, NY; London.
  * Krenn and Zeilinger (2020) Krenn, M., Zeilinger, A., 2020\.  Predicting research trends with semantic and neural networks with an application in quantum physics.  Proceedings of the National Academy of Sciences 117, 1910–1916.  doi:10.1073/pnas.1914370116.
  * Langacker (2017) Langacker, P., 2017.  The Standard Model and Beyond.  CRC Press, Boca Raton, FL.  doi:10.1201/b22175.
  * Lebedev (2005) Lebedev, A.V., 2005.  Maxima of independent sums in the presence of heavy tails.  Theory of Probability & Its Applications 49, 700–703.  doi:10.1137/S0040585X9798138X.
  * Leydesdorff and Nerghes (2017) Leydesdorff, L., Nerghes, A., 2017\.  Co-word maps and topic modeling: A comparison using small and medium-sized corpora $(n<1{,}000)$.  Journal of the Association for Information Science and Technology 68, 1024–1035.  doi:10.1002/asi.23740.
  * Li et al. (2016) Li, H., An, H., Wang, Y., Huang, J., Gao, X., 2016.  Evolutionary features of academic articles co-keyword network and keywords co-occurrence network: Based on two-mode affiliation network.  Physica A: Statistical Mechanics and its Applications 450, 657–669.  doi:10.1016/j.physa.2016.01.017.
  * Liu et al. (2019) Liu, W., Saganowski, S., Kazienko, P., Cheong, S.A., 2019\.  Predicting the evolution of physics research from a complex network perspective.  Entropy 21, 1152\.  doi:10.3390/e21121152.
  * López-Ruiz et al. (1995) López-Ruiz, R., Mancini, H.L., Calbet, X., 1995.  A statistical measure of complexity.  Physics Letters A 209, 321–326.  doi:10.1016/0375-9601(95)00867-5.
  * Lozano et al. (2019) Lozano, S., Calzada-Infante, L., Adenso-Díaz, B., García, S., 2019\.  Complex network analysis of keywords co-occurrence in the recent efficiency analysis literature.  Scientometrics 120, 609–629.  doi:10.1007/s11192-019-03132-w.
  * Martin et al. (2006) Martin, M.T., Plastino, A., Rosso, O.A., 2006.  Generalized statistical complexity measures: Geometrical and analytical properties.  Physica A: Statistical Mechanics and its Applications 369, 439–462.  doi:10.1016/j.physa.2005.11.053.
  * Masuda et al. (2017) Masuda, N., Porter, M.A., Lambiotte, R., 2017.  Random walks and diffusion on networks.  Physics Reports 716–717, 1–58.  doi:10.1016/j.physrep.2017.07.007.
  * McKeown et al. (2016) McKeown, K., Daume, H., Chaturvedi, S., Paparrizos, J., Thadani, K., Barrio, P., Biran, O., Bothe, S., Collins, M., Fleischmann, K.R., Gravano, L., Jha, R., King, B., McInerney, K., Moon, T., Neelakantan, A., O’Seaghdha, D., Radev, D., Templeton, C., Teufel, S., 2016\.  Predicting the impact of scientific concepts using full-text features.  Journal of the Association for Information Science and Technology 67, 2684–2696.  doi:10.1002/asi.23612.
  * Milojević (2015) Milojević, S., 2015.  Quantifying the cognitive extent of science.  Journal of Informetrics 9, 962–973.  doi:10.1016/j.joi.2015.10.005.
  * Mingers and Leydesdorff (2015) Mingers, J., Leydesdorff, L., 2015\.  A review of theory and practice in scientometrics.  European Journal of Operational Research 246, 1–19.  doi:10.1016/j.ejor.2015.04.002.
  * Newman (2018) Newman, M., 2018.  Networks.  Oxford University Press, Oxford.  doi:10.1093/oso/9780198805090.001.0001.
  * Palchykov et al. (2016) Palchykov, V., Gemmetto, V., Boyarsky, A., Garlaschelli, D., 2016\.  Ground truth? Concept-based communities versus the external classification of physics manuscripts.  EPJ Data Science 5.  doi:10.1140/epjds/s13688-016-0090-4.
  * Palchykov et al. (2021) Palchykov, V., Krasnytska, M., Mryglod, O., Holovatch, Y., 2021\.  Network of scientific concepts: Empirical analysis and modeling.  Advances in Complex Systems 24, 24001\.  doi:10.1142/S0219525921400014.
  * Palmucci et al. (2020) Palmucci, A., Liao, H., Napoletano, A., Zaccaria, A., 2020\.  Where is your field going? A machine learning approach to study the relative motion of the domains of physics.  PLoS ONE 15, e0233997.  doi:10.1371/journal.pone.0233997.
  * Pan et al. (2012) Pan, R.K., Sinha, S., Kaski, K., Saramäki, J., 2012\.  The evolution of interdisciplinarity in physics research.  Scientific Reports 2, 551\.  doi:10.1038/srep00551.
  * Patania et al. (2017) Patania, A., Petri, G., Vaccarino, F., 2017.  The shape of collaborations.  EPJ Data Science 6, 18\.  doi:10.1140/epjds/s13688-017-0114-8.
  * Perc (2013) Perc, M., 2013.  Self-organization of progress across the century of physics.  Scientific Reports 3, 1720\.  doi:10.1038/srep01720.
  * Petersen (2018) Petersen, A.M., 2018.  Multiscale impact of researcher mobility.  Journal of The Royal Society Interface 15, 20180580.  doi:10.1098/rsif.2018.0580.
  * Pluchino et al. (2019) Pluchino, A., Burgio, G., Rapisarda, A., Biondo, A.E., Pulvirenti, A., Ferro, A., Giorgino, T., 2019.  Exploring the role of interdisciplinarity in physics: Success, talent and luck.  PLoS ONE 14, e0218793.  doi:10.1371/journal.pone.0218793.
  * Radhakrishnan et al. (2017) Radhakrishnan, S., Erbis, S., Isaacs, J.A., Kamarthi, S., 2017\.  Novel keyword co-occurrence network-based methods to foster systematic reviews of scientific literature.  PLoS ONE 12, e0185771.  doi:10.1371/journal.pone.0172778.
  * Renoust et al. (2017) Renoust, B., Claver, V., Baffier, J.F., 2017.  Flows of Knowledge in Citation Networks, in: Cherifi, H., Gaito, S., Quattrociocchi, W., Sala, A. (Eds.), Complex Networks & Their Applications V, Springer, Cham. pp. 159–170.  doi:10.1007/978-3-319-50901-3_13.
  * Rosso et al. (2007) Rosso, O.A., Larrondo, H.A., Martin, M.T., Plastino, A., Fuentes, M.A., 2007.  Distinguishing noise from chaos.  Phys. Rev. Lett. 99, 154102\.  doi:10.1103/PhysRevLett.99.154102.
  * Salnikov et al. (2018) Salnikov, V., Cassese, D., Lambiotte, R., Jones, N.S., 2018\.  Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge.  Applied Network Science 3, 37\.  doi:10.1007/s41109-018-0074-3.
  * Silva et al. (2019) Silva, F.S.V., Schulz, P.A., Noyons, E.C.M., 2019.  Co-authorship networks and research impact in large research facilities: Benchmarking internal reports and bibliometric databases.  Scientometrics 118, 93–108.  doi:10.1007/s11192-018-2967-4.
  * Sinatra et al. (2015) Sinatra, R., Deville, P., Szell, M., Wang, D., Barabási, A.L., 2015.  A century of physics.  Nature Physics 11, 791–796.  doi:10.1038/nphys3494.
  * Singh (2019) Singh, T.B., 2019.  Introduction to Topology.  Springer Singapore, Singapore.  doi:10.1007/978-981-13-6954-4.
  * Strumia and Torre (2019) Strumia, A., Torre, R., 2019\.  Biblioranking fundamental physics.  Journal of Informetrics 13, 515–539.  doi:10.1016/j.joi.2019.01.011.
  * Sun and Latora (2020) Sun, Y., Latora, V., 2020.  The evolution of knowledge within and across fields in modern physics.  Scientific Reports 10, 12097\.  doi:10.1038/s41598-020-68774-w.
  * Tripodi et al. (2020) Tripodi, G., Chiaromonte, F., Lillo, F., 2020.  Knowledge and social relatedness shape research portfolio diversification.  Scientific Reports 10, 14232\.  doi:10.1038/s41598-020-71009-7.
  * van Eck and Waltman (2014) van Eck, N.J., Waltman, L., 2014\.  Measuring Scholarly Impact: Methods and Practice. Springer International Publishing, Cham. chapter Visualizing Bibliometric Networks.  pp. 285–320.  doi:10.1007/978-3-319-10377-8_13.
  * van Raan (2019) van Raan, A., 2019.  Springer Handbook of Science and Technology Indicators. Springer International Publishing, Cham. chapter Measuring Science: Basic Principles and Application of Advanced Bibliometrics.  pp. 237–280.  doi:10.1007/978-3-030-02511-3_10.
  * Wang et al. (2021) Wang, X., Wang, H., Huang, H., 2021.  Evolutionary exploration and comparative analysis of the research topic networks in information disciplines.  Scientometrics 126, 4991–5017.  doi:10.1007/s11192-021-03963-6.
  * Wen et al. (2017) Wen, B., Horlings, E., van der Zouwen, M., van den Besselaar, P., 2017.  Mapping science through bibliometric triangulation: An experimental approach applied to water research.  Journal of the Association for Information Science and Technology 68, 724–738.  doi:10.1002/asi.23696.
  * Xing et al. (2019) Xing, Y., Zeng, A., Fan, Y., Di, Z., 2019.  The strong nonlinear effect in academic dropout.  Scientometrics 120, 793–805.  doi:10.1007/s11192-019-03135-7.
  * Xu et al. (2020a) Xu, S., Hao, L., An, X., Pang, H., Li, T., 2020a.  Review on emerging research topics with key-route main path analysis.  Scientometrics 122, 607–624.  doi:10.1007/s11192-019-03288-5.
  * Xu et al. (2020b) Xu, S., Mariani, M.S., Lü, L., Medo, M., 2020b.  Unbiased evaluation of ranking metrics reveals consistent performance in science and technology citation data.  Journal of Informetrics 14, 101005\.  doi:10.1016/j.joi.2019.101005.
  * Yi and Choi (2012) Yi, S., Choi, J., 2012.  The organization of scientific knowledge: the structural characteristics of keyword networks.  Scientometrics 90, 1015–1026.  doi:10.1007/s11192-011-0560-1.
  * You et al. (2021) You, T., Yoon, J., Kwon, O., Jung, W., 2021.  Tracing the evolution of physics with a keyword co-occurrence network.  Journal of the Korean Physical Society 78, 236–243.  doi:10.1007/s40042-020-00051-5.
  * Zeng et al. (2019) Zeng, A., Shen, Z., Zhou, J., Fan, Y., Di, Z., Wang, Y., Stanley, H.E., Havlin, S., 2019.  Increasing trend of scientists to switch between topics.  Nature Communications 10, 3439\.  doi:10.1038/s41467-019-11401-8.
  * Zeng et al. (2017) Zeng, A., Shen, Z., Zhou, J., Wu, J., Fan, Y., Wang, Y., Stanley, H.E., 2017\.  The science of science: From the perspective of complex systems.  Physics Reports 714-715, 1–73.  doi:10.1016/j.physrep.2017.10.001.
  * Zhang et al. (2013) Zhang, Q., Perra, N., Gonçalves, B., Ciulla, F., Vespignani, A., 2013.  Characterizing scientific production and consumption in Physics.  Scientific Reports 3, 1640\.  doi:10.1038/srep01640.
  * Zingg et al. (2020) Zingg, C., Nanumyan, V., Schweitzer, F., 2020.  Citations driven by social connections? A multi-layer representation of coauthorship networks.  Quantitative Science Studies 1, 1493–1509.  doi:10.1162/qss_a_00092.