# A Perturbation-Based Approach to Identifying Potentially Superfluous Network
Constituents

Timo Bröhl<EMAIL_ADDRESS>Department of Epileptology, University of
Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany  Helmholtz
Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee
14–16, 53115 Bonn, Germany  Klaus Lehnertz  Department of Epileptology,
University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany
Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn,
Nussallee 14–16, 53115 Bonn, Germany  Interdisciplinary Center for Complex
Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany

###### Abstract

Constructing networks from empirical time series data is often faced with the
as yet unsolved issue of how to avoid potentially superfluous network
constituents. Such constituents can result, e.g., from spatial and temporal
oversampling of the system’s dynamics, and neglecting them can lead to severe
misinterpretations of network characteristics ranging from global to local
scale. We derive a perturbation-based method to identify potentially
superfluous network constituents that makes use of vertex and edge centrality
concepts. We investigate the suitability of our approach through analyses of
weighted small-world, scale-free, random, and complete networks.

> Understanding complex dynamical systems such as climate and brain profits
> from the network approach. Deriving networks from measurements of the
> systems’ dynamics, however, can lead to spurious indications of network
> properties, depending on the employed sampling strategies and time-series
> analysis techniques to define networks constituents. This, together with
> limitations in knowledge about the system’s actual structural organization,
> calls for approaches to identify potentially superfluous network
> constituents. Here, we present such an approach. It is based on minuscule
> and elementary perturbations targeting single network constituents.
> Constituents are deemed potentially superfluous if the perturbations lead to
> no or only negligible changes of network characteristics, covering the local
> to global scale. We test our approach on various paradigmatic network
> models.

##  I Introduction

Complex network approaches have been repeatedly shown to provide deeper
insights into structure and dynamics of spatially extended complex systems in
diverse areas of science. In many natural and man-made networked systems,
access to the underlying coupling structure may be restricted or even
impossible Boccaletti  _et al._ (2006); Arenas  _et al._ (2008); Bullmore and
Sporns (2009); Donges  _et al._ (2009a); Allen and Elston (2011); Barthélemy
(2011); Barabási, Gulbahce, and Loscalzo (2011); Newman (2012); Baronchelli
_et al._ (2013); Lehnertz  _et al._ (2014); Heckmann, Schwanghart, and
Phillips (2015); Gao, Barzel, and Barabási (2016). Nevertheless, in such cases
can the system be described by an interaction network with vertices
representing subsystems or elementary units and edges representing
interactions between them. This ansatz has been successfully applied e.g. in
the study of (functional) brain networks Bullmore and Sporns (2009); Lehnertz
_et al._ (2014), climate networks Donges  _et al._ (2009b); Zhou  _et al._
(2015), protein-protein interactions Uetz  _et al._ (2000), gene interactions
Tyler  _et al._ (2009), plant-pollinator interactions Hegland  _et al._
(2009); Olesen  _et al._ (2011); Halekotte and Feudel (2020), food-webs Delmas
_et al._ (2019), or communication and social networks Onnela  _et al._ (2007);
Palla, Barabási, and Vicsek (2007).

When it comes to analyzing real-world complex systems, lacking explicit
knowledge of the structural organization of the dynamical system under study
is quite common. Hence, vertices of the related interaction network are
commonly associated with sensors that are placed to sufficiently capture a
subsystems’ dynamics. Deriving edges from the system’s dynamics is usually
based on a data-driven quantification of interaction properties, namely
strength, direction, and coupling function. Given that interactions can
manifest themselves in various aspects of the dynamics (amplitudes,
frequencies, phases, etc.), a large number of time series analysis techniques
is now available. The reliability of techniques, however, may be influenced by
the mostly unavoidable finiteness of noisy field data which can lead to
erroneous estimates of interaction properties. Together with the fact that
there is by now no commonly accepted method to derive binary or weighted (or
weighted and directed) networks from interaction properties, this represents a
source for severe misinterpretations of network properties Bialonski,
Horstmann, and Lehnertz (2010); Hlinka, Hartman, and Paluš (2012); Porz, Kiel,
and Lehnertz (2014); Wens (2015); Gastner and Ódor (2016); Papo  _et al._
(2016); Hlinka  _et al._ (2017); Zanin  _et al._ (2018).

Yet, these issues are influenced and preceded by a more general problem:
choosing the right number of sensors and placing them in a meaningful way.
Arrangement and placement of sensors is highly non-trivial and most often
leads to a spatial over- or undersampling of a system. These issues translate
to the presence of additional and potentially superfluous constituents or the
absence of potentially relevant constituents and may cause severe
misinterpretations of network properties Bialonski, Horstmann, and Lehnertz
(2010); Bialonski, Wendler, and Lehnertz (2011); Hlinka, Hartman, and Paluš
(2012); Chung  _et al._ (2012); Porz, Kiel, and Lehnertz (2014); Wens (2015);
Gastner and Ódor (2016); Papo  _et al._ (2016); Hlinka  _et al._ (2017); Zanin
_et al._ (2018). When investigating real-world systems, aiming to gather as
much information as possible is rather common practice but bears the risk of
oversampling the system. Hence, there is a strong need for methods that allow
to identify redundant or superfluous vertices and edges. For edges, a vast
plethora of methods has been proposed Lü and Zhou (2011); Lü  _et al._ (2015);
Liao  _et al._ (2017); Kramer  _et al._ (2009); Yan  _et al._ (2018), although
their suitability continues to be matter of debate Zeng and Cimini (2012);
Zhang, Li, and Deng (2018); Kumar  _et al._ (2020); Cantwell  _et al._ (2020).
Interestingly, the issue of identifying superfluous vertices has so far been
addressed only rarely Frantz, Cataldo, and Carley (2009); Bellingeri  _et al._
(2020).

We here propose a perturbation-based approach to identify potentially
superfluous network constituents (vertices and edges), employing elemental and
minuscule perturbations that directly target single constituents. With the
premise that targeting constituents of potentially superfluous nature has
little to no effect on the characteristics of networks, exactly these are
compared prior and after perturbation. We test the suitability of this method
on weighted small-world, scale-free, random, and complete networks.

##  II Methods

There are several ways to perturb a network, with different types of
perturbation potentially leading to different outcomes. Almost all
perturbations, however, can be viewed as an accumulation, superposition, or
interplay of the following, straightforward and elementary perturbations:

  1. 1.

vertex/edge removal: one or more constituents are removed from the network. In
case of removing a vertex, its connected edges are removed along with it. The
removal of constituents can have a significant impact on the network’s
connectedness, and it can lead to the fragmentation of the network into
smaller disconnected components;

  2. 2.

vertex/edge addition: constituents are added to the network. This can increase
the network’s connectedness and can facilitate the exchange of information or
resources in different ways;

  3. 3.

rewiring: an edge is randomly rewired, possibly leading to changes in the
network’s topology. Rewiring can alter the network’s characteristics,
especially path-structural aspects;

  4. 4.

weight changes: the weight of an edge is altered, possibly influencing local
up to global network characteristics.

However, the influences of structurally minuscule perturbations are hard to
determine and even harder to control on the level of the complex system.
Previous research has shown that random perturbations can have major influence
on very specific macroscopic network characteristics Latora and Marchiori
(2005); Ghoshal and Barabási (2011); Ceci and Barbarossa (2018). Nonetheless,
and especially when it comes to the investigation of real-world systems, it
remains unclear, how on a general basis minuscule perturbations targeting
single constituents change the respective networks. Thus far, comparing
networks is a notorious difficult task, particularly for networks of different
sizes (and changes in network sizes often go hand in hand with these
perturbations) and there is no commonly accepted and sufficient way to do so
Tantardini  _et al._ (2019); Mheich, Wendling, and Hassan (2020). Hence, we
can only focus on network metrics Rings, Bröhl, and Lehnertz (2022) that, in
total, describe the network somewhat comprehensively (cf. Sect. II.2 and Table
1).

The principal idea behind our perturbation-based approach now is, that if the
targeted perturbation of a network constituent (cf. Sect. II.3) does not alter
network characteristics assessed with the various metrics, or only to a small
negligible extent, the targeted constituent can be deemed potentially
superfluous.

###  II.1 Employed perturbations

Of the above listed four elementary perturbations, only the first two are
universally applicable in any kind of network, independent of its topology and
definitions of edges (regarding weight, direction or multiple edges). Based on
these, we further differentiate between the following three perturbations that
we will employ in the wake of our perturbation-based method to identify
potentially superfluous constituents:

  * •

vertex removal: a vertex $v$ and its connected edges $\\{v,j\\}$ are
discarded, with $j$ denoting vertices adjacent to $v$;

  * •

vertex cloning: a vertex $v$ – that is already present in the network – is
duplicated, including its connected edges $\\{v,j\\}$ by adding a vertex
$v^{\prime}$ and adding the respective edges $\\{v^{\prime},j\\}$, with $j$
denoting vertices adjacent to $v$; cloned vertex $v$ and clone $v^{\prime}$
are not directly connected;

  * •

edge removal: an edge is discarded from the network.

Perturbations directly targeting a single vertex, hence may indirectly affect
edges connected to the respective vertex in the course of the perturbation.

###  II.2 Network metrics

We generally differentiate between global and local network characteristics
that are evaluated by network metrics (cf. Table 1). Global network metrics
describe the network as a whole, often associating a network characteristic
with a single quantity. Local network metrics focus on aspects of single
network constituents (vertices and edges) or groups of such. While some of
these local metrics might still depend on the composition of the network as a
whole, others merely depend on the direct neighborhood of the respective
constituent.

A network consists of a set of vertices $\mathscr{V}$
($v_{i}\in\mathscr{V},i=1,\dots,V$; $V=|\mathscr{V}|$) and a set of edges
($e_{n}\in\mathscr{E},n=1,\dots,E$; $E=|\mathscr{E}|$) with an edge connecting
two vertices ($e_{ij}={v_{i},v_{j}}$). The network can be described by its
adjacency matrix $\mathscr{A}\in\\{0,1\\}^{V\times V}$, with
$\mathscr{A}_{ij}=1$ if edge $e_{ij}$ exists between vertices $i$ and $j$, and
$\mathscr{A}_{ij}=0$ otherwise. Complementarily, for weighted networks, we
define the weight matrix $\mathscr{W}\in[0,1]^{V\times V}$, with
$\mathscr{W}_{ij}$ referring to the edge weight (strength of interaction)
between vertices $i$ and $j$.

####  II.2.1 Global network metrics

Some metrics of networks describe certain aspects of the network as a whole.
This not only can allow the classification of network topologies, but also
allows the comparison of other characteristics of networks, such as network
size, path-structure, degree-correlations, robustness, and stability.

The (pseudo) diameter $D$ is the length of the longest shortest path between
any pair of vertices in a network. The length of a path is chosen as the sum
of the inverse of all edge weights on that path.

The average shortest path length $L$ quantifies the average length of a path
$\psi$ between any two vertices ($\\{z,l\\}\in\mathscr{V}$) in a network:

$L=\frac{\sum_{zl}\psi_{zl}}{V(V-1)}.$

The global clustering coefficient $G$ quantifies to what extend network
vertices tend to cluster together. For a weighted network, the global
clustering coefficient is defined as:

$G=\frac{\mathrm{Tr}\mathscr{W^{3}}}{\sum_{z\neq l}[\mathscr{W}^{2}]_{zl}}.$

Assortativity $A$ characterizes how vertices with (dis)similar properties
(here: strength, being the sum of the weights of the attached edges) are
preferentially connected amongst themselves Newman (2002); Bialonski and
Lehnertz (2013). To calculate $A$, we estimate the (Pearson) correlation
coefficient between the strengths of connected vertices:

$A=\sum_{xy}xy(q_{xy}-a_{x}b_{y})/\sigma_{a}\sigma_{b},$

with $x$ and $y$ representing strength values and $q_{xy}$ representing the
fraction of edges that connect a vertex with strength $x$ to a vertex of
strength $y$. Then $q_{xy}$ satisfies the sum rules: $\sum_{xy}q_{xy}=1$,
$\sum_{y}q_{xy}=a_{x}$, $\sum_{x}q_{xy}=b_{y}$. $\sigma_{a}$ and $\sigma_{b}$
are the standard deviations of the distributions $a_{x}$ and $b_{y}$.

Synchronizability $S$ of a network describes the stability of its globally
synchronized state Barahona and Pecora (2002); Atay, Bıyıkoğlu, and Jost
(2006). We here characterize it by the eigenratio $S=\lambda_{V}/\lambda_{2}$.
$\lambda_{V}$ denotes the largest eigenvalue and $\lambda_{2}$ the smallest
non-vanishing eigenvalue of the network’s Laplacian matrix
$L_{zl}=s_{z}\delta_{zl}-\mathscr{W}_{zl}$ ($\delta$ is the Kronecker delta,
$s_{z}$ denotes the strength of vertex $z$; see below).

####  II.2.2 Local network metrics

The concept of centrality has been introduced in many different fields and
contexts Beauchamp (1965); Sabidussi (1966); Freeman (1977, 1979); Bonacich
(1987); Wuchty and Stadler (2003); Koschützki  _et al._ (2005); Borgatti and
Everett (2006); Estrada and Higham (2010); Valente and Fujimoto (2010); Chen
_et al._ (2012); Costa  _et al._ (2015); Lawyer (2015); Wu, Tian, and Liu
(2018); Bröhl and Lehnertz (2019); Zhao  _et al._ (2020); Bröhl and Lehnertz
(2022). The general idea is to quantify a constituent’s role or importance in
the larger network based on certain characteristics, primarily focusing on the
integration of a constituent in the network due to specific aspects. Hence
centrality metrics can be utilized to measure importance yielding an
importance ranking Ghoshal and Barabási (2011); Lü  _et al._ (2016); Iñiguez
_et al._ (2022).

The degree of a vertex $z$ is the sum of edges connected to this vertex:
$d_{z}=\sum_{l=0}^{V}\mathscr{A}_{zl}$. Analogously the strength (or strength
centrality) of a vertex is the sum of edge weights of all edges connected to
this vertex:
$s_{z}=\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}(z)=\sum_{l=0}^{V}\mathscr{W}_{zl}$.

Similarly, with nearest-neighbor centrality Bröhl and Lehnertz (2022), an edge
is considered to be more central the larger its weight and the more similar
and the higher the strengths of the vertices which are connected by that edge.
Nearest-neighbor edge centrality of an edge $z$ between vertices $a$ and $b$
is defined as Bröhl and Lehnertz (2022)

$\pazocal{C}^{\mathrm{N}}_{\mathrm{e}}(z)=\frac{\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}(a)+\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}(b)-2w_{z}}{|\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}(a)-\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}(b)|+1}\;w_{z},$

where $w_{z}=\mathscr{W}_{ab}$ denotes the edge weight and
$z\in\left\\{1,\ldots,E\right\\}$ and $(a,b)\in\\{1,\dots,V\\}$. Hence,
nearest-neighbor centrality can be considered as a strength-based edge
centrality concept. Much like the strength of a vertex, the nearest-neighbor
centrality value of an edge is only influenced by its adjacent constituents.

Eigenvector centrality considers the influence of a vertex/edge (v/e) on the
network as a whole. A constituent is regarded as central if adjacent
constituents are also central. This centrality is defined as Bonacich (1972);
Bröhl and Lehnertz (2019)

$\pazocal{C}^{\mathrm{E}}_{\mathrm{v,e}}(z)=\frac{1}{\lambda_{\max}}\sum_{l}M_{zl}\,\pazocal{C}^{\mathrm{E}}_{\mathrm{v,e}}(l).$

In case of vertices, $\left\\{z,l\right\\}\in\mathscr{V}$ and ${\bf M}$
denotes the weight matrix $\mathscr{W}^{\rm(v)}\in[0,1]^{V\times V}$, with
$\mathscr{W}^{\rm(v)}_{zl}$ denoting the weight of an edge between vertices
$z$ and $l$. We define
$\mathscr{W}^{\rm(v)}_{zz}\mathrel{\mathop{:}}=0\,\forall\,z$ with
$z\in\left\\{1,\ldots,V\right\\}$. In case of edges,
$\left\\{z,l\right\\}\in\mathscr{E}$ and ${\bf M}$ denotes the weight matrix
$\mathscr{W}^{\rm(e)}\in[0,1]^{E\times E}$ whose entries
$\mathscr{W}^{\rm(e)}_{zl}$ are assigned the average weight of edges $z$ and
$l$ if these edges are connected to a same vertex, and 0 otherwise. As above,
we define $\mathscr{W}^{\rm(e)}_{zz}\mathrel{\mathop{:}}=0\,\forall\,z$ with
$z\in\left\\{1,\ldots,E\right\\}$. The aforementioned definition is applied
iteratively until eigenvector centrality values remain stable. Eigenvector
centrality can be considered as a strength-based centrality concept, which,
due to its recursive definition, relates a constituent to all the other
constituents in the network.

Closeness centrality considers the distance between a constituent and all
other constituent in a network. A constituent with a high closeness centrality
is central as information from this constituent can reach all other
constituents in the network via short paths, and so the constituent can exert
a more direct influence over the network. Closeness centrality of vertex $z$
is defined asBavelas (1950):

$\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(z)=\frac{V-1}{\sum_{l}{d_{zl}}},$

with $(z,l)\in\left\\{1,\ldots,V\right\\}$ and where $d_{zl}$ is the length of
the shortest path between vertices $z$ and $l$, calculated as the sum of the
inverse of all edge weights on the path. Closeness centrality of edge $z$
between vertices $a$ and $b$ can be defined asBröhl and Lehnertz (2019):

$\displaystyle\pazocal{C}^{\mathrm{C}}_{\mathrm{e}}(z)$
$\displaystyle=\frac{E-1}{\sum_{l}{(d_{la}+d_{lb})}}=\frac{E-1}{\frac{1}{\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(a)}+\frac{1}{\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(b)}}$
$\displaystyle=(E-1)\frac{\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(a)\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(b)}{\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(a)+\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}(b)},$

with $z\in\left\\{1,\ldots,E\right\\}$ and $(a,b,l)\in\\{1,\dots,V\\}$. Hence,
closeness centrality can be considered as a path-based centrality concept,
which is therefore influenced by the network as whole.

Betweenness centrality is a measure of how frequently a shortest path
traverses a given constituent. A constituent with a high betweenness
centrality value is central because it acts as a bridge between other parts of
the network. Vertex/edge betweenness centrality (of vertex/edge $z$) can be
defined as Freeman (1977); Brandes (2001); Girvan and Newman (2002); Bröhl and
Lehnertz (2019)

$\pazocal{C}^{\mathrm{B}}_{\mathrm{v,e}}(z)=\frac{2}{F}\sum_{l\neq
m}\frac{q_{lm}(z)}{G_{lm}},$

where $z\in\left\\{1,\ldots,V\right\\}$ (for vertices), resp.
$z\in\left\\{1,\ldots,E\right\\}$ (for edges),
$\left\\{l,m\right\\}\in\left\\{1,\ldots,V\right\\}$, $q_{lm}(z)$ is the
number of shortest paths between vertices $l$ and $m$ running through
vertex/edge $z$, and $G_{lm}$ is the total number of shortest paths between
vertices $l$ and $m$. Again, the length of a path is chosen as the sum of the
inverse of all edge weights on that path. The normalization factor is
$F=(V-1)(V-2)$ in case of vertices and $F=V(V-1)$ in case of edges.
Betweenness centrality can be considered as a path-based centrality concept,
which is therefore influenced by the network as whole.

In order to be able to compare results yielded by different centrality
concepts qualitatively, we introduce a centrality-value-based ranking of the
networks constituents. A vertex/edge is assigned rank 1 if the largest
centrality value is associated with it. The rank increases in increments of 1
for the vertex/edge with second largest centrality value, third largest
centrality value etc., yielding an increasing rank as centrality values
decrease. We abstain from assigning two or more constituents the same rank and
rank in order of appearance for equal centrality values.

|  |  characteristics | metrics
---|---|---|---
global | network |  length of the longest shortest path between any pair of vertices in a network | (pseudo) diameter $D$
average length of a path between any two vertices in a network | average shortest path length $L$
extent to which vertices in the network tend to cluster together | global clustering coefficient $G$
extent to which vertices with (dis-)similar properties are preferentially connected amongst themselves | assortativity $A$
stability of the network’s globally synchronized state | synchronizability $S$
local | vertices |  intergration of a vertex in its direct neighborhood (binary network) | degree $d$
intergration of a vertex in its direct neighborhood (weighted network) | strength centrality $\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}$
extent to which a strongly integrated vertex is connected to other strongly integrated vertices | eigenvector centrality $\pazocal{C}^{\mathrm{E}}_{\mathrm{v}}$
extent to which a vertex can reach any other vertex in the network via shortest paths | closeness centrality $\pazocal{C}^{\mathrm{C}}_{\mathrm{v}}$
extent to which a vertex connects otherwise distant regions in the network | betweenness centrality $\pazocal{C}^{\mathrm{B}}_{\mathrm{v}}$
edges |  integration of an edge in its direct neighborhood | nearest-neighbor centrality $\pazocal{C}^{\mathrm{N}}_{\mathrm{e}}$
extent to which a strongly integrated edge is connected to other strongly integrated edges | eigenvector centrality $\pazocal{C}^{\mathrm{E}}_{\mathrm{e}}$
extent to which an edge can reach any other edge in the network via shortest paths | closeness centrality $\pazocal{C}^{\mathrm{C}}_{\mathrm{e}}$
extent to which an edge connects otherwise distant regions in the network | betweenness centrality $\pazocal{C}^{\mathrm{B}}_{\mathrm{e}}$

Table 1: Overview of global and local network characteristics and their
respective quantifying metrics.

###  II.3 Perturbations of network constituents

For a given realization of a network (out of $N_{\rm r}$ realizations),

  * •

we estimate local and global network metrics (cf. Sect. II.2) and deduce an
initial ranking of the network’s constituents based on their centrality
values. The latter allows to identify a constituent, based on its rank, in the
different realizations;

  * •

we iterate over all network constituents and

    * –

employ the respective perturbation (cf. Sect. II.1) to the test if the
respective constituent $\gamma$ is potentially superfluous,

    * –

estimate local and global network metrics for this perturbed network,

    * –

quantify the influence of the perturbation by comparing local and global
network metrics prior and after the perturbation (cf. Sect. II.4).

###  II.4 Quantifying influences of network perturbations

In case of the global network characteristics, we track the perturbed
constituent $\gamma$ throughout the realizations of a given network via its
respective rank $r_{\rm u}(\gamma)$ prior to perturbation (estimated with
$\pazocal{C}^{\mathrm{S}}_{\mathrm{v}}$ for vertices and with
$\pazocal{C}^{\mathrm{N}}_{\mathrm{e}}$ for edges). Generally, we expect the
structurally minuscule perturbations to also have a negligible impact on any
global network characteristics, if such does not strongly depend on the number
of vertices or the number of edges. We calculate the average percentage change
(from $N_{\rm r}$ realizations) of each metric resulting from the perturbation
as

$\overline{\delta\mu^{(r_{\rm u}(\gamma))}}=\frac{1}{N_{\rm
r}}\sum_{i=1}^{N_{\rm r}}100\frac{\mu_{i,\rm u}-\mu_{i,\rm p}^{(r_{\rm
u}(\gamma))}}{\mu_{i,\rm u}}.$

Here $\mu\in\\{D,L,G,A,S\\}$, $\mu_{i,\rm u/p}$ denotes the global metric of
the unperturbed/perturbed network (of realization $i$) and $r_{\rm
u}(\gamma)\in\left\\{1,\ldots,V\right\\}$ (for vertices), resp. $r_{\rm
u}(\gamma)\in\left\\{1,\ldots,E\right\\}$ (for edges) is the rank of the
perturbed constituent $(\gamma)$ in the unperturbed network.

In case of the local network characteristics and when investigating a change
in centrality values, it is important to recognize that there is no one true
centrality concept. Each of the centrality concepts employed here does focus
on different topological aspects of the network. However, we can generally
divide the centrality concepts into strength-based (nearest-neighbor
centrality and eigenvector centrality) and path-based (betweenness centrality
and closeness centrality) concepts. Hence, it is not necessarily to be
expected to observe perturbation-induced changes in the distribution of
strength-based centrality values, when observing a perturbation-induced change
in the distribution of path-based centrality values (or vice versa).
Nevertheless, in order for a targeted constituent to be considered potentially
superfluous, the respective perturbation should not lead to meaningful changes
in either distribution. For the employed perturbations, we would expect
largely comparable distributions of centrality values for a given network
prior and after perturbation (independent of the centrality concept). The
values of a given centrality metric, for a given network prior to and after
perturbation, can therefore be considered to be drawn from the same
distribution, and we test this null hypothesis using the Kolmogorov-Smirnov
test. The null hypothesis is rejected for $p<0.05$. Hence, if the null
hypothesis can be rejected for a certain perturbation, and with regard to any
of the centrality concepts, the perturbed network constituent can not be
considered potentially superfluous under this perturbation. However opposite
reasoning, deeming a constituent as potentially superfluous is not valid if
the respective perturbation did not lead to significant changes between the
distributions of centrality values of the unperturbed and perturbed network.
Still it can be a further indication of the targeted constituent being
potentially superfluous. Nonetheless, the specific local changes in the
importance ranking of constituents can be abundant and meaningful in a greater
context, while the distribution of centrality values is left unaltered.

In case of the importance hierarchies, as deduced from the centrality-based
rankings of a networks’ constituents (vertices and edges, respectively), we
quantify the local impact of a perturbation of constituent $\gamma$ by
considering the following metric, calculating the difference
$\Delta^{\bullet}_{\mathrm{v/e}}$between the ranks Saavedra  _et al._ (2011)
$r_{\rm u}(z)$ and $r_{\rm p}(z)$ of any constituent $z\neq\gamma$
($z\in\mathscr{V}$ for vertices (v) and $z\in\mathscr{E}$ for edges (e)) for
the unperturbed and the perturbed network:

$\Delta^{\bullet}_{\mathrm{v/e}}(z)=|r_{\rm u}(z)-r_{\rm p}(z)|,$

where $\bullet\in\\{\text{S,E,C,B}\\}$ denotes the centrality concept employed
for the ranking. $\Delta^{\bullet}_{\mathrm{v/e}}(z)\to 0$ can be considered
as a further indication for constituent $\gamma$ to be potentially
superfluous. The median value of $\Delta^{\bullet}_{\mathrm{v/e}}(z)$ is
expected to be rather small.

Overall, deeming a network constituent potentially superfluous can not be
considered an absolute truth, but is rather an assessment of a number of
indications based on observed changes in network characteristics due to
specific perturbations. The more qualifying observations can be made regarding
these network characteristics – for the respective perturbation – the more
considerable is a potential superfluous nature of the targeted constituent.

We thus set three criteria that indicate a constituent targeted by the
respective perturbations to be considered potentially superfluous: negligible
changes in a number of global network metrics, negligible changes in the
distributions of centrality values (based on multiple centrality concepts),
and negligible changes in the actual importance hierarchies of the
constituents.

With regard to potentially dependencies of the aforementioned criteria such as
network topology, size and edge density, and thus with an eye on potential
applications in the analyses of real-world systems, we investigate the
suitability of these criteria analyzing various networks with preset
properties.

##  III Networks investigated

When approximating real-world complex systems with networks, the latter are
often associated with specific topologies (or combinations of such).
Independent of the underlying complex systems, these topologies can have quite
distinct properties that may not only influence network characteristics
substantially but also can induce superfluous constituents, at least in some
of their realizations. As an example, it may naively to be expected to find
superfluous information in a very dense and large network (e.g., large random
networks) while it is rather less likely to find such in sparse networks
(e.g., small-world or scale-free networks) or in networks with regular
structures (e.g., lattices or rings).

We here consider undirected, weighted and connected networks without self-
loops or multiple edges. In the following, edge weights are drawn from a
uniform distribution $U(0,1)$. We investigate four different topologies and
networks of different sizes ($V\in\\{20,50,100,200,500,1000\\}$) and different
edge densities.  Each network type of the following four topologies will be
realized $N_{\rm r}=100$ times:

  * •

small-world networks Watts and Strogatz (1998) with rewiring probabilities
$p\in\\{0.01,0.1,0.2,0.3\\}$ starting from a ring with $m\in\\{4,8\\}$
nearest-neighbors being connected,

  * •

random networks Erdős and Rényi (1959); Batagelj and Brandes (2005) with edge
creation probabilities $q\in\\{0.05,0.1,0.2,0.3,0.5,0.7\\}$,

  * •

scale-free networks Albert and Barabási (2002) with the newly added vertices
preferentially attached to existing vertices of high degree with
$k\in\\{4,6,10\\}$ edges,

  * •

complete networks.

The edge density for a network is then defined as:

$\varrho=E/E_{\mathrm{max}}=E/\binom{V}{2}=\frac{2E}{V(V-1)}.$

##  IV Results

###  IV.1 Impact of perturbations on global network characteristics

For each of the three types of perturbation and predominantly independent of
the network topology, we observe – overall and on overage – perturbation-
induced changes of global network metrics in the order of a few percent
($<5\,\%$, cf. Table 2).

perturbation metrics | $<\overline{\delta D}>$ | $\sigma(\overline{\delta D})$ | $\max(|\delta D|)$ | $<\overline{\delta L}>$ | $\sigma(\overline{\delta L})$ | $\max(|\delta L|)$ | $<\overline{\delta G}>$ | $\sigma(\overline{\delta G})$ | $\max(|\delta G|)$ | $<\overline{\delta A}>$ | $\sigma(\overline{\delta A})$ | $\max(|\delta A|)$ | $<\overline{\delta S}>$ | $\sigma(\overline{\delta S})$ | $\max(|\delta S|)$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
vertex removal | -1.5 | 4.85 | 16.66 | -0.81 | 1.47 | 29.33 | -0.19 | 2.01 | 15.93 | -1.01 | 12.44 | 39.63 | -4.87 | 19.83 | 1047.42
vertex cloning | -0.01 | 0.06 | 1.76 | 0.03 | 0.24 | 5.69 | 0.91 | 1.55 | 9.73 | -0.37 | 13.74 | 53.97 | -1.62 | 3.23 | 21.8
edge removal | -0.56 | 2.02 | 9.46 | -0.42 | 0.71 | 28.37 | 0.37 | 1.19 | 6.65 | -0.39 | 4.6 | 24.98 | -1.72 | 8.73 | 1047.74

Table 2:  Percentage increase/decrease (as well as standard deviation and
maximum value) of global network characteristics under respective perturbation
and averaged over all constituents and all investigated networks
($<\bullet>$). $D$ diameter; $L$ average shortest path length; $G$ global
clustering coefficient; $A$ assortativity; $S$ synchronizability.

It is to be mentioned that while, on average, pointing to rather negligible
changes, we do observe large maximum changes in few, specific networks –
primarily observed for assortativity (up to $\sim 50\,\%$) and
synchronizability (up to $\sim 1000\,\%$) – the latter pointing towards the
existence of some rare but seemingly vital constituents regarding the
stability of a synchronized state particularly in random and complete
networks.

Furthermore, we do observe dependencies regarding the importance $r_{\rm u}$
of the constituent targeted by the perturbation, for at least some of the
global network metrics. Yet, these dependencies vary regarding the network
topology. Especially for those network topologies that are less random and of
more regular structure (small-world and scale-free networks) the magnitude of
changes of global clustering coefficient and average shortest path length (but
also of assortativity and synchronizability; data not shown) depend on the
importance $r_{\rm u}$ of the removed/cloned constituent. Generally, largest
total changes of metrics can be observed when removing/cloning most and least
important constituents (cf. Figure 1).

Figure 1: Averages of percentage changes of (pseudo) diameter
$\overline{\delta D^{(r_{\rm u})}}$, of global clustering coefficient
$\overline{\delta G^{(r_{\rm u})}}$, and of average shortest path length
$\overline{\delta L^{(r_{\rm u})}}$ under the respective perturbation of a
targeted constituent with certain rank $r_{\rm u}$. Vertex ranks estimated
with strength centrality and edge ranks estimated with nearest-neighbor
centrality prior to perturbation. Averages from 100 realizations each of four
exemplary network types (small-world network: $V=100$, $m=4$, $p=0.2$ — random
network: $V=100$, $q=0.05$ — scale-free network: $V=100$, $k=4$ — complete
network: $V=100$).

Surprisingly, we also observe not the removal of an edge but the cloning of a
vertex to have smallest average impact on four of the five global network
metrics, even though removing a single edge is arguably a smaller structural
network perturbation. It can be concluded that the employed network
perturbations overall lead to minuscule changes of global network
characteristics. Nevertheless, it is vital to realize that the observe changes
can depend on the targeted constituents’ importance. Our results indicate that
particularly those constituents at neither end of the importance ranking can
be deemed potentially superfluous.

###  IV.2 Impact of perturbations on local network characteristics

The observed dependencies of changes of global network characteristics
regarding the targeted constituent’s importance $r_{\rm u}$ and the type of
perturbation indicate that similar changes and dependencies can be observed on
smaller network scales as well. In case of local network metrics, we first
investigate whether deviations in the distribution of centrality values – for
any of the four centrality concepts (vertex and edge centralities,
respectively) – can be observed after perturbation (cf. Sect. II.4). If
centrality values from an unperturbed and perturbed network can generally be
considered to be drawn from different distributions, the removed/cloned
vertex/edge can hence not be deemed potentially superfluous. Naively viewed,
it is still apparent that of the three perturbations, removing a single edge
is the smallest structural network alteration, as removing/cloning a vertex
would also include the removal/cloning of attached edges. Hence, it is
generally to be expected that removing an edge has not only the smallest
impact but potentially no impact at all on the distribution of centrality
values compared to removing or cloning a vertex.

Figure 2: Percentage of network realizations (color-coded) that show no
significant change in either of the centrality distributions (cf. Section
II.4) when removing/cloning a constituent of a given rank $r_{\rm u}$
(estimated with strength/nearest-neighbor centrality, in the respective
network) (SW small-world network; SF scale-free network; R random network; C
complete network). Empty cells – due to differences in network size – are
colored in grey.

Figure 2 shows that the respective perturbations did not lead to a significant
change in the distribution of centrality values in the majority of
investigated networks. We do, however, observe the changes to strongly depend
on network topology, type of perturbation, and rank $r_{\rm u}$ of
removed/cloned constituent. In case of removing an edge, we only observe very
few significant changes whatsoever in small-world networks with small rewiring
probability ($p=0.01$). Their almost regular structure explains why removing
only important edges still can lead to changes in the distribution of
centrality values in at least some of the realizations of networks (<$40\%$),
as removing such an important edge from the network will lead to large changes
in the path-structure and thus will greatly affect centrality values estimated
with path-based centrality metrics. Similar effects, that can be explained in
an analogous way, regarding these specific small-world networks (the 12 small-
world network types with $p=0.01$) are observed when removing or cloning a
vertex. The more regular the network the stronger is the alteration of the
regular structure when introducing or removing a vertex and its respective
edges. Furthermore, as the ranking in case of the vertices is done via the
strength centrality, it can be deduced that the more important the vertex, the
larger is its integration in its direct neighborhood within the network (cf.
Table 1). Therefore the more important the removed/cloned vertex, the larger
the alteration of the network’s structure.

Especially the aforementioned small-world networks that are large in size and
have a high edge density ($V\in\left\\{500,1000\right\\}$, $m=8$) show the
largest amount of significant changes. This is, otherwise, only observed for
networks with largest possible edge densities (fully connected networks) or
comparably large random networks ($V=500$). For these large and/or dense
networks, cloning a vertex will consequently result in adding a large amount
of edges, due to cloned vertex’ high degree. Likewise, removing a vertex
includes removing a large amount of edges in these networks. Both
perturbations hence result in large changes of the distributions of centrality
values.

In case of the less regular and less dense networks (random networks with
$V<500$, small-world networks with $0.01<p<1.0$, scale-free networks), the
observed changes in centrality values highly depend on which vertex was
removed/cloned. The amount of network realizations with significant changes in
the distributions of centrality values (regarding the respective perturbation)
decreases with increasing rank (decreasing importance) of the targeted vertex.
This once again is likely explained by the high interconnectedness (high
degree) of important vertices. Cloning the most important vertex always – in
$100\,\%$ of the realizations – led to significant changes of the distribution
of centrality values with regard to the unperturbed network. Furthermore, and
especially in small-world networks, even cloning less important vertices much
more often led to significant changes than removing said vertices.

We can conclude, that the structurally smallest perturbation, namely removing
a single edge, has the overall smallest influence on the distribution of
centrality values and that removing a vertex less often leads to significant
changes than cloning said vertex. Furthermore, removing/cloning an important
constituent has a stronger impact than removing a less important constituent.
Almost regular as well as dense and large networks are most strongly affected
by perturbations targeting vertices. Our results here, and in contrast to
those reported on in Sect. IV.1, indicate that particularly those constituents
identified as less important prior to perturbation (high rank), may qualifies
as potentially superfluous.

###  IV.3 Impact of perturbations on importance hierarchy

Having observed mostly insignificant changes in the distribution of centrality
values, at first glance, points toward a weak and minimal alteration of the
network. Yet, the importance hierarchies of network constituents might have
changed greatly. As a most extreme example: the constituents with respectively
highest and smallest centrality value prior to perturbation exchange their
positions in the ranking as a result of the perturbation. A constituent with
little importance prior to perturbation is now, after the employed
perturbation, considered the most important constituent and vice versa, while
the distribution of centrality values remained the same. Furthermore, changes
in the importance hierarchies can be used to identify if constituents that are
directly affected by the perturbation (e.g., removed or cloned) can be deemed
potentially superfluous. To this end, we investigate the changes in
constituents ranks due to the respective perturbations (cf. Sect. II.4).

Figure 3: Percentage of constituents with rank changes $\geq\theta$ of the
maximum possible rank change in dependency of the rank of the constituent
targeted by the respective perturbation (rows). We here chose $\theta\geq 5\%$
as this amounts to $\Delta^{\bullet}_{\mathrm{v/e}}\geq 1$ in case of the
smallest investigated networks with $V=20$. Ranking is done via different
centrality concepts (color-coded). Data is averaged over 100 realizations of
each of four exemplary network types (columns): small-world network ($V=100$,
$m=4$, $p=0.2$), random network ($V=100$, $q=0.05$), scale-free network
($V=100$, $k=4$), and complete network ($V=100$). The black horizontal line is
for eye guidance only and represents $5\%$ of possible constituents showing a
rank change $\Delta^{\bullet}_{\mathrm{v/e}}\geq 1$. Similar relations are
observed for $20\geq V\geq 1000$.

It can be deduced from Figure 3 that, largely independent of the employed
perturbation, the changes in the importance hierarchies depend on the
constituent’s rank $r_{\rm u}$ targeted by the perturbation. Perturbing
constituents with small rank (high importance) in comparison to those with
high ranks (low importance) tends not only to lead to greater rank changes of
single constituents but also to more constituents showing such changes
($\Delta^{\bullet}_{\mathrm{v/e}}\gg 0$) overall. Widely independent of the
networks’ topologies, we observe that the smaller the rank $r_{\rm u}$ (the
higher the importance) of the perturbed constituent, the larger are the
changes in the total ranking of the constituents. This general relationship
can be observed with each of the employed centrality concepts, while the
precise functional relationship depends on multiple factors such as network
size, network topology, type of perturbation, and centrality concept.

The quantitative nature of these relationships regarding perturbation and
centrality concept are exemplary depicted in the Appendix (cf. Figures 5-8)
for some network models, highlighting not only the overall stronger effect on
the ranking when targeting important constituents, but also the fact that the
rankings of all the other constituents are affected very distinctly depending
on their respective rank prior to perturbation (schematically depicted in
Figure 4). Especially in case of removing vertices, we observe that
constituents at either end of the importance hierarchy are affected less by
this perturbation than constituents with median rank.

Figure 4: Schematic depiction of changes in rank for each constituent (with
importance hierarchy prior to perturbation) in dependency of the rank of the
targeted constituent.

Our results here are in line with those reported in Sect. IV.2 and indicate
that particularly those constituents identified as less important prior to
perturbation can be deemed potentially superfluous.

###  IV.4 Concluding Remarks

It is to conclude that while aspects such as the path-structure, degree-
correlations, robustness, and stability are, largely and on average, left
unaltered by the employed perturbations, we do observe high dependency of
changes of metrics $\overline{\delta\mu}$ regarding the rank of the
constituent targeted by the perturbation. This points towards a possible
intrinsic existence of potentially superfluous constituents in the networks
investigated here. Observing changes in the rankings of constituents (as
determined with different centrality concepts) consolidates these findings.
Especially targeting constituents with low rank (high importance) in
comparison to targeting constituents with high rank (low importance) led to
greater changes in these rankings and revealed dependencies regarding the
network topology Saavedra  _et al._ (2011). This shows that – arguably
contrary to expectation – less dense networks and also less regular networks
(following no trivial geometric arrangements such as a ring or a lattice) can
contain more potentially superfluous constituents than for example very dense
and even complete network. For the investigated networks, we can conclude that
the three criteria point toward constituents of tendentially low (but not
least) importance, to be potentially superfluous.

##  V Discussion

We here proposed a perturbation-based method in order to tackle the extensive
problem of identifying potentially superfluous network constituents. We
formulated the premise that the instant absence or additional presence of a
potentially superfluous network constituent should lead to negligible changes
in network characteristics only, that not trivially depend on even the
smallest change in network size. Making use of minuscule and elemental
perturbations, targeting single constituents directly, we investigated whether
such perturbations lead to changes of global as well as local metrics that
describe the investigated networks rather comprehensively. The less changes we
observe for the metrics when perturbing the respective network constituent the
more this is an indication for this constituent to be of potentially
superfluous nature. We formulated three criteria, which can provide important
information when it comes to identifying potentially superfluous constituents.

It is generally to be expected that, independent of the investigated real-
world system, certain network topologies may contain superfluous constituents,
simply due to their structural makeup. Following this line of thinking, it is
to be expected that the sheer size and density of complete networks should
provide great possibility for the existence of such potentially superfluous
constituents. Likewise, it is easy to understand that certain regular
structures, e.g., a ring or a lattice with a large amount of nearest- and
next-nearest-neighbors connections, are more likely to contain potentially
superfluous constituents than a ring/lattice with only nearest-neighbors
connections.

We could confirm that – even though generally shown to have small influence on
local as well as global characteristics Holme  _et al._ (2002); Platig, Ott,
and Girvan (2013); Lekha and Balakrishnan (2020); Rings, Bröhl, and Lehnertz
(2022) – the here investigated effects of employed perturbations indeed
largely differ for different network topologies as revealed with the three
criteria. While we could show that size and edge density affect the values of
global network metrics, the here employed minuscule perturbations, on average,
led to negligible changes of these values. On the other hand, regular
structures, whether it be a ring/lattice or a complete network, were
especially prone to be influenced in their local characteristics.

Although in almost regular networks an importance hierarchy is dominated by
edge weights, said importance hierarchy in less regular networks might be
influenced by their distinct topological makeup (small-world or scale-free
networks). This makes it rather hard to get an intuitive feeling about the
existence of potentially superfluous constituents in these complex network
topologies. However, contrary to expectation, our perturbation-based approach
points to far less potentially superfluous constituents in complete and
regular networks than in more complex topologies such as small-world and
scale-free networks.

Nevertheless, independent of the networks’ topology, if the perturbation
targeted a more important constituent, the changes in local network
characteristics (distribution of centrality values and centrality-based
rankings) were also larger, in comparison to targeting a less important
constituent. This also shows that a priori knowledge about the importance
hierarchy of the networks’ constituents might not only be highly useful but in
some cases even necessary to end up with a satisfactory and somewhat accurate
representation of a real-world complex system. In addition, and especially in
those cases for which a priori knowledge about the network’s actual structure
is either not accessible or very limited, our perturbation-based approach can
aid in identifying potentially superfluous and likewise indispensable network
constituents.

Future studies should focus on employing the presented approach to identify
potentially superfluous constituents either in networks with built-in
superfluous constituents or in networks constructed from empirical
observations prone to have superfluous constituents. Further investigations
considering scenarios from real-world issues (like noise contamination and
other measurement errors Martin and Niemeyer (2019)) might aid in a more
accurate modeling of real complex (dynamical) systems. This could mean taking
into account not necessarily the exact cloning of network constituents but a
combination of cloning and perturbations regarding the edge weights of cloned
edges.

###### Acknowledgements.

The authors would like to thank Thorsten Rings for interesting discussions and
for critical comments on earlier versions of the manuscript.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Appendix

Figure 5: Changes in rank for each constituent (with importance hierarchy
prior to perturbation) in dependency of the rank of the targeted constituent
in a small-world network with $V=100$, $m=4$, $p=0.01$. Figure 6: Same as Fig.
5 but for a random network with $V=100$, $q=0.05$. Figure 7: Same as Fig. 5
but for a scale-free network with $V=100$, $k=4$. Figure 8: Same as Fig. 5 but
for a complete network with $V=100$.

## References

  * Boccaletti  _et al._ (2006) S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, “Complex networks: Structure and dynamics,” Phys. Rep. 424, 175–308 (2006).
  * Arenas  _et al._ (2008) A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469, 93–153 (2008).
  * Bullmore and Sporns (2009) E. Bullmore and O. Sporns, “Complex brain networks: graph theoretical analysis of structural and functional systems,” Nat. Rev. Neurosci. 10, 186–198 (2009).
  * Donges  _et al._ (2009a) J. F. Donges, Y. Zou, N. Marwan, and J. Kurths, “The backbone of the climate network,” Europhys. Lett. 87, 48007 (2009a).
  * Allen and Elston (2011) R. J. Allen and T. C. Elston, “From physics to pharmacology?” Rep. Prog. Phys. 74, 016601 (2011).
  * Barthélemy (2011) M. Barthélemy, “Spatial networks,” Phys. Rep. 499, 1–101 (2011).
  * Barabási, Gulbahce, and Loscalzo (2011) A.-L. Barabási, N. Gulbahce, and J. Loscalzo, “Network medicine: a network-based approach to human disease,” Nat. Rev. Genet. 12, 56–68 (2011).
  * Newman (2012) M. E. J. Newman, “Communities, modules and large-scale structure in networks,” Nat. Phys. 8, 25–31 (2012).
  * Baronchelli  _et al._ (2013) A. Baronchelli, R. Ferrer-i-Cancho, R. Pastor-Satorras, N. Chater, and M. H. Christiansen, “Networks in cognitive science,” Trends Cogn. Sci. 17, 348–360 (2013).
  * Lehnertz  _et al._ (2014) K. Lehnertz, G. Ansmann, S. Bialonski, H. Dickten, C. Geier, and S. Porz, “Evolving networks in the human epileptic brain,” Physica D 267, 7–15 (2014).
  * Heckmann, Schwanghart, and Phillips (2015) T. Heckmann, W. Schwanghart, and J. D. Phillips, “Graph theory – recent developments of its application in geomorphology,” Geomorphology 243, 130–146 (2015).
  * Gao, Barzel, and Barabási (2016) J. Gao, B. Barzel, and A.-L. Barabási, “Universal resilience patterns in complex networks,” Nature 530, 307 (2016).
  * Donges  _et al._ (2009b) J. F. Donges, Y. Zou, N. Marwan, and J. Kurths, “Complex networks in climate dynamics,” Eur. Phys. J.-Spec. Top. 174, 157–179 (2009b).
  * Zhou  _et al._ (2015) D. Zhou, A. Gozolchiani, Y. Ashkenazy, and S. Havlin, “Teleconnection paths via climate network direct link detection,” Phys. Rev. Lett. 115, 268501 (2015).
  * Uetz  _et al._ (2000) P. Uetz, L. Giot, G. Cagney, T. A. Mansfield, R. S. Judson, J. R. Knight, D. Lockshon, V. Narayan, M. Srinivasan, P. Pochart, A. Qureshi-Emili, Y. Li, B. Godwin, D. Conover, T. Kalbfleisch, G. Vijayadamodar, M. Yang, M. Johnston, S. Fields, and J. M. Rothberg, “A comprehensive analysis of protein–protein interactions in Saccharomyces cerevisiae,” Nature 403, 623–627 (2000).
  * Tyler  _et al._ (2009) A. L. Tyler, F. W. Asselbergs, S. M. Williams, and J. H. Moore, “Shadows of complexity: what biological networks reveal about epistasis and pleiotropy,” Bioessays 31, 220–227 (2009).
  * Hegland  _et al._ (2009) S. J. Hegland, A. Nielsen, A. Lázaro, A.-L. Bjerknes, and Ø. Totland, “How does climate warming affect plant-pollinator interactions?” Ecol. Lett. 12, 184–195 (2009).
  * Olesen  _et al._ (2011) J. M. Olesen, J. Bascompte, Y. L. Dupont, H. Elberling, C. Rasmussen, and P. Jordano, “Missing and forbidden links in mutualistic networks,” Proc. Roy. Soc. B: Biological Sciences 278, 725–732 (2011).
  * Halekotte and Feudel (2020) L. Halekotte and U. Feudel, “Minimal fatal shocks in multistable complex networks,” Sci. Rep. 10, 11783 (2020).
  * Delmas  _et al._ (2019) E. Delmas, M. Besson, M.-H. Brice, L. A. Burkle, G. V. Dalla Riva, M.-J. Fortin, D. Gravel, P. R. Guimarães Jr, D. H. Hembry, E. A. Newman, J. M. Olesen, M. M. Pires, J. D. Yeakel, and T. Poisot, “Analysing ecological networks of species interactions,” Biol. Rev. 94, 16–36 (2019).
  * Onnela  _et al._ (2007) J. P. Onnela, J. Saramäki, J. Hyvönen, G. Szábo, D. Lazer, K. Kaski, J. Kertész, and A.-L. Barabási, “Structure and tie strengths in mobile communication networks,” Proc. Natl. Acad. Sci. U.S.A. 104, 7332–7336 (2007).
  * Palla, Barabási, and Vicsek (2007) G. Palla, A.-L. Barabási, and T. Vicsek, “Quantifying social group evolution,” Nature 446, 664–667 (2007).
  * Bialonski, Horstmann, and Lehnertz (2010) S. Bialonski, M. Horstmann, and K. Lehnertz, “From brain to earth and climate systems: Small-world interaction networks or not?” Chaos 20, 013134 (2010).
  * Hlinka, Hartman, and Paluš (2012) J. Hlinka, D. Hartman, and M. Paluš, “Small-world topology of functional connectivity in randomly connected dynamical systems,” Chaos 22, 033107 (2012).
  * Porz, Kiel, and Lehnertz (2014) S. Porz, M. Kiel, and K. Lehnertz, “Can spurious indications for phase synchronization due to superimposed signals be avoided?” Chaos 24, 033112 (2014).
  * Wens (2015) V. Wens, “Investigating complex networks with inverse models: Analytical aspects of spatial leakage and connectivity estimation,” Phys. Rev. E 91, 012823 (2015).
  * Gastner and Ódor (2016) M. T. Gastner and G. Ódor, “The topology of large open connectome networks for the human brain,” Sci. Rep. 6, 27249 (2016).
  * Papo  _et al._ (2016) D. Papo, M. Zanin, J. H. Martínez, and J. M. Buldú, “Beware of the small-world neuroscientist!” Front. Hum. Neurosci. 10, 96 (2016).
  * Hlinka  _et al._ (2017) J. Hlinka, D. Hartman, N. Jajcay, D. Tomeček, J. Tintěra, and M. Paluš, “Small-world bias of correlation networks: From brain to climate,” Chaos 27, 035812 (2017).
  * Zanin  _et al._ (2018) M. Zanin, S. Belkoura, J. Gomez, C. Alfaro, and J. Cano, “Topological structures are consistently overestimated in functional complex networks,” Sci. Rep. 8, 11980 (2018).
  * Bialonski, Wendler, and Lehnertz (2011) S. Bialonski, M. Wendler, and K. Lehnertz, “Unraveling spurious properties of interaction networks with tailored random networks,” PLoS ONE 6, e22826 (2011).
  * Chung  _et al._ (2012) N. N. Chung, L. Y. Chew, J. Zhou, and C. H. Lai, “Impact of edge removal on the centrality betweenness of the best spreaders,” EPL (Europhysics Letters) 98, 58004 (2012).
  * Lü and Zhou (2011) L. Lü and T. Zhou, “Link prediction in complex networks: A survey,” Physica A 390, 1150–1170 (2011).
  * Lü  _et al._ (2015) L. Lü, L. Pan, T. Zhou, Y.-C. Zhang, and H. E. Stanley, “Toward link predictability of complex networks,” Proc. Natl. Acad. Sci. 112, 2325–2330 (2015).
  * Liao  _et al._ (2017) H. Liao, M. S. Mariani, M.Medo, Y.-C. Zhang, and M.-Y. Zhou, “Ranking in evolving complex networks,” Phys. Rep. 689, 1–54 (2017).
  * Kramer  _et al._ (2009) M. A. Kramer, U. T. Eden, S. S. Cash, and E. D. Kolaczyk, “Network inference with confidence from multivariate time series,” Phys. Rev. E 79, 061916 (2009).
  * Yan  _et al._ (2018) X. Yan, L. G. S. Jeub, A. Flammini, F. Radicchi, and S. Fortunato, “Weight thresholding on complex networks,” Phys. Rev. E 98, 042304 (2018).
  * Zeng and Cimini (2012) A. Zeng and G. Cimini, “Removing spurious interactions in complex networks,” Phys. Rev. E 85, 036101 (2012).
  * Zhang, Li, and Deng (2018) Q. Zhang, M. Li, and Y. Deng, “Measure the structure similarity of nodes in complex networks based on relative entropy,” Physica A 491, 749–763 (2018).
  * Kumar  _et al._ (2020) A. Kumar, S. S. Singh, K. Singh, and B. Biswas, “Link prediction techniques, applications, and performance: A survey,” Physica A 553, 124289 (2020).
  * Cantwell  _et al._ (2020) G. T. Cantwell, Y. Liu, B. F. Maier, A. C. Schwarze, C. A. Serván, J. Snyder, and G. St-Onge, “Thresholding normally distributed data creates complex networks,” Phys. Rev. E 101, 062302 (2020).
  * Frantz, Cataldo, and Carley (2009) T. L. Frantz, M. Cataldo, and K. M. Carley, “Robustness of centrality measures under uncertainty: Examining the role of network topology,” Comput. Math. Organ. Theor. 15, 303–328 (2009).
  * Bellingeri  _et al._ (2020) M. Bellingeri, D. Bevacqua, F. Scotognella, R. Alfieri, Q. Nguyen, D. Montepietra, and D. Cassi, “Link and node removal in real social networks: a review,” Front. Phys. 8, 228 (2020).
  * Latora and Marchiori (2005) V. Latora and M. Marchiori, “Vulnerability and protection of infrastructure networks,” Phys. Rev. E 71, 015103 (2005).
  * Ghoshal and Barabási (2011) G. Ghoshal and A.-L. Barabási, “Ranking stability and super-stable nodes in complex networks,” Nat. Commun. 2, 394 (2011).
  * Ceci and Barbarossa (2018) E. Ceci and S. Barbarossa, “Small perturbation analysis of network topologies,” in  _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ (2018) pp. 4194–4198.
  * Tantardini  _et al._ (2019) M. Tantardini, F. Ieva, L. Tajoli, and C. Piccardi, “Comparing methods for comparing networks,” Sci. Rep. 9, 17557 (2019).
  * Mheich, Wendling, and Hassan (2020) A. Mheich, F. Wendling, and M. Hassan, “Brain network similarity: methods and applications,” Network Neurosci. 4, 507–527 (2020).
  * Rings, Bröhl, and Lehnertz (2022) T. Rings, T. Bröhl, and K. Lehnertz, “Network structure from a characterization of interactions in complex systems,” Sci. Rep. 12, 11742 (2022).
  * Newman (2002) M. E. J. Newman, “Spread of epidemic disease on networks,” Phys. Rev. E 66, 016128 (2002).
  * Bialonski and Lehnertz (2013) S. Bialonski and K. Lehnertz, “Assortative mixing in functional brain networks during epileptic seizures,” Chaos 23, 033139 (2013).
  * Barahona and Pecora (2002) M. Barahona and L. M. Pecora, “Synchronization in small-world systems,” Phys. Rev. Lett. 89, 054101 (2002).
  * Atay, Bıyıkoğlu, and Jost (2006) F. M. Atay, T. Bıyıkoğlu, and J. Jost, “Network synchronization: Spectral versus statistical properties,” Physica D 224, 35–41 (2006).
  * Beauchamp (1965) M. A. Beauchamp, “An improved index of centrality,” Behav. Sci. 10, 161–163 (1965).
  * Sabidussi (1966) G. Sabidussi, “The centrality index of a graph,” Psychometrika 31, 581–603 (1966).
  * Freeman (1977) L. C. Freeman, “A set of measures of centrality based on betweenness,” Sociometry 40, 35–41 (1977).
  * Freeman (1979) L. C. Freeman, “Centrality in social networks: Conceptual clarification,” Soc. Networks 1, 215–239 (1979).
  * Bonacich (1987) P. Bonacich, “Power and centrality: A family of measures,” Am. J. Sociol. 92, 1170–1182 (1987).
  * Wuchty and Stadler (2003) S. Wuchty and P. F. Stadler, “Centers of complex networks,” J. Theor. Biol. 223, 45–53 (2003).
  * Koschützki  _et al._ (2005) D. Koschützki, K. Lehmann, L. Peeters, S. Richter, D. Tenfelde-Podehl, and O. Zlotowski, “Centrality indices,” in  _Network Analysis_, Lecture Notes in Computer Science, Vol. 3418, edited by U. Brandes and T. Erlebach (Springer, Berlin, Heidelberg, 2005) pp. 16–61.
  * Borgatti and Everett (2006) S. P. Borgatti and M. G. Everett, “A graph-theoretic perspective on centrality,” Soc. Networks 28, 466–484 (2006).
  * Estrada and Higham (2010) E. Estrada and D. J. Higham, “Network properties revealed through matrix functions,” SIAM Rev. 52, 696–714 (2010).
  * Valente and Fujimoto (2010) T. W. Valente and K. Fujimoto, “Bridging: locating critical connectors in a network,” Soc. Netw. 32, 212–220 (2010).
  * Chen  _et al._ (2012) D. Chen, L. Lü, M.-S. Shang, Y.-C. Zhang, and T. Zhou, “Identifying influential nodes in complex networks,” Physica A 391, 1777–1787 (2012).
  * Costa  _et al._ (2015) E. C. Costa, A. B. Vieira, K. Wehmuth, A. Ziviani, and A. P. C. Da Silva, “Time centrality in dynamic complex networks,” Adv. Complex Syst. 18, 1550023 (2015).
  * Lawyer (2015) G. Lawyer, “Understanding the influence of all nodes in a network,” Sci. Rep. 5, 8665 (2015).
  * Wu, Tian, and Liu (2018) A.-K. Wu, L. Tian, and Y.-Y. Liu, “Bridges in complex networks,” Phys. Rev. E 97, 012307 (2018).
  * Bröhl and Lehnertz (2019) T. Bröhl and K. Lehnertz, “Centrality-based identification of important edges in complex networks,” Chaos 29, 033115 (2019).
  * Zhao  _et al._ (2020) N. Zhao, J. Li, J. Wang, T. Li, Y. Yu, and T. Zhou, “Identifying significant edges via neighborhood information,” Physica A 548, 123877 (2020).
  * Bröhl and Lehnertz (2022) T. Bröhl and K. Lehnertz, “A straightforward edge centrality concept derived from generalizing degree and strength,” Sci. Rep. 12, 4407 (2022).
  * Lü  _et al._ (2016) L. Lü, D. Chen, X.-L. Ren, Q.-M. Zhang, Y.-C. Zhang, and T. Zho, “Vital nodes identification in complex networks,” Phys. Rep. 650, 1–63 (2016).
  * Iñiguez  _et al._ (2022) G. Iñiguez, C. Pineda, C. Gershenson, and A.-L. Barabási, “Dynamics of ranking,” Nat. Commun. 13, 1646 (2022).
  * Bonacich (1972) P. Bonacich, “Factoring and weighting approaches to status scores and clique identification,” J. Math. Sociol. 2, 113–120 (1972).
  * Bavelas (1950) A. Bavelas, “Communication patterns in task-oriented groups,” J. Acoust. Soc. Am. 22, 725–730 (1950).
  * Brandes (2001) U. Brandes, “A faster algorithm for betweenness centrality,” J. Math. Sociol. 25, 163–177 (2001).
  * Girvan and Newman (2002) M. Girvan and M. E. J. Newman, “Community structure in social and biological networks,” Proc. Natl. Acad. Sci. U.S.A. 99, 7821–7826 (2002).
  * Saavedra  _et al._ (2011) S. Saavedra, D. B. Stouffer, B. Uzzi, and J. Bascompte, “Strong contributors to network persistence are the most vulnerable to extinction,” Nature 478, 233–235 (2011).
  * Watts and Strogatz (1998) D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393, 440–442 (1998).
  * Erdős and Rényi (1959) P. Erdős and A. Rényi, “On random graphs I,” Publ. Math. Debrecen 6, 290–297 (1959).
  * Batagelj and Brandes (2005) V. Batagelj and U. Brandes, “Efficient generation of large random networks,” Phys. Rev. E 71, 036113 (2005).
  * Albert and Barabási (2002) R. Albert and A.-L. Barabási, “Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002).
  * Holme  _et al._ (2002) P. Holme, B. J. Kim, C. N. Yoon, and S. K. Han, “Attack vulnerability of complex networks,” Phys. Rev. E 65, 056109 (2002).
  * Platig, Ott, and Girvan (2013) J. Platig, E. Ott, and M. Girvan, “Robustness of network measures to link errors,” Phys. Rev. E 88, 062812 (2013).
  * Lekha and Balakrishnan (2020) D. S. Lekha and K. Balakrishnan, “Central attacks in complex networks: a revisit with new fallback strategy,” Physica A 549, 124347 (2020).
  * Martin and Niemeyer (2019) C. Martin and P. Niemeyer, “Influence of measurement errors on networks: Estimating the robustness of centrality measures,” Netw. Sci. 7, 180–195 (2019).


# SCHWARZSCHILD SOLUTION OF THE MODIFIED EINSTEIN FIELD EQUATIONS

Carringtone Kinyanjui, Dismas S. Wamalwa

###### Abstract

A reformulation of the Schwarzschild solution of the linearized Einstein field
equations in post-Riemannian Finsler spacetime is derived. The solution is
constructed in three stages: the exterior solution, the event-horizon solution
and the interior solution. It is shown that the exterior solution is
asymptotically similar to Newtonian gravity at large distances implying that
Newtonian gravity is a low energy approximation of the solution. Application
of Eddington-Finklestein coordinates is shown to reproduce the results
obtained from standard general relativity at the event horizon. Further
application of Kruskal-Szekeres coordinates reveals that the interior solution
contains maximally extensible geodesics.

AMS Subject Classification: 83C05, 83C57, 83C75

Key Words and Phrases: Einstein field equations, Schwarzschild metric, Black
holes, Finsler spaces

Department of Physics

University of Nairobi

Chiromo, Nairobi, 30197-0100, KENYA

##  1 Introduction

The general theory of relativity as proposed by Einstein [1] relates the
curvature of spacetime to gravity. The theory describes the relation between
the curvature of spacetime to the energy of an object. This can succinctly be
described by Einstein field equations (stated without the cosmological
constant) [2]:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G}{c^{4}}T_{\mu\nu}$

where $R_{\mu\nu}$ is the Ricci curvature tensor, $R$ is the Ricci scalar,
$g_{\mu\nu}$ is the metric, G is the gravitational constant, $T_{\mu\nu}$ is
the stress energy tensor and $c$ is the speed of light. The theory was soon
tested observationally by Eddington in 1919 and found to be correct [3]. The
simplest analytical solution to the field equations is the solution for a
static uncharged and spherical mass. The solution was proposed by Karl
Schwarzschild in 1916 [4]. The Schwarzschild metric for paths along radial
lines is given by [2]:

$d\mathit{s}^{2}=\left(1-\frac{2m}{\mathit{r}}\right)^{-1}d\mathit{t}^{2}+\left(1-\frac{2m}{\mathit{r}}\right)d\mathit{r}^{2}$
(1)

It can be seen that the metric is singular at $\mathit{r}=2m$ and
$\mathit{r}=0$ [5]. However, a change of coordinates particularly proposed by
Eddington [6] and later by Finklestein [7], showed that the singularity at
$\mathit{r}=2m$ can be removed. However, the curvature singularity due to
spacetime structure at $\mathit{r}=0$ persists and can not be removed [5].

The interior solution ($\mathit{r}<2m$) describes objects called “blackholes”
that “swallow” objects that come too close to them ($\mathit{r}=2m$) and have
escape velocities greater than the speed of light. This means that at
$\mathit{r}<2m$, there is no possibility of escape [2]. While scientific
speculation on the existence of black holes predates general relativity [8],
they seem implicit within the description of the theory. Initially, blackholes
were not accepted as physically feasible objects. According to Einstein, the
solution was in fact only a mathematical curiosity and of no astrophysical
importance [9]. Chandrasekhar [10], showed that beyond some mass limit, stars
at the endpoints of stellar evolution and therefore undergoing gravitational
collapse can not be held back by electron degeneracy. Later work by
Oppenheimer, Volkoff and Tolmann generalized Chandrasekhar’s work [11] and
proved that collapse to a blackhole is astrophysically feasible. Since then,
the reality of astrophysical black holes has been confirmed [12], [13].

Important theoretical work in the description of blackholes, both from general
relativity and from a quantum field theoretic perspective has been done by
Hawking et. al. [14], [15]. It is hoped that quantum gravity will resolve the
singularity problem in the centre of stationary blackholes. Gambini and Pullin
[16], [17] have proposed that Loop Quantum Gravity can lead to the description
of a non-singular quantised Schwarzschild metric . In this work, we shall take
a different approach from the (relatively) standard gravity quantisation
procedure as a correction of gravity at the scale of black hole energies. We
shall rely on the results of the model formulated within the frame work of
finsler spaces. Finsler spaces [18], [19] can be thought of as generalizations
of Riemann spaces. In Particular, we adopt the extended Einstein field
equations in Finsler geometry [20]:

$\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)+\left(S_{\mu\nu}-\frac{1}{2}g_{\mu\nu}S\right)=\frac{8\pi
G}{c^{4}}\left(T_{\mu\nu}+\tau_{\mu\nu}\right)$ (2)

where $S_{\mu\nu}$ and $S$ are additional Ricci tensor and Ricci scalar terms
respectively. Furthermore, $S_{\mu\nu}$ and $R_{\mu\nu}$ are functions of
position, $\mathit{x}$ and velocity, $\mathit{y}$ respectively. We hold the
velocity terms constant so that
$R_{\mu\nu}(\mathit{x,y})=R_{\mu\nu}(\mathit{x})$ and
$S_{\mu\nu}(\mathit{x,y})=S_{\mu\nu}(\mathit{x})$. It should be noted that the
field equations in equation (2) above are difficult to solve. We shall
therefore proceed to solve the above field equations with an extra constraint
of symmetry and hence develop a mathematical model of Schwarzschild blackholes
appropriately. In this respect, we shall assume that components of the tensors
$R_{\mu\nu}$ and $S_{\mu\nu}$ are related through a symmetrical linear
transformation to be described later.

##  2 The Vacuum Field Equations

We can rewrite Einstein field equations in equation (2) above as:

$Z^{\alpha\beta}-\frac{1}{2}Zg^{\alpha\beta}=kQ^{\alpha\beta}$ (3)

where

$Z^{\alpha\beta}=R^{\alpha\beta}+S^{\alpha\beta}$ (4) $Z=R+S$ (5)
$Q^{\alpha\beta}=T^{\alpha\beta}+\tau^{\alpha\beta}$ (6)

Transforming equation (3) into covariant form, we obtain

$Z_{\mu\nu}-\frac{1}{2}Zg_{\mu\nu}=kQ_{\mu\nu}$ (7)

We shall now proceed to find the vacuum field equations corresponding to
equation (7). Setting $Q_{\mu\nu}=0$ and performing contraction with metric
tensor, it can be shown that

$R_{\mu\nu}+S_{\mu\nu}=0$ (8)

These are our desired vacuum field equations. We next consider a solution of
the vacuum field equations for the Schwarzschild metric.

###  2.1 Solution of the vacuum field equations

The difficulty of solving Finsler extended Einstein field equations is
evident. In this paper, we simply introduce a further constraint of symmetry
by demanding that the tensor terms $R_{\mu\nu}$ and $S_{\mu\nu}$ are linearly
related. While this of course limits the richness of the theory of Finsler
spaces, it helps us develop a physically relevant and realistic model. In
order to solve equation (8), we follow the standard procedure of finding the
actual form of the metric by constraining its functional parameters. Rewriting
equation (8) for $\mu=\nu=\theta$, we have:

$\displaystyle R_{\hat{\theta}\hat{\theta}}+S_{\hat{\theta}\hat{\theta}}=0$
(9)

It has been shown that the value of $R_{\hat{\theta}\hat{\theta}}$ [2] is:

$\displaystyle
R_{\hat{\theta}\hat{\theta}}=\frac{2}{\mathit{r}}\frac{d\lambda}{d\mathit{r}}e^{-2\lambda}+\frac{1-e^{-2\lambda}}{\mathit{r}^{2}}$
(10)

But $R_{\hat{\theta}\hat{\theta}}$ and $S_{\hat{\theta}\hat{\theta}}$ are
linearly related, and therefore similar in structure. Indeed given two
differential equations $D_{1}(x)$ and $D_{2}(y)$ such that:

$\displaystyle
D_{1}(\mathit{x})+D_{2}(\mathit{y})=0;\quad\mathit{y}=\alpha\mathit{x}$
$\displaystyle\mbox{or}\;\;D_{1}(\mathit{x})+D_{2}(\alpha\mathit{x})=0$
$\displaystyle\mbox{therefore,}\;\;D_{1}(\mathit{x})=-\alpha
D_{2}(\mathit{x})$

implying that:

$\frac{D_{1}(\mathit{x})}{D_{2}(\mathit{x})}=-\alpha=\gamma$ (11)

It thus, follows that $D_{1}(\mathit{x})$, $D_{2}(\mathit{x})$ are linearly
related and similar in structure. Differential equations can be modelled as
matrix eigenvalue problems whereby the differential operator becomes the
matrix and the solution the eigenfunction. Taking into account that for any
eigenvalue problem

$\hat{A}\textbf{X}=\beta\textbf{X},$

and vector $\textbf{Y}=g\textbf{X}$, we have

$\hat{A}\textbf{Y}=\beta\textbf{Y}$

where g is a constant. Implication of equation (11) is that the structure of
$S_{\hat{\theta}\hat{\theta}}$ arises as:

$S_{\hat{\theta}\hat{\theta}}=\frac{2}{\mathit{r}}\frac{d\mathit{L}}{dr}e^{-2\mathit{L}}+\frac{1-e^{-2\mathit{L}}}{\mathit{r}^{2}}$
(12)

where $\mathit{L}$ is a function in space that is linearly related to
$\lambda$. Equation (9) is, therefore, rewritten as:

$\frac{2}{\mathit{r}}\frac{d\lambda}{d\mathit{r}}e^{-2\lambda}+\frac{1-e^{-2\lambda}}{\mathit{r}^{2}}+\frac{2}{\mathit{r}}\frac{d\mathit{L}}{d\mathit{r}}e^{-2\mathit{L}}+\frac{1-e^{-2\mathit{L}}}{\mathit{r}^{2}}=0$
(13)

Or

$2\mathit{r}\frac{d\lambda}{d\mathit{r}}e^{-2\lambda}+1-e^{-2\lambda}+2\mathit{r}\frac{d\mathit{L}}{d\mathit{r}}e^{-2\mathit{L}}+1-e^{-2\mathit{L}}=0$
(14)

Application of chain rule

$\frac{d\mathit{L}}{d\lambda}=\frac{d\mathit{L}}{d\lambda}\frac{d\lambda}{d\mathit{r}}$
(15)

and trying to find $\frac{dL}{d\lambda}$ by assuming the most general case of
“pseudopolynomial” functions, we have:

$\mathit{L(r)}=A\mathit{r}^{n}+B\mathit{r}^{n-1}+C\mathit{r}^{n-2}+\cdots+D\mathit{r}+E+F\mathit{r}^{-1}+G\mathit{r}^{-2}+H\mathit{r}^{-3}+\cdots+I\mathit{r}^{-n}$
$\lambda(\mathit{r})=J\mathit{r}^{n}+K\mathit{r}^{n-1}+L\mathit{r}^{n-2}+\cdots+M\mathit{r}+N+O\mathit{r}^{-1}+P\mathit{r}^{-2}+Q\mathit{r}^{-3}+\cdots+R\mathit{r}^{-n}$

where $A$ to $R$ are distinct constants. Therefore,

$\frac{d\mathit{L}}{d\lambda}=\frac{A^{\prime}\mathit{r}^{n-1}+B^{\prime}\mathit{r}^{n-2}+C^{\prime}\mathit{r}^{n-3}+\cdots+D+F^{\prime}\mathit{r}^{-2}+G^{\prime}\mathit{r}^{-3}+H^{\prime}\mathit{r}^{-4}+\cdots
I^{\prime}\mathit{r}^{-n-1}}{J^{\prime}\mathit{r}^{n-1}+K^{\prime}\mathit{r}^{n-2}+L^{\prime}\mathit{r}^{n-3}+\cdots+M+O^{\prime}\mathit{r}^{-2}+P^{\prime}\mathit{r}^{-3}+Q^{\prime}\mathit{r}^{-4}+\cdots+R^{\prime}\mathit{r}^{-n-1}}$

where primed constants are new constants obtained after differentiation.
Neglecting higher order terms for large $\mathit{r}$, we get:

$\frac{d\mathit{L}}{d\lambda}=\frac{A^{\prime}\mathit{r}^{n-1}+B^{\prime}\mathit{r}^{n-2}+C^{\prime}\mathit{r}^{n-3}+\cdots+D+F^{\prime}\mathit{r}^{-1}}{J^{\prime}\mathit{r}^{n-1}+K^{\prime}\mathit{r}^{n-2}+L^{\prime}\mathit{r}^{n-3}+\cdots+M}$

We can express $\frac{d\mathit{L}}{d\lambda}$ as:

$\frac{d\mathit{L}}{d\lambda}=\frac{b(\mathit{r})}{\mathit{r}}$ (16)

where

$b(\mathit{r})=\frac{A^{\prime}\mathit{r}^{n}+B^{\prime}\mathit{r}^{n-1}+C^{\prime}\mathit{r}^{n-2}+\cdots+D\mathit{r}+E^{\prime}\mathit{r}^{-1}}{J^{\prime}\mathit{r}^{n-1}+K^{\prime}\mathit{r}^{n-2}+L^{\prime}\mathit{r}^{n-3}+\cdots+M}$
(17)

Equations (16) when used in equation (13) yields:

$2-e^{-2\lambda}+2\mathit{r}\frac{d\lambda}{d\mathit{r}}e^{-2\lambda}+-e^{-2\mathit{L}}+2\mathit{r}\frac{b}{\mathit{r}}\frac{d\lambda}{d\mathit{r}}e^{-2\mathit{L}}=0$
(18)

Since $\mathit{L}=k\lambda$, we have equation (18) as:

$2-e^{-2\lambda}+2\mathit{r}\frac{d\lambda}{d\mathit{r}}e^{-2\lambda}+-e^{-2k\lambda}+2\mathit{r}\frac{b}{\mathit{r}}\frac{d\lambda}{d\mathit{r}}e^{-k2\lambda}=0$
(19)

or expressing $k$ as the logarithm of a certain constant $a$, we may write

$e^{-k2\lambda}=\left(e^{-2\lambda}\right)^{k}=ae^{-2\lambda}$ (20)

Using equation (20) in equation (19) and regarding the small constant $a$ as
unity, we have:

$-\frac{d\lambda}{d\mathit{r}}2e^{-2\lambda}(\mathit{r}+b)+2e^{-2\lambda}=2$
(21)

Equation (21) can easily be integrated to give:

$e^{-2\lambda}=\left(1-\frac{2m}{\mathit{r}+b}\right)$ (22)

Where $m=\frac{GM}{c^{2}}$

##  3 Modified Schwarzschild Metric

The standard Schwarzschild metric is written as:

$d\mathit{s}^{2}=e^{-2\lambda}d\mathit{t}^{2}-e^{2\lambda}d\mathit{r}^{2}-r^{2}(d\theta^{2}+sin^{2}\phi^{2})$
(23)

If we consider paths along radial lines for light cones around the singularity
$\mathit{r}=2m$, the above metric reduces to:

$d\mathit{s}^{2}=e^{-2\lambda}d\mathit{t}^{2}-e^{2\lambda}d\mathit{r}^{2}$
(24)

Using equation (22) in equation (24) yields:

$d\mathit{s}^{2}=\left(1-\frac{2m}{\mathit{r}+b}\right)d\mathit{t}^{2}-\left(1-\frac{2m}{\mathit{r}+b}\right)^{-1}d\mathit{r}^{2}$
(25)

Comparison with equation (1) shows that as a result of our computation, there
is a correction to the term $2m/\mathit{r}$ present in equation (25).

Let us now consider the asymptotic behaviour of equation (25) at the
Schwarzschild radius, at the centre of the black hole and at large distances.

###  3.1 Asymptotic behaviour of the metric

####  3.1.1 Behaviour of the metric at r=2m

At $\mathit{r}=2m$, a radius is defined such that nothing that goes into the
blackhole ever gets out [4]. In relativistic terms, the light cones of test
particles are completely tipped over such that the geodesics are pointing
towards the center of the black hole [6] i.e.,

$d\mathit{s}^{2}=0$ (26)

Singularity at $\mathit{r}=2m$ is due to poor choice of coordinate system
hence, is called coordinate singularity. Transformation into Eddington-
Finklestein coordinates removes the singularity. We now proceed to transform
the metric into Eddington-Finklestein coordinates. These coordinates describe
spacetime at the event horizon. They are derived from null geodesics where the
metric is set to zero (equation 26). Combining equations (26) and (25), we
obtain:

$\left(1-\frac{2m}{\mathit{r}+b}\right)d\mathit{t}^{2}=\left(1-\frac{2m}{\mathit{r}+b}\right)^{-1}d\mathit{r}^{2}$

This equation can be solved to give:

$\displaystyle t=\mathit{r}+2m\;ln(\frac{\mathit{r}+b}{2m}-1)+c$

where $c=b-2m+2m\;ln\;2m$ is a constant. Relabelling $\mathit{t}$ as
$\mathit{r*}$, we have:

$\mathit{r*}=\mathit{r}+2m\;ln(\frac{\mathit{r}+b}{2m}-1)+c$ (27)

so that

$\frac{d\mathit{r}*}{d\mathit{r}}=\frac{\mathit{r}+b}{\mathit{r}+b-2m}$ (28)

Introducing “tortoise” coordinates coordinates [2] of the form:

$\displaystyle\mathit{v}=\mathit{t}+\mathit{r*}$ (29)
$\displaystyle\mathit{u}=\mathit{t}-\mathit{r*},$ (30)

it can easily be shown that:

$d\mathit{r}^{2}=\frac{1}{4}\left(\frac{\mathit{r}+b-2m}{\mathit{r}+b}\right)^{2}(d\mathit{v}^{2}-2d\mathit{u}d\mathit{v}+d\mathit{v}^{2})\\\
$ (31)

and

$d\mathit{t}^{2}=\frac{1}{4}(d\mathit{u}^{2}+2d\mathit{u}d\mathit{v}+d\mathit{v}^{2})$
(32)

Using equations (22), (31) and (32) in equation (25), we obtain:

$d\mathit{s}^{2}=\left(1-\frac{2m}{\mathit{r}+b}\right)\cdot\frac{1}{4}(d\mathit{u}^{2}+2d\mathit{u}d\mathit{v}+d\mathit{v}^{2})-$
$\left(1-\frac{2m}{\mathit{r}+b}\right)^{-1}\frac{1}{4}\left(1-\frac{2m}{\mathit{r}+b}\right)^{2}(d\mathit{v}^{2}-2d\mathit{u}d\mathit{v}+d\mathit{v}^{2})$
(33)

Equation (33) can be reduced to:

$\displaystyle d\mathit{s}^{2}=\left(1-\frac{2m}{\mathit{r}+b}\right)dudv$
(34)

Equation (34) is the metric restated in Eddington-Finklestein coordinates.
Taking asymptotic behaviour at $\mathit{r}=2m$ and noting that $2m>>b$ yields:

$d\mathit{s}^{2}|_{\mathit{r}=2m}=0$ (35)

which is consistent with equation (25). The result is also consistent with the
behaviour of the standard Schwarzschild metric [2] and hence, with the
mathematical predictions of standard general relativity. Therefore, the theory
is asymptotically similar to the standard theory outside the black hole and at
the event horizon.

####  3.1.2 Asymptotic behaviour inside the Scwhwarzschild black hole

By inspection, it can easily be seen that the metric in equation (25) is non-
singular at $\mathit{r}=0$ i.e.,

$d\mathit{s}^{2}|_{\mathit{r}=0}=(1-\frac{2m}{b})d\mathit{t}^{2}+(1-\frac{2m}{b})^{-1}d\mathit{r}^{2}$
(36)

However, to be completely sure that we have eliminated the singularity, we
need to transform the metric into Kruskal-Szekeres coordinates [5]. Using
equations (29) and (30) in equation (27), we get:

$\mathit{r*}=\mathit{r}+2m\;ln(\frac{\mathit{r}+b}{2m}-1)+c=\frac{1}{2}(\mathit{v}-\mathit{u})$
(37)

Dividing equation (37) by $2m$, we obtain:

$r+b-2m=2m\;e^{{\frac{1}{4m}(v-u)}{-\frac{1}{2m}(r+c)}}$ (38)

Rearranging equation (34) and applying equation (38), we obtain:

$d\mathit{s}^{2}=\frac{2m}{\mathit{r}+b}e^{\frac{(\mathit{v}-\mathit{u})}{4m}}e^{-\frac{(\mathit{r}+c)}{2m}}d\mathit{u}d\mathit{v}$

Using transformations of the form:

$\mathit{U}=-e^{-\frac{\mathit{u}}{4m}};\qquad\mathit{V}=e^{-\frac{\mathit{v}}{4m}},$

it is easy to show that:

$\displaystyle
d\mathit{s}^{2}=\frac{32m^{3}}{\mathit{r}+b}e^{-\frac{(\mathit{r}+c)}{2m}}d\mathit{U}d\mathit{V}$

so that application of cooordinate transformation suggested by Kruskal [4]
gives:

$d\mathit{s}^{2}=\frac{32m^{3}}{\mathit{r}+b}e^{-\frac{(\mathit{r}+c)}{2m}}(d\mathit{T}^{2}-d\mathit{X}^{2})$
(39)

Taking asymptotic behaviour at $\mathit{r}=0$, we obtain

$d\mathit{s}^{2}|_{\mathit{r}=0}=\frac{32m^{3}}{b}e^{-\frac{\mathit{r}+c}{2m}}(d\mathit{T}^{2}-d\mathit{X}^{2})$
(40)

showing that the metric has non-singular behaviour at $\mathit{r}=0$.

####  3.1.3 Asymptotic behaviour at large distances

At large distances, equation (25) reduces to:

$d\mathit{s}^{2}=d\mathit{t}^{2}-d\mathit{r}^{2}$ (41)

Equation (41) is just the Minkowski spacetime [5]. At large distances from the
mass, curvature is minimised, the general relativistic curvature corrections
are absent and thus the metric is asymptotically flat and similar to the
standard metric.

###  3.2 Conclusion

The metric corresponding to a Schwarzschild solution for the extended Einstein
field equations has been derived. The metric has been shown to have the
external Schwarzschild solution as an aymptotic extension at long distances.
At the event horizon, the metric is shown to be equal to the standard
Schwarzschild metric. Further, and more interestingly, the metric is shown to
be non-singular at $\mathit{r}=0$. We invite further exploration of the work
presented, including the calculation of the Kretschmann invariant and possible
modification of the Kerr metric. We hope in later work to explore a
unification scheme based on the extended field equations that will assist in
the determination of the constant b.

## References

  * [1] A. Einstein,The Field Equations of Gravitation, Proceedings of The Prussian Academy of Sciences, 2, (1915), 884-887
  * [2] D. McMmahon, Relativity Demystified, McGraw-Hill, USA(2006).
  * [3] F.W. Dyson, A.S. Eddington, C.R. Davidson, A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total eclipse of May 29, 1919, Philosophical Transactions of The Royal Society of London 220, (1919), 291-333, doi:10.1098/rsta.1920.0009.
  * [4] K. Schwarzschild, On the Gravitational Field of a Mass Point by Einstein’s Theory, Proceedings of The Prussian Academy of Sciences, (1916), 189-196 , arXiv:physics/9905030 [physics.hist-ph]
  * [5] M. P. Hobson , G. P. Efstathiou, A. N. Lasenby, General Relativity: An Intro duction for Physicists, Cambridge University Press, UK(2006), 196-220.

  * [6] A. Eddington, A Comparison of Whitehead’s and Einstein’s Formulae, Nature
(Letters to The Editor), 113, 2832(1924), 192.

  * [7] D. Finkelstein, Past-Future Asymmetry of the Gravitational Field of a Point Particle, Physical Review Journal, 110, N0.4(1958), doi:10.1103/PhysRev.110.965.

  * [8] C.Montgomery, W.Orchiston, I. Whittingham, Michell, Laplace and the Origin of the Black Hole Concept, Journal of Astronomical History and Heritage, 12, 2(2009), 90-96

  * [9] M. Bartusiak, Blackholes: How an Idea abandoned by Newtonians, hated by Einstein, and gambled on by Hawking, became loved, Yale University Press, USA(2016)

  * [10] S. Chandrasekhar, The Maximum Mass of Ideal White Dwarfs, Astrophysical Journal, 74, (1931), 81-82
  * [11] J.R. Oppenheimer, G.M. Volkoff, On Massive Neutron Cores, Physical Review, 55, 4(1939), 374–381, doi:10.1103/PhysRev.55.374
  * [12] B.L. Webster,P. Murdin, Cygnus X-1—a Spectroscopic Binary with a Heavy Companion?, it Nature(Letters to Nature), 235, 37-38(1972), 235.
  * [13] R. Johnston, List of black hole candidates, http://www.johnstonsarchive.net/
relativity/bhctable.html, (2016), Accessed on 12th July 2016

  * [14] L.H. Ford, Quantum Field Theory in Curved Spacetime, arXiv:gr-qc/9707062, (1997).
  * [15] J.M.M. Senovilla, Singularity Theorems in General Relativity: Achievements and Open Questions, http://arxiv.org/abs/physics/0605007v1, (2006).
  * [16] R. Gambini, J. Pullin, Loop Quantisation of The Schwarzschild Blackhole,Physical Review Letters, 110, 211301(2013), doi:10.1103/PhysRevLett.110.211301
  * [17] D. McMmahon, String Theory Demystified, McGraw-Hill, New York(2009), 239-255
  * [18] Z. Shen, Lectures on Finsler Geometry, Indiana University-Purdue University Indianapolis, USA(2001)
  * [19] G.S. Asanov, Finsler Geometry, Relativity and Gauge Theories, D Reidei Publishing Company, Holland(1985)
  * [20] D.S.Wamalwa, J.A. Omolo, Generalised Relativistic Dynamics In A Non-Inertial Frame of Reference, Indian Journal of Physics, 84, N0.9(2010), 1241-1255


* (146) D. Cooper, C. D. Hodgson, and S. P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, vol. 5 of MSJ Mem. Tokyo: Mathematical Society of Japan (MSJ), 2000.
  * (147) F. C. Caramello Jr, Introduction to orbifolds, arXiv:1909.08699.
  * (148) K. Behrend and B. Noohi, Uniformization of Deligne-Mumford curves, math/0504309.
  * (149) R. C. Gunning, Complex analytic varieties, in Proceedings of 12th Summer Institute for Theoretical Physics (Boulder, CO, USA, 1969): Mathematical Methods in Field Theory and Complex Analytic Varieties. (K. T. Mahanthappa and W. E. Brittin, eds.), vol. 12C of Lectures in Theoretical Physics, 1971.
  * (150) K. G. Charles Boyer, Sasakian Geometry.  Oxford Mathematical Monographs. Oxford University Press, USA, 0 ed., 2008\.
  * (151) T. Kawasaki, The Riemann-Roch theorem for complex V-manifolds, Osaka journal of mathematics 16 (1979), no. 1 151–159.
  * (152) F. Campana, Orbifolds, special varieties and classification theory, Annales de l’institut Fourier 54 (2004), no. 3 499–630.
  * (153) J. W. Milnor, Dynamics in one complex variable: Introductory lectures, math/9201272.
  * (154) A. M. Uludağ, Orbifolds and their uniformization, in Arithmetic and geometry around hypergeometric functions, pp. 373–406.  Springer, 2007.
  * (155) A. Adem, J. Leida, and Y. Ruan, Orbifolds and stringy topology, vol. 171.  Cambridge University Press, 2007.
  * (156) R. C. Gunning, Lectures on complex analytic varieties: The local parametrization theorem.  [Princeton University Press], 1970.
  * (157) H. Grauert and R. Remmert, Coherent Analytic Sheaves.  Grundlehren der mathematischen Wissenschaften No.265. Springer, 1 ed., 1984.
  * (158) O. Riemenschneider, Quotient singularities: General theory, in Singular Points of complex Analytic Surfaces, ch. 8, pp. 231–256.  Hamburg, 2011.  Available online at https://www.math.uni-hamburg.de/home/riemenschneider/bookchap08.pdf (accessed on Jan. 14, 2023).
  * (159) M. Namba, On finite Galois covering germs, Osaka Journal of Mathematics 28 (1991), no. 1 27–35.
  * (160) M. Namba, Finite branched coverings of complex manifold, Sugaku expositions 5 (1992), no. 2 193–211.
  * (161) T. Suwa, Introduction to complex analytic geometry, in Singularities in geometry and topology, pp. 161–220.  World Scientific, 2007.
  * (162) D. N. Akhiezer, A. L. Onishchik, and V. Palamodov, Several complex variables IV: algebraic aspects of complex analysis, vol. 10.  Springer Science & Business Media, 2012.
  * (163) J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization.  Springer Science & Business Media, 2007.
  * (164) C. Voisin, Hodge Theory and Complex Algebraic Geometry I, vol. 1 of Cambridge Studies in Advanced Mathematics 76.  Cambridge University Press, 1 ed., 2003.
  * (165) D. Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967) 375–386.
  * (166) E. Artal Bartolo, J. Martín-Morales, and J. Ortigas-Galindo, Cartier and Weil divisors on varieties with quotient singularities, International Journal of Mathematics 25 (2014), no. 11 1450100.
  * (167) W.-m. Chen and Y.-b. Ruan, A New cohomology theory for orbifold, Commun. Math. Phys. 248 (2004) 1–31, [math/0004129].
  * (168) I. Moerdijk and D. A. Pronk, Orbifolds, sheaves and groupoids, K-theory 12 (1997), no. 1 3–21.
  * (169) B. Nasatyr and B. Steer, Orbifold Riemann surfaces and the Yang-Mills-Higgs equations, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 22 no. 4 595–643, [alg-geom/9504015].
  * (170) V. Erlandsson and J. Souto, Counting curves on orbifolds, arXiv:2012.14018.
  * (171) M. Boileau, S. Maillot, and J. Porti, Three-dimensional orbifolds and their geometric structures, vol. 15.  Société mathématique de France Paris, 2003.
  * (172) F. Hirzebruch and T. Höfer, On the Euler number of an orbifold, Mathematische Annalen 286 (1990) 255–260.
  * (173) P. Griffiths and J. Harris, Principles of algebraic geometry.  John Wiley & Sons, 2014.
  * (174) I. Moerdijk and J. Mrcun, Introduction to Foliations and Lie Groupoids.  Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2003.
  * (175) M. Troyanov, LES SURFACES EUCLIDIENNES À SINGULARITÉS CONIQUES, Fondation L’Enseignement Mathématique (1986).
  * (176) S. A. Wolpert, The hyperbolic metric and the geometry of the universal curve, Journal of Differential Geometry 31 (1990) 417–472.
  * (177) S. A. Wolpert, Cusps and the family hyperbolic metric, Duke Mathematical Journal 138 (2005) 423–443, [math/0508470].
  * (178) K. Obitsu, W.-K. To, and L. Weng, The Asymptotic Behavior of the Takhtajan-Zograf Metric, Communications in Mathematical Physics 284 (2008) 227–261.
  * (179) C. M. Judge, On the Existence of Maass Cusp Forms on Hyperbolic Surfaces with Cone Points, Journal of the American Mathematical Society 8 (1995) 715–759.
  * (180) A. McIntyre and L. A. Takhtajan, Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of kronecker’s first limit formula, Analysis 16 (2006) 1291, [math/0410294].
  * (181) C. Ehresmann, Sur les espaces localement homogenes, Lens.Math 35 (1936), no. 1 317–335.
  * (182) W. M. Goldman, Locally homogeneous geometric manifolds, in Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, pp. 717–744, World Scientific, 2010.  arXiv:1003.2759.
  * (183) W. Goldman, Geometric structures and varieties of representation, in The Geometry of Group Representations, ed. by W. Goldman, A. Magid. Summer Conference, 1987.
  * (184) W. M. Goldman, Higgs bundles and geometric structures on surfaces, arXiv:0805.1793.
  * (185) S. Choi, Geometric structures on orbifolds and holonomy representations, Geometriae Dedicata 104 (2004), no. 1 161–199, [math/0107172].
  * (186) S. Choi, Geometric structures on 2-orbifolds. Exploration of discrete symmetry, vol. 27 of MSJ Mem. Tokyo: Mathematical Society of Japan, 2012.
  * (187) P. Ares-Gastesi and I. Biswas, On the symplectic structure over a moduli space of orbifold projective structures, arXiv:1308.3353.
  * (188) Z. ShengYuan, A note on holomorphic families of complex projective structures, .
  * (189) G. Faltings, Real projective structures on Riemann surfaces, Compositio Mathematica 48 (1983), no. 2 223–269.
  * (190) F. Luo, Monodromy groups of projective structures on punctured surfaces, Inventiones mathematicae 111 (1993), no. 1 541–555.
  * (191) S. Gupta, Monodromy groups of $\mathbb{CP}^{1}$-structures on punctured surfaces, Journal of Topology 14 (2021), no. 2 538–559, [arXiv:1909.10771].
  * (192) J. Lehner, Discontinuous Groups and Automorphic Functions (Mathematical Surveys).  Mathematical Surveys. American Mathematical Society, 1964.