#  Schrödinger Equation on Fractals Curves Imbedding in $R^{3}$

Alireza Khalili Golmankhaneh a†
Dumitru Baleanu b,c,d 111Tel:+903122844500, Fax:+903122868962
E-mail addresses<EMAIL_ADDRESS>
aDepartment of Physics, Islamic Azad University, Urmia Branch,
PO Box 969, Urmia, Iran
<EMAIL_ADDRESS>
bDepartment of Mathematics and Computer Science
$\c{C}ankaya$ University, 06530 Ankara, Turkey
cInstitute of Space Sciences,
P.O.BOX, MG-23, R76900, Magurele-Bucharest, Romania
dDepartment of Chemical and Materials Engineering, Faculty of Engineering,
King Abdulaziz University, P.O. Box: 80204, Jeddah, 21589, Saudi Arabia

###### Abstract

In this paper we have generalized the quantum mechanics on fractal time-space.
The time is changing on Cantor-set like but space is considered as fractal
curve like Von-Koch curve. The Feynman path method in quantum mechanics has
been suggested on fractal curve. Using $F^{\alpha}$-calculus and Feynman path
method we found the Schrëdinger on fractal time-space. The Hamiltonian
operator and momentum operator has been derived. More, the continuity equation
and the probability density is given in generalized formulation.

Keywords:Feynman path method, Schrëdinger on fractal time-space,continuity
equation

##  1 Introduction

Fractal is objects that are very fragmented and irregular at all scales. Their
important properties are non-differentiability and having non-integer
dimension. Fractal has topological dimension less than Hausdorff-Besicovitch,
box-counting, and similarity dimensions. In general, dimension of fractal can
be integer or not well-defined dimension[1, 2, 3, 4, 5, 6, 7]. Fractional
local calculus and nonlocal has applied to model the process with memory and
fractal structure[8, 9, 10, 11, 12, 13, 14]. The electric and magnetic fields
are derived using fractional integrals as a approximation method on fractals
[15]. The quantum space-time on the basis of relativity principle and
geometrical concept of fractals is introduced [16].The probability density of
quantum wave function with by Dirichlet boundary conditions in a D-dimensional
spaces has been studied [17]. The fractal concept to quantum physics and the
relationships between fractional integral and Feynman path integral method is
developed [18, 19]. The generalized wave functions is introduced to fractal
dimension, a wide class of quantum problems, including the infinite potential
well, harmonic oscillator, linear potential, and free particle [20].Fratal
path in quantum mechanics and their contributing in Feynman path integral is
investigated [21]. The classical mechanics is derived without the need of the
least-action principle using path-integral approach [22]. The calculus on the
fractals has been studied in different methods like probabilistic approach
method, sequence of discrete Laplacians, measure-theoretical method, time
scale calculus [23].Riemann integration like method has been studied since
that is useful and algorithmic [24, 25, 26, 27, 28, 29].Using the calculus on
fractals the Newtonian mechanics, Lagrange and Hamilton mechanics, and Maxwell
equations has been generalized [30, 31, 32]. As a pursue theses research we
generalized the quantum mechanics on fractals.
The plan of this paper is as following:
Section 2 we review the fractal calculus. In section 3 we defined the
gradient, divergent and Laplacian on fractal space. Section 4 is explained the
quantum mechanics on fractals curves. In section 5 we suggested the
probability density and continuity equation on the generalized quantum
formalism. Finally, section 6 is devoted to conclusion.

##  2 A Summery of the calculus on fractal curves

We review the $F^{\alpha}$-calculus on fractal curves [24, 25, 26, 27, 28, 29,
30, 31, 32]. Suppose fractal curve $F\subset R^{3}$ which is continuously
parameterizable i.e there exists a function
$\textbf{w}:[a_{0},b_{0}]\rightarrow F\subset R^{3}$ which is continuous. We
also assume w to be invertible. A subdivision $P_{[a,b]}$ of interval
$[a,b],a<b,$ is a finite set of points $\\{a=v_{0}<v_{1},...<v_{n}=b\\}$. For
$a_{0}\leq a<b<b_{0}$ and appropriate $\alpha$ to be chosen, therefore let

$\gamma^{\alpha}(F,a,b)=\lim_{\delta\rightarrow
0}\inf_{\\{P_{[a,b]}:|P|\leq\delta\\}}\sum_{i=0}^{n-1}\frac{|\textbf{w}(v_{i+1})-\textbf{w}(v_{i})|^{\alpha}}{\Gamma(\alpha+1)},$
(1)

where $|.|$ denotes the Euclidean norm on $R^{3}$ and
$|P|=\max\\{v_{i+1}-v_{i};i=0,...,n-1\\}$. A $\gamma$-dimension of $F$, which
is defined as

$\textmd{dim}_{\gamma}(F)=\inf\\{\alpha:\gamma^{\alpha}(F,a,b)=0\\}=\sup\\{\alpha:~{}\gamma^{\alpha}(F,a,b)=\infty\\}.$
(2)

After this defintion $\alpha$ is equal to $dim_{\gamma}(F).$ The staircase
function $S_{F}^{\alpha}:[a_{0},b_{0}]\rightarrow R$ of order $\alpha$ for a
set $F$, is defined as

$S_{F}^{\alpha}(v)=\begin{cases}\gamma^{\alpha}(F,p_{0},v)~{}~{}~{}v\geq
p_{0}\\\ -\gamma^{\alpha}(F,v,p_{0})~{}~{}~{}v<p_{0},\end{cases}$ (3)

where $a_{0}\leq p_{0}\leq b_{0}$ is arbitrary but fixed, and
$v\in[a_{0},b_{0}].$ It is monotonic function. The $\theta=\textbf{w}(v),$
denote a point on fractal curve $F$

$J(\theta)=S_{F}^{\alpha}(\textbf{w}^{-1}(\theta)),~{}~{}~{}\theta\in F.$ (4)

We suppose that fractal curves whose $S_{F}^{\alpha}$ is finite and invertible
on $[a,b]$. The $F^{\alpha}$-derivative of the bounded function
$f:F\rightarrow R$ $(f\in B(F))$ at $\theta\in F$ is defined.

Then the directional $F^{\alpha}$-derivative of function $f$ at $\theta\in F$
is defined as

$^{w_{j}}\mathfrak{D}_{F}^{\alpha}f(\textbf{w}(v))=F-\lim_{t^{\prime}\rightarrow
t}\frac{f(w_{1}(v),w_{2}(v),...w_{j}(v^{\prime}),...w_{i}(v))-f(\textbf{w}(v))}{S_{F}^{\alpha}(v^{\prime})-S_{F}^{\alpha}(v)},$
(5)

where $w_{j}$ is shows direction of $F^{\alpha}$-derivative, if the limit
exists [27].
Let $f\in B(F)$ is an $F$-continuous function on $C(a,b)$ which is the segment
$\\{\textbf{w}(v):v\in[a,b]\\}$ of $F$. Now let $g:f\rightarrow R$ be define
as

$g(w(v))=\int_{C(a,v)}f(\theta)d_{F}^{\alpha}\theta,$ (6)

for all $v\in[a,b]$. So that

$\mathfrak{D}_{F}^{\alpha}g(\theta)=f(\theta)$ (7)

Note: Let $\gamma^{\alpha}(F,a,b)$ be finite and $f(\theta)=1$, $\theta\in F$
denote the constant function. Then

$\int_{C(a,b)}f(\theta)d_{F}^{\alpha}\theta=\int_{C(a,b)}1d_{F}^{\alpha}\theta=S_{F}^{\alpha}(b)-S_{F}^{\alpha}(a)=J((w(b))-J((w(a)).$
(8)

Remark: $F^{\alpha}$-derivative and $F^{\alpha}$-integral is a linear
operation.
1) Let $f:F\rightarrow R$, $f(\theta)=k\in R$ then
$\mathfrak{D}_{F}^{\alpha}f=0$.
2) IF $f:F\rightarrow R$ be a $F$-continuous function such that
$\mathfrak{D}_{F}^{\alpha}f=0$. Then $f=k$ where $k$ on $C(a,b).$
Suppose $f:F\rightarrow R$ be $F^{\alpha}$-differentiable function and
$h:F\rightarrow R$ be $F$-continuous such that
$h(\theta)=\mathfrak{D}_{F}^{\alpha}f(\theta)$, so

$\int_{C(a,b)}h(\theta)d_{F}^{\alpha}\theta=f(w(b))-f(w(a)).$ (9)

Analogue Taylor series is defined for $h(\theta)\in B(F)$ as

$f(\textbf{w}(v))=\sum_{n=0}^{\infty}\frac{(S_{F}^{\alpha}(v)-S_{F}^{\alpha}(v^{\prime}))^{n}}{n!}(\mathfrak{D}_{F}^{\alpha})^{n}f(\textbf{w}(v^{\prime})),$
(10)

where $h(\theta)$ is $F^{\alpha}$-differentiable any number of times on
$C(a,b)$. That is $(\mathfrak{D}_{F}^{\alpha})^{n}h\in B(F)$, $\forall n>0$.

##  3 Gradient, Divergent, Curl and Laplacian on Fractal Curves

In this section we generalized the $F^{\alpha}$-calculus by defining the
gradient, divergent, curl and Laplacian on fractal curves imbedding in
$R^{3}$.

###  3.1 Gradient on fractal curves

Let us consider the $f\in B(F)$ as an $F$-continuous function on
$C(a,b)\subset F$ and $\textbf{w}(v,w_{i}(v)):R\rightarrow R^{3},i=1,2,3$, so
the gradient of the $f(\textbf{w}):F\rightarrow R$ is

$\mathfrak{\nabla}_{F}^{\alpha}f(\textbf{w})=~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha}f(\textbf{w})\hat{e}^{i}~{}~{}~{}i=1,2,3,$
(11)

where the $\hat{e}^{i}$ is the basis of $R^{n}$.

###  3.2 Divergent on fractal curves

Let the
$\textbf{f}(\textbf{w}(v))=f_{i}(\textbf{w}(v))~{}\hat{e}^{i}~{}~{}i=1,2,3$,
be a vector field on fractal curve. Then we define the divergent of the
$\textbf{f}:F\rightarrow R^{n}$ as follows:

$\mathfrak{\nabla}_{F}^{\alpha}.\textbf{f}(\textbf{w}(v))=~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha}f_{i}(\textbf{w}(v)),$
(12)

where $f_{i}(\textbf{w}(v))$ are components of vector field.

###  3.3 Laplacian on fractal curves

Consider the $\textbf{w}(v,w_{i}(v)):R\rightarrow R^{3}$ on the fractal curve,
therefore the Laplacian is defined as

$\triangle_{F}^{\alpha}f=(\mathfrak{\nabla}_{F}^{\alpha})^{2}f=(^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}f(\textbf{w}(v))$
(13)

where the $\triangle_{F}^{\alpha}$ is called Laplacian on fractal curve.

##  4 Quantum mechanics on fractal curve

The classical mechanics is reformulated in terms of a minimum principle. The
Euler-Lagrange equations of motion is derived from the least action. The
Feynman paths for a particle in quantum mechanics are fractals with dimension
2 [33]. In this section, we obtain the Schrödinger equation on fractal curves.

###  4.1 Generalized Feynman path integral method

Feynman method for studying quantum mechanics using classical Lagrangian and
action is presented in Ref [34, 35]. Now we want to generalized Feynman method
using Lagrangian on fractals curves. Consider generalized action as

$\mathfrak{A}^{\alpha}_{F}=\int_{t_{1}}^{t_{2}}L_{F}^{\alpha}(t,\textbf{w}(v),~{}^{t}D_{F}^{\alpha}\textbf{w}(v))d_{F}^{\alpha}v~{}d_{F}^{\alpha}t~{}~{}~{}~{}L_{F}^{\alpha}:F\times
F\times F\rightarrow R.$ (14)

In view of Feynman method, if wave function on fractal in
$t_{1},\textbf{w}_{a}(v_{1})$ is
$\psi_{F}^{\alpha}(t_{1},\textbf{w}_{a}(v_{1}))$. So it gives the total
probability amplitude at $t_{2},\textbf{ w}_{b}(v_{2})$ as

$\psi_{F}^{\alpha}(t_{2},\textbf{w}_{b}(v_{2}))=\int_{-\infty}^{\infty}K_{F}^{\alpha}(t_{2},\textbf{w}_{b}(v_{2}),t_{1},\textbf{w}_{a}(v_{1}))(\psi_{F}^{\alpha}(t_{1},\textbf{w}_{a}(v_{1}))d_{F}^{\alpha}\textbf{w}(v),$
(15)

where $K_{F}^{\alpha}$ is the propagator which is defined as follows:

$K_{F}^{\alpha}(t_{2},\textbf{w}_{b}(v_{2}),t_{1},\textbf{w}_{a}(v_{1}))=\int_{w_{a}}^{w{b}}\exp[\frac{i}{\hbar}\mathfrak{A}^{\alpha}_{F}]\mathcal{D}_{F}^{\alpha}\textbf{w}(v).$
(16)

Where symbol $\mathcal{D}_{F}^{\alpha}$ indicates the integration over all
fractal paths from $\textbf{w}_{a}(v_{1})$ to $\textbf{w}_{b}(v_{2})$.
Now we derive the Schrödinger equation for a free particle on fractal curve,
which is describes the evolution of the wave function from
$\textbf{w}_{a}(v_{1})$ to $\textbf{w}_{b}(v_{2})$ , when $t_{2}$ differs from
$t_{1}$ an infinitesimal amount $\epsilon$. Supposing
$S_{F}^{\alpha}(v_{2})=S_{F}^{\alpha}(v_{1})+\epsilon$, leads to Lagrangian
for free particle as

$L_{F}^{\alpha}(t,\textbf{w}(v),~{}^{t}D_{F}^{\alpha}\textbf{w}(v))\simeq\frac{m(\textbf{w}(v)-\textbf{w}(v_{0}))^{2}}{2(S_{F}^{\alpha}(v_{2})-S_{F}^{\alpha}(v_{1}))}.$
(17)

The generalized action on fractal $\mathfrak{A}^{\alpha}_{F}$ is approximately

$\mathfrak{A}^{\alpha}_{F}\sim\epsilon
L_{F}^{\alpha}=\frac{m(\textbf{w}(v)-\textbf{w}(v_{0}))^{2}}{2\epsilon}.$ (18)

As a consequence, we obtain

$\psi_{F}^{\alpha}(t+\epsilon,\textbf{w}(v))=\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m(\textbf{w}(v)-\textbf{w}_{0}(v_{0}))^{2}}{2\epsilon}]\psi_{F}^{\alpha}(t,\textbf{w}_{0}(v_{0}))\mathcal{D}_{F}^{\alpha}\textbf{w}_{0}(v_{0}).$
(19)

Here, because of properties of exponential function only those fractal paths
give significant contributions which are very close to $\textbf{w}(v)$.
Changing the variable in the integral
$\delta=\textbf{w}(v)-\textbf{w}_{0}(v_{0})$ we have
$\psi_{F}^{\alpha}(t,\textbf{w}_{0}(v_{0}))=\psi_{F}^{\alpha}(t,\textbf{w}(v)+\delta)$.
Since both $\epsilon$ and $\delta$ are small quantities, so that
$\psi_{F}^{\alpha}(t,\textbf{w}(v)+\delta)$ and
$\psi_{F}^{\alpha}(t+\epsilon,\textbf{w}(v))$ can be expanded using Eq. (10).
We only keep up to terms of second order of the $\epsilon$ and $\delta$. As a
result we get

$\displaystyle\psi_{F}^{\alpha}(t,\textbf{w}(v))+\epsilon(~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))\simeq\chi_{F}(t)\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}](\psi_{F}^{\alpha}(t,\textbf{w}(v))+\delta(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v)))$
$\displaystyle+\frac{\delta^{2}}{2}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))d_{F}^{\alpha}\delta,$
(20)

where $\chi_{F}(t)$ is the characteristic function for Cantor like sets. The
second term in the right hand side vanishes on integration. It follows by
equating the leading terms on both sides we obtain

$\psi_{F}^{\alpha}(t,\textbf{w}(v))=\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}]\psi_{F}^{\alpha}(t,\textbf{w}(v))d_{F}^{\alpha}\delta.$
(21)

Also, we arrive at

$A=\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}]d_{F}^{\alpha}\delta=\sqrt{\frac{2i\pi\hbar\epsilon}{m}},$
(22)

and

$\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}](\frac{\delta^{2}}{2}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))d_{F}^{\alpha}\delta=\epsilon\frac{i\hbar}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v))).$
(23)

Finally, equating the remaining terms, we get Schrödinger equation on fractal
curves for free particle as

$(i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))=~{}\chi_{F}(t)\frac{-\hbar^{2}}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v))).$
(24)

The Eq. (24) leads to the definition of the Hamiltonian and momentum operator
on fractal curves as

$\hat{H}_{F}^{\alpha}=i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha}~{}~{}~{}\hat{P}_{F}^{\alpha}=-i\hbar\mathfrak{\nabla}_{F}^{\alpha}$
(25)

The solution of Eq. (24) can be find using conjugate equation as

$i\hbar~{}\frac{\partial\theta(t,\xi)}{\partial
t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}}{\partial\xi^{2}}\theta(t,\xi)~{}~{}~{}\theta(\xi,t)=\phi[\psi_{F}^{\alpha}(t,\textbf{w}(v)))].$
(26)

Since the solution Eq. (26) is

$\theta(t,\xi)=(Ae^{ik\xi}+Be^{-ik\xi})e^{-i\beta t},$ (27)

where $k=\frac{\sqrt{2mE}}{\hbar}$ and $\omega=\frac{E}{\hbar}$ are constants.
Now by applying $\phi^{-1}$ we have the solutions as

$\psi_{F}^{\alpha}(t,\textbf{w}(v)))=(Ae^{ikS_{F}^{\alpha}(v)}+Be^{-ikS_{F}^{\alpha}(v)})e^{-i\beta
S_{F}^{\alpha}(t)}.$ (28)

It is straight forward to extended to the case of a free particle to the
motion involving the potential. In this case the Lagrangian will be
$L_{F}^{\alpha}=T_{F}^{\alpha}-V_{F}^{\alpha}(t,\textbf{w}(v))$. By
substituting the Lagrangian in the Eq. (4.1) one can derive the Schrödinger
equation as

$\displaystyle\psi_{F}^{\alpha}(t,\textbf{w}(v))+\epsilon(~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))\simeq\chi_{F}(t)\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}][1-\frac{i\epsilon}{\hbar}V_{F}^{\alpha}(t,\textbf{w}(v))](\psi_{F}^{\alpha}(t,\textbf{w}(v))$
$\displaystyle+\delta(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v)))+\frac{\delta^{2}}{2}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))d_{F}^{\alpha}\delta.$
(29)

The same manner we worked out above the Eq. (4.1) becomes

$(i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))=~{}\chi_{F}(t)\frac{-\hbar^{2}}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))+\chi_{F}(t)V_{F}^{\alpha}(t,\textbf{w}(v))\psi_{F}^{\alpha}(t,\textbf{w}(v))$
(30)

##  5  Continuity equation and probability on fractal

It is well known that the continuity equation is a important concept in
quantum mechanics. Therefor, the probability density on the fractal for a
particle is defined as

$\rho_{F}^{\alpha}(t,\textbf{w}(v))=(~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v)))~{}\psi_{F}^{\alpha}(t,\textbf{w}(v)).$
(31)

The complex conjugate wave function of Eq. (30) is

$(-i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha})~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))=~{}\chi_{F}(t)\frac{-\hbar^{2}}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v)))+\chi_{F}(t)V(t,\textbf{w}(v))~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))$
(32)

where
$V_{F}^{\alpha}(t,\textbf{w}(v))=~{}^{*}V_{F}^{\alpha}(t,\textbf{w}(v))$.
Applying this identity is given below

$^{t}\mathfrak{D}_{F}^{\alpha}(\psi_{F}^{\alpha}(t,\textbf{w}(v))~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v)))=~{}^{t}\mathfrak{D}_{F}^{\alpha}(\psi_{F}^{\alpha}(t,\textbf{w}(v)))^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))+\psi_{F}^{\alpha}(t,\textbf{w}(v))~{}^{t}\mathfrak{D}_{F}^{\alpha}(~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))),$
(33)

and substituting Eq. (30) and Eq. (32), into Eq.(33) yield us

$\displaystyle
i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha}\rho_{F}^{\alpha}(t,\textbf{w}(v))=\chi_{F}(t)\frac{\hbar^{2}}{2m}[\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))$
$\displaystyle-~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}\psi_{F}^{\alpha}(t,\textbf{w}(v))].$
(34)

As a consequence the definition of a probability current density on fractal
curve is

$\displaystyle
J_{F}^{\alpha}(t,\textbf{w}(v))=\chi_{F}(t)\frac{\hbar}{2mi}[\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))$
$\displaystyle-~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}\psi_{F}^{\alpha}(t,\textbf{w}(v))]$
(35)

In the following table correspondence between standard quantum mechanics and
generalized quantum framework is presented.

Comparison between Standard Quantum and Quantum on Fractals |
---|---
Postulates |  Standard Quantum | Quantum on Fractals
State |  $\psi(t,x)$ | $\psi_{F}^{\alpha}(t,\textbf{w}(v))$
Hamiltonian |  $i\hbar\frac{\partial}{\partial x}$ | $i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha}$
Momentum |  $-i\hbar\nabla$ | $-i\hbar\mathfrak{\nabla}_{F}^{\alpha}$

##  6 Conclusion

The calculus on sets, vector space and manifold is used in the classical,
quantum mechanics and general relativity respectively. The geometry has
important role in this generalization and modeling the physical phenomena.
Recently, fractal geometry has been suggested by Mandelbrot. So the calculus
on them has been suggest by many researcher but it is still an open problem.
In this work we have studied the calculus on fractal curves. Since the path
integral in Feynman formulation is fractal so that is motivated us to suggest
this generalization. This framework can suggest correct way for obtaining
Schrödinger equation from Fyenman path quantum mechanics.

## Acknowledgements

One of the authors (AKG) would like to thank Professor A. D. Gangal for useful
discussion on this topic during the period of time he was in Pune University.

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11institutetext: Department of Electrical Engineering and Computer Science
Indian Institute of Technology, Bhilai
Chhattisgarh, India.
11email<EMAIL_ADDRESS>
# Temporal Analysis of Worldwide War

Devansh Bajpai  Rishi Ranjan Singh

###### Abstract

Analysis of wars and conflicts between regions has been an important topic of
interest throughout the history of humankind. In the latter part of the
$20^{th}$ century, in the aftermath of two World Wars and the shadow of
nuclear, biological, and chemical holocaust, more was written on the subject
than ever before. Wars have a negative impact on a country’s economy, social
order, infrastructure, and public health. In this paper, we study the wars
fought in history and draw conclusions from that. We explore the participation
of countries in wars and the nature of relationships between various countries
during different timelines. A big part of today’s wars is fought against
terrorism. Therefore, this study also attempts to shed light on different
countries’ exposure to terrorist encounters and analyses the impact of wars on
a country’s economy in terms of change in GDP.

###### Keywords:

Temporal Network Analysis, War Networks, Data Visualization

##  1 Introduction

Wars are one of the most important factors in deciding the state of the world.
These are the major turning points in any country or empire’s history. Several
literature study about cause [19, 12, 17], damage [18, 9] and outcomes [16,
21] associated with interstate and civil wars. In the network of military
alliances, wars and international trade, the relation between the
international trade and wars happening among the countries have been studied
[10, 13]. In [14], authors have done a neural network based analysis of
militarized disputes for a timeline from 1885 till 1992. Network theoretic
analysis for international relations is done in [10]. A similar work to this
paper [11], analysed temporal network of international relations based on the
wars fought between 1816 and 2007. Since the events are historical we already
know the facts. Therefore, it becomes easier to connect the reported results
with the cause that triggered it.

In this paper, we carry out a temporal analysis of the wars that have been
fought in the history and report the inter-country relationships during
different timelines. We focus only on the wars fought after 1500 CE due to
availability of data and with the assumption that wars fought before 1500 CE
might have only minor effects on the present inter-country relationships. We
use temporal multi-graph to analyze the wars fought during 1500-2020 timeline.
Some basic graph properties are used for the analysis of temporal data. The
distribution of wars over the timeline, countries participating in most of the
wars in different timelines, rival relationship between countries, countries
which fought most wars along the same side during different timelines are few
of the focal points of this study. The relevant statistics and crucial
conclusions are reported. Some of the major events related to wars in the
history are marked and also mapped to respective roots and causes. Terrorist
organizations are also considered as nodes in the temporal graphs to analyze
the terrorist activities and encounters with different countries. Finally, the
impact of wars on a country’s economy has been studied in terms of change in
GDP.

##  2 Data Collection

One of the main reasons for limited related research is the limited
availability of data-sets. A data-set maintained by Sarkees and Wayman[22]
covers the wars from 1816 - 2007 timeline. It only covers a small portion of
time-line that we plan to cover. Their study also did not cover the impact of
wars on economy of countries. Therefore, we create a data-set for more wider
timeline which we collect based on the data available on Wikipedia pages [1].
These set of pages list war history of 165 different countries. Each country
has a different page listing wars and conflicts that the country has
participated in. The temporal multi-graph was built based on the data
available on these pages. Requests and Beautiful Soup libraries available with
Python3 are used for scraping the web pages. These libraries can fetch the
content written inside HTML tags. A python script is used which processes
every country’s page and scrapes the details of the wars that country has
participated in. Most of the pages have the same format and contain
information such as, name of wars, countries participated in wars and
timelines of wars. The raw data from most of the pages is retrieved and stored
in a CSV file. Next, the raw data collected in CSV file is used to create a
temporal multi-graph from it.

The MPD [4, 8, 7, 5, 3, 23, 20, 15, 2, 6] is a freely available excel spread-
sheet (Maddison, 2016) that provides per capita income for 184 countries upto
2016 CE. We have used the MPD data to study the impact of wars on country’s
economic growth. The data for years 1850-2016 has been used since most of the
values were missing for years before 1850.

###  2.1 Challenges in Network Formation

Out of 165 around 25 pages had a different format. The difference was in the
formatting of available data. Since the scraping script is automated for
general format, therefore, it failed to process those 25 pages. After some
further modifications in the script and manually writing some entries directly
in CSV file, the complete data was collected.

Entries in the CSV file with different formats made it difficult to directly
read data from the CSV file and create the network. For example, different
formats used for representing timeline shows why temporal network could not be
made directly: 1948, 1948-1948, May 1948 September 1948, 5th May 1948-9th
September 1948. 18th century, Early 17th century to 19th century.
A data processing script is used to process each input and convert it into a
standard format. Python’s regular expression library RE and some string
processing are used in this script. Next, an integer id is assigned to each
opponent, as well as a node is added for it and the adjacency list is created.

Some countries are known by many names and a separate node was created for all
of them. For example, Argentina was mentioned as: Argentine Republic,
Argentine Armed Forces, Argentine Air Force, Exiled Argentines, Argentine
Army, Argentine Navy, Argentine Unitarians, Government of Argentina and
Argentine Confederation i.e. 9 different names. To counter this issue, another
processing layer was used before assigning an id to an opponent as well as
adding a node for it, which replaced each opponent’s name from a map entry.
Map contained standard names for countries which were present with multiple
names in our data-set. The map was created manually after observing entries in
the data-set.

##  3 Network Description

Nodes in the network represent the fighting entity which could be a
nation(e.g. India), an empire (e.g. Ottoman empire), countries or empires
which existed in history (e.g. Saxony), terrorist organisations (e.g. ISIS),
inter-governmental country alliances (e.g. NATO). Edges of the network
represent the wars fought between nodes. Network is in the form of a temporal
multi-graph with a total of 3000 nodes and 27721 edges. The average degree of
the graph, considering edges over the whole timeline, is 18.5. The maximum
degree is 1321 and the minimum degree is 1. Here, the degree of a node
represents the number of wars that country/empire/terrorist organization was a
part of.

Figure 1: Degree Distribution of The Network

In Figure 1, the degree distribution has been plotted, where the $x$-axis
represent the degree of a node, i.e., the number of wars a
country/empire/terrorist organization was a part of. The $y$-axis denotes the
number of nodes with degree $x$, i.e., the number of countries/empires/
terrorist organizations with $x$ number of war participation.

Table 1: Degree Distribution Total Nodes | 3000
---|---
Number of nodes with degree 1 | 682
Number of nodes with degree $\leq$ 10 | 2205
Number of nodes with degree $\leq$ 50 | 2832
Number of nodes with degree $\leq$ 100 | 2896
Number of nodes with degree $\leq$ 300 | 2975
Number of nodes with degree $\geq$ 301 | 25

Degree distribution of the network in the presence of all edges is skewed. The
degree distribution curve follows power law. Table 1 summarises the cumulative
degree for some of degree values. It is clear from the table that only
$0.833\%$ of the total number of node have more than 300. $73.5\%$ of the
nodes are with at most 10 degree. Very few entities have been there that have
been engaging in a lot many number of wars. A large number of
countries/empires/ terrorist organizations do not participate in several
wars.In next section, we identify these nodes with high degree over different
timelines.

##  4 Results and Discussion

In this section, we note down all the statistical findings as well as
conclusions drawn from these over the considered timeline.

###  4.1 Timeline analysis:

We have considered wars that have happened between 1500 CE and 2020 CE. It is
a period of 520 years. Analysis of the whole timeline at once doesn’t give us
good results as an ally in the fifteenth century might be enemy in the twenty
first century. It is also difficult to do analysis on each year separately as
there are a total of 520 years. To decide the segments, we created a plot
Figure 2(a) between years and number of wars fought in a specific year. In
Figure 2(a), It can be observed there is a huge increase in the number of wars
after 2001. In particular, the height of the curve at year 2003 is 3895 which
is more than twice of the cumulative number of wars during the second world
war. We found that many countries had around 30 edges every year because of
encounters with different terrorist organisations like Tahrik-i-Taliban
Pakistan, HI-Khalis, Islamic Emirates, Boko Haram etc. We drew, another plot,
given in Figure 2(b), similar to Figure 2(a), but without the terrorist
encounters. In the new plot total wars fought reduced heavily, e.g. wars
fought in 2016, 2017, 2018 and 2019 reduced from 3351, 3370, 3153, 3148 to
1008, 994, 871 and 870 respectively. Which are less than 2/3 of the wars
fought during the second world war. In Figure 2(b), it is observed that the
number of wars are almost constant before 1800. After, 1800 spikes start
occurring in the curve. There are 3 big spikes after 1800 and before 2000.
First spike is at year 1914 which is in the timeline of the first world war,
and there are 1411 edges in that year. Second spike is at year 1945 which is
in timeline of the second world war, and there are 1560 edges in that year.
Third spike is at year 1987. After analyzing the edges, it was found out that
this spike is because of the different individual wars which happened in 1987
e.g. Iran-Iraq war, Cold War, Baltic states vs Soviet Union etc.

(a) Considering all nodes

(b) Ignoring terrorist units

(c) For each continent

Figure 2: Year-wise number of wars

After analyzing the plot in Figure 2(a) and Figure 2(b), we concluded to
divide the timeline from 1500 CE to 2020 CE into three parts and study all of
them separately as well.

  1. 1.

Early wars[1500-1800]: This is the part in the graph which has almost same
number of wars. The wars are mostly dominated by European countries. 4413 wars
started in this timeline, which is least of the three timelines.

  2. 2.

During and pre-world wars[1801-1945]: This timeline covers the incidents of
World wars. Both the world wars fall in this timeline, 9064 wars started in
this timeline and total 9510 war were fought in this timeline.

  3. 3.

Post world wars[1946-2020]: Post world war era. This timeline’s results are
most effective in influencing the present state of different countries. 14244
wars started in this timeline and 14358 wars were fought in this timeline.

Wars in different Continents: We also plot the number of wars fought by
countries from different continents. The plot is given in Figure 2(c) where
the $x$-axis denotes the years and the $y$-axis denotes the number of wars
fought by countries from a continent. For each continent, a curve with a
different color is plotted. The nodes having degree more than 40 are only
considered for respective continents. We have only plotted curve for Asia,
Europe, Africa, North America, South America and Australia. We haven’t
included Antarctica as none of the country with degree more than 40 belonged
to it. We also added a continent with name Euro-Asia to cover the countries
which exist partially in Asia and partially in Europe. From Figure 2(c), we
can observe that European countries dominated the world in terms of
participating in wars between 1500 to 1800. 1800-1900 period had relatively
lesser number of war. This period wasn’t solely dominated by European
countries as other continents’ degree had increased. Number of wars increased
significantly during the world war timelines and European nations had the
highest degree. Asia, Europe and Africa continents have mostly participated in
wars after 2000.

###  4.2 High Degree Nodes

The nodes having high degrees in the constructed network correspond to the
countries or empires fighting most number of wars in that timeline. The
powerful countries generally become condescending and fight more wars. So, the
high degree nodes in the graph represent the powerful countries of the
timeline. Table 2 shows the list of top 10 countries/empires that participated
in maximum number of wars during different timelines.

Table 2: High Degree Nodes

[1500 – 1800] | [1801 – 1945] | [1946 – 2020] | [1500 – 2020]
---|---|---|---
Node | Degree | Node | Degree | Node | Degree | Node | Degree
Ottoman Empire | 376 | Russia | 608 | United States | 632 | Russia | 1279
France | 364 | Germany | 587 | Russia | 455 | United Kindom | 1276
United Kindom | 334 | Italy | 564 | United Kindom | 433 | France | 1253
Spain | 333 | France | 543 | Al-Qaeda | 410 | United States | 1065
Russia | 237 | United Kindom | 526 | France | 351 | Germany | 999
Holy Roman Empire | 207 | United States | 424 | Australia | 334 | Spain | 865
Turkey | 199 | Ottoman Empire | 405 | Germany | 292 | Italy | 848
Dutch Republic | 191 | Japan | 330 | China | 287 | Ottoman Empire | 781
Sweden | 185 | Spain | 324 | Iraq | 285 | Romania | 583
Portugal | 161 | Bulgaria | 316 | Italy | 250 | Bulgaria | 495

Timeline 1: Early Wars[1500-1800]: This was an era where countries had to
fight against each other to establish their control and show dominance. We can
observe in the [1500-1800] section of Table 2 Ottoman Empire is the node with
highest degree. During 16th and 17th centuries, the Ottoman Empire was a
multinational, multilingual empire controlling most of Southeast Europe, parts
of Central Europe, Western Asia, parts of Eastern Europe and the Caucasus and
North Africa. The presence of its name at the top of the list reflects its
dominance in the timeline. Most of the nodes in the list are European
countries as they were participating in a lot of wars in trying to establish
their colonies.

Timeline 2: Pre-world wars [1801-1945]: In the first world war, Germany,
Ottoman Empire and Bulgaria were the major participants of the first side
which were opposed by Russia, United Kingdom. In the second world war,
Germany, Italy and Japan were major participants of the first side which were
opposed by France, United Kingdom, Russia and United States. Clearly all the
nodes except Spain in the [1801-1945] section of Table 2 are among the major
participants of one of the world war. Russia, Germany, Italy, France and
United Kingdom actively participated in both the world wars so they are at the
top of list, where as United States, Ottoman Empire and Bulgaria were active
in one of the world wars so they are towards the bottom of list. Spain is
present in the list as it also participated in WW2 and also fought wars during
1800-1900.

Timeline 3: Post-World wars[1946-2020]: United States is at the top of the
[1946-2020] section of Table 2 as it is known for its efforts in trying to
mitigate terrorism, US army is fighting against terrorism in various areas
like Afghanistan, Iraq and Syria. USA also fought the cold war against Russia.
It is followed by other countries like Russia, United Kingdom, France,
Australia etc, all of these nations are undoubtedly powerful nations of the
present era. This is the only timeline in which we see a terrorist
organisation (al-Qaeda) present in the list, which supports the fact that this
timeline is dominated by the wars against terrorism.

Overall [1500-2020]: Now, since the timeline is much bigger the nodes present
in the list could be the node which were powerful throughout the timeline or
the nodes which dominated a specific timeline. Nodes like Russia, United
Kingdom, France, Germany are the ones which were powerful in all three
timelines as we have seen in the timeline specific analysis. Whereas nodes
like United States, Ottoman Empire and Bulgaria were not very strong in all
three timelines. For example, USA was not active in the first timeline but
dominated the next two, Ottoman Empire on the other side was powerful in the
first two timelines and disappeared from the list in the third timeline.

###  4.3 Rival Countries

The nodes having highest number of edges between them in our graph corresponds
to the pair of countries fighting most number of wars against each other
during a timeline. List of pair of nodes with highest number of edges between
them gives us the list of countries with strong rivalries between them. Table
3 shows the list of such rival countries.

Timeline 1: Early Wars[1500-1800]: We can observe in [1500-1800] section of
Table 3 that all of these rivalries are among the European Countries. United
Kingdom-France is at the top of the list. Anglo-French wars which were
happening for almost throughout this timeline are example of conflicts between
England and France. Similarly all other pairs are also of European countries
fighting for dominance.

Timeline 2: Pre-world wars [1801-1945]: Germany, Ottoman Empire and Bulgaria
fought against Russia, United Kingdom in world war 1. Germany, Italy and Japan
fought against France, United Kingdom, Russia and United States. United
kingdom and Russia fought against Germany in both the world wars rooting for
the first two entries of [1801-1945] section of Table 3. United kingdom and
Russia fought against Italy in the second world war rooting for the next two
entries of table. Indonesian national revolution was a major armed conflict
that happened in 1945-1949 which included a number of military encounters
between Dutch and Indonesia rooting for the presence of last entry in the
list.

Timeline 3: Post-World wars[1946-2020]: As mentioned earlier this timeline is
the most recent and stats of this timeline are most effective in deciding the
current situations. All the rival nodes present in the [1946-2020] section of
Table 3 are well known rivals of present era. United States and Russia fought
long cold war against each other. United states-china, Thailand-Vietnam,
China-Vietnam our well known conflicts. Al-Qaeda attacked on United States in
2001 and straight after that USA declared war against it which is still going
on.

Overall [1500-2020]: The [1500-2020] section of Table 3 shows the pairs of
nodes which constantly fought against each other or the pairs of nodes which
fought many battles against each other in specific timeline. We can see that
United Kingdom- France is at the top of the list because they fought against
each other in the first timeline and in the beginning of second timeline.
Whereas Russia and United states fought against each other majorly in the
third timeline.

Table 3: Rival Countries

[1500 – 1800] | [1801 – 1945]
---|---
Node1 | Node2 | Edge Count | Node1 | Node2 | Edge Count
United Kingdom | France | 43 | United Kingdom | Germany | 22
France | Spain | 40 | Russia | Germany | 20
France | Holy Roman Empire | 36 | Russia | Italy | 21
United Kingdom | Spain | 36 | United Kingdom | Italy | 21
Ottoman Empire | Spain | 30 | Indonesia | Dutch East Indies | 21
[1946 – 2020] | [1500 – 2020]
Node1 | Node2 | Edge Count | Node1 | Node2 | Edge Count
Russia | United States | 23 | United Kingdom | France | 61
United States | China | 18 | United Kingdom | Spain | 50
Al-Qaeda | United States | 17 | France | Spain | 46
Thailand | Vietnam | 16 | France | Holy Roman Empire | 37
China | Vietnam | 16 | Russia | France | 35

###  4.4 Countries With highest number of wars fighting against common enemy

Next we analyzed the countries who fought the most number of wars against the
same side at the same time. If a pair of node is present in the rival country
list and also in this list, this implies that the pair of nodes was fighting
against each other and also against a common enemy at the same time. This
would represent a clique like situation where many countries were fighting
against each other. Whereas if two countries have fought most wars fighting
against the same side and very few or no war against each other it give us a
sense of friendship among the two. Table 4 shows the list of such nodes.

Timeline 1: Early Wars[1500-1800]: Pair of France and Spain was present in the
[1500-1800] section of Table 3, and is also present in the same section for
Table 4. This follows the first scenario and shows that the countries were
fighting against each other. This follows the reasoning that we gave while
introducing the timeline that many European Countries were fighting against
each other.

Timeline 2: Pre-world wars [1801-1945]: United Kingdom-France, Russia-United
Kingdom, United Kingdom-United States all have very small number of wars
against each others and large number of wars along the same side so this
reflects friendship among these pair of nodes. This is also evident from the
world war facts as United Kingdom, France and Russia were part of ally
countries during second world wars. USA also joined ally countries during
later part of second world war. India helped the United kingdom in the world
wars so India-United Kingdom pair is also present in the list. We can see
these stats clearly reflect the dominance of world wars in this timeline.

Timeline 3: Post-World wars[1946-2020]: All the nodes present in [1946-2020]
section of Table 4 are powerful nations. We will see in the later section that
these are the nodes which are fighting most against the different terrorist
organizations as allies.

Table 4: War Against Common Enemy

[1500 – 1800] | [1801 – 1945]
---|---
Node1 | Node2 | Number of Wars | Node1 | Node2 | Number of Wars
Spain | Holy Roman Empire | 44 | United Kingdom | France | 35
Ottoman Empire | Turkey | 26 | United Kingdom | India | 29
Portugal | State of Brazil | 23 | Russia | United Kindom | 28
Spain | Papal States | 22 | United Kingdom | United States | 26
France | Spain | 20 | Germany | Italy | 24
[1946 – 2020] | [1500 – 2020]
Node1 | Node2 | Number of Wars | Node1 | Node2 | Number of Wars
United Kingdom | United States | 47 | United Kingdom | France | 79
United States | Australia | 39 | United Kingdom | United States | 73
United States France | France | 37 | United States France | France | 57
United Kingdom | Australia | 33 | United Kingdom | Italy | 37
United Kingdom | France | 32 | France | Spain | 35

Overall Analysis [1500-2020]:The reasoning given in the beginning of Section
4.4 fails here. Recall, we noted that if a pair of node is present in the
rival country list and also in this list, this shows that the pair of nodes
were fighting against each other and also against a common enemy at the same
time. This represents a clique like situation where many countries are
fighting against each other. Where as if two countries have fought most wars
fighting against the same side and very few or no against each other it gives
us a sense of friendship among the two. As the size of timeline is much
larger, there is a possibility that two countries fought many wars against
each other at some time and fought along the same side at a different time and
this would not imply a clique like situation. Table 4 shows the list of such
nodes. Pairs at the top of the list are the pairs which fought together during
the world wars and are also fighting against terrorism in the present era e.g
United Kingdom-France, United Kingdom-United States, United states-France.

###  4.5 Inter-country Relations

We studied the inter-country relations between different countries by
analyzing the number of wars they fought against each other and number of wars
they fought along the same side. To analyze this we plotted two lines for each
pair of nodes. The blue line indicates if the two nodes fought a war along the
same side in that year, where as the orange line indicates if two nodes fought
against each other in that year. In the graph we plot both the lines and
depending on the values of lines 4 scenarios are possible. Orange line is at 1
and blue line is at 0: Countries were fighting against each other in that
particular year. Orange line is at 1 and Blue line is at 1: Countries were
fighting against each other and also against some common opponent in that
particular year. Orange line is at 0 and Blue line is at 1: Countries were
fighting against some common opponent in that particular year. Orange line is
at 0 and Blue line is at 0: Countries did not have any direct interaction with
each other. Analysis of some inter-country relationship pairs are plotted in
Figure 3.

(a) UK and France

(b) USA and Russia

(c) UK and USA

(d) UK and Germany

(e) UK and India

(f) USA and China

Figure 3: Relations between two countries

United Kingdom and France: Figure 3(a) shows the plot of the relationship
between United Kingdom and France based on the wars fought between them and
against common enemies. Orange line has value 1 for almost all the time
between 1500 CE and 1820 CE. This is because of the Anglo-French war which
happened during that time. Blue line has value 1 for some time between 1550 CE
to 1650 CE. This is because of the 80 years long war in which France and UK
both fought against Spain from 1568 CE to 1648 CE. In 1830 France accepted
Britain as ally so orange line doesn’t touch y=1 value for long duration after
that. Blue line touches y=1 constantly after 1970 as both developed nations
were helping other countries in fighting civil wars and in mitigating
terrorism.

United Kingdom and Germany: Figure 3(d) plots the relationship between United
Kingdom and Germany. Germany didn’t fight many wars against other countries
between 1500 CE to 1600 CE as a result except for one spike both orange and
blue lines move along y=0 during that time. Between 1615 CE and 1775 CE blue
and orange line move along each other at value 1 because during this timeline,
European countries were fighting against each other for dominance. In later
part of $18^{th}$ century, Britain and Prussia became allies, so only blue
line touches y=1 in that area. Both blue and orange lines are at y=0 between
1800 CE and 1900 CE indicating not much direct interaction among two
countries. Both the lines start to touch y=1 after 1900 as pre-world war
activities were happening. Orange line has value 1 around 1915 and 1945
indicating the world war in which both the countries fought against each
other. Blue line touches y=1 constantly after somewhere around 1980.

United States and Russia: Figure 3(b) plots the relationship between United
States and Russia. There is no interaction between countries before 1900 CE
except for a small period around 1770 CE which is because of involvement of
Russia in United States’ revolution. Blue lines touches y=1 around 1914 and
1945 as countries participated together in world wars. Orange line touches y=1
after 1946 up to 1991 because of the cold war. Blue line touches y=1 and
remains there afterward as both nations were helping other countries in
fighting civil wars and in mitigating terrorism.

United Kingdom and India: Figure 3(e) plots the relationship between United
Kingdom and India. Their is no interaction between countries before 1700 CE.
Orange line touches y=1 many times between 1700 CE and 1800 CE because of
multiple Anglo-Indian wars. Orange line touches y=1 several times between 1800
CE and 1950 CE e.g. at 1847 CE because of the Revolutionary war of India.
After 1800, Britain started gaining control over India and Indian troops
fought along with them in certain wars. As a result we see blue line touching
y=1 from around 1800 CE to 1950 CE, as India got freedom in 1947. Blue line
again touches y=1 as both India and United Kingdom were fighting against
terrorism.

United States and China: Figure 3(f) plots the relationship between United
States and China. Orange line touches y=1 for the first time at 1900 CE as in
1900 CE, China fought against USA during Boxer Rebellion. Blue line touches
y=1 around the timeline of world wars as both USA and China fought on the same
side. Orange line touches y=1 after the second world war, as USA and China
supported opposing countries in different wars. Korean war is one such example
in which USA supported South Korea and China supported North Korea. Blue line
again touches y=1 after 1990.

United States and United Kingdom:Figure 3(c) plots the relationship between
United States and United Kingdom. Choctaw (Native American) had conflict with
British government in 1702 CE -1713 CE due to which Orange line touches y=1 at
that time. Orange line touches y=1 again at around 1777 CE because of USA’s
revolutionary war. After the revolution, two countries became ally so we can
see blue line touches y=1 again. Conflict among countries started again in
1812 CE (War of 1812) After that both countries remain ally for most of the
time from 1812 CE to 2020 CE excluding some small duration. Some dispute
happened between these. countries due to which orange line also touches y=1
for small intervals. Both USA and UK fought along the same sides in world war
and remained allies afterward.

###  4.6 Analysis of Countries’ History

This sections sheds light on the year-wise number of war-fronts in which some
countries and empires were involved in. In these analysis, wars involving
terrorist organizations are ignored to have a clear understanding of inter-
country wars. The analysis for considered countries is plotted in Figure 4.

(a) United States

(b) United Kingdom

(c) Ottoman Empire

(d) Germany

(e) China

(f) India

Figure 4: Number of wars participated by countries/Empires over timeline

United States: Figure 4(a) plots year-wise number of wars fought by United
States. First significant increase in the curve is at 1702 CE - 1712 CE
because of conflict between Native Americans and Britain. United States’
revolution happened in 1776 CE so it hasn’t participated in any war before
that. After 1775 CE, curve starts increasing because of the Revolutionary
War(1776). Curve shows significant increase around 1861 because of United
States’ Civil War. Two spikes in curve in the timeline of world wars. United
States participated in various different wars during 1970 CE to 2000 CE e.g.
Cold war, Gulf war, Tanker war, Tobruk wars etc. due to which curve attains
its peak value around this time. United States is still sending its forces to
help countries in need, therefore, curve remains at a significant value even
after 2000 CE.

Germany: Figure 4(d) plots year-wise number of wars fought by Germany. First
Considerable value is around 1610 CE to 1640 CE because of the 30 years
war(1618-1658). Huge Spike in 1914 CE because of the first world war. Curve
almost touches 0 after 1918 CE because of the restrictions imposed on Germany
after the first world war. Another huge spike in 1945 CE because of the second
world war. Number of wars significantly reduce after that.

United Kingdom: Figure 4(b) plots year-wise number of wars fought by United
Kingdom. UK has been a strong country throughout the 520 years period in terms
of fighting most number of wars. First significant increase in 1700 CE as
United Kingdom participated in two big wars: Great Northern war, and War of
Spanish succession at the same time. The curve shows huge increase around the
timeline of world wars reflecting the active participation of United Kingdom
in it. United Kingdom participated in various different wars during 1970 CE to
2000 CE e.g. Bosnian war, Gulf war, Falklands wars, Cod wars etc.

China: Figure 4(e) plots year-wise number of wars fought by China. The
republic of china was formed in 1912 CE. Curve takes significant values after
1912. The curve takes value approximately 15 in 1914 CE reflecting China’s
participation in the first world war. Curve achieves its peak value around
1945-1955 period, as China fought against Japan in the second world war and
also participated in battle of Chamdo, Korean war and the war against Taiwan.
Curve has significant value throughout second half of 20th century reflecting
its participation in Taiwan Crisis, Tibetan Uprising, Xinjiang conflict, Sino-
Indian war, Vietnam war etc. As China is still sending its forces to help
countries like Somalia as a result curve keeps a significant value even after
2000s.

India: Figure 4(f) plots year-wise number of wars fought by India. The curve
has lower values as compared to the above 4 plots. Small number of wars up to
1850 CE, because of wars between different empires (Maurya, Maratha, Sikkha,
British Rule). First spike in 1857 CE because of Indian revolution movement
during that time. Curve attains largest value around 1945 CE - 1955 CE period
as India participated in the second world war to help Britain and got freedom
in 1947 CE. India fought with Pakistan in 1947-48 CE for the state of Jammu
and Kashmir. India also conducted military operations for annexation of
Hyderabad and Junagadh. India had a conflict with Portugal for annexation of
Dadra and Nagar Haveli in 1954. Curve doesn’t vanish even though India isn’t
participating in war directly. It is because India is part of NATO which is
helping different countries in fighting their wars.

Ottoman Empire: Figure 4(c) plots year-wise number of wars fought by Ottoman
Empire. We can observe that the curve takes significant value after 1520 CE,
as Ottoman Empire came in the total power after 1520 CE. This curve gains
significantly larger value as compared to rest of 5 plots from 1520 CE to 1700
CE. Empire starts loosing power as the number of war starts reducing after
1710 CE. Curve has a huge spike in 1914 CE reflecting its participation in the
first world war. The Empire was defeated in 1918 and officially ended in 1922
as a result the curve vanishes after that.

###  4.7 Wars involving Terrorist Organisations

In this section, we analyse the wars involving terrorist organization. These
wars consider both: the attacks by terrorist organization on a country and
attack by nations on the hubs of terrorist organizations. All the plots in
this section is for timeline 1950 CE to 2020 CE because before 1950 CE, there
is very limited number of such wars. Year-wise number of wars involving
terrorist organization is plotted in Figure 5(a). It is observed that the
total number of wars involving terrorist organization are close to zero in the
begin. The curve increases slowly till 2000. Curve increases sharply and
attains its maximum value in 2001. The number of terrorist encounters have
decreased since then. Still the count is high.

(a) Against all countries

(b) For most involved terrorist organization

(c) Against most exposed countries

Figure 5: Year-wise number of wars involving terrorist organizations

The list of the top 5 terrorist organisations which have been involved in most
wars. It reports that Al-Qaeda has been the most active terrorist
organization. Table 5 also lists the top 5 countries with maximum engagement
with various terrorist organisations in wars along with the number of wars.
The timeline analysis for these top five entities is given in Figure 5(b). It
becomes clear from these plots, when these organization became active and when
they are at peak for getting involved in wars.

Table 5: Wars involving Terrorist Organisations

Most Active Terrorist Organizations | Countries with Most Terrorist Encounters
---|---
Terrorist Organisation | Total Number of Wars | Country | Number of terrorist Encounters
Al-Qaeda | 422 | United States | 212
Taliban | 235 | United Kingdom | 201
Islamic States of Iraq | 208 | Australia | 168
ISIL | 199 | France | 149
Hezbollah | 148 | Denmark | 135

Table 6 contains the number of engagement a country had in wars with each
specific terrorist organizations. It turns out that Al-qaeda and United States
are the top engaging nodes. Al-qaeda is the terrorist organisation in all top
five node pairs that exhibits the reach and engagement of this organization
across several continent. Al-Qaeda has been the most active terrorist
organization. Further, it also helps us to analyse the names of the countries
which are fighting most in trying to mitigate the terrorism. It reports that
United States has participated most in the wars against terrorist
organizations. It becomes clear from these plots, when these countries were
first exposed against terrorism and when these get most engaged in wars
against terrorist organization. Finally, we analyse the number of engagement a
country had in wars with each specific terrorist organizations. It turns out
that Al-qaeda and United States are the top engaging nodes. Al-qaeda is the
terrorist organisation in all top five node pairs that exhibits the reach and
engagement of this organization across several continent.

Table 6: Most wars for Country-Terrorist Org. pairs:

Country | Terrorist Organisation | Number of encounters
---|---|---
United States | Al-Qaeda | 18
United Kingdom | Al-Qaeda | 13
Australia | Al-Qaeda | 12
Netherland | Al-Qaeda | 11
Denmark | Al-Qaeda | 10

###  4.8 Economic Effects of War

We have plotted the number of wars fought by each country and their GDP value
and we can clearly see the correlation among the two. We have plotted the two
stats represented by blue and orange line across the timeline (X axis). The
blue line represents the GDP of the country in that particular year. We have
scaled it down by dividing with a constant to show correlation properly. The
Orange line represents the number of wars fought by the country in past 3
years. We have plotted the cumulative sum of wars for past 3 years instead of
the wars in that particular year since wars impact the GDP of upcoming years
as well. We observe that the spikes in the orange lines are always followed by
a dip in the blue curve for all countries. We have attached few of those plots
in the Figure 6. The dips are deeper for few countries and shallower for the
others. Note that all the countries have a dip around 1945-50 because of the
second world war but the size of dip (relative to their GDP) for United States
and United Kingdom is smaller in comparison to that of Germany and Japan. It
seems that the countries which won the wars tend to have smaller dips. The
impact of outcome of the wars require further investigation.

(a) Germany

(b) United Kingdom

(c) USA

Figure 6: Economic Effects of War

##  5 Conclusion

Temporal analysis of wars fought after 1500 CE is done by curating data
available on Wikipedia. The analysis results comply with the historical
events. The powerful and dominating nations are identified during different
timelines. Rivalries and allies based on wars explain the inter-country
relationships during different timelines. Most active terrorist organizations
and most active countries against terrorism were also identified using this
analysis. The data did not include the damage associated with wars, due to
which all wars were treated equally. Analysis incorporating damage associated
with wars by assigning weights on the edges in temporal graph and considering
the decision of war will give better insights. Impact on economy has been
analysed in terms of change in GDP. After 1500, european countries began to
move from feudalism to capitalist societies, and domestic revolutions played a
crucial role in the future development of the countries. Taking into account
the industrial revolution and the impact of domestic revolutionary movements
on GDP is another future direction.

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