# Analytical solutions of the Arrhenius-Semenov problem
for constant volume burn

Galen T. Craven  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, New Mexico, USA

###### Abstract

Analytical solutions to the Semenov thermal ignition problem for constant
volume burn governed by Arrhenius reaction kinetics are derived. Specifically,
an approximate analytical solution technique for the Arrhenius-Semenov
differential equation is derived for reaction orders $n\in\mathbb{R}_{>0}$ and
exact solutions are also constructed for reaction orders $n\in\mathbb{N}:n\leq
3$. The approximation technique relies on expansion of the respective
nondominant terms in the differential equation at the lower and upper bounds
of the reaction progress variable in order to create a pair of integrable
series. The two integrated series are then connected to create a single
continuous analytical solution. Excellent agreement is observed between the
analytical approximation and solutions obtained numerically. The presented
approximation constitutes a simple and robust strategy for solving the
Arrhenius-Semenov problem analytically.

##  I Introduction

Theories describing equilibriumUzer  _et al._ (2002); Pollak and Talkner
(2005); Bartsch  _et al._ (2005) and nonequilibriumBazant (2013); Craven and
Hernandez (2015); Craven and Nitzan (2016); Matyushov (2016); Craven and
Hernandez (2016); Craven  _et al._ (2014); Santamaría-Holek and Pérez-Madrid
(2020) chemical kinetics play a fundamental role in computational treatments
of shock-induced chemical reactions and high explosive detonations Tarver  _et
al._ (2007); Menikoff (2009); Kunova  _et al._ (2016); Aslam (2018); Menikoff
(2015, 2019); Peterson and Coe (2020); Huber  _et al._ (2020). The principal
motivations for developing these theories are to (a) determine chemical
reaction rates, (b) to predict reaction products, and (c) to elucidate complex
molecular behavior that arises due to energy and charge redistribution that
occurs as the reaction proceeds. While various advanced computational methods
Ceriotti  _et al._ (2021); Butler  _et al._ (2018) can provide high-level
insight into a system’s properties Houston  _et al._ (2019); Komp and Valleau
(2020); Greaves  _et al._ (2021); Liu  _et al._ (2019); Craven  _et al._
(2020), it is often advantageous to describe molecular processes using
analytical methods, for example, to develop physical intuition about the
system or for computational efficiency. In complex systems, however,
nonlinearities typically preclude exact analytical solutions to a system’s
dynamical equations and, therefore, approximation methods must be employed. In
this report, we develop an analytical approximation method to solve the
kinetic equation describing a constant volume burn governed by Arrhenius
reaction kinetics.

Constant volume burn occurs when a thermal explosion takes place in a rigid
reaction vessel over a fast enough time scale such that no heat is exchanged
across the vessel boundary and the chemical and thermodynamic properties of
the system are spatially uniform Menikoff (2017); Boddington  _et al._ (1977).
Following Menikoff’s mathematical exposition for constant volume burn from
Ref. 26, the ordinary differential equation (ODE) that describes the
temperature evolution of a constant volume burn is Menikoff (2017)

$\frac{dT}{dt}=\frac{\mathcal{Q}}{C_{v}(\lambda,T)}\mathcal{R}(\lambda,T),$
(1)

where $\lambda\in\left[0,1\right]$ is a reaction progress variable,
$\mathcal{R}$ is the reaction rate, $\mathcal{Q}$ is the specific energy
released, and $C_{v}$ is the specific heat. When $\lambda=0$ the system
consists of completely unburned reactant and when $\lambda=1$ the reactant is
completely burned. The initial ($\lambda=0$) system temperature is $T_{0}$ and
the final ($\lambda=1$) system temperature at the end of the reaction is
$T_{1}$. Equation (1) is coupled with an equation that describes the time
evolution of the reaction progress variable

$\frac{d\lambda}{dt}=\mathcal{R}(\lambda,T).$ (2)

Here, the reaction rate is assumed to take the Arrhenius form

$\mathcal{R}(\lambda,T)=(1-\lambda)^{n}k\exp\left[-\frac{T_{a}}{T(\lambda)}\right],$
(3)

where $n$ is the reaction order, $k$ is a pre-exponential frequency factor,
and $T_{a}$ is the activation temperature. Assuming that $\mathcal{Q}$ and
$C_{v}$ are constant, the temperature of the system as a function of reaction
progress is

$T(\lambda)=T_{0}+(\mathcal{Q}/C_{v})\lambda=T_{0}+(T_{1}-T_{0})\lambda,$ (4)

with $T_{1}=T_{0}+\mathcal{Q}/C_{v}$. The time-dependence of $T$ in Eq. (4)
arises through the reaction progress variable $\lambda\rightarrow\lambda(t)$.
Under these conditions, the time evolution of the reaction can be described by
the single ODE

$\frac{d\lambda}{dt}=(1-\lambda)^{n}k\exp\left[-\frac{T_{a}}{T_{0}+(\mathcal{Q}/C_{v})\lambda}\right]=(1-\lambda)^{n}k\exp\left[-\frac{T_{a}}{T_{0}+(T_{1}-T_{0})\lambda}\right].$
(5)

Our aim is to develop analytical techniques that can be used to solve Eq. (5)
over a broad class of reaction orders. Note that this ODE is separable,

$\int
kdt=\int(1-\lambda)^{-n}\exp\left[\frac{T_{a}}{T_{0}+(T_{1}-T_{0})\lambda}\right]d\lambda,$
(6)

and, therefore, the Semenov ignition problem in the presence of Arrhenius rate
kinetics can in essence be solved by evaluating the integral on the RHS of Eq.
(6):

$I_{n}(\lambda)\equiv\int(1-\lambda)^{-n}\exp\left[\frac{T_{a}}{T_{0}+(T_{1}-T_{0})\lambda}\right]d\lambda.$
(7)

Below, in Sec. II we illustrate an analytical technique for approximating
$I_{n}$ and, therefore, for solving the Semenov problem. In Sec. III we derive
exact solutions for reaction orders $n\in\mathbb{N}:n\leq 3$. Section IV
contains a comparison between the developed approximation and numerical
solutions for various reaction conditions.

##  II Approximate Solution for $n\in\mathbb{R}_{>0}$

An exact solution of $I_{n}$ is not known for
$n\in\mathbb{R}^{+}\setminus\mathbb{N}$. Below we give an approximate
analytical solution for $n\in\mathbb{R}_{>0}$ which relies on series expansion
of different terms in the integrand at different ends of the
$\lambda\in\left[0,1\right]$ interval.

As $\lambda\to 1$, the term $(1-\lambda)^{-n}$ dominates the integrand for
$n>0$:

$\lim_{\lambda\to
1}\frac{\exp\left[T_{a}/T(\lambda)\right]}{(1-\lambda)^{-n}}=0.$ (8)

In this fully-burned limit, we therefore expand
$\exp\left[T_{a}/T(\lambda)\right]$ about $\lambda=1$ to obtain the
approximation

$\int(1-\lambda)^{-n}\exp\left[\frac{T_{a}}{T(\lambda)}\right]d\lambda\approx\int\sum^{\alpha}_{m=0}(1-\lambda)^{m-n}\frac{D^{m}_{\lambda}\exp\left[\frac{T_{a}}{T(\lambda)}\right]\bigg{|}_{\lambda=1}}{m!}\,d\lambda,$
(9)

where $\alpha$ is the series expansion order and $D^{m}_{\lambda}$ is $m$-th
order derivative operator with respect to $\lambda$. The integrals in the
series approximation up to third order ($\alpha=3$) are

$\displaystyle I^{+}_{n}(\lambda)$
$\displaystyle\approx\underbrace{\int(1-\lambda)^{-n}\exp\left[\frac{T_{a}}{T_{1}}\right]d\lambda}_{I^{(0+)}_{n}}$
(10)
$\displaystyle+\underbrace{\int(1-\lambda)^{1-n}\exp\left[\frac{T_{a}}{T_{1}}\right]\left(\frac{\mathcal{Q}}{C_{v}}\right)\frac{T_{a}}{T^{2}_{1}}\,d\lambda}_{I^{(1+)}_{n}}$
$\displaystyle+\underbrace{\int(1-\lambda)^{2-n}\exp\left[\frac{T_{a}}{T_{1}}\right]\left(\frac{\mathcal{Q}}{C_{v}}\right)^{2}\frac{T_{a}(T_{a}+2T_{1})}{2T^{4}_{1}}\,d\lambda}_{I^{(2+)}_{n}}$
$\displaystyle+\underbrace{\int(1-\lambda)^{3-n}\exp\left[\frac{T_{a}}{T_{1}}\right]\left(\frac{\mathcal{Q}}{C_{v}}\right)^{3}\frac{T_{a}(T^{2}_{a}+6T_{1}^{2}+6T_{a}T_{1})}{6T^{6}_{1}}\,d\lambda}_{I^{(3+)}_{n}},$

where the superscript “$+$” in $I^{+}_{n}$ denotes that the approximate
solution is approached from the upper bound of $\lambda$ and $I^{(j+)}_{n}$
denotes the $j$th order term in the upper-bound expansion. Evaluating each
integral in the series yields

$\displaystyle I^{(0+)}_{n}$
$\displaystyle=\frac{(1-\lambda)^{1-n}}{n-1}\exp\left[\frac{T_{a}}{T_{1}}\right],$
(11) $\displaystyle I^{(1+)}_{n}$
$\displaystyle=\frac{(1-\lambda)^{2-n}}{n-2}\exp\left[\frac{T_{a}}{T_{1}}\right]\left(\frac{\mathcal{Q}}{C_{v}}\right)\frac{T_{a}}{T^{2}_{1}},$
(12) $\displaystyle I^{(2+)}_{n}$
$\displaystyle=\frac{(1-\lambda)^{3-n}}{n-3}\exp\left[\frac{T_{a}}{T_{1}}\right]\left(\frac{\mathcal{Q}}{C_{v}}\right)^{2}\frac{T_{a}(T_{a}+2T_{1})}{2T^{4}_{1}},$
(13) $\displaystyle I^{(3+)}_{n}$
$\displaystyle=\frac{(1-\lambda)^{4-n}}{n-4}\exp\left[\frac{T_{a}}{T_{1}}\right]\left(\frac{\mathcal{Q}}{C_{v}}\right)^{3}\frac{T_{a}(T^{2}_{a}+6T^{2}_{1}+6T_{a}T_{1})}{6T^{6}_{1}}.$
(14)

Note that each of the $I^{(j+)}_{n}$ functions above is vertically asymptotic
for $n=j+1$. However, as illustrated below in Sec. II.1, after applying the
initial value to the full solution, there is no asymptote with respect to
variation of $n$.

In the limit $\lambda\to 0$, the ratio between the two functions in the
integrand of $I_{n}$ is

$\lim_{\lambda\to
0}\frac{\exp\left[T_{a}/T(\lambda)\right]}{(1-\lambda)^{-n}}=\exp\left[\frac{T_{a}}{T_{0}}\right].$
(15)

In this limit, we approximate the integral by expanding the $(1-\lambda)^{-n}$
term about $\lambda=0$:

$\int(1-\lambda)^{-n}\exp\left[\frac{T_{a}}{T(\lambda)}\right]d\lambda\approx\int\sum^{\alpha}_{m=0}\exp\left[\frac{T_{a}}{T(\lambda)}\right]\lambda^{m}\frac{D^{m}_{\lambda}(1-\lambda)^{-n}\bigg{|}_{\lambda=0}}{m!}\,d\lambda.$
(16)

Expanding the RHS of Eq. (16) up to third order gives the following
approximation for $I_{n}$:

$\displaystyle I^{-}_{n}(\lambda)$
$\displaystyle\approx\underbrace{\int\exp\left[\frac{T_{a}}{T(\lambda)}\right]d\lambda}_{I^{(0-)}_{n}}$
(17)
$\displaystyle+\underbrace{\int\exp\left[\frac{T_{a}}{T(\lambda)}\right]n\lambda\,d\lambda}_{I^{(1-)}_{n}}$
$\displaystyle+\underbrace{\int\exp\left[\frac{T_{a}}{T(\lambda)}\right]\frac{(n+n^{2})}{2}\lambda^{2}\,d\lambda}_{I^{(2-)}_{n}}$
$\displaystyle+\underbrace{\int\exp\left[\frac{T_{a}}{T(\lambda)}\right]\frac{(2n+3n^{2}+n^{3})}{6}\lambda^{3}\,d\lambda}_{I^{(3-)}_{n}},$

where the superscript “$-$” denotes that the approximate solution is
approached from the lower bound of $\lambda$ and $I^{(j-)}_{n}$ denotes the
$j$th-order term in the lower-bound expansion.

Evaluating these integrals gives

$\displaystyle I^{(0-)}_{n}$
$\displaystyle=\left(\frac{C_{v}}{\mathcal{Q}}\right)\Bigg{[}T(\lambda)\exp\Big{[}h(\lambda)\Big{]}-T_{a}\text{Ei}\Big{[}h(\lambda)\Big{]}\Bigg{]},$
(18) $\displaystyle I^{(1-)}_{n}$
$\displaystyle=\frac{n}{2}\left(\frac{C_{v}}{\mathcal{Q}}\right)^{2}\Bigg{[}T(\lambda)\left(\frac{\mathcal{Q}}{C_{v}}\lambda-
T_{0}+T_{a}\right)\exp\Big{[}h(\lambda)\Big{]}+T_{a}(2T_{0}-T_{a})\text{Ei}\Big{[}h(\lambda)\Big{]}\Bigg{]},$
(19) $\displaystyle I^{(2-)}_{n}$
$\displaystyle=\frac{n+n^{2}}{12}\left(\frac{C_{v}}{\mathcal{Q}}\right)^{3}$
$\displaystyle\quad\times\Bigg{[}T(\lambda)\left(2\left(\frac{\mathcal{Q}}{C_{v}}\right)^{2}\lambda^{2}+\left(\frac{\mathcal{Q}}{C_{v}}T_{a}-2T_{0}T_{1}+2T_{0}^{2}\right)\lambda+\left(T_{a}^{2}-5T_{a}T_{0}+2T_{0}^{2}\right)\right)$
$\displaystyle\quad\times\exp\Big{[}h(\lambda)\Big{]}-T_{a}\left(T_{a}^{2}-6T_{a}T_{0}+6T_{0}^{2}\right)\text{Ei}\Big{[}h(\lambda)\Big{]}\Bigg{]},$
(20) $\displaystyle I^{(3-)}_{n}$
$\displaystyle=\frac{(2n+3n^{2}+n^{3})}{144}\left(\frac{C_{v}}{\mathcal{Q}}\right)^{4}\Bigg{[}\Bigg{(}6\left(\frac{\mathcal{Q}}{C_{v}}\right)^{4}\lambda^{4}+2T_{a}\left(\frac{\mathcal{Q}}{C_{v}}\right)^{3}\lambda^{3}-\left(\frac{\mathcal{Q}}{C_{v}}\right)^{2}T_{a}(6T_{0}-T_{a})\lambda^{2}$
$\displaystyle\quad+\left(\frac{\mathcal{Q}}{C_{v}}\right)T_{a}(T_{a}^{2}-10T_{a}T_{0}+18T_{0}^{2})\lambda-
T_{0}(3T_{0}-T_{a})(T_{a}^{2}-8T_{a}T_{0}+2T_{0}^{2})\Bigg{)}\exp\Big{[}h(\lambda)\Big{]}$
$\displaystyle\quad-
T_{a}(T_{a}^{3}-12T_{a}^{2}T_{0}+36T_{a}T_{0}^{2}-24T_{0}^{3})\text{Ei}\Big{[}h(\lambda)\Big{]}\Bigg{]}.$
(21)

where

$h(\lambda)=\frac{T_{a}}{T(\lambda)}.$ (22)

We now enforce the condition that the complete solution must be continuous in
$t$. First, we apply the inital value $\lambda(0)=0$ to both solution branches
leading to the relations

$kt^{+}(\lambda)=I^{+}_{n}(\lambda)-I^{+}_{n}(0),$ (23)

for the upper-bound approximation and

$kt^{-}(\lambda)=I^{-}_{n}(\lambda)-I^{-}_{n}(0),$ (24)

for the lower-bound approximation, where $t^{\pm}(\lambda)$ denotes time as a
function of $\lambda$ given by the respective “$\pm$” solution. We want to
enforce the condition $t^{+}(\lambda^{*})=t^{-}(\lambda^{*})$ at some point
$\lambda^{*}$ in the interval $\left[0,1\right]$ and then to connect the two
series using a continuous piecewise function that shifts between branches at
$\lambda^{*}$. The method we apply to construct this continuous function is to
shift the upper-bound approximation $I^{+}_{n}(\lambda)-I^{+}_{n}(0)$ by a
factor $S$ such that the lower-bound approximation
$I^{-}_{n}(\lambda)-I^{-}_{n}(0)$ and the shifted upper-bound approximation
$I^{+}_{n}(\lambda)-I^{+}_{n}(0)+S$ are equal at $\lambda^{*}$. The shift is
obtained by solving the equation

$I^{+}_{n}(\lambda^{*})-I^{+}_{n}(0)+S=I^{-}_{n}(\lambda^{*})-I^{-}_{n}(0),$
(25)

for $S$. An empirically-motivated and simple choice for $\lambda^{*}$ is to
take $\lambda^{*}=1/2$, i.e., to connect the “$+$” and “$-$” approximations at
the point where the reaction is half burned. With $\lambda^{*}=1/2$, a
complete solution to the Arrenhius-Semenov ODE that is continuous in $\lambda$
is given by

$kt=I_{n}(\lambda)-I_{n}(0)\approx\left\\{\begin{array}[]{lc}I^{-}_{n}(\lambda)-I^{-}_{n}(0),&\lambda<1/2\\\
I^{+}_{n}(\lambda)-I^{+}_{n}(1/2)+I^{-}_{n}(1/2)-I^{-}_{n}(0),&\lambda\geq
1/2\end{array}\;.\right.$ (26)

Equation (26) is the primary result in this report.

It is important to note that the analytical approximation in Eq. (26) is not a
smooth function and contains a derivative discontinuity at $t(\lambda=1/2)$.
This lack of smoothness will not be significant in situations in which
$\lambda(t)$ is the desired quantity, but it may limit the applicability of
the present approach when $\tfrac{d\lambda}{dt}$ is explicitly needed. We have
found that, under typically physical conditions, the derivative discontinuity
is small in comparison to the magnitude of the derivative itself and, as
expected, the discontinuity magnitude decreases with increasing series
expansion order.

###  II.1 Approximate Solutions for $n\in\mathbb{N}$

In the limit $n\to j+1$ with $j\in\mathbb{N}$ (where $n$ is the reaction
order), the functions $I^{(j+)}_{n}(\lambda)$ given in Eqs. (11)-(14) are
vertically asymptotic, and therefore, before applying an initial value
$\lambda(t=0)=0$, Eq. (10) is divergent for $n\in\mathbb{N}$ if $n\leq\alpha$.
This can be seen in the general expression for these functions:

$I^{(j+)}_{n}(\lambda)=\frac{(1-\lambda)^{j+1-n}}{n-j+1}C^{(j+)}_{n},$ (27)

where $C^{(j+)}_{n}$ is a nonzero constant. However, when the $j$th order term
in the full solution $f^{(j+)}_{n}(\lambda)$ is constructed that includes the
initial value:

$f^{(j+)}_{n}(\lambda)=I^{(j+)}_{n}(\lambda)-I^{(j+)}_{n}(0),$ (28)

we find that

$\lim_{n\to j+1}f^{(j+)}_{n}(\lambda)=-C^{(j+)}_{n}\ln\left(1-\lambda\right).$
(29)

This shows that the approximate solution does not have a vertical asymptote
with respect to variation of $n$.

##  III Exact Solutions for $n\in\mathbb{N}$

In this section, we derive exact analytical solutions for the RHS of Eq. (6)
for $n\leq 3:n\in\mathbb{N}$. In all cases the initial value is
$\lambda(t=0)=0$.

###  III.1 $n=0$

For $n=0$, the problem reduces to evaluating the integral

$I_{0}=\int\exp\left[\frac{T_{a}}{T_{0}+\mathcal{Q}/C_{v}\lambda}\right]d\lambda.$
(30)

With $u=T_{0}+\mathcal{Q}/C_{v}\lambda$, the integral takes the form

$I_{0}=\frac{C_{v}}{\mathcal{Q}}\int\exp\left[\frac{T_{a}}{u}\right]du.$ (31)

Now, let $v=u/T_{a}=1/h(\lambda)$, and

$I_{0}=\frac{C_{v}T_{a}}{\mathcal{Q}}\int\exp\left[\frac{1}{v}\right]dv,$ (32)

which is a known integral form resulting in

$I_{0}=\frac{C_{v}T_{a}}{\mathcal{Q}}\Bigg{[}v\exp\left[\frac{1}{v}\right]-\text{Ei}\left[\frac{1}{v}\right]\Bigg{]}$
(33)

where Ei is the exponential integral. After substitution for $v$ we obtain

$I_{0}=\frac{C_{v}}{\mathcal{Q}}\Bigg{[}T(\lambda)\exp\Big{[}h(\lambda)\Big{]}-T_{a}\text{Ei}\Big{[}h(\lambda)\Big{]}\Bigg{]}.$
(34)

The final solution for $n=0$ is

$\displaystyle kt$
$\displaystyle=\left(\frac{C_{v}}{\mathcal{Q}}\right)\Bigg{[}T(\lambda)\exp\Big{[}h(\lambda)\Big{]}-T_{a}\text{Ei}\Big{[}h(\lambda)\Big{]}-T_{0}\exp\left[\frac{T_{a}}{T_{0}}\right]+T_{a}\text{Ei}\left[\frac{T_{a}}{T_{0}}\right]\Bigg{]},$
(35)

with $h(\lambda)$ defined in Eq. (22) above.

###  III.2 $n=1$

An exact analytical solution for the $n=1$ case is known (see Menikoff’s
derivation in Section 3 of Ref. 26). Solving $I_{1}$ gives

$\displaystyle kt$
$\displaystyle=\text{Ei}\Big{[}h(\lambda)\Big{]}-\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(\lambda)\Big{]}-\text{Ei}\left[\frac{T_{a}}{T_{0}}\right]+\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(0)\Big{]},$
(36)

with

$g(\lambda)=h(\lambda)-\frac{T_{a}}{T_{1}}.$ (37)

###  III.3 $n=2$

For $n=2$, the integral to evaluate in order to construct the exact solution
is

$I_{2}=\int(1-\lambda)^{-2}\exp\left[\frac{T_{a}}{T_{0}+\mathcal{Q}/C_{v}\lambda}\right]d\lambda.$
(38)

Let $u=\tfrac{T_{0}+\mathcal{Q}/C_{v}\lambda}{T_{a}}=1/h(\lambda)$. The
integral now takes the form

$I_{2}=\frac{C_{v}T_{a}}{\mathcal{Q}}\int\exp\left[\frac{1}{u}\right]\left(1+\frac{C_{v}T_{0}}{\mathcal{Q}}-\frac{C_{v}T_{a}}{\mathcal{Q}}u\right)^{-2}du.$
(39)

Now, let $v=1/u=h(\lambda)$, then,

$I_{2}=-\frac{C_{v}T_{a}}{\mathcal{Q}}\int\exp\left[v\right]\Bigg{(}\frac{C_{v}T_{1}}{\mathcal{Q}}v-\frac{C_{v}T_{a}}{\mathcal{Q}}\Bigg{)}^{-2}dv.$
(40)

Integration by parts and algebraic manipulation gives

$I_{2}=\frac{\mathcal{Q}T_{a}}{C_{v}}\Bigg{[}\frac{\exp\left[v\right]}{\left(T_{1}^{2}v-T_{1}T_{a}\right)}-\int\frac{\exp\left[v\right]}{\left(T_{1}^{2}v-T_{1}T_{a}\right)}dv\Bigg{]}.$
(41)

Finally, let $w=v-T_{a}/T_{1}=g(\lambda)$ in the integral in previous
equation, leading to

$I_{2}=\frac{\mathcal{Q}T_{a}}{C_{v}}\Bigg{[}\frac{\exp\left[v\right]}{\left(T_{1}^{2}v-T_{1}T_{a}\right)}-\frac{1}{T_{1}^{2}}\exp\left[\frac{T_{a}}{T_{1}}\right]\int\frac{\exp\left[w\right]}{w}dw\Bigg{]}.$
(42)

The integral in the second term is the exponential integral:

$I_{2}=\frac{\mathcal{Q}T_{a}}{C_{v}}\Bigg{[}\frac{\exp\left[v\right]}{\left(T_{1}^{2}v-T_{1}T_{a}\right)}-\frac{1}{T_{1}^{2}}\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\left[w\right]\Bigg{]}.$
(43)

After substitution for $w$ and $v$ we obtain

$I_{2}=\frac{T(\lambda)}{(1-\lambda)T_{1}}\exp\Big{[}h(\lambda)\Big{]}-\frac{\mathcal{Q}T_{a}}{C_{v}T_{1}^{2}}\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(\lambda)\Big{]}.$
(44)

The final solution for $n=2$ is

$\displaystyle kt$
$\displaystyle=\frac{T(\lambda)}{T_{1}(1-\lambda)}\exp\Big{[}h(\lambda)\Big{]}-\frac{\mathcal{Q}T_{a}}{C_{v}T_{1}^{2}}\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(\lambda)\Big{]}$
(45)
$\displaystyle\quad-\frac{T_{0}}{T_{1}}\exp\left[\frac{T_{a}}{T_{0}}\right]+\frac{\mathcal{Q}T_{a}}{C_{v}T_{1}^{2}}\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(0)\Big{]}.$

###  III.4 $n=3$

For $n=3$, the solution is derived using similar techniques to the cases
described above. After some algebraic manipulation, the solution can be
expressed as

$\displaystyle kt$
$\displaystyle=\frac{\mathcal{Q}^{2}T_{a}}{2C_{v}^{2}T_{1}^{3}g(\lambda)}\left(2+\frac{T_{a}}{T_{1}}+\frac{T_{a}}{T_{1}g(\lambda)}\right)\exp\Big{[}h(\lambda)\Big{]}-\frac{\mathcal{Q}^{2}T_{a}\left(T_{a}+2T_{1}\right)}{2C_{v}^{2}T_{1}^{4}}\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(\lambda)\Big{]}$
(46)
$\displaystyle\quad-\frac{\mathcal{Q}^{2}T_{a}}{2C_{v}^{2}T_{1}^{3}g(0)}\left(2+\frac{T_{a}}{T_{1}}+\frac{T_{a}}{T_{1}g(0)}\right)\exp\left[\frac{T_{a}}{T_{0}}\right]+\frac{\mathcal{Q}^{2}T_{a}\left(T_{a}+2T_{1}\right)}{2C_{v}^{2}T_{1}^{4}}\exp\left[\frac{T_{a}}{T_{1}}\right]\text{Ei}\Big{[}g(0)\Big{]}.$

##  IV Results

Figure 1:  Reaction progress variable $\lambda$ as a function of time $t$
(units of $t_{\text{adb}}$) for varying reaction orders $n$ shown in the
legend. The system parameters are $T_{0}=800\,\text{K}$,
$T_{1}=4000\,\text{K}$, and $T_{a}=6000\,\text{K}$. The dashed black curves
are numerical solutions and the solid curves are the results given by the
analytical approximation in Eq. (26) with expansion order $\alpha=3$.

Figure 1 illustrates a comparison between the solution given by the
approximation in Eq. (26) and the corresponding solution generated numerically
for various reaction orders. In all cases we define time $t$ in units of the
adiabatic induction time Menikoff (2017)

$t_{\text{adb}}=\frac{C_{v}T^{2}_{0}}{k\mathcal{Q}T_{a}}\exp\left[\frac{T_{a}}{T_{0}}\right],$
(47)

and, therefore, the frequency factor $k$ does not enter into the presented
results. All numerical solutions are obtained using the Mathematica program
NDSolve function. The different color curves correspond to different reaction
orders. The system is characterized by three temperatures: $T_{0}$, $T_{1}$,
and $T_{a}$. In this Figure, the value of these parameters are
$T_{0}=800\,\text{K}$, $T_{1}=4000\,\text{K}$, and $T_{a}=6000\,\text{K}$
which gives rise to the three stages of ignition observed in a system with
large activation temperature $T_{a}\gg\tfrac{C_{v}T_{0}^{2}}{\mathcal{Q}}$ :
induction, thermal runway, and burn-out. Over each stage of the burn process,
the analytical approximation agrees with the numerical solutions. The accuracy
of approximation decreases as $\lambda$ approaches $1/2$. The cause of this
accuracy decrease is that we expand the upper-bound solution about $\lambda=1$
and the lower-bond solution about $\lambda=0$, and then connect those
solutions at $\lambda=1/2$. This means that in the complete piecewise
solution, $\lambda=1/2$ is the point furthest from the respective expansion
point for each solution branch, and therefore each series generates the
greatest error at this point. In general, the accuracy of the analytical
approximation increases as $n\to 0$ and decreases as $n$ is increased. In
order to illustrate the effectiveness of present approach for relatively small
expansion order, we have used $\alpha=3$ in all analytical calculations. For
$n\gg 1$, the analytical approximation can also lead to accurate solutions by
increasing the expansion order as illustrated later.

Figure 2:  Reaction progress variable $\lambda$ as a function of time $t$
(units of $t_{\text{adb}}$) for varying reaction orders $n$ shown in the
legend. The system parameters are $T_{0}=3500\,\text{K}$,
$T_{1}=4000\,\text{K}$, and $T_{a}=8000\,\text{K}$. The dashed black curves
are numerical solutions and the solid curves are the results given by the
analytical approximation in Eq. (26) with expansion order $\alpha=3$.

A comparison between numerical and analytical solutions for a constant volume
burn with small energy release $\mathcal{Q}\ll\tfrac{C_{v}T_{0}^{2}}{T_{a}}$
is shown in Fig. 2 for various reaction orders. The system parameters in this
Figure are $T_{0}=3500\,\text{K}$, $T_{1}=4000\,\text{K}$, and
$T_{a}=8000\,\text{K}$. As before, the numerical and analytical results are in
agreement, illustrating that the presented approximation is robust with
respect to system parameters.

Figure 3:  Progress variable as a function of time for the same system
parameters as Fig. 1 with the analytical series expansion order increased to
$\alpha=6$. The different color curves correspond to different reaction orders
shown in the legend.

For simplicity, we have only explicitly shown the algebra for analytical
expansions up to third order. Greater solution accuracy, however, can be
obtained by increasing $\alpha$. Shown in Fig. 3 is a comparison between
analytical and numerical solutions for the same system parameters used in Fig.
1 with expansion order increased to $\alpha=6$. The increased value of
$\alpha$ results in excellent agreement between solutions for all values of
$n$ shown. In fact, for small $n$, the analytical result is essentially
indistinguishable from the numerical solution at the shown level of visual
fidelity. This illustrates that the analytical method developed here
approaches the exact solution in a relatively small number of expansion terms.
Note that we have added a curve for $n=2.5$ in this plot (cf. Fig. 1) in order
to illustrate that accurate solutions for relatively large reaction orders can
be constructed by increasing $\alpha$.

##  V Acknowledgments

We thank Josh Coe for insightful discussions. This work was supported by Los
Alamos National Laboratory (LANL) Directed Research and Development funds
(LDRD).

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_K_ eywords mobile phone data; call detail records; type allocation code; data
analysis; human mobility; large social event; social sensing; socioeconomic
status

# Analyzing the Behavior and Financial Status of Soccer Fans from a Mobile
Phone Network Perspective: Euro 2016, a Case Study

Gergő Pintér
John von Neumann Faculty of Informatics,
Óbuda University,
Bécsi út 96/B, 1034 Budapest, Hungary
<EMAIL_ADDRESS>&Imre Felde
John von Neumann Faculty of Informatics,
Óbuda University,
Bécsi út 96/B, 1034 Budapest, Hungary
<EMAIL_ADDRESS>0000-0003-4731-3816 0000-0003-4126-2480

###### Abstract

In this study, Call Detail Records (CDRs), covering Budapest, for the month of
June in 2016 has been analyzed. During this observation period, the 2016 UEFA
European Football Championship took place, which affected significantly the
habit of the residents, despite the fact that not a single match was played in
the city. We evaluated the fans’ behavior in Budapest, during and after the
Hungarian matches, and found that the mobile phone network activity reflects
the football fans’ behavior, demonstrating the potential of mobile phone
network data within a social sensing system. The Call Detail Records are
enriched with mobile phone properties to analyze the subscribers’ devices.
Applying the device information (Type Allocation Code) from the activity
records, the Subscriber Identity Modules, that do not operate in cell phones
are omitted from mobility analyses, allowing to focus on people. The mobile
phone price is proposed and evaluated as a socioeconomic indicator, and
correlation between the phone price and the mobility customs have been found.
We also found that, beside the cell phone price, the subscriber age and the
subscription type also have an effect on the mobility. On the other hand,
these do not seem to affect the interest in football.

##  1 Introduction

Football is one of the most popular sports worldwide and European or World
Championships, especially the finals, are among the most watched sporting
events. The Euro 2016 Final was watched by more than 20 million people in
France [19], or the Germany vs. France semifinal was watched by almost 30
million people in Germany [19]. But, what about Hungary?

According to the MTVA (Media Services and Support Trust Fund), that operates
the television channel M4 Sport, the first Hungarian match was watched by
about 1.734 million people, the second by about 1.976 million and the third
group match by about 2.318 million people111According to the Hungarian Central
Statistical Office (KSH), the population of Hungary was about 9.83 million in
2016 [22].. With these ratings, the M4 Sport, turned out to be the most
watched television channel in Hungary, during those days [17]. The whole
participation of the Hungarian national football team was beyond expectations
and raised interest, even among those, who generally, do not follow football
matches. But, is it possible to measure/correlate this interest, with a mobile
phone network?

Mobile phones can function as sensors, that detect the whereabouts and
movement of their carrier. In this day and age, practically everyone has a
mobile phone, that makes it possible to use large scale analyses. With enough
data, the general mobility customs and reactions to events can also be
studied. The first step is to prepare the data and select the appropriate
individuals for the study. Filtering the subscribers of the CDR data sets is
always a crucial step. Not just to eliminate the inactive users: a subscriber,
who only appears a few times in a data set, cannot be used for mobility
analysis, but the abnormally active subscribers can also bias the result.
Especially if their location does not change, as CDR data may not only contain
records for cell phones, that are carried by people.

Csáji et al. took into account subscribers who had at least 10 activity during
the observation period (15 months) [6]. Xu et al. chose to use those
subscribers, who had at least one activity record at least half of the days
during the observation period [47]. Pappalardo et al. discarded the
subscribers who had only one location, and the individuals have at least half
as many calls as hours are in the data set. Furthermore, the abnormally active
(more than $300$ calls per day) SIM cards are excluded [33]. In [34], we
selected the SIM cards, that have activity at least 20 days (out of 30), the
daily mean activity number is at least 40 on workdays and at least 20 on
weekends, but not more than $1000$. The upper limit is especially important to
remove SIM cards, that possibly operate in mobile broadband modems, for
example. Filtering by activity is not necessarily sufficient to keep only
individuals in the data set. Type Allocation Codes (TAC), on the other hand,
can determine the type of the device and the exact model of a cell phone.

After the right subscribers have been selected, it is common to determine the
home and work locations [43, 25, 32], then between these two crucial
locations, the commuting trends can be identified. The commuting is studied
between cities [23, 48, 25, 1] or within a city [7, 20, 10, 8, 28, 14, 13,
34].

Apart from commuting and connectivity analysis, CDR processing is often used
[41, 46, 24, 11, 26, 36, 38, 16] for large social event detection. When
thousands of people are on the same place at the same time, they generate a
significant ‘anomaly’ in the data, whereas small groups usually do not stand
out from the ‘noise’. This is especially true when the passive, transparent
communication between the mobile phone device and the cell are not included in
the data, but only the active communication (voice calls, text messages and
data transfer) are recorded.

In [36] and [38], mass protests are analyzed via mobile phone network data. In
[41, 24, 46] and [16], the authors examined the location of stadiums, where
the football matches took place. Traag et al. [41] and Hiir et al. [16] also
found that the mobile phone activity of the attendees decreased significantly.
In [41], z-score is also used to express the activity deviation during the
social event from the average. Xavier et al. compared the reported number of
attendees of these events with the detected ones. Furletti et al. also
analyzed sociopolitical events, football matches and concerts, in Rome [11].
This paper focuses on football matches, that however, took place in a remote
country (France), and the fans’ activity are studied in Budapest.

Mobility indicators, such as Radius of Gyration or Entropy, are often
calculated [33, 47] to describe and classify the subscribers’ mobility
customs. Furthermore, using mobility to infer about Social Economic Status
(SES) is a current direction of mobility analysis [47, 5, 1, 34]. Cottineau et
al. [5] explored the relationship between mobile phone data and traditional
socioeconomic information from the national census in French cities. Barbosa
et al. found significant differences in the average travel distance between
the low and high income groups in Brazil [1]. Xu et al. [47] found opposite
travel tendencies in mobility of Singapore and Boston. In our previous work
[34], we showed that the real estate price of the home and work locations
characterize the mobility and validated our results with census data. In this
paper, the price and the age of the subscribers’ mobile phones are proposed as
a source of the socioeconomic indicator. While Blumenstock et al. used the
call history as a factor of socioeconomic status [2], Sultan et al. [40]
applied mobile phone prices as socioeconomic indicator and identified areas
where more expensive phones appear more often, however, only manually
collected market prices were used.

Mobile phone network data is also used to analyze the human mobility during
COVID-19 pandemic and the effectiveness of the restrictions. Willberg et al.
identified a significant decrease of the population presence in the largest
cities of Finland after the lockdown compared to a usual week [45]. Bushman et
al. analyzed the compliance to social distancing in the US using mobile phone
data [3]. Gao et al. found negative correlation in stay-at-home distancing and
COVID-19 increase rate [12]. Still, these analyses might not be common enough.
Oliver et al. asked: ‘Why is the use of mobile phone data not widespread, or a
standard, in tackling epidemics?’ [30]. This, however, is not within the scope
of this paper.

In this study, we analyzed the mobile phone network activity before, during
and after the matches of the Hungarian national football team. The Call Detail
Records (CDR), analyzed in this study, have been recorded Budapest, however
the matches took place in France. We present another example of social
sensing, using CDR s, in an indirect and a direct way. Indirectly, as the
mobile phone activity of the sport fans, residing in Budapest, are studied
during matches played in France. Directly, as the spontaneous festival on the
streets of Budapest after the third match, and the welcome event at the
Heroes’ Square are presented from a data perspective.

The Call Detail Records are filtered by the Type Allocation Codes (TAC) to
remove those Subscriber Identity Module (SIM) cards, that do not operate in
mobile phones, thus not used by actual people. The price and age of the cell
phones are also analyzed in contrast of the subscribers’ age and mobility
customs.

The contributions of this paper are summarized briefly as follows:

  1. 1.

Fusing CDR data set with mobile phone prices and release dates.

  2. 2.

Filtering out SIM cards, that do not operate in mobile phones.

  3. 3.

Demonstrating connection between the phone price and the mobility customs.

  4. 4.

Proposing mobile phone price as a SES indicator.

  5. 5.

Attendees of the large social events are compared to the rest of the
subscribers based on their mobility and SES.

The rest of this paper is organized as follows. The utilized data is described
in Section 2, then, in Section 3, the applied methodology is summarized, and
in Section 4, the results of this study are introduced. Finally, in Section 5,
the findings of the paper are summarized and concluded.

##  2 Materials

Vodafone Hungary, one of the three mobile phone operators providing services
in Hungary, provided anonymized CDR data for this study. The observation area
was Budapest, capital of Hungary and its agglomeration, and the observation
period is one month (June 2016). In 2016 Q2, the nationwide market share of
Vodafone Hungary was 25.3% [27]. This data set contains $2291246932$ records
from $2063005$ unique SIM cards, and does not specify the type of the activity
(voice calls, text messages or data transfers).

Figure 1: SIM cards categorized by the number of activity records. The SIM
cards with more than 1000 activity records (26.98% of the SIM cards) provide
the majority (91.31%) of the activity.

Figure 1, shows the activity distribution between the activity categories of
the SIM cards. The dominance of the last category, SIM cards with more than
1000 activity records, is even more significant. This almost 27% of the SIM
cards produce the more the 91% of the activity.

Figure 2, shows the SIM card distribution by the number of active days. Only
the 34.59% of the SIM cards have activity on at least 21 different days. There
were $241824$ SIM cards (11.72%), that have appearance at least two days, but
the difference between the first and the last activity is not more the seven
days. This may indicate the presence of tourists. High tourism is usual during
this part of the year.

Figure 2: SIM card distribution by the number of active days.

The obtained data was in a ‘wide’ format, and contained a SIM ID, a timestamp,
cell ID, the base station (site) coordinates in WGS 84 projection, the
subscriber (age, sex) and subscription details (consumer/business and
prepaid/postpaid) and the Type Allocation Code (TAC) of the device. The TAC is
the first 8 digits of the International Mobile Equipment Identity (IMEI)
number, allocated by the GSM Association and uniquely identifies the mobile
phone model.

The Type Allocation Codes are provided for every record, because a subscriber
can change their device at any time. Naturally, most of the subscribers
($95.71$%) use only one device during the whole observation period, but there
are some subscribers, maybe mobile phone repair shops, who use multiple
devices (see Figure 4(a)). As a part of the data cleaning, the wide format has
been normalized. The CDR table contains only the SIM ID, the timestamp and the
cell ID. A table is formed from the subscriber and the subscription details,
and another table to track the device changes of the subscriber.

Figure 3: Number of daily activity records, during two weeks of June 2016. The
matches of the Hungarian national football team took place from June 14 to
June 26.

While the subscription details are available for every SIM cards, the
subscriber information is missing in slightly more than 40% of the cases,
presumably because of the subscribers’ preferences of personal data usability.
Figure 4(b), shows the age distribution of the subscribers, whose data is
available ($58.65$%), in respect of the subscription type. Note that, this may
not represent the age distribution of the population, not even the customers
of Vodafone Hungary, as one is allowed to have multiple subscription and the
actual user of the phone may differ from the owner of the subscription.
Nevertheless, it is still clear that among the elderly people, the prepaid
subscriptions are more popular.

Figure 3, shows number of daily activity records during the second half of the
month. Weekends (brown bars) show significantly fewer activity, hence the
activity during the matches compared to the weekday or weekend activity
average, respectively to the day of the match.

Although the data contains cell IDs, only the base station locations are
known, where the cell antennas are located. As a base station usually serve
multiple cells, these cells has been merged by the serving base stations.
After the merge, 665 locations (sites) remained with known geographic
locations. To estimate the covered area of these sites, the Voronoi
Tessellation, has been performed on the locations. This is a common practice
[31, 6, 44, 4, 29, 42] for CDR processing.

(a) Number of used phones.

(b) Subscribers’ age distribution.

Figure 4: The number of different TAC s used by the subscribers, and the
subscriber’ age distribution in respect of the subscription type.

###  2.1 Resolving Type Allocation Codes

The socioeconomic status SES of the members in the celebrating crowd have been
intended to characterize by the mobile device they use. The preliminary
assumption was that the price of the mobile phone represents the SES of a
person.

According to our knowledge, there is no publicly available TAC database to
resolve the TAC s to manufacturer and model, although some vendors (e.g.,
Apple, Nokia) publishes the TAC s of their products. The exact model of the
phone is required to know how recent and expensive a mobile phone is. Although
this is not even enough to determine how much the cell phone costed for the
subscriber as they could have bought it on sale or discount via the operator
in exchange for signing an x-year contract. Still, the consumer price should
designate the order of magnitude of the phone price.

The dataset of TAC s provided by “51Degrees” has been used, representing the
model information with three columns: ‘HardwareVendor’, ‘HardwareFamily’ and
‘HardwareModel’. The company mostly deals with smartphones that can browse the
web, so feature phones and other GSM-capable devices are usually not covered
by the data set. Release date and inflated price columns are also included,
but these are usually not known, making the data unsuitable to use on its own.

Although it cannot be separated by type, but the CDR data contains not only
call and text message records, but data transfer as well. Furthermore, some
SIM cards do not operate in phones, but in other – often immobile – devices
like a 3G router or a modem. 51Degrees managed to annotate several TAC s as
modem or other not phone devices. This was extended by manual search on the
most frequent TAC s. There were $324793$ SIM cards that uses only one device
during the observation period and operates in a non-phone device.

###  2.2 Fusing Databases

For a more extensive mobile phone price database, a scarped GSMArena
database[39] has been used. GSMArena222https://www.gsmarena.com/ has a large
and respectable database, that is also used in other studies[37, 49]. The
concatenation of the brand and model fields of the GSMArena database could
serve as an identifier for the database fusion. 51Degrees stores the hardware
vendor, family and model, where hardware family is often contains a marketing
name (e.g., [Apple, iPhone 7, A1778]). As these fields are not always properly
distinguished, the concatenation of the three fields may contain duplications
(e.g., [Microsoft, Nokia Lumia 820, Lumia 820]). So, for the 51Degrees
records, three identifiers are built using the concatenation of fields (i)
vendor + family, (ii) vendor + model and (iii) vendor + family + model, and
all the three versions are matched against the GSMArena records.

Another step of the data cleaning is to correct the name changes. For example,
BlackBerries were manufactured by RIM (e.g., [RIM, BlackBerry Bold 9700,
RCM71UW]), but later, the company name was changed to BlackBerry and the
database records are not always consistent in this matter. The same situation
occurs due to the Nokia acquisition by Microsoft.

To match these composite identifiers, the simple string equality cannot be
used, due to writing distinction, so Fuzzy String match is applied using the
FuzzyWuzzy Python package, that uses Levenshtein Distance to calculate the
differences between strings. This method is applied for all the three
identifiers from the 51Degrees data set and the duplicated matches (e.g., when
the family and the model is the same) were removed. Mapping the GSMArena
database to the 51Degrees adds phone price and release date information to the
TAC s, that can merged with the CDR s.

From the GSMArena data, two indicators have been extracted: (i) price of the
phone (in EUR), and (ii) the relative age of the phone (in months). The phone
price was left intact without taking into consideration the depreciation, and
the relative age of the phone is calculated as the difference of the date of
the CDR data set (June 2016) and the release date of the phone.

##  3 Methodology

The framework, introduced in our earlier work[34], has been applied to process
the mobile phone network data. The CDR s are normalized, cleaned and the
mobility metrics (Section 3.1) are determined for every subscriber. The
records can be filtered spatially and temporally, both of these filtering is
applied for this work. Additionally, a group of SIM cards can be selected from
the activity records.

Only temporal filtering is applied to visualize the activity trends during the
football matches. Figures 9, 10, 11, 12, 14 and 17), illustrate the activity
of the subscribers in the whole observation area during the matches, including
the two hours before and after the matches. For the celebration after the
Hungary vs. Portugal match, spatial and temporal filtering is applied to
select the area of interest (Budapest downtown) in the given time interval.

To determine the activity levels for the map, Figure 13, the match-day
activity, the average weekdays activity (without the match-day) and the
Z-scores 333The standard score (or z-score) is defined as
${z=\frac{x-\mu}{\sigma}}$, where $\mu$ is the mean and $\sigma$ is the
standard deviation. are determined for the sites of the area of interest
(downtown), in the selected time interval (20:15–20:20). We observed that the
standard deviation would be higher, without removing the target-day activity
from the reference average, consequently the Z-score would be lower and the
relative differences less consistent. The histogram of the Z-score were
generated for the selected sites (Figure 5) to determine the activity
categories. Zero value means that, the activity level equals to the average,
but a wider interval (between $-2$ and $2$) is considered average to allow
some variation. Sites with Z-score between $2$ and $8$ are considered having
high activity during the given time interval. There are sites with either low
(below $-2$) or very high activity (over $8$). The same method is applied for
the map of Figure 18(b), but as the area of interest and the event differs,
the thresholds are not the same (see Section 4.5).

Figure 5: Z-score distribution of the downtown sites, with the activity level
thresholds at $-2$, $2$ and $8$, using the same colors as in Figure 13.

The groups of football fans are formed from the subscribers based on only the
activity during the Hungary vs. Portugal match. The owner of those non-phone
SIM cards, that were active after at least two goals are considered active
football fans. The properties of these subscribers, including the age,
mobility metrics, phone age and price are compared to the rest of the
subscribers (Figure 15).

###  3.1 Mobility Metrics

The metrics of Radius of Gyration and Entropy has been used to characterize
human mobility. These indicators are determined for every subscriber, omitting
those SIM cards, that operate in non-phone devices. In this study, locations
are represented by the base stations.

The Radius of Gyration [15] is the radius of a circle, where an individual
(represented by a SIM card) can usually be found. It was originally defined in
Equation (1), where $L$ is the set of locations visited by the individual,
$r_{cm}$ is the center of mass of these locations, $n_{i}$ is the number of
visits at the i-th location.

$r_{g}=\sqrt{\frac{1}{N}\sum_{i\in L}{n_{i}(r_{i}-r_{cm})^{2}}}$ (1)

The entropy characterizes the diversity of the visited locations of an
individual’s movements, defined as Equation (2), where $L$ is the set of
locations visited by an individual, $l$ represents a single location, $p(l)$
is the probability of an individual being active at a location $l$ and $N$ is
the total number of activities of an individual [31, 5].

$e=-\frac{\sum_{l\in L}{p(l)\log p}}{\log N}$ (2)

###  3.2 Socioeconomic Status

In our earlier work [34], the real estate price of the subscribers’ home
locations were used to describe the socioeconomic status. In this study, the
CDR s are enriched by phone prices and the phone price is assumed to apply as
a socioeconomic indicator. To demonstrate the applicability of the mobile
phone price as a socioeconomic indicator, it was examined in respect of the
mobility indicators, applying Principal Component Analysis (PCA).

The SIM cards are aggregated by the subscriber age categories (5-year steps
between 20 and 80) and the phone price categories (100 EUR steps to 700 EUR),
the Radius of Gyration and Entropy categories. For the Radius of Gyration, 0.5
km distance ranges are used between 0.5 and 20 km, and the Entropy values are
divided into twelve bins between $0.05$ and $1.00$. The structure of the data
used for the Principal Component Analysis defined as follows.

A table has been generated where, every row consists of 40 columns,
representing 40 Radius of Gyration bins between 0.5 and 20 km and 20 columns
representing 20 Entropy bins, between $0.05$ and $1.00$. The subscribers,
belonging to each bin are counted, and the cardinality have been normalized by
metrics to be able to compare them. The categories are not explicitly labeled
by them, so the subscriber age and the phone price descriptor columns are not
provided to the PCA algorithm. The same table is constructed using
weekend/holiday metrics and its rows are appended after the weekdays ones.

When the PCA is applied, the 60-dimension vector is reduced to two dimensions
based on the mobility customs, where the bins are weighted by the number of
subscribers. The cumulative variance of the two best components is about 61%
(see Figure 7(b)). The bins, representing the two new dimensions (PC1 and PC2)
are plotted (see Figure 6) and the markers are colored by the phone price,
marker sizes indicate the subscriber age category, using larger markers for
younger subscribers.

##  4 Results and Discussion

As Figure 6 shows, the markers are clustered by color, in other words, the
phone price, that is proportional to PC1, but inversely proportional to PC2.
Within each phone price group, the younger subscribers (larger markers) are
closer to the origin, indicating that the mobility custom of the younger
subscribers differs from the elders, although this difference is smaller
within the higher price categories. This finding coincides with [9], where
Fernando et al. found correlation between subscribers’ age and mobility
metrics.

To give context to Figure 6, Figure 7(a), shows the phone price distribution:
most of the phones are within the 50–200 EUR range. Note that, there are only
a few phones over 550 EUR, but the owners of those have significantly
different mobility patterns.

Figure 6 does not only show that the phone price forms clusters, but also
reveals the effect of the subscription type to the mobility. Within the phone
price categories, except the highest with only a very few subscribers, the
postpaid groups are usually closer to the origin. Prepaid subscriptions are
usually for those, who do not use their mobile phone extensively, and it seems
that people with a prepaid subscription have similar mobility customs as
people with less expensive phones but postpaid subscription. That is most
notable at (-6, 2) and (-5, -1).

Figure 6: Scatter plot of the 2-component PCA. Marker sizes indicate
subscriber age category, the color represents the phone price category and the
subscription type (Prepaid/Postpaid) is distinguished by the marker type.

(a) Phone price distribution.

(b) The Pareto histogram for the PCA.

Figure 7: Phone price distribution and the Pareto histogram for the 60
components of the Principal Component Analysis.

Sultan et al. identified areas in Jhelum, Pakistan, where more expensive
phones appear more often [40]. Using the same method, Budapest and its
agglomeration was evaluated: the average phone prices from the activity
records are determined for every site. The ground truth is that the real
estate prices are higher on Buda side (West of river Danube) of Budapest and
downtown [34], and this tendency can be clearly seen in Figure 8. The airport
area has a significantly higher average than its surroundings, that is not
surprising. The spatial tendencies of the mobile phone price, along with the
result of the PCA (Figure 6), clearly demonstrates the expressiveness of the
phone price as a socioeconomic indicator.

Figure 8: Average price (in EUR) of the mobile phones, that generated the
activity records in each site, during the whole observation period (June
2016).

The rest of this section examines the results, in the time order of the
Hungarian Euro 2016 matches.

###  4.1 Austria vs. Hungary

The first match against Austria (Figure 9) was started at 18:00, on Tuesday,
June 14, 2016. Before the match, the activity level was significantly higher
than the average of the weekdays, and later decreased until the half-time.
During the second half, the activity level dropped to the average, which
indicated that more people started to follow the match. Right after the
Hungarian goals, there are two significant peaks have been observed in the
activity, which exactly indicates increased attention and the massive usage of
mobile devices during the match.

As the data source cannot distinguish the mobile phone activities by type, it
cannot be examined what kind of activities caused the peaks. It is supposed
that the activity was mostly data transfer or text messages, not phone calls.
It simply does not seem to be lifelike to call someone during the match just
because of a goal, but sending a line of text via one of the popular instant
messaging services, is very feasible.

Figure 9: Mobile phone activity during and after the Austria–Hungary Euro 2016
match, in comparison with the average activity of the weekdays.

###  4.2 Iceland vs. Hungary

The match against Iceland was played on Saturday, June 18, 2016. Figure 10,
shows the mobile phone activity levels before, during and after the match. As
the weekend activity is generally lower (see Figure 3), the average of the
weekdays are used as a reference. The match began at 18:00, and from that
point, the activity level was significantly below the average, except the
half-time break and, again, the peak after the Hungarian goal. Interestingly,
the Icelandic goal does not result such a significant peak, only a very
moderate one can be seen in the time series.

Traag et al. [41] also found activity drop during a game, but in that case the
area of the stadium was analyzed, where the match was played and there was no
peak during the match.

Figure 10: Mobile phone activity during and after the Iceland–Hungary Euro
2016 match, in comparison with the average activity of the weekends.

###  4.3 Hungary vs. Portugal

On Wednesday, June 22, 2016, as the third match of the group state of the 2016
UEFA European Football Championship, Hungary played draw with Portugal. Both
teams scored three goals and with this result, Hungary won their group and
qualified for the knockout phase. During the match, the mobile phone activity
dropped below the average, but the goals against Portugal resulted significant
peaks, especially the first one (see Figure 11). On the other hand, the
Portuguese equalizer goal(s) did not cause significant mark in the activity.
In the second half, the teams scored four goals in a relatively short time
period, but only the Hungarian goals resulted in peaks. This observation
suggests that the football fans had notable influence on the mobile network
traffic.

After the match, the activity level is over the average, that might represent
the spontaneous festival in downtown Budapest. According to the MTI (Hungarian
news agency), thousands of people celebrated in the streets, starting from the
fan zones, mainly from Erzsébet square (Figure 13 a), Margaret Island (Figure
13 b) and Erzsébet square (Figure 13 c) direction Budapest Nyugati railway
station. The Grand Boulevard was completely occupied by the celebrating crowd
and the public transportation was disrupted along those affected lines.

This social event is comparable to mass protests from a mobile phone network
perspective. In an earlier work [35], we have analyzed the mobile phone
activity at the route of a mass protest. The activity of the cells were
significantly high when the protesters passed through the cell. In this case,
however, the affected area were smaller and the sites along the Grand
Boulevard were very busy at the same time after the game.

Figure 11: Mobile phone activity during and after the Hungary–Portugal Euro
2016 match, in comparison with the average activity of the weekdays.

The activities of the sites (multiple cells aggregated by the base stations),
in Budapest downtown, are illustrated on Figure 12. The highlighted site
covers mostly Szabadság square (for the location, see Figure 13 a), where one
of the main fan zones was set up with a big screen. The activity curve
actually follows the trends of the whole data set (see Figure 11). There is
high activity before the match, during half-time and, for a short period,
after the match. During the match, the activity decreased except four, not so
significant, peaks around the goals.

In the highlighted site, in Figure 12, almost 7 thousand SIM cards had been
detected between 17:00 and 20:00. The data shows that 53.57% of the
subscribers were between 20 and 50 years old, while 33.49%, had no age data.

After the match, there is a significant increase in the activity in some other
sites. These sites are (mostly) around the Grand Boulevard, where the fans
marched and celebrated the advancement of the national football team to the
knockout phase.

Figure 13, shows the spatial distribution of this social event, using Voronoi
polygons generated around the base stations locations. The polygons are
colored by the mobile phone network activity increase at 20:15, compared to
average of the weekday activity. For the comparison, the standard score was
determined for every base station with a 5-minute temporal aggregation. The
darker colors indicate the higher activity surplus in an area. The figure also
denotes the three main fan zones in the area, routes of the fans by arrows,
and the affected streets are emphasized.

Figure 12: Site activities, in Budapest downtown, on the day of the Hungary
vs. Portugal football match (June 22, 2016). The highlighted site covers
mostly the Szabadság Square, where one of the main fan zones was set up.
Figure 13: After the Hungary vs. Portugal football match, the fans, delirious
with joy, filled the streets. The arrows show their route from the main fan
zones to and along the Grand Boulevard. Voronoi polygons colored by the mobile
phone network activity at the peak of the event, at 20:15.

#### Who are responsible for the peaks?

There were three Hungarian goals during the match, hence there were three
peaks, starting at 18:18, 19:02 and 19:18. All of them had about 5-minute
fall-times. To answer this question, the SIM cards that were active during any
two of the peaks were selected. Selecting SIM cards that were active during
any of the peaks, would also include many subscribers, that cannot be
considered as football fan. The participation of all the three peaks, on the
other hand, would be too restrictive.

Figure 14(a), presents the activity of the selected $44646$ SIM cards and the
owner of these cards, which may belong to the football fans. Removing these
SIM cards from the data set, should result an activity curve without peaks,
and at the same time similar, in tendency, to the average activity. However,
as Figure 14(b) shows, the activity still drops during the match. Therefore,
the ‘football fan’ category should be divided to ‘active’ and ‘passive’ fans,
from the mobile phone network perspective. Active fans are assumed to express
their joy using the mobile phone network (presumably to access the social
media) and cause the peaks. It seems that the passive fans ceased the other
activities and watched the game, that caused some lack of activity, compared
to the average. By removing the active fans from the observed set of SIM
cards, the activity level decreased in general (Figure 14(b)). However, this
is not surprising, as these people reacted to the goals, they must often use
the mobile phone network. There are also some negative peaks, indicating that
the selection is not perfect.

Is there any difference between the active fans regarding the phone age and
price compared to the other subscribers? Figure 15(a), shows the relative age
of the phones in respect of the subscribers’ behavior after the goals. No
significant difference has been realized between the active fans and other
subscribers, the median of the phone relative age is about two years, and
there are some much older (nearly ten years old) phones in use. It should be
noted that older devices are used by elderly people. The price of the phones
show opposite tendency: the younger subscribers own more expensive phones
(Figure 15(b)).

Naturally, not all of these $169089$ SIM card (without the ones operating non-
phone devices) generated activity after all the goals. $83352$ devices were
active after the first goal, $70603$ after the second and $68882$ after the
third. After at least two goals $44646$, and after all the three goals only
$9102$ devices had activity, within 5 minutes.

(a) Activity of fans.

(b) Activity without the fans.

Figure 14: Mobile phone network activity of the SIM cards (fans), that had
activity right after any two of the Hungarian goals, and the mobile phone
activity of the other SIM cards.

Why would they use the mobile phone network to access social media? If they
were at home, they would have used the wired connection, via Wi-Fi for mobile
devices. In Hungary, the $79.2$% of the households had wired internet
connection, according to the KSH [21], and it could be even higher in
Budapest. However, if they were at fan zones, for example in Szabadság Square,
using the mobile network is more obvious.

(a) Relative age of the phones.

(b) Price of the Phones.

Figure 15: Mobile phone relative age and the price distributions in different
age categories, comparing the fans, who had activity right after any two of
the Hungarian goals, and the rest of the SIM cards.

As Figure 15 shows, there is no significant difference in the phone age
between the active football fans and the rest of the subscribers. The medians
are almost the same within the young adult and the middle-age categories, but
elders tend to use older devices, especially those, who did not react to the
goals. The active football fans’ median phone price is 180 EUR, in contrast of
the 160 EUR median of the rest of the subscribers. However, the older
subscribers tend to use less expensive phones. This tendency is also present
within the football fans, but stronger within the other group.

Figure 16, illustrates the mobility metrics in different age categories, also
comparing the football fans and the rest of the subscribers. The Radius of
Gyration median is almost the same in all the age categories and groups. The
Entropy medians have a notable difference between the two groups, but do not
really change between the age categories. This means, that the mobility
customs of the football fans, who use the mobile phone network more actively,
are similar, regardless of the subscribers’ age.

(a) Radius of Gyration of the subscribers.

(b) Entropy of the subscribers.

Figure 16: Radius of Gyration and Entropy distributions in different age
categories, comparing the fans, who had activity right after any two of the
Hungarian goals, and the rest of the SIM cards.

###  4.4 Hungary vs. Belgium

On Sunday, June 26, 2016, Hungary played the fourth and last Euro 2016 match
against Belgium. Figure 17, shows the mobile phone network activity before,
during and after the match. During the match, the activity level was below the
weekend average. The activity after the match was slightly higher than
average, since the match ended late on Sunday, when the activity average is
usually very low. This activity surplus may only indicate that the fans were
simply leaving the fan zones and going home.

Figure 17: Mobile phone activity during and after the Hungary–Belgium Euro
2016 match, in comparison with the average activity of the weekends.

###  4.5 Homecoming

The Hungarian national football team returned to Budapest, on June 27, 2016. A
welcome event at the Heroes’ Square have been held, where the football fans
can greet the national football team. According to the M4 Sport television
channel, approximately 20 thousand people attended to the event [18]. Between
18:00 and 19:30, there were $4246$ unique, non-phone SIM cards active in the
site, that covers the Heroes’ Square. $3425$ are known to use smartphone,
based on the operating system column of the GSMArena data set.

The cells of this base station cover a larger area, so not all of these
subscribers actually attended to the event, but on the other hand, it is not
compulsory to use the mobile phones during this event. Supposing that the
mobile phone operator preferences among the attendees corresponded to the
nationwide trends in 2016, there could even be about 17 thousand people, as
the data provider had $25.3$% market share [27].

Figure 18(b) shows, a part of District 6 and the City Park with the Heroes’
Square and the Voronoi polygons of the area are colored according to the
Z-score values, to indicate the mobile phone activity in the area, at 18:35.
The activity is considered low below $-1$, average between $-1$ and $1$, high
between $1$ and $2.5$ and very high above $2.5$. Figure 18(a) shows, the
mobile phone network activity (upper), and the Z-score (bottom) of the site,
covering Heroes’ Square. It is clear, that during the event, the activity is
significantly higher than the weekday average, and the Z-score values are also
follows that.

(a) Activity and Z-score of the site, at Heroes’ square.

(b) Spatial view of the activity at 18:35.

Figure 18: Mobile phone network activity at Heroes’ Square and its
neighborhood, during the welcoming event of the Hungarian national football
team.

###  4.6 Limitations

We associated subscribers’ SES with the release price of their cell phones,
however, it is not necessary for them to buy their phones at that price. Many
people buy their phone on sale or discount via the operator in exchange for
signing an x-year contract.

Also, subscribers can change their phone devices at any time. We have taken
into consideration only those subscribers, who had used only one device during
the observation period, or had a dominant device that generated most of the
activity records of the given subscriber.

We have fused three data sets to exclude the non-phone SIM cards, but the
identified devices are not complete. There remained devices, that models are
unknown and there are phones, that release date and price are unknown. It is
not possible to determine SES of these subscribers with the proposed solution.

###  4.7 Future Work

Although, the current solution to select the football fans’ SIM cards, in
other words, the SIM cards, that caused the peaks gives a reasonable result,
but could be improved by analyzing the activity during the whole observation
period. For example, applying a machine learning technique.

Extending the list of the non-phone TAC s could also help to refine the
results, and combining the mobile phone prices with the real estate prices of
the home location would most certainly enhance the socioeconomic
characterization.

The relative age of the cell phone might be used as a weight for the phone
price, when applied as SES indicator to distinguish between the phone price
categories. As an expensive, but older phone is not worth as much as a newer
one with the same price.

##  5 Conclusions

In this study, we demonstrated that mobile phone network activity shadows
precisely the football fans’ behavior, even if the matches are played in
another country. This analysis focused the people followed the matches on TV
(at home) or big screens at the fan zones, but not in the stadium, where the
matches were actually played. The mobile phone network data and the mobile
phone specification database has been applied to characterize the SES of the
football fans. The data fusion allowed us to remove a considerable number of
SIM cards from the examination that certainly operates in other devices than
mobile phones. Although, there are some still unidentified TAC s in the data
set, but this way, the activity records, involved in this study, have a
significantly higher possibility to used by an actual person during the
events.

The time series of mobile network traffic clearly show that the activity was
below the average during the matches, indicating that many people followed
their team. This observation coincides with other studies [41, 24, 46, 16],
where the activity of the cells at the stadium were analyzed. We also
demonstrated that a remote football match can also have notable effect on the
mobile phone network. Moreover, the joy felt after the Hungarian goals, is
clearly manifested in the data, as sudden activity peaks. The CDR data is
certainly capable of social sensing.

The spontaneous festival after the Hungary vs. Portugal match and the
welcoming event at the Heroes’ Square are direct applications of social
sensing and comparable to mass protests from a data perspective. During the
events, the mobile phone network activity was significantly higher than the
average in affected areas.

The price of the mobile phone was proved to be an expressive socioeconomic
indicator. It is capable not only to cluster the areas of a city, but also to
distinguish the subscribers by mobility customs. On the other hand, it does
not seem to affect the interest in football.

## Author Contributions

Conceptualization, G.P; methodology, G.P.; software, G.P.; validation, G.P.;
formal analysis, G.P.; investigation, G.P.; resources, G.P. and I.F.; data
curation, G.P.; writing—original draft preparation, G.P.; writing—review and
editing, G.P. and I.F.; visualization, G.P.; supervision, I.F.; project
administration, I.F.; funding acquisition, I.F. All authors have read and
agreed to the published version of the manuscript.

## Funding

This research supported by the project 2019-1.3.1-KK-2019-00007 and by the
Eötvös Loránd Research Network Secretariat under grant agreement no. ELKH
KÖ-40/2020.

## Acknowledgments

The authors would like to thank Vodafone Hungary and 51Degrees for providing
the Call Detail Records and the Type Allocation Code database for this study.

For plotting the map, OpenStreetMap data is used, that is copyrighted by the
OpenStreetMap contributors and licensed under the Open Data Commons Open
Database License (ODbL).

## Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the
design of the study; in the collection, analyses, or interpretation of data;
in the writing of the manuscript, or in the decision to publish the results.

## Glossary

CDR

    Call Detail Record
IMEI

    International Mobile Equipment Identity
KSH

    Központi Statisztikai Hivatal, Hungarian Central Statistical Office
MTI

    Magyar Távirati Iroda, Hungarian news agency
PCA

    Principal Component Analysis
SES

    Social Economic Status
SIM

    Subscriber Identity Module
TAC

    Type Allocation Code
WGS 84

    World Geodetic System, also known as EPSG:4326

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