# Entropic Spatial Auto-correlation of Voter Uncertainty and Voter Transitions
in Parliamentary Elections

Omar El Deeb Department of Natural Sciences, Lebanese American University

###### Abstract

This paper studies a novel spatial auto-correlation model of voter uncertainty
across districts. We use the Moran $I$ index to measure the auto-correlation
of Shannon, relative Shannon, Tsallis and relative Tsallis entropies of
regional electoral outcomes with respect to geographic adjacency, proximity
and sectarian adjacency. Using data from the Lebanese parliamentary elections,
we find strong geographic and gravitational adjacency correlations. More
importantly, there is a notably strong correlation in sectarian adjacency in
both $2018$ and $2022$ elections, with a very high level of confidence. This
result asserts the dominance of the sectarian factor in Lebanese politics. We
also introduce the method of maximized general entropy estimation that allows
us to determine the Markov transition matrix of voters between consecutive
elections.

##  1 Introduction

Entropy is a fundamental concept in thermodynamics and statistical physics
that measures disorder or randomness in a system. In thermodynamics, it is a
measure of the amount of energy that is unavailable to do work. In information
theory, it is a measure of the amount of uncertainty or randomness in a
message. Entropy has applications in a wide range of fields, including
physics, chemistry, biology, and engineering. In physics, entropy is used to
explain the behavior of gases, as well as the behavior of energy and matter on
a molecular level. In chemistry, entropy is used to predict the direction of
chemical reactions and the distribution of energy among particles in a system.
In biology, entropy is used to study the behavior of complex systems, such as
the evolution of species while engineering, entropy is used to design
efficient systems for storing, transmitting, and processing information. It
has been also widely employed in multidisciplinary fields ranging from
information sciences, machine learning, social sciences and economics to
astrophysical systems, heavy ion collisions, election studies and voter
transitions [1, 2, 3, 4, 5, 6, 7, 8].

In election studies, entropy can be used to measure the degree of uncertainty
or randomness in an election. This could be useful for predicting the outcome
of an election or for analyzing the effectiveness of different voting systems.
Entropy can also be used to measure the diversity of preferences among voters,
which could be useful for identifying trends or patterns in voter behavior.
Additionally, entropy can be used to investigate the fairness of an election,
by comparing the distribution of votes across different candidates, parties or
coalitions. Particularly in elections, higher entropy signifies a more uniform
spread of votes to various lists, while lower entropy indicates more
concentration of votes for less lists. A maximum entropy indicates equal
distribution of votes for all lists while a zero entropy implies that one list
has one all of the votes.

After it was initially proposed by Claude Shannon in [9], the entropy equation
was heavily and successfully used in information theory and all subsequent
related fields. Other formulations of entropy measures were later proposed and
implemented in social systems, like the Tsallis entropy. Tsallis entropy and
Shannon entropy are both measures of entropy, but they are based on different
mathematical definitions and have different applications. In general, Tsallis
entropy is more versatile and applicable to a wider range of systems than
Shannon entropy [10, 11]. Shannon’s entropy in this regard could be considered
as a special case of the more general Tsallis entropy [12].

An important tool that we utilize in this paper is the spatial auto-
correlation measure known by Moran’s $I$ index of a spatial distribution [13].
It is an inferential statistic used to measure the spatial auto-correlation
between locations and feature values simultaneously [14, 15]. The Moran index
is a measure of spatial auto-correlation, which is the degree to which the
values of a variable are clustered together in space. It can be used to
analyze a wide range of spatial data, including data on population density,
economic activity, and environmental factors. It is calculated by comparing
the observed values of a variable to the values that would be expected if the
variable were randomly distributed in space. A positive Moran index indicates
that similar values tend to be located near each other, while a negative Moran
index indicates that dissimilar values tend to be located near each other. The
Moran index is commonly used in geography, sociology, and other fields to
study the patterns and trends in spatial data [16, 17].

Entropy maximization is the process of finding the distribution of a random
variable that has the maximum entropy, subject to certain constraints. By
finding the distribution with the maximum entropy, researchers can make
predictions about the behavior of the system and its properties. It is a
powerful tool for analyzing complex systems and predicting their behavior.
Entropy maximization is applied in various fields, such as information theory,
where it can be used to design efficient data compression algorithms or to
study the reliability of communication systems. Entropic models were heavily
used in electoral studies to analyze cases as the uncertainty among voters and
correlations between increased uncertainty and voter turnout. The method of
entropy maximization was also employed to analyze voter transitions between
consecutive elections [18, 19, 20, 21]. Voter transition refers to the process
of voters switching their support from one candidate or political party to
another. This can happen for many reasons, such as changes in the political
landscape, shifts in voter attitudes and priorities, or the influence of
external factors, such as the media or campaign tactics. Voter transitions are
a key aspect of elections and political analysis, as they can help to predict
the outcome of an election and to understand the factors that drive voter
behavior. In general, voter transitions are studied using techniques from
political science and sociology, such as survey research, statistical
analysis, and network analysis. These methods are used to identify trends and
patterns in voter behavior and to predict the outcome of an election based on
voter preferences and voting patterns [22, 23, 24, 25].

Spatial auto-correlation methods were used in numerous fields of research
including geography, plant populations, traffic crashes, landscape patterns,
drought propagation, satellite data, migration flows, tectonics, image
segmentation, COVID-19 spread patterns, thermal patterns and in almost all
fields of applied sciences [26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Many
models from statistical physics were used in the analysis of spatial relations
between electoral turnouts and outcomes [36, 37, 38, 39], however, there is a
gap in the literature in utilizing spatial auto-correlation techniques in the
study of entropy distribution in electoral outcomes.

In this paper, we develop a novel entropic spatial auto-correlation method by
applying the Moran’s index of spatial auto-correlation on various entropic
measures across districts. This method allows us to be able to spatially
relate voter behavior across districts or across other feature values. We
apply the method on the outcomes of the Lebanese parliamentary elections and
we use it as a demonstrational protocase. The most recent two parliamentary
elections were held in May, 2018 and May, 2022 respectively, adopting same
electoral law, district division and seat allocation. The country is divided
into $15$ electoral areas comprising $26$ regional districts, and $128$ seats
are allocated at the district level. Lebanon is a democratic country and holds
regular parliamentary elections. The political system of Lebanon is based on a
multi-party system, with several political parties representing different
ideologies and interest groups. The parliament of Lebanon is made up of 128
members, who are elected through a system of proportional representation.
Overall, the democratic elections in Lebanon are an important part of the
country’s political system and its commitment to democratic principles.
Several studies have qualitatively investigated some recent elections in
Lebanon [40, 41, 42, 43] but there is a huge gap in quantitative analysis of
its elections in general and very scarce academic studies about the latest
couple of them.

Entropic measures provide a way to quantify how evenly the votes are
distributed among different lists in a district. A greater level of entropy
indicates greater uniformity and distribution of votes among the lists. This
means that a higher entropy measure can indicate a shift in electoral
dominance from fewer lists to a more even distribution among multiple lists.
This measure of entropy also indicates that there is more uncertainty in the
results. The spatial correlation of entropy measures proves that electoral
voting behaviors are strongly correlated between adjacent districts. Spatial
correlation has been applied to other schemes than elections in the
literature. However, our novel contribution is to specifically apply these
techniques on entropic measures on the district level. This combination of two
known techniques creates a new tool for the spatial study of voting behavior.
We also introduce the concept of sectarian adjacency to inspect the effect of
this factor on voting outcomes. Furthermore, we apply concept of gravitational
adjacency which is commonly used in economics and econophysics models, to
analyze correlations of voting outcomes.

Using the methods of entropy and spatial auto-correlations, we show that the
uncertainty outcomes across districts are spatially auto-correlated by
geographic adjacency, voter gravitational adjacency and most apparently by
sectarian adjacency. Voters from physically neighboring districts mostly have
correlated voting uncertainty patterns. Voters from regions with higher
numbers of voters and closer distances between their centers mostly have
correlated voting uncertainty patterns. We also show that voters from
districts that have similar sectarian majorities have correlated voting
uncertainty patterns.

In this paper, we also study voter transitions among lists between $2018$ and
$2022$ parliamentary election, and determine the Markov transition matrices of
voting outcomes in some selected districts. We make use of the significant
advancement in the theoretical understanding of the generalized maximum
entropy (GME) estimator in recent years. Additionally, the use of the GME
estimator has become more widespread with the use of dedicated packages in
statistical or computational software like in SAS, MATLAB, STATA,…. [44, 45,
46]. We employ the GME estimator package in SAS in order to estimate voter
transitions among lists between the $2018$ and $2022$ Lebanese parliamentary
election.

The structure of the paper is as follows: In section 2, we introduce Shannon’s
and Tsallis’ entropy, Spatial auto-correlation and Moran’s Index and introduce
our model. We also define our methods used in voter migration analysis. In
section 3, we present our results and discuss them, then we conclude in
section 4.

##  2 Model

###  2.1 Shannon’s and Tsallis’ Entropy

Entropy is a measure of a state disorder. In elections outcomes, entropy
represents the dispersion of the voting patterns for different lists, parties
or coalitions. We use the definition of Shannon’s entropy, as defined in
information theory to be

$H_{j}=-\sum_{i=1}^{N_{j}}p_{i,j}\log(p_{i,j})$ (1)

where $H_{j}$ is Shannon’s entropy in district $j$, $p_{i,j}$ is the relative
number of votes of list $i$ in district $j$ and $\sum_{i=1}^{N_{j}}p_{i,j}=1$
and $N_{j}$ is the number of lists in the $j$-th district. It is a non-
negative measure of uncertainty. Shannon’s entropy is an extensive quantity,
as the entropies are additive. The maximum entropy value occurs when the
distribution is uniform or when all events are equiprobable, hence when
$p_{i,j}=\frac{1}{N_{j}}$ while its minimum occurs when there is certainty
about one particular event, corresponding to a single list gaining $100\%$ of
the votes in its district. In this sense,

$H_{\max_{j}}=-N_{j}\log\left(\frac{1}{N_{j}}\right)=N_{j}\log N_{j}$ (2)

We define the relative entropy $H_{R_{j}}$ of the vote distribution in a
district $j$ as the normalized Shannon’s entropy with respect to its maximal
value $H_{\max_{j}}$, hence it is given by

$H_{R_{j}}=\frac{H_{j}}{H_{\max_{j}}}=-\frac{\sum_{i=1}^{N_{j}}p_{i,j}\log(p_{i,j})}{N_{j}\log
N_{j}}$ (3)

Figure 1: A summary of the entropy data of the Lebanese Parliamentary
elections in 2018 and 2022. The upper figures show Shannon’s Entropy and
Relative Shannon’s entropy in each electoral district, while the lower figures
show the values of Tsallis and relative Tsallis’ entropies of entropic index
$2$ in the same districts.

An alternative non-additive description of entropy is that defined by the
Tsallis entropy, and it is widely used in the study of complex system. The
Tsallis entropy of a system on $N_{j}$ events is given by:

$S_{q_{j}}=-\frac{1-\sum_{i=1}^{N_{j}}p_{i,j}^{q}}{1-q}$ (4)

where the $q$ is the entropic index, and can be any real parameter. In
contrast with Shannon’s entropy, Tsallis entropy is a non-extensive quantity
as entropies of different systems are not additive according to equation (4).
In the limit $q\rightarrow 1$, Tsallis entropy reduces to the Shannon’s
entropy defined in (1). The maximum value of $S_{q_{j}}$ occurs when all
events are equiprobable so that

$S_{q\max_{j}}=-\frac{1-N_{j}^{1-q}}{1-q}$ (5)

and the relative Tsallis entropy is:

$S_{qR_{j}}=\frac{1-\sum_{i=1}^{N_{j}}p_{i,j}^{q}}{1-N_{j}^{1-q}}$ (6)

Choosing the entropic index in Tsallis entropy involves considering various
factors, such as the nature of the system under study, the characteristics of
the data, and the desired properties of the entropy measure. For $q>1$,
Tsallis entropy is more sensitive to rare events: Increasing the value of $q$
makes Tsallis entropy more sensitive to rare events in the system, however,
there is no universal best value for $q$. In many cases, the value of $q=2$
has been found to be useful in describing the statistical properties of
systems. In complex systems, the value of $q=2$ was found to be particularly
useful in capturing the hierarchical structure of networks and biological
systems. More importantly, in nonlinear dynamical systems, it was found to be
useful in characterizing the long-range correlations and spatiotemporal
complexity [47, 48, 49, 50]. In this paper, we study Tsallis entropy with
entropic index $q=2$, hence we specifically use

$S_{j}=1-\sum_{i=1}^{N_{j}}p_{i,j}^{2}$ (7)

and its normalized relative value

$S_{R_{j}}=\frac{1-\sum_{i=1}^{N}p_{i,j}^{2}}{1-\frac{1}{N_{j}}}$ (8)

###  2.2 Spatial auto-correlation

In this paper, we introduce the idea of measuring the spatial auto-correlation
of election entropies across districts in relation to their geographic
adjacency, and the driving distance between their centers.

Moran’s $I$ index is an inferential statistic used to measure the spatial
auto-correlation based both on locations and feature values simultaneously. It
is given by:

$I=\frac{N\Sigma_{ij}W_{ij}(X_{i}-\bar{X})(X_{j}-\bar{X})}{\Sigma_{ij}W_{ij}\Sigma_{i}(X_{i}-\bar{X})^{2}}$
(9)

where $W_{ij}$ resembles a row-standardized location value matrix.

We consider $3$ different parameterizations corresponding to geographic
adjacency, gravitational voter’s weight and sectarian adjacency between
districts. They are defined as follows:

  * •

First, $W_{ij}$ is taken to be the geographic adjacency matrix between
districts. Based on the district division in an election, districts that share
a common border are defined to be adjacent such that:

$W_{ij}=\begin{cases}1&\text{if districts $i\text{ and $j$ are adjacent}$}\\\
0&\text{Otherwise}\end{cases}$ (10)

  * •

Second, $W_{ij}$ is defined in terms of the gravitational voter weight which
is directly proportional to the numbers of voters $v_{i}$ and $v_{j}$ and
inversely proportional to the square of the driving distances $d_{ij}$ between
the centers of districts $i$ and $j$. This leads to:

$W_{ij}=\frac{v_{i}v_{j}}{d_{ij}^{2}}$ (11)

  * •

Third, we set $W_{ij}$ to be an adjacency matrix that relates districts
according to the sectarian majority in each. In this regard, two districts are
considered adjacent if they have the same majority sect and non-adjacent
otherwise.

$W_{ij}=\begin{cases}1&\text{if districts $i\text{ and $j$ have same majoirty
sect}$}\\\ 0&\text{Otherwise}\end{cases}$

In all these cases, the diagonal terms are set to be zero, $W_{ii}=0$, and the
rows are standardized such that $\sum_{j=1}^{N_{j}}W_{ij}=1\ \forall i$.

In equation (9), $N$ is the number of regions under consideration and $X_{i}$
and $X_{j}$ represent the entropies in regions $i$ and $j$ respectively.
$\bar{X}$is the average entropy, and it is given by
$\bar{X}=\frac{\Sigma_{i}X_{i}}{N}$. We consider four different
parameterizations of $X$ corresponding to Shannon’s entropy $H_{j}$, relative
Shannon’s entropy $H_{R}$, Tsallis entropy $S$ (of entropic index $2$) and its
relative value $S_{R}$ defined in equations (1), (3), (7) and (8)
respectively.

The numerical outcome of $I$ falls between $-1$ and $+1$ and it indicates
whether a distribution is dispersed, random or clustered. A value of $I$ close
to $0$ indicates a random distribution, while positive values indicate
clustered spatial distribution and negative values indicate dispersion. Larger
values of $|I|$ nearer to 1 mean stronger clustering (positive I) or stronger
dispersion (negative I).

The $z_{I}$-score associated to these statistics is defined by:

$z_{I}=\frac{I-E[I]}{\sqrt{V[I]}}$ (12)

where $E[I]$ is the expected value and $V[I]$ is the variance, which are
defined explicitly in the Appendix.

The $z$-score or its corresponding $p$-value of the statistic are used to
reject the null hypothesis and eliminate the possibility of a random pattern
leading to the obtained value of the Moran $I$ statistic. In this paper, we
take a $95\%$ confidence level corresponding to $z_{I}>1.96$ or equivalently
to $p<0.05$ in order to confirm the outcome of clustering or dispersion of our
electoral spatial data under consideration. When the $p$-value is
statistically significant, and based on the value of $I$ we can determine the
pattern of the distribution.

###  2.3 Voter Transitions

We use the method of "maximum entropy” in order to estimate the voter
transitions among lists between consecutive elections. The maximum entropy
estimator utilizes a set of imposed constraints and available information in
order to find the most uniform estimate the maximizes the entropy.

Parliament Elections 2018 |  Statistic |  Entropy |  Relative Entropy |  Tsallis Entropy |  Relative Tsallis Entropy
---|---|---|---|---|---
|  |  |  |
Adjacency auto-correlation |  Moran I |  $0.473$ |  $0.464$ |  $0.551$ |  $0.526$
p-value |  $9.46\times 10^{-5}$ |  $1.22\times 10^{-4}$ |  $8.56\times 10^{-6}$ |  $1.90\times 10^{-6}$
Gravitational auto-correlation |  Moran I |  $0.406$ |  $0.289$ |  $0.437$ |  $0.375$
p-value |  $6.15\times 10^{-6}$ |  $6.32\times 10^{-4}$ |  $1.41\times 10^{-6}$ |  $2.37\times 10^{-5}$
Sectarian auto-correlation |  Moran I |  $0.683$ |  $0.712$ |  $0.792$ |  $0.804$
p-value |  $1.36\times 10^{-9}$ |  $3.06\times 10^{-10}$ |  $3.79\times 10^{-12}$ |  $1.84\times 10^{-12}$
Table 1: Spatial adjacency, gravitational and sectarian auto-correlation of
entropy, relative entropy, Tsallis entropy and relative Tsallis entropy among
Lebanese electoral districts in the May 2018 Parliamentary elections with
their p-values, using the Moran I index.

We consider two consecutive elections held in district $j$, and we designate
the number of lists competing in the two elections in that district by $i$ and
$k$ respectively. We define $x_{ji}$to be the fraction of votes gained in the
first election by list $i$ in district $j$ and $y_{jk}$ to be its counterpart
in the following elections. We obviously have the condition that:

$\sum_{i}x_{ji}=\sum_{k}y_{jk}=1\ \forall j$ (13)

To study the voter transitions between lists, we need to estimate the Markov
transition matrix $\boldsymbol{p}_{j}$ whose elements $p_{j,ik}$ measure the
probability that an individual voted for list $k$ in the last elections given
that he/she has voted for list $i$ in the previous elections. This is a pure
Markov inverse problem. It is solved by maximizing the entropy of voter
migration defined by:

$H(\boldsymbol{p}_{j})=-\sum_{i}\sum_{k}p_{j,ik}\log(p_{j,ik})$ (14)

subject to several conditions that include (13), and the following
constraints:

$\begin{cases}y_{jk}=\sum_{i}x_{ji}p_{j,ik}&\forall j\forall k\\\
\sum_{k}p_{j,ik}=1&\forall j\forall i\\\ p_{j,ik}\geq 0&\forall j\forall
i\forall k\end{cases}$ (15)

In addition to to these constraints, any information available from exit polls
or trusted estimates can be imposed on $p_{j,ik}$ as additional constraints,
and would improve the accuracy of this estimate. The model is a maximization
problem with constraints, and can be solved by Lagrangian methods. However,
closed form analytic solution are not available, so the problem is solved
numerically.

##  3 Results and discussion

###  3.1 Entropic Auto-correlation

The $2018$ Lebanese parliamentary elections were held on May $6,2018$, and
were the first parliamentary elections to be held in the country in nine years
[51]. The elections were held under a new proportional representation
electoral law with one preferential vote, which aimed to make the electoral
process more inclusive and transparent. The country is divided into $15$
electoral districts, which contain $26$ sub-districts. The lists run on the
district level and the seats are allocated proportionally with respect to
their votes in these districts, but simultaneously reserving fixed quotas for
the sub-districts and for the sectarian constituents of the districts. In this
sense, all seats across all districts are allocated according to a predefined
sectarian distribution. The preferential votes are used to rank the candidates
of all lists that attain the minimum threshold for winning, then the winning
candidates are declared in decreasing rank, up until the sectarian quota of
each sect is fulfilled. Once a sectarian quota is reached, all candidates from
the same sect are eliminated and the process continues until all the dedicated
seats of the winning lists are reached. The electoral threshold is set to be
the percentage of votes needed to achieve one whole seat. In this context, it
varies from one district to another, varying between $20\%$ in the smallest
district that is represented by $5$ seats and $7.7\%$ in the largest district
represented by $13$ seats.

The main political forces that participated in the $2018$ Lebanese elections
were the sectarian parties that controlled the power during the last three
decades: The Free Patriotic Current (FPC), The Future Movement (FM), The
Lebanese Forces (LF), Hizbullah, Amal (Hope) Movement and the Progressive
Socialist Party (PSP), in addition to minor allies of them and several
opposition parties and groups which couldn’t break their hegemony. The
coalition of March $8$, including the FPC, Hizbullah and Amal held the
majority with some of their minor allies.

Parliament Elections 2022 |  Statistic |  Entropy |  Relative Entropy |  Tsallis Entropy |  Relative Tsallis Entropy
---|---|---|---|---|---
|  |  |  |
Adjacency auto-correlation |  Moran I |  $0.494$ |  $0.402$ |  $0.529$ |  $0.503$
p-value |  $5.21\times 10^{-5}$ |  $6.55\times 10^{-4}$ |  $1.76\times 10^{-5}$ |  $3.96\times 10^{-4}$
Gravitational auto-correlation |  Moran I |  $0.491$ |  $0.416$ |  $0.490$ |  $0.453$
p-value |  $8.10\times 10^{-8}$ |  $3.36\times 10^{-6}$ |  $8.45\times 10^{-8}$ |  $5.57\times 10^{-7}$
Sectarian auto-correlation |  Moran I |  $0.760$ |  $0.665$ |  $0.846$ |  $0.804$
p-value |  $2.32\times 10^{-11}$ |  $3.22\times 10^{-9}$ |  $1.53\times 10^{-13}$ |  $1.91\times 10^{-12}$
Table 2: Spatial adjacency, gravitational and sectarian auto-correlation of
entropy, relative entropy, Tsallis entropy and relative Tsallis entropy among
Lebanese electoral districts in the May 2022 Parliamentary elections with
their p-values, using the Moran I index.

The $2022$ elections were held on May $15,2022$ under the same proportional
confessional electoral law. The main parties mentioned above were challenged
this time by lists that were based on coalition between new and emerging
opposition force and existing secular forces. The elections resulted in a hung
parliament were March $8$ coalition was still the most numerous group but
without a majority [52, 53]. About $19$ seats were won by candidates of the
secular opposition lists.

In both elections, most parties ran in coalition lists with other parties and
independent candidates, and the coalitions varied from one district to
another, ranging from political to pragmatic coalitions to guarantee winning
seats under the existing proportional law.

Figure (1) summarizes the values of Shannon’s entropy, relative Shannon’s
entropy, Tsallis entropy and relative Tsallis entropy of entropic index $2$ on
the level of the electoral district in the elections of $2018$ and $2022$.
They are calculated using equations (1), (3), (4) and (6). We can notice that,
for most districts, there was an increase in entropy between the two
elections.

In order to better understand possible correlation between different
districts, we applied the Moran I index tool as defined in (9) to measure the
auto-correlation between local district voter entropies with geographical
district adjacency, voter gravitational adjacency and sectarian adjacency
using the results of the $2018$ and the $2022$ election results that publicly
available in [43, 54, 55]. We found that there is a positive auto-correlation
among all those values with confidence level greater than $99.9\%$ for all
measured Moran indices. This means that we can reject the null hypothesis with
a very high confidence level in all these cases, and quantify the positive
correlations between the entropies and the analyzed geographic, demographic
and political features.

Table (1) shows that in the $2018$ Lebanese parliamentary elections, there is
strong entropy-adjacency auto-correlation, where the Tsallis entropy is the
most auto-correlated with adjacency. This means that geographically
neighboring districts have correlated patterns of voter distribution
uncertainty. Gravitational auto-correlation is positive, but is weaker than
adjacency for all entropic measures. This indicates that the effect of the
ratio of voter population to the inverse of the squared distance between
districts is weaker than that of geographic adjacency. Finally, the strongest
auto-correlation occurs for sectarian adjacency, meaning that districts with
similar sectarian majorities will have the strongest correlation between their
voter distributions uncertainty, reflecting the dominant weight of the
sectarian identity in Lebanese politics.

In the $2022$ Lebanese parliamentary elections, there also was a strong
relationship between the distribution of voter uncertainty and geographic
proximity, with neighboring districts showing correlated patterns in this
regard, as shown in Table (2). The impact of the ratio of voter population to
the inverse of the squared distance between districts was similar to that of
geographic proximity, hence we could see a stronger pattern of geographic
auto-correlation in this elections that in the previous one. Sectarian
similarity again had the strongest influence on the correlation between voter
uncertainty in different districts, with districts with similar sectarian
majorities showing the strongest correlations in this regard. This reflects
the continued significant role that sectarian identity plays in Lebanese
politics.

Our findings about geographic adjacency and proximity are in line with the
outcomes of other studies that analyzed spatial correlation of turnout rates
and found out that the fraction of winning votes decays logarithmically with
the distance between towns [36], using a different theoretical framework.

Sectarianism refers to the division of society or politics along religious or
ethnic lines. It can have a significant impact on voting patterns because
people may be more likely to vote for candidates or parties that align with
their religious or ethnic identity. This can lead to a situation where certain
groups are more likely to support certain political parties or candidates,
which can create divisions within a society. Sectarianism can also lead to
political polarization and can make it more difficult for politicians to build
coalitions or reach compromise. It can also create an environment where
politicians feel pressure to take positions or adopt policies that appeal to
their particular sectarian group, rather than considering the interests of the
entire society.

Origin 2018 | Beirut II Voter Destination 2022
---|---
List Name |  Beirut Unity |  Beirut for Change |  This is Beirut |  Bei. needs a Heart |  Beirut Confronts |  Others |  Non Voters
|  |  |  |  |  |  |
Future for Beirut |  $0.015\%$ |  $21.2\%$ |  $16.0\%$ |  $5.13\%$ |  $16.6\%$ |  $17.6\%$ |  $23.5\%$
Beirut Unity |  $74.4\%$ |  $5.28\%$ |  $4.13\%$ |  $1.65\%$ |  $4.26\%$ |  $4.48\%$ |  $5.81\%$
Lebanon’s Worth it |  $0.434\%$ |  $2.76\%$ |  $2.54\%$ |  $86.2\%$ |  $2.57\%$ |  $2.61\%$ |  $2.85\%$
Beirut is Homeland |  $3.99\%$ |  $9.19\%$ |  $51.9\%$ |  $7.72\%$ |  $8.90\%$ |  $8.97\%$ |  $9.32\%$
All for Beirut |  $1.24\%$ |  $86.5\%$ |  $2.48\%$ |  $2.19\%$ |  $2.49\%$ |  $2.50\%$ |  $2.59\%$
Others |  $10.8\%$ |  $15.1\%$ |  $14.9\%$ |  $14.1\%$ |  $14.9\%$ |  $15.0\%$ |  $15.2\%$
Non-Voters |  $0.00\%$ |  $4.17\%$ |  $1.32\%$ |  $0.018\%$ |  $1.53\%$ |  $1.94\%$ |  $91.0\%$
Table 3: GME estimate of the voter transitions between lists in Beirut II
district between May 2018 and May 2022 parliamentary elections.

###  3.2 Voter Transitions

Changes in the voting patterns of citizens during elections can be affected by
multiple factors, and the significance of each factor may vary based on the
election and the electorate’s preferences. In the case of the Lebanese
parliamentary elections, various factors, such as significant economic,
social, security, and political developments that happened in Lebanon from
$2018$ to $2022$, might have contributed to the changes in voting behavior of
the people. Lebanon has been grappling with a severe economic crisis since
$2019$, which has caused an economic collapse. The country also witnessed a
large-scale uprising between $2019$ and $2020$, with significant protests
expressing widespread dissatisfaction [52]. Former Prime Minister Saad Hariri
resigned, left the country, and subsequently withdrew from politics, including
not running in the $2022$ elections [56]. The $2020$ explosion at the Beirut
port had devastating consequences for the city and its surrounding areas,
directly impacting the lives of hundreds of thousands of residents [57].
Security and economic factors are important variables that can affect election
results in various ways. In particular, they can influence voter behavior,
such as turnout, and campaign messaging. In this context, the aforementioned
developments could have influenced voter transitions, possibly into opposition
lists, in several districts. In addition, the results obtained in Figure (1)
reveal a small yet observable increase in entropy in $2022$, indicating a more
uniform spread of votes away from dominant lists. These qualitative and
quantitative factors combined could provide a logical validation for the
actual voter transitions between $2018$ and $2022$ elections.

We apply the GME estimation method to determine the voter transitions between
lists in the Lebanese parliamentary elections of $2018$ and $2022$. We show in
this paper the corresponding voter transitions in $3$ electoral districts with
different sectarian and political compositions. In particular, we choose to
show the voter transitionss in the districts of Beirut II, South III and Mount
Lebanon IV.

#### Beirut II

The Lebanese capital Beirut is divided into two electoral districts in the
current election law adopted since $2018$. The second district (Beirut II) has
$11$ seats, distributed according to a sectarian quota that assigns $6$ seats
for Muslim Sunnis, $2$ seats for Muslim Shia, $1$ seat for Druze, $1$ seat for
Christian Orthodox and $1$ seat for Christian evangelicals. The main political
parties in this district are the Future Movement, Hizbullah, Amal movement,
the Progressive Socialist Party, Free Patriotic Current, and several other
smaller sectarian sectarian parties. In addition, a coalition of opposition
secular forces established itself as an alternative force, especially in the
$2022$ elections. In $2018,$ the main electoral lists were: "Future for
Beirut”, a coalition between the Future movement, Progressive Socialist Party
and other minor groups; "Beirut Unity”, a coalition between Hizbullah, Amal
movement, Free patriotic current, the Islamic Project Association and other
smaller groups; and several other lists. In $2022$, the Future movement
boycotted the elections, and several lists, mainly "This is Beirut” and
"Beirut Confronts” were formed by its formal members and allies. The "Beirut
Unity” continued, with the Islamic Project Association forming its own list.
"Lebanon’s Worth it” list of the former elections re-branded itself as "Beirut
needs Heart”. The main challenge came from the new secular opposition list,
"Beirut for Change”.

The GME estimator shows that the $2018$ voters of "Future for Beirut” migrated
in $2022$ elections into non-voters ($23.5\%)$, "Beirut for Change”
($21.2\%$), Others ($17.6\%$), "Beirut Confronts” ($16.6\%$) and "This is
Beirut” ($16\%$). On the other hand, $74.4\%$ of the voters of "Beirut Unity”
were retained by the list, while the others votes migrated to the other list.
"Beirut needs a heart” retained $86.2\%$ of its former voters of "Lebanon’s
Worth it” list. "Beirut for Change” also inherited about $86.5\%$ of the votes
of "All for Beirut”, a secular list of the $2018$ elections. $91\%$ of non-
voters from the previous elections remained as non-voters, while around
$4.2\%$ of them voted for "Beirut for Change” in the $2022$ elections. The
full transition estimates are shown in Table 3.

#### Mount Lebanon IV

Mount Lebanon IV is the district with highest number of representatives in the
Lebanese parliament, with $13$ seats. It is represented according to the
sectarian quota: $5$ seats for Christian Maronites, $1$ seat for Christian
Catholics, $1$ seat for Christian Orthodox, $4$ seats for Druze and $2$ seats
for Muslim Sunni. The main political parties in the district are the
Progressive Socialist Party, the Lebanese Forces, the Free Patriotic Movement,
the Lebanese Democratic Party, Future Movement, the Arab Unity Party and other
smaller parties and groups, in addition to a broad spectrum of secular
opposition parties and groups. In the $2018$ elections, the main lists were
the "Reconciliation” list, formed by the coalition of the Progressive
Socialist Party with the Future Movement and the Lebanese Forces; "Mountain
Guarantee” list formed by the Lebanese Democratic Party, the Free Patriotic
Current and their allies; "National Unity” formed by the Arab Unity Party; and
"All for my Country” which was a coalition of secular and left forces. In
$2022$, the former components of the "Mountain Guarantee” and "National Unity”
list merged in one list called "The Mountain” list, while the constituents of
the "Reconciliation” list formed the "Will and Participation” list, retaining
all of its components except the Future Movement. All secular, left and civil
opposition forces collaborated in a broad alliance list that they called
"Unified for Change”.

The GME estimates shown in Table 4 reveal that $81.2\%$ of the voters of the
"Reconciliation” list of $2018$ voted for the "Will and Participation” in
$2022$, while about $17\%$ of its voters migrated their votes to "Unified for
Change”. About $80\%$ of the votes of "Mountain Guarantee” and "National
Unity” were retained by the merger list "The Mountain” while $18.2\%$ and
$11.4\%$ of their votes respectively migrated to the "Unified for Change”
list. About "$93.6\%$ of the voters of "All for my Country” were retained by
the "Unified for Change” list. The vast majority ($96.4\%$) of non-voters in
$2018$ did not vote in $2022$ as well, while about $2.5\%$ of these former
non-voters voted for "Unified for Change”.

Origin 2018 | Mount Lebanon IV Voter Destination 2022
---|---
List Name |  Will & Participation |  Unified for Change |  The Mountain |  Others |  Non Voters
|  |  |  |  |
Reconciliation |  $81.2\%$ |  $16.9\%$ |  $1.48\%$ |  $0.24\%$ |  $0.11\%$
Mountain Guarantee |  $1.77\%$ |  $18.2\%$ |  $79.6\%$ |  $0.19\%$ |  $0.23\%$
National Unity |  $5.31\%$ |  $11.4\%$ |  $79.8\%$ |  $2.17\%$ |  $1.33\%$
All for my Country |  $2.28\%$ |  $93.6\%$ |  $3.01\%$ |  $1.13\%$ |  $0.76\%$
Others |  $20.7\%$ |  $31.0\%$ |  $26.4\%$ |  $12.5\%$ |  $9.33\%$
Non-Voters |  $0.08\%$ |  $2.51\%$ |  $0.108\%$ |  $0.01\%$ |  $96.4\%$
Table 4: GME estimate of the voter transitions between lists in Mount Lebanon
IV district between May 2018 and May 2022 parliamentary elections.

#### South III

South III electoral district is represented by $11$ members of parliament that
are assigned according to the following sectarian quota: $8$ seats for Muslim
Shia, $1$seat for Muslim Sunni, $1$ seat for Druze and $1$ seat for Christian
Orthodox. The main political parties are Hizbullah, Amal Movement, the
Lebanese Communist Party, Future movement, Lebanese Forces and Free Patriotic
Current, in addition to other minor parties and groups. In the $2018$
elections, the main lists were the "Hope and Loyalty”, a coalition between
Hizbullah, Amal movement and other minor forces; "The South Deserves”, a
coalition between Future movement and the Free Patriotic Current; "One Vote
for Change”, formed by the Lebanese Communist Party; and other minor lists. In
the $2022$ elections, "Hope and Loyalty” continued as an alliance list of all
of its former constituents, while all secular and progressive forces and
groups formed the "Together for Change” list in alliance with the Lebanese
Communist Party. The Future movement boycotted the elections, while the Free
Patriotic Current and the Lebanese Forces did not take part in any list.

Based on the GME estimator method, we show in Table 5 the voter transitions
among these lists between the $2018$ and $2022$ elections. We find out that
the "Hope and Loyalty” list retained $96.6\%$ of its voters, while only about
$1.7\%$ of its former voters supported "Together for Change” in $2022$. On the
other hand, "The South Deserves” saw most of its voters supporting "Together
for Change” in $2022$, while about $19.7\%$ of its voters transitioned into
non-voting, in line with the call of the Future Movement for election boycott.
Most of the voters of "One Vote for Change” voted for "Together for Change” in
$2022$, with a transition estimated at $95.3\%$, while more than $3\%$ of its
former voters became non-voters in $2022$. We also estimate that the majority
of the voters of other lists in $2018$ either voted for "Together for Change”
or did not vote. More than $96\%$ of non-voters in $2018$ did not vote in
$2022$ while about $2.1\%$ of them voted for "Hope and Loyalty” and $1.4\%$
for "Together for Change”.

Origin 2018 | South III Voter Destination 2022
---|---
List Name |  Hope & Loyalty |  Together for Change |  Voice of South |  Non Voters
|  |  |  |
Hope & Loyalty |  $96.6\%$ |  $1.73\%$ |  $0.049\%$ |  $0.51\%$
South Deserves |  $8.31\%$ |  $71.7\%$ |  $0.050\%$ |  $19.7\%$
One Vote for Change |  $0.92\%$ |  $95.3\%$ |  $0.23\%$ |  $3.35\%$
Fed up with Rhetoric |  $5.11\%$ |  $62.6\%$ |  $0.25\%$ |  $32.2\%$
Others |  $6.96\%$ |  $84.9\%$ |  $0.24\%$ |  $8.1\%$
Non-Voters |  $2.11\%$ |  $1.41\%$ |  $0.049\%$ |  $96.2\%$
Table 5: GME estimate of the voter transitions between lists in South III
district between May 2018 and May 2022 parliamentary elections.

###  3.3 Discussion and Limitations

In our analysis of voter transitions, we considered the percentage of votes
attained by each list, without reflecting that on the total number of votes.
One of the main issues in such analysis is the number of voters that varies
from one election to another due to inflows of new eligible voters mainly due
to young people reaching the voting age and naturalized citizen, and outflows
of dead or emigrating voters. The change in the total number of voters puts
some challenges on the interpretation of transition probabilities, that mainly
represent the probabilities of changing an electoral preference from one
election to another. However, the inflows and outflows are not taken into
account in such an analysis. In this paper, we have assumed the total voting
population to be fixed, and we ignored the small relative changes in this
population over a course of four years.

The second point of interest is that the percentage of non-voters is
relatively very high in Lebanon due to the fact that high proportion of the
enlisted voters on the official records may have already left the country for
good. Recent estimates put the number of Lebanese emigrants at around one
third of the population, but they would remain enlisted on the records [58,
59], leading to participation rates varying historically around $40-50\%$ of
the registered voters on the national level [43, 54, 55]. This complicates the
question of accurately determining the actual number of registered voters who
might eventually want to vote in the elections. For practical purposes, we
have classified all the non-voting population as non-voters regardless of
their residence/immigration statuses, in addition to all blank and invalid
votes, taken together as a single block.

##  4 Conclusions

In this paper, we defined different measures of entropy, namely the Shannon,
relative Shannon, Tsallis and relative Tsallis entropies of entropic index
two. We introduced the Moran I index and used it to measure the spatial auto-
correlation of these entropies with respect to geographic adjacency,
gravitational adjacency and sectarian adjacency. The various adjacency
features were quantified through the adjacency matrix whose elements are
defined as: non-zero when two districts are geographically adjacent, the
product of the number of voters in each two districts over the square of the
distance between their centers, and non-zero when two districts have same
sectarian majority, respectively. This is the main novel model of our paper,
and it allowed us to study the corresponding correlations between entropies of
election outcomes on the district level. We found out that, with a very high
degree of confidence, there are relatively strong geographic and gravitational
adjacency correlations, but most importantly and most strongly, there is a
very apparent auto-correlation in the sectarian adjacency feature, in both
parliamentary elections in Lebanon in $2018$ and $2022$. The result presents
an important assertion on the high impact of the sectarian effect in Lebanese
politics and elections.

We also introduced the method of the maximized general entropy estimation that
allows us to determine the Markov transition matrix of list voters between
consecutive elections. We employed this method to determine the voting
transitions in three different key electoral districts between the $2018$ and
$2022$ Lebanese parliamentary elections. We also pointed out some limitations
and related assumptions.

Without loss of generality, we applied the model on data from Lebanese
elections, but they are universally valid. Future work can extend the entropic
spatial auto-correlation model to cover additional features of interest, and
applications on other elections.

## Acknowledgments

The author thanks Werner Antweiler for his useful comments about the
literature on the maximum entropy estimation of transition probabilities and
related software packages for its calculation.

## Appendix

The $z_{I}$-score is defined to be:

$z_{I}=\frac{I-E[I]}{\sqrt{V[I]}}$

where $E[I]$ is the expected value and $V[I]$ is the variance. The expected
value of Moran’s index is explicitly given by

$E[I]=\frac{-1}{N-1}$

while the variance is

$V[I]=E[I^{2}]-E[I]^{2}$

with

$E[I^{2}]=\frac{A-B}{C}$

$A$, $B$ and $C$ are given by:

$A=N\left[2\left(N^{2}-3N+3\right)\Sigma_{ij}W_{ij}^{2}-2N\Sigma_{i}\left(\Sigma_{j}W_{ij}\right)^{2}+3\left(\Sigma_{ij}W_{ij}\right)^{2}\right]$

$B=\frac{2\Sigma_{i}(X_{i}-\bar{X})^{4}}{\left(\Sigma_{i}(X_{i}-\bar{X})^{2}\right)^{2}}\left[\left(N^{2}-N\right)\Sigma_{ij}W_{ij}^{2}-2N\Sigma_{i}\left(\Sigma_{j}W_{ij}\right)^{2}+3\left(\Sigma_{ij}W_{ij}\right)^{2}\right]$

$C=\left(N-1\right)\left(N-2\right)\left(N-3\right)\left(\Sigma_{ij}W_{ij}\right)^{2}$

## References

  * [1] Ullah A., "Entropy, divergence and distance measures with econometric applications.” (1993), Journal of Statistical Planning and Inference 49(1), 137-162, https://doi.org/10.1016/0378-3758(95)00034-8.
  * [2] Karmeshu, J., "Entropy measures, maximum entropy principle and emerging applications." (2003), Studies in Fuzziness and Soft Computing, Springer-Verlag Berlin.
  * [3] Antweiler, W., "Estimating voter migration in Canada using generalized maximum entropy” (2007), Electoral Studies 26(4), 756-771, https://doi.org/10.1016/j.electstud.2007.07.005
  * [4] Zhou, R., Cai R., & Tong G., "Applications of Entropy in Finance: A Review” (2013), Entropy 15(11), 4909-4931, https://doi.org/10.3390/e15114909.
  * [5] Proops, J., "Entropy, Information and Confusion in the Social Sciences” (1987), Journal of Interdisciplinary Economics 1(4), https://doi.org/10.1177/02601079X8700100403.
  * [6] Gill, J., "An entropy measure of uncertainty in vote choice” (2005), Electoral Studies 24(3), 371-392, https://doi.org/10.1016/j.electstud.2004.10.009.
  * [7] Marmani S., Ficcadenti V., Kaur P., & Dhesi G., "Entropic Analysis of Votes Expressed in Italian Elections between 1948 and 2018” (2020), Entropy 22(5), 523, https://doi.org/10.3390/e22050523.
  * [8] Ma C.W. & Ma Y.G., "Shannon information entropy in heavy-ion collisions” (2018), Progress in Particle and Nuclear Physics 99, 120-158, https://doi.org/10.1016/j.ppnp.2018.01.002.
  * [9] Shannon, C., "A mathematical theory of communication” (1948), Bell System Technical Journal 27(3), 379-423, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
  * [10] Tsallis, C., Mendes, R., & Plastino, A.R., "The role of constraints within generalized nonextensive statistics.” (1998), Physica A: Statistical Mechanics and its Applications 261(3-4), 534-554, https://doi.org/10.1016/S0378-4371(98)00437-3.
  * [11] Martinez, S., Nicolas, F., Pennini, F., & Plastino A., Tsallis’ entropy maximization procedure revisited.” (2000), Physica A: Statistical Mechanics and its Applications 286(3-4), 489-502, https://doi.org/10.1016/S0378-4371(00)00359-9.
  * [12] Dos Santos, R., "Generalization of Shannon’s theorem for Tsallis entropy” (1997), Journal of Mathematical Physics 38(8), https://doi.org/10.1063/1.532107.
  * [13] Moran, P. A., "Notes on Continuous Stochastic Phenomena" (1950), Biometrika 37 (1): 17–23. https://doi.org/10.2307/2332142
  * [14] Cliff, A., & Ord, J. Spatial auto-correlation, London: Pion (1973).
  * [15] Cliff, A., & Ord, J. Spatial processes: models and applications, London: Pion (1981).
  * [16] Brown, A., & Chung S.Y., "Spatial segregation, segregation indices and the geographical perspective” (2006), Population, Space and Place 12(2), 125-143, https://doi.org/10.1002/psp.403.
  * [17] Fotheringham, S., Brunsdon, C., & Charlton, M., Quantitative Geography: Perspectives on Spatial Data Analysis, Sage Publications, London (2000).
  * [18] Antweiler, W., "Estimating voter migration in Canada using generalized maximum entropy” (2007), Electoral Studies 26 (4), 756-771, https://doi.org/10.1016/j.electstud.2007.07.005.
  * [19] Berg, S., "Spatial influence on voter transitions in Swedish elections: An application of Johnston’s maximum entropy method” (1988), Electoral Studies 7(3), 233-250, https://doi.org/10.1016/0261-3794(88)90003-0.
  * [20] Ghosh, A., Shreya, P., & Basu, B., "Maximum entropy framework for a universal rank order distribution with socio-economic applications” (2021), Physica A: Statistical Mechanics and its Applications 563,125433, https://doi.org/10.1016/j.physa.2020.125433.
  * [21] Cardoso, M., Silva, L.M.C., Neli, R.R. & Souza, W.E., "Electorate involvement disorder: Universal relationship between the amplitude and electorate size in second round of Brazilian Presidential Election” (2022), Physica A: Statistical Mechanics and its Applications 591, 126778, https://doi.org/10.1016/j.physa.2021.126778.
  * [22] Andreadis, I., & Chadjipadelis, T., "A Method for the Estimation of Voter Transition Rates” (2009), Journal of Elections, Public Opinion and Parties 19(2), 203-218, https://doi.org/10.1080/17457280902799089.
  * [23] Klima, A., Thurner, P., Molnar C., et al, "Estimation of voter transitions based on ecological inference: an empirical assessment of different approaches” (2015), AStA Adv Stat Anal 100, 133–159, https://doi.org/10.1007/s10182-015-0254-8.
  * [24] Fule, E., "Estimating voter transitions by ecological regression” (1994), Electoral Studies 13(4), 313-330, https://doi.org/10.1016/0261-3794(94)90043-4.
  * [25] Khalil, N., & Toral R., "The noisy voter model under the influence of contrarians” (2019), Physica A: Statistical Mechanics and its Applications 515, 81-92, https://doi.org/10.1016/j.physa.2018.09.178.
  * [26] Soltani, A., Askari, S., "Exploring spatial auto-correlation of traffic crashes based on severity” (2017), Injury 48 (3), 637-647, https://doi.org/10.1016/j.injury.2017.01.032.
  * [27] Qi, Y., & Wu, j., "Effects of changing spatial resolution on the results of landscape pattern analysis using spatial auto-correlation indices” (1996), Landscape Ecol. 11, 39–49, https://doi.org/10.1007/BF02087112.
  * [28] Li, R., Chen, N., Zhang X., et al., "Quantitative analysis of agricultural drought propagation process in the Yangtze River Basin by using cross wavelet analysis and spatial auto-correlation” (2020), Agricultural and Forest Meteorology 280, 107809 https://doi.org/10.1016/j.agrformet.2019.107809.
  * [29] Das, M., & Ghosh, S., "A cost-efficient approach for measuring Moran’s index of spatial auto-correlation in geostationary satellite data” (2016), IEEE International Geoscience and Remote Sensing Symposium (IGARSS), 5913-5916, https://doi.org/10.1109/IGARSS.2016.7730545.
  * [30] Scardaccione, G., Scorza, F., Las Casas, G., & Murgante, B., "Spatial Autocorrelation Analysis for the Evaluation of Migration Flows: The Italian Case” (2010), Computational Science and Its Applications – ICCSA 2010, Lecture Notes in Computer Science, 6016\. Springer, Berlin, https://doi.org/10.1007/978-3-642-12156-2_5.
  * [31] Lanorte, A., Danese, M., Lasaponara, R., & Murgante B., "Multiscale mapping of burn area and severity using multisensor satellite data and spatial auto-correlation analysis” (2013), International Journal of Applied Earth Observation and Geoinformation, 20, 42-51, https://doi.org/10.1016/j.jag.2011.09.005.
  * [32] El Deeb, O., "Spatial Autocorrelation and the Dynamics of the Mean Center of COVID-19 Infections in Lebanon” (2021), Front. Appl. Math. Stat,. 6:620064, https://doi.org/10.3389/fams.2020.620064.
  * [33] Perez-Pena, J.V., Azanon, J.M. Booth-Rea, G., et al., "Differentiating geology and tectonics using a spatial auto-correlation technique for the hypsometric integral” (2009), J. Geophys. Res., 114, F02018, https://doi.org/10.1029/2008JF001092.
  * [34] Fan, C., & Myint, S., "A comparison of spatial auto-correlation indices and landscape metrics in measuring urban landscape fragmentation” (2014), Landscape and Urban Planning, 121, 117-128, https://doi.org/10.1016/j.landurbplan.2013.10.002.
  * [35] Ghalhari, G., & Roudbari, A., "An investigation on thermal patterns in Iran based on spatial auto-correlation” (2018), Theoretical and Applied Climatology, 131, 865–876 https://doi.org/10.1007/s00704-016-2015-3.
  * [36] Borghesi, C., & Bouchaud, J.P., "Spatial correlations in vote statistics: a diffusive field model for decision-making" (2010), The European Physical Journal B , 75, 395-404, https://doi.org/10.1140/epjb/e2010-00151-1.
  * [37] Michaud, J., & Szilva, A., "Social influence with recurrent mobility and multiple options" (2018), Physical Review E , 97, 062313, https://doi.org/10.1103/PhysRevE.97.062313.
  * [38] Fernandez-Gracia, J., Suchecki, K., Ramasco, J., et al., "Is the Voter Model a Model for Voters?" (2014), Physical Review Letters , 112, 158701, https://doi.org/10.1103/PhysRevLett.112.158701.
  * [39] Stadler, B.M.R. , "Abstention in dynamical models of spatial voting" (2000), Physica A: Statistical Mechanics and its Applications, 287(3-4), 660-668, https://doi.org/10.1016/S0378-4371(00)00400-3.
  * [40] Salloukh, B., "The Limits of Electoral Engineering in Divided Societies: Elections in Postwar Lebanon” (2006), Canadian Journal of Political Science, 39(3), 635-655, https://doi.org/10.1017/S0008423906060185.
  * [41] Haddad, S., "The Lebanese Parliamentary Elections of 2005” (2005), The Journal of social, political, and economic studies, 30(3):305-332.
  * [42] Diss, M., & Steffen, F., "The Distribution of Power in the Lebanese Parliament Revisited” (2017), GATE WP 1723, http://dx.doi.org/10.2139/ssrn.3036848.
  * [43] El Deeb, O., & Juraidini, F., "Lebanese Parliamentary Elections 2018: A numerical and analytical study” (2018), Al-Farabi, Beirut, (In Arabic). Available at https://www.researchgate.net/publication/335517575.
  * [44] SAS Institute (2004) SAS/ETS 9.1 User’s Guide, Chapter 15: The Entropy Procedure, Cary, NC: SAS Publishing.
  * [45] Corral, P., Kuehn, D., & Jabir, E., "Generalized maximum entropy estimation of linear models” (2017), The Stata Journal 17(1), 240–249, https://doi.org/10.1177/1536867X1701700113.
  * [46] Van der Straeten, E., "Maximum entropy estimation of transition probabilities of reversible Markov chains " (2009), Entropy, 11(4), 867-887, https://doi.org/10.3390/e11040867.
  * [47] Malacarne, L., Mendes, R., & Lenzi, E., "Average entropy of a subsystem from its average Tsallis entropy" (2002), Physical Review E, 65(4), 046131, https://doi.org/10.1103/physreve.65.046131.
  * [48] Xiaozhuan, G., Fan L., Lipeng, P., et al., "Uncertainty measure based on Tsallis entropy in evidence theory" (2019), International Journal of Intelligent Systems, 34(11), 3105-3120, https://doi.org/10.1002/int.22185.
  * [49] O’Malley, D., Vesselinov, V., & Cushman, J., "Diffusive mixing and Tsallis entropy" (2015), Physical Review E, 9(4), 042143, https://doi.org/10.1103/physreve.91.042143.
  * [50] Johal, R., "Tsallis entropy and q-analogue of information" (1999), Physics Letters A, 253(1-2), 47-49, https://doi.org/10.1016/s0375-9601(99)00031-6.
  * [51] Political Party Mapping in Lebanon Ahead of the 2018 Elections, Konrad Adenauer Stiftung, available at: laur.lau.edu.lb:8443/xmlui/bitstream/handle/10725/7513/Political%20Party%20Mapping%20in%20Lebanon%20Ahead%20of%20the%202018%20Elections.pdf?sequence=1.
  * [52] D., Khatib, "17 October (2019) Revolution in Lebanon: A Preliminary Analysis” (2022), New Wave of Revolutions in the MENA Region. Perspectives on Development in the Middle East and North Africa (MENA) Region. Springer, https://doi.org/10.1007/978-3-031-15135-4_4.
  * [53] Abed, D., Sawaya, R., & Tabbal, N., "Analyzing Voter Turnout in Lebanon: Political Change in Times of Crisis” (2022), Oxfam, https://doi.org/10.21201/2022.8823.
  * [54] El Deeb, O., & Juraidini, F., "Lebanese Parliamentary Elections 2022: Detailed data and Results” (2022), Al-Farabi, Beirut, (In Arabic). Available at https://https://www.researchgate.net/publication/366275211.
  * [55] Lebanon Elections 22, Lebanese Minsitry of Interior Official Elections website, available at: https://elections.gov.lb/.
  * [56] Perry, T., & Bassam, L., "Lebanon’s Hariri steps away from politics, upending election landscape” (2022), Reuters, Available at https://www.reuters.com/world/middle-east/lebanons-hariri-expected-announce-election-boycott-party-members-say-2022-01-24/.
  * [57] Yu, G., Wang, Y., Zheng, L., et al., "Comprehensive study on the catastrophic explosion of ammonium nitrate stored in the warehouse of Beirut port” (2021), Process Safety and Environmental Protection 152, 201–219, https://doi.org/10.1016/j.psep.2021.05.030.
  * [58] Abdelhady, D., "Representing the Homeland: Lebanese Diasporic Notions of Home and Return in a Global Context” (2008), Cultural Dynamics, 20(1), 53–72, https://doi.org/10.1177/0921374007088055.
  * [59] Verdeil, E., & Dewailly, B., International Migration and the Lebanese Diaspora, Atlas of Lebanon (2019), Presses de l’Ifpo, CNRS Liban, https://doi.org/10.4000/books.ifpo.13178.


# Asymmetrical post quench transport in an embedded parity time symmetric Su-
Schrieffer-Heeger system

A. Ghosh  School of Physics, University of Melbourne, Melbourne, 3010,
Australia  Indian Institute of Technology Kharagpur, Kharagpur 721302, West
Bengal, India  A. M. Martin  School of Physics, University of Melbourne,
Melbourne, 3010, Australia

###### Abstract

We study the effect of PT-symmetric non-Hermiticity on the transport of edge
state probability density arising as a result of a quench. A hybrid system
involving a PT-symmetric SSH region sandwiched between two plain SSH systems
is designed to study the dynamics. Geometrical arguments and numerical
calculations were made to ascertain the nature of edge states. We then compute
the quench dynamics numerically and demonstrate that the post-quench
probability density light cones exhibit contrasting shapes as a result of
asymmetrical reflections from the non-Hermitian part of the system depending
on the direction of propagation of the transporting wave and, hence, on the
initial localization of the edge state.

##  I Introduction

The pioneering theoretical works of C.M. Bender and colleagues [1, 2, 3]
demonstrated that non-Hermitian operators can characterize real eigenvalues if
they obey PT symmetry. Interesting theoretical works on the possibility of the
physical implementation of the PT-symmetric Hamiltonian in optical settings
followed [4, 5, 6, 7, 8]. The last decade has seen many exotic phenomena
involving PT-symmetric Hamiltonian and exceptional points [5, 8, 9, 10, 11].
It has been shown that many of the exotic effects of PT-symmetric systems are
not only limited to quantum systems but also to classical ones [12, 13]. The
exotic effects of PT-symmetric systems have been demonstrated on various
platforms, including photonic systems [14, 11, 15], atomic ensembles [16],
topoelectrical systems [17], mechanical systems[12], and acoustic systems
[18]. Some examples of these exciting effects of exceptional points in non-
Hermitian PT-symmetric systems constitute unidirectional invisibility [8, 14,
18], Bloch oscillations [19], and loss enhanced transmission through
waveguides [10].

The concept of non-Hermiticity has also been implemented in the field of
topological physics, and some interesting phenomenon has been observed in
topological models perturbed with non-Hermitian elements [20, 21, 22].
Interesting studies have been undertaken on the topology of quasiparticles
with finite time that arise from the non-Hermiticity of the self-energy
arising from electron-phonon coupling [23]. It has been established that the
Su-Schrieffer-Heeger(SSH) model [24, 25], which, under non-Hermitian
conditions and obeying PT symmetry, is a periodic structure with alternating
imaginary on-site potentials [20]. Very recently, it has been shown that an
SSH system, after a quench from one topological regime to another,
characterizes a post-quench transport that forms a light cone [26].

Non-Hermiticity’s impact on post-quench information spread or light cones
needs to be explored, as little has been studied on this topic. In this work,
we shed light on the effect of non-Hermiticity with PT symmetry on the quench
dynamics. We consider a hybrid system to investigate the impact of non-
Hermiticity on post-quench transport. The system consists of a non-Hermitian
PT-symmetric SSH system sandwiched between two Hermitian SSH systems. The
hybrid system has been designed by considering that the edge states in the SSH
system are localized at either end of the chain, and the post-quench transport
takes place from the place of initial localization to the other end [26]. It
is found that the quench dynamics is determined by the direction of
propagation of the transport wave, and hence on the initial state
corresponding to the quench since the initial state determines the direction
of propagation. Depending upon the initial edge state, there can be two
directions for the post-quench transport and we have found that for two
directions of post-quench transport; in one case, there is negligible
scattering from the middle PT-symmetric part, and in another, there is
significant scattering, giving rise to a very different dynamics.
This paper is organized as follows. The first part of Section 2 introduces the
schematics of the model. The Hamiltonian is introduced, and by geometric
considerations, it is apprehended that there might be zero energy edge states
at the edges of the hybrid system. In part B of this section, we numerically
demonstrate the topological nature of the edge states by considering the
stability of the spatial structure of the edge states by varying the hopping
parameter. The energy band structure is also shown to indicate the topological
nature corresponding to PT symmetry. We then move forward to the next section,
where we discuss the results. In part A of this section, we perform a
numerical analysis of the quench dynamics. Here, we study and compare the
light cone diagrams for the two quenches with different initial edge states.
In part B, we model the system using real and imaginary potential blocks and
then proceed to calculate the scattering matrix considering a simple plane
wave with fixed energy. Based on the analysis of the results, it is
demonstrated that for different transport directions, there is a contrasting
asymmetry in their properties. Conclusions and summaries were discussed in the
final section.

##  II Introduction to the system: Embeded non- Hermitian PT symmetric SSH

###  II.1 Schematics of the Model

Figure 1: Schematics of (a) a plain SSH system, with staggered hopping
amplitudes $v$ and $w$. (b) shows a non-Hermitian PT-symmetric SSH system with
alternating complex on-site potentials, $u$ and $u^{*}$. The hybrid system
comprising of a non-Hermitian PT symmetric SSH system sandwiched between two
plain SSH systems is shown in (c).

A SSH system can be made non-Hermitian by a number of ways [20]. For example,
the introduction of complex hopping amplitudes, keeping the onsite potentials
at zero, introduces non-Hermiticity by preserving the chiral symmetry. On the
other hand, a staggered complex onsite potential with real hopping amplitudes
introduces non-Hermiticity whilst maintaining PT symmetry and breaking the
chiral symmetry (refer to Figure 1(b)). In the non-hermitian PT-symmetric SSH
model, two regions of the unbroken PT-symmetric phase are separated by an
extended region of the broken PT-symmetric phase, making the topological
transition very different from that of chiral symmetric ones [20]. We consider
a hybrid system that comprises a PT-symmetric non-Hermitian SSH system
sandwiched between two simple SSH systems as shown in Figure 1(c). The
staggered potential on the middle PT symmetric part is complex, i.e., has the
form $u(u^{*})=u_{Re}-(+)iu_{Im}$. The Hamiltonian of our system is of the
form:

$\hat{H}=H_{SSH}+H_{PT},$ (1)

i.e.,

$\displaystyle\hat{H}$
$\displaystyle=v\Sigma_{n=1}^{N}\ket{A,n}\bra{B,n}+w\Sigma_{n=1}^{N-1}\ket{A,n}\bra{B,n+1}$
(2)
$\displaystyle+\Sigma_{n=n1}^{n=n2}u\ket{A,n}\bra{A,n}+u^{*}\ket{B,n}\bra{B,n}.$

In the Equation 2, $v$ represents the intracellular hopping, and $w$
represents the intercellular hopping; the non-Hermitian PT-symmetric region
extends from the unit cell $n1$ to $n2$. Even without numerical consideration,
a geometrical inspection of the system under the extreme dimerized case of
$v=0$ suggests the presence of zero energy edge states, as shown in Figure 2.

Figure 2: Schematic of the model under an extreme case. This dimerized case of
intracellular hopping ($v=0$) reveals that there are two zero energy edge
states present in the system.

In this case, the edge states are zero energy because we have assumed that the
on-site potential at the sandwiching layers is zero.

###  II.2 Electronic properties of the system

To explore information about the nature of edge states, it is illuminating to
write the Hamiltonian in matrix form.

$\hat{H}=\begin{bmatrix}0&v&0&0&0&0&0&\dots&0\\\ v&0&w&0&0&0&0&\dots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots&\dots&0\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\vdots&\vdots&\vdots\\\
0&\dots&\dots&w&u&v&\dots&\dots&0\\\ 0&\dots&\dots&0&v&u^{*}&w&\dots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&\dots&\dots&\dots&\dots&\dots&w&0&v\\\
0&\dots&\dots&\dots&\dots&\dots&\dots&v&0\end{bmatrix}.$ (3)

The Hamiltonian in Equation 3 is a sparse tri-diagonal matrix. The diagonal
element is zero except for the elements from $(m_{1},m_{1})$ to
$(m_{2},m_{2})$, which has stagered potentials $u$ and $u^{*}$ representing
the non-Hermitian PT symmetric region.

There should be states at the boundaries if the Hamiltonian is associated with
a topological property. A look at the spatial spread of the edge-state wave
functions confirms their localized nature. The topological nature leads to the
fact that the edge states are robust to changes in the Hamiltonian.

Figure 3: This graph illustrates the spatial spread (amplitude vs. site index)
of the two edge-state wavefunctions corresponding to the different
configurations of the hybrid system of 220 sites with $u(u^{*})=-0.3-(+)0.1i$
from lattice site $m_{1}=101$ to $m_{2}=120$, and $u=0$ elsewhere. (a), and
(b) represents the real components of the edge states for the case ($v=0.1$;
$w=0:4$), the imaginary components being zero. (c) and (d) represents the real
part of the wavefunction corresponding to ($v=0:35$; $w=0:4$), whereas (e) and
(f) represent the complex part. The tendency of the edge state to delocalise
as $v\rightarrow 0.4$, with $w=0.4$ can be seen in (c–f).

It is evident from Figure 3, that the localized nature of the edge-state
wavefunctions begin to lose their localized character as we approach the point
$v=0.4$, $w=0.4$ from the point $v=0.1$, $0.4$. This reveals the topological
nature of the states. Moreover, the overlap of the edge-state wavefunctions of
the PT symmetric hybrid system and the plain SSH system, i.e.
$\bra{\psi_{\text{edge}}^{\text{Hybrid}}}\ket{\psi_{\text{edge}}^{\text{SSH}}}$,
remains close to $1$ over a range of values of $v$. Hence, the edge state
structure of the hybrid system does not change significantly from that of the
plain SSH system. In order to learn more about the nature of the edge states,
we need to examine the band structure of the system.

Figure 4:  shows the Energy vs. the intra-cellular hopping parameter $v$. (a)
and (b) shows the real and imaginary component of the energy, respectively.
The inter-cellular hopping is fixed at $w=0.4$. Among lattice sites in the
$m_{1}=101$ to $m_{2}=120$ range, $u(u^{*})=-0.3-(+0.1i)$, and u=0 for cells
outside this range. There is the presence of clearly defined zero energy edge
states on the real energy axis till the mark $v=0.4$, on crossing which, the
states deviate from their zero energy nature.

In spite of having somewhat unusual characteristics due to the hybrid PT
symmetric nature, the energy diagrams do reveal the presence of the robust
zero energy edge states, characterizing the topological nature. The real
component of the energy shows a clear signature of topological transition at
$v=0.4$, i.e, for the real part of the energy, there is a band closure at the
point when the zero energy states losses its character. The imaginary part of
the energy at this point raises from its near zero value before again going to
near zero value. This resembles the energy characteristics of a non-Hermitian
PT symmetric SSH system in the context of exceptional point [20]. We now
proceed forward to study the quench dynamics in this system.

##  III Results and Discussions

###  III.1 Initial state localization dependent post quench transport

We have seen in Section 2 that the zero energy states are localized in the
ends of the chain i.e, the right end and/or the left end. Now, a quench in a
SSH system with an edge state as the initial state $\psi_{i}$ from a regime
(characterized by parameter say $\chi_{1}$) that holds edge states to a regime
(characterized by parameter say $\chi_{2}$) that is without edge states leads
to a post quench transport of the edge state [26].

$\ket{\psi_{i}(t)}=\sum_{f=1}^{N}e^{-iE_{f}(\chi_{2})t}\ket{\psi_{f}(\chi_{2})}\bra{{\psi_{f}(\chi_{2})}}\ket{\psi_{i}(\chi_{1})}.$
(4)

This transport which essentially is a result of quantum evolution (Equation 4)
of states, is from the side which holds the initial edge state to the other
side. The system is modeled in such a way that the post-quench probability
density wave will face the PT-symmetric region in the middle in a direction
that is opposite for the two cases of initial edge states. The middle PT-
symmetric layer suggests some asymmetry in the transport characteristics for
the two kinds of transport.

Figure 5: Demonstration of the asymmetry in quench dynamics based on the
initial state localization. The hybrid system with the PT-symmetric non-
Hermitian region having the alternating potentials $u(u*)=-0.3-(+)0.1i$ from
$n1=101$ to $n2=120$ is considered. The quench is performed by the sudden
change of the intra-cellular hopping $v$, from $0.1$ to $0.5$, keeping the
inter-cellular hopping $w$ fixed at $0.4$. (a) and (b) are the light cone
diagram corresponding to the post quench transport, taking the initial edge
state to be localized at the left and right end respectively. The marked
branching in (b), around the non-Hermitian region from $m_{1}=101$ to
$m_{2}=120$, demonstrates a higher reflection as compared to (a). (c) and (d)
are the plots of probability density with respect to the time at the left and
the right end, respectively. The magnitude of the re-emergence of the
probability density after the initial drop, which indicates the amount of
reflection present, is significantly higher in (d) than in (c).

It is interesting to note that Figure. 5 reveals a significant aspect of post-
quench transport. The post-quench transport of the probability density
displays a stark asymmetry in reflection from the middle PT symmetric layer
depending upon the direction in which it is traveling. If the initial edge
state is localized on the right-hand side of the system(Figure. 3(b)), i.e.,
the post-quench transport is towards the left side, then there is a
significant reflection of the probability density from the middle layer(see
Figure 5(b)), on the contrary, if the initial edge state is the one as shown
in Figure. 3(a) that leads to transport of the probability density from the
left edge to the right edge, the reflection from the middle layer is almost
zero(as shown in Figure 5(a)), similar to unidirectional invisibility as
observed in [8]. The light cone diagrams in Figures 5(a) and 5(b) correspond
to the two cases of post-quench transport in which a sub-cone due to the
reflection is prominent in Figure 5(b), and it is nearly absent in 5(a),
denoting a reflection-less transport. Measuring the probability density with
time at the end site where the initial state was initially localized elegantly
indicates the magnitude of reflection as is depicted in Figure 5(c) and 5(d).
The region of the dip in the probability density corresponds to the time when
the probability density is traveling, and the re-emergence of it after a
particular time denotes the scenario when the probability density, which is
reflected back from the middle layer, has reached the end. By the magnitude of
re-emergent probability densities, we are able to quantify the reflection
amplitude, and as a result, it is evident that the reflection amplitude is
negligible for 5(c) and is substantial for 5(d). We will now proceed to
explain this asymmetrical quench dynamics using transfer matrix method [27,
28].

###  III.2 A transfer matrix analysis.

The post quench transport is reflection-less if the system is entirely an SSH
system, i.e., without a middle PT-symmetric non-Hermitian region. It is the
introduction of the middle PT-symmetric non-Hermitian layer which gives rise
to the scattering of the traveling probability density. Modeling the system
for transfer matrix calculation requires consideration of the alternating on-
site potentials $u$ and $u^{*}$ in the middle layer and the hopping amplitudes
associated with them. Figure 6(a) shows the schematics of modeling the unit
potential block $AB$ for transfer matrix analysis; there is a uniform real
potential $u_{Re}$ spread over the two sites, and the complex potential
$iu_{Im}$ has opposite polarities and different widths $L_{A}$ and $L_{B}$ for
A and B sites, respectively. The whole scattering part model, as shown in
Figure 6(b), is thus a periodic repetition of the individual potential blocks
in Figure 6(a) The transfer matrix is given by,

$T=[t_{R}]^{-1}[\tilde{T}_{A}][\tilde{T}_{B}]....[\tilde{T}_{A}][\tilde{T}_{B}][t_{L}]$
(5)

where,

$\tilde{T}_{\text{A(B)}}=\begin{bmatrix}1&1\\\
k^{\text{A(B)}}_{+}&k^{\text{A(B)}}_{-}\end{bmatrix}\begin{bmatrix}e^{ik_{+}L_{\text{A(B)}}}&0\\\
0&e^{ik_{-}L_{\text{A(B)}}}\end{bmatrix}\begin{bmatrix}1&1\\\
k^{\text{A(B)}}_{+}&k^{\text{A(B)}}_{-}\end{bmatrix}^{-1}$ (6)

The matrix $\tilde{T}_{\text{A(B)}}$ constitutes information corresponding to
the potential block A(B). It is important to note that, here, $t_{R}$ and
$t_{L}$ correspond to the potential free regions at right and left ends of the
stack, respectively. If T has the form $T=\begin{bmatrix}\alpha&\beta\\\
\gamma&\sigma\end{bmatrix}$, then the sacttering matrix will be,

$S=\begin{bmatrix}-\sigma^{-1}\gamma&\sigma^{-1}\\\
\alpha-\beta\sigma^{-1}\gamma&\beta\sigma^{-1}\end{bmatrix}$ (7)

Figure 6: This figure illustrates the schematics of the potential block
modelling for the middle PT symmetric non-Hermitian part employed in the
transfer matrix calculations. (a) shows the spatial variation of the potential
components of each unit block $AB$. There are two imaginary potential
components with reversing polarity,$-u_{Im}$ and $u_{Im}$, and spatial spread
$L_{A}$ and $L_{B}$, respectively, corresponding to the two sub-lattices A and
B for the middle scattering part. There is also a uniform real
potential($u_{Re}$) spread through out the two sub-lattices A and B (b) The
stacking structure of the potentials represents the whole PT-symmetric middle
layer. This diagram is not to scale.

In Equation 7 for $S$, the diagonal elements $S_{11}$ and $S_{22}$ represents
the reflection coefficients for the transport probability wave coming from
left and right respectively.

Figure 7: Demonstration of the left and right reflection amplitudes, $R_{L}$
and $R_{R}$, arising out of transfer matrix calculations corresponding to $10$
units of potential block $AB$ for the cases when, (a) the scattering part is
hermitian with uniform on-site potentials $u=-0.3$ and, (b) the scattering
part is PT-symmetric non-hermitian with the alternating on-site potential
$u(u^{*})=-0.3-(+)0.1i$ and $L_{A(B)}=6(10)$. In (a), the reflection
amplitudes $R_{L}$ and $R_{R}$ are the same, and hence, only one color is
prominent as one overlaps the other.

By considering the periodic repetition of potential block $AB$ as a model for
the scattering medium, Figure 7(b) shows that the transfer matrix analysis
leads to asymmetric left and right reflection amplitudes ($R_{L}=|S_{11}|^{2}$
and $R_{R}=|S_{22}|^{2}$) of the transport probability density. Figure 7(a),
as a reference, shows that a purely real scattering part without the
alternating imaginary component($u_{Im}=0$) leads to perfectly symmetric left
and right ($R_{L}$ and $R_{R}$) reflections, as one might have expected. It is
important to note that, in our transfer matrix calculation, this
unidirectionality phenomenon arises even under the approximation that the
incoming wave is a simple plane wave with particular energy. In the actual
post-quench transport considered, this probability wave consists of a series
of energy values depending upon the quench, as can be understood from Equation
4.

##  IV Summary and conclusions

We considered a hybrid model designed to exhibit highly anisotropic post-
quench transport. The system comprises a non-Hermitian PT-symmetric SSH system
sandwiched between two ordinary SSH systems. This system is the first
demonstrated to have topological characteristics and is able to host robust,
well-defined edge states. An extreme dimerized case was used to demonstrate
the presence of edge states geometrically, and then the spatial structure of
the edge states was studied, and it was found that the localization character
of the edge state is maintained for a range of hopping parameters, revealing
its topological character. The interesting character of the states is revealed
by examining the system’s energy vs. the parameters, which included zero
energy states and exceptional point like behaviour.

We then proceeded forward to study the post-quench transport. The study of the
”light cone” diagrams reveals the directional dependence of the post-quench
transport. The direction of the transport of the probability density depends
upon the end at which the initial wave -function was localized. It is found
that in one direction, there is almost no reflection of the traveling
probability density wave, whereas, in the other direction, there is a
considerable reflection from the middle PT-symmetric region. In order to a
deeper understanding of this unidirectional post-quench transport, we model
the system with an assembly of potentials ordered in a repeated fashion to
mimic the middle PT-symmetric part of our hybrid system. Using transfer matrix
calculations, it is apparent that there is an asymmetry between left and right
reflection amplitudes. In cases where the energy is too large or too low, both
of the reflection amplitudes approach the same value. Summarily, the post-
quench transport for this hybrid system depends on the initial edge state
localization, i.e., the initial localization (right end or left end)
determines the direction of post-quench transport, which determines quench
dynamics.

## ACKNOWLEDGMENTS

Melbourne Research Scholarship and N.D Goldsworthy Scholarship fund this
project. We thank the University of Melbourne and the Indian Institute of
Technology Kharagpur for providing a supportive research environment.

## References

  * Bender and Boettcher [1998] Carl M Bender and Stefan Boettcher.  Real spectra in non-hermitian hamiltonians having p t symmetry.  _Physical review letters_ , 80(24):5243, 1998\.
  * Bender [2005] Carl M Bender.  Introduction to pt-symmetric quantum theory.  _Contemporary physics_ , 46(4):277–292, 2005\.
  * Bender [2007] Carl M Bender.  Making sense of non-hermitian hamiltonians.  _Reports on Progress in Physics_ , 70(6):947, 2007\.
  * Klaiman et al. [2008] Shachar Klaiman, Uwe Günther, and Nimrod Moiseyev.  Visualization of branch points in p t-symmetric waveguides.  _Physical review letters_ , 101(8):080402, 2008\.
  * Makris et al. [2008] Konstantinos G Makris, R El-Ganainy, DN Christodoulides, and Ziad H Musslimani.  Beam dynamics in p t symmetric optical lattices.  _Physical Review Letters_ , 100(10):103904, 2008\.
  * El-Ganainy et al. [2007] R El-Ganainy, KG Makris, DN Christodoulides, and Ziad H Musslimani.  Theory of coupled optical pt-symmetric structures.  _Optics letters_ , 32(17):2632–2634, 2007.
  * Mostafazadeh [2009] Ali Mostafazadeh.  Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies.  _Physical review letters_ , 102(22):220402, 2009\.
  * Lin et al. [2011] Zin Lin, Hamidreza Ramezani, Toni Eichelkraut, Tsampikos Kottos, Hui Cao, and Demetrios N Christodoulides.  Unidirectional invisibility induced by p t-symmetric periodic structures.  _Physical Review Letters_ , 106(21):213901, 2011\.
  * El-Ganainy et al. [2018] Ramy El-Ganainy, Konstantinos G Makris, Mercedeh Khajavikhan, Ziad H Musslimani, Stefan Rotter, and Demetrios N Christodoulides.  Non-hermitian physics and pt symmetry.  _Nature Physics_ , 14(1):11–19, 2018.
  * Guo et al. [2009] A Guo, GJ Salamo, D Duchesne, R Morandotti, M Volatier-Ravat, V Aimez, GA Siviloglou, and DN Christodoulides.  Observation of p t-symmetry breaking in complex optical potentials.  _Physical review letters_ , 103(9):093902, 2009\.
  * Rüter et al. [2010] Christian E Rüter, Konstantinos G Makris, Ramy El-Ganainy, Demetrios N Christodoulides, Mordechai Segev, and Detlef Kip.  Observation of parity-time symmetry in optical systems.  _Optics and Photonics News_ , 21(12):47–47, 2010\.
  * Huber [2016] Sebastian D Huber.  Topological mechanics.  _Nature Physics_ , 12(7):621–623, 2016.
  * Anandwade et al. [2023] Ritika Anandwade, Yaashnaa Singhal, Sai Naga Manoj Paladugu, Enrico Martello, Michael Castle, Shraddha Agrawal, Ellen Carlson, Cait Battle-McDonald, Tomoki Ozawa, Hannah M Price, et al.  Synthetic mechanical lattices with synthetic interactions.  _Physical Review A_ , 108(1):012221, 2023.
  * Regensburger et al. [2012] Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov, Demetrios N Christodoulides, and Ulf Peschel.  Parity–time synthetic photonic lattices.  _Nature_ , 488(7410):167–171, 2012.
  * Miri and Alu [2019] Mohammad-Ali Miri and Andrea Alu.  Exceptional points in optics and photonics.  _Science_ , 363(6422):eaar7709, 2019.
  * Zhang et al. [2016] Zhaoyang Zhang, Yiqi Zhang, Jiteng Sheng, Liu Yang, Mohammad-Ali Miri, Demetrios N Christodoulides, Bing He, Yanpeng Zhang, and Min Xiao.  Observation of parity-time symmetry in optically induced atomic lattices.  _Physical review letters_ , 117(12):123601, 2016\.
  * Gupta and Jayannavar [2021] Nitish Kumar Gupta and Arun M Jayannavar.  Non-hermitian topoelectrical circuits: Expedient tools for topological state engineering with gain-loss modulation.  _arXiv preprint arXiv:2108.11587_ , 2021.
  * Fleury et al. [2015] Romain Fleury, Dimitrios Sounas, and Andrea Alù.  An invisible acoustic sensor based on parity-time symmetry.  _Nature communications_ , 6(1):5905, 2015.
  * Longhi [2009] Stefano Longhi.  Bloch oscillations in complex crystals with p t symmetry.  _Physical review letters_ , 103(12):123601, 2009\.
  * Lieu [2018] Simon Lieu.  Topological phases in the non-hermitian su-schrieffer-heeger model.  _Physical Review B_ , 97(4):045106, 2018.
  * Yao and Wang [2018] Shunyu Yao and Zhong Wang.  Edge states and topological invariants of non-hermitian systems.  _Physical review letters_ , 121(8):086803, 2018\.
  * Zhang et al. [2020] Kai Zhang, Zhesen Yang, and Chen Fang.  Correspondence between winding numbers and skin modes in non-hermitian systems.  _Physical Review Letters_ , 125(12):126402, 2020\.
  * Kozii and Fu [2017] Vladyslav Kozii and Liang Fu.  Non-hermitian topological theory of finite-lifetime quasiparticles: prediction of bulk fermi arc due to exceptional point.  _arXiv preprint arXiv:1708.05841_ , 2017.
  * Su et al. [1979] Wu-Pei Su, John Robert Schrieffer, and Alan J Heeger.  Solitons in polyacetylene.  _Physical review letters_ , 42(25):1698, 1979\.
  * Asbóth et al. [2016] János K Asbóth, László Oroszlány, and András Pályi.  A short course on topological insulators.  _Lecture notes in physics_ , 919:166, 2016.
  * Ghosh et al. [2023] A. Ghosh, A. M. Martin, and S. Majumder.  Quench dynamics of edge states in a finite extended su-schrieffer-heeger system.  _Phys. Rev. E_ , 108:034102, 2023.
  * Markos and Soukoulis [2008] Peter Markos and Costas M. Soukoulis.  _Wave Propagation_.  Princeton University Press, Princeton, 2008.  ISBN 9781400835676.
  * Walker and Gathright [1992] James S Walker and J Gathright.  A transfer-matrix approach to one-dimensional quantum mechanics using mathematica.  _Computers in Physics_ , 6(4):393–399, 1992.