# Road Surface Friction Estimation for Winter Conditions Utilising General
Visual Features

Risto Ojala1 and Eerik Alamikkotervo2 This work was funded by Henry Ford
Foundation Finland.The authors are with the Department of Mechanical
Engineering, Aalto University, 00076 Aalto,
Finland.1risto.j.ojala<EMAIL_ADDRESS>
###### Abstract

In below freezing winter conditions, road surface friction can greatly vary
based on the mixture of snow, ice, and water on the road. Friction between the
road and vehicle tyres is a critical parameter defining vehicle dynamics, and
therefore road surface friction information is essential to acquire for
several intelligent transportation applications, such as safe control of
automated vehicles or alerting drivers of slippery road conditions. This paper
explores computer vision-based evaluation of road surface friction from
roadside cameras. Previous studies have extensively investigated the
application of convolutional neural networks for the task of evaluating the
road surface condition from images. Here, we propose a hybrid deep learning
architecture, WCamNet, consisting of a pretrained visual transformer model and
convolutional blocks. The motivation of the architecture is to combine general
visual features provided by the transformer model, as well as fine-tuned
feature extraction properties of the convolutional blocks. To benchmark the
approach, an extensive dataset was gathered from national Finnish road
infrastructure network of roadside cameras and optical road surface friction
sensors. Acquired results highlight that the proposed WCamNet outperforms
previous approaches in the task of predicting the road surface friction from
the roadside camera images.

##  I Introduction

Winter conditions pose several challenges for operation of road vehicles. A
key challenge is safe manoeuvring in the varying road conditions caused by
below freezing winter temperatures. Studies have shown that winter road
conditions have an immense impact on the accident rates of road traffic [1]
[2]. Different amounts and mixtures of snow, ice, and water on the road
notably affect the road surface friction, i.e. the slipperiness of the road.
Friction between the road and vehicle tyres is a key parameter defining the
vehicle dynamics, largely deciding the limits of safe vehicle control and
manoeuvring. Tyre-road friction is naturally dependent on both road and tyre
friction properties, yet the tyre properties are mostly constant, whereas
active monitoring is needed for the highly varying road surface friction.
Consequently, evaluation of road surface friction enables different
intelligent transportation applications for improving traffic safety. Such
applications can range from adjusting automated vehicle control parameters to
alerting drivers of slippery conditions.

Several different approaches have been developed for predicting road surface
friction, such as evaluation of vehicle tyre-slip [3], and special optical
sensors utilising infrared spectroscopy [4]. In recent years, computer vision
methods with regular visible light cameras have gained notable traction.
Cameras are inexpensive sensors, and deep learning approaches have enabled
increased accuracy evaluation of the road surface condition. Development has
been aimed at both vehicle on-board cameras, as well as roadside cameras.
Roadside cameras have the benefit of better visibility of the road due to
higher installation position, as well as more convenient integration into
centralised intelligent transportation data services.

This paper advances roadside camera-based road surface friction prediction by
proposing a novel deep learning model, WCamNet, for the task. The model
consists a hybrid architecture, combining a pretrained visual transformer
(ViT), DINOv2, with convolutional blocks. Previous studies have investigated
applying convolutional neural networks (CNNs) as well as the ViT model for
task. Our hybrid architecure feeds general visual features from the pretrained
DINOv2 model into a CNN processing pipeline. This novel approach offers clear
accuracy benefits compared to previous state-of-the-art, as indicated by the
acquired results.

##  II Related Work

###  II-A Computer Vision-based Road Surface Condition Monitoring

Several previous studies have developed methodologies for analysing the road
surface condition with camera-based methods, both with on-board cameras and
roadside cameras. Majority of previous studies propose classification methods
for the task, providing a general label for the road, e.g. snowy, icy, wet, or
dry, which can be used to roughly estimate the road surface friction. Other
methods however aim for higher resolution by directly quantifying a continuous
variable as the friction prediction. Another key characteristic
differentiating related works is that several studies are limited to summer
conditions, whereas other studies also consider winter conditions and the
encompassed highly varying road surface conditions.

Roadside camera-based method for winter conditions was proposed by Jonsson
[5]. The proposed method utilised weather information as well as camera images
to perform classification of the road surface condition. Several manually
selected features were extracted from images, and principal component analysis
was applied to acquire relevant features for the prediction model. A simple
linear classifier was utilised for prediction. More recently, Ozcan et al. [6]
applied deep learning, a VGG16 [7] CNN model for classifying road condition
from roadside cameras in winter conditions. Similarly, Grabowski et al. [8]
evaluated several CNN models for the same task. In their tests, DenseNet201
[9] demonstrated the best performance, with VGG19 achieving the second best
results.

In summer conditions, with the evaluation limited to dry and wet roads,
Sirirattanapol et al. [10] have proposed a custom CNN architecture for
classifying the road condition from a roadside camera. Similarly, Abdelraouf
et al. [11] classified dry and wet roads from a roadside camera with deep
learning. However, they applied a ViT model for the task, and they also
proposed a novel architecture for fusing inputs from several cameras
monitoring the same area.

Development has also been active around computer vision methods aimed for on-
board use, utilising a forward facing camera installed on a vehicle. In the
study of Nolte et al. [12], ResNet50 [13] and InceptionV3 [14] CNN models were
applied to classify road surface condition. Winter conditions were included in
the analysis. Wang et al. [15] also developed a model for classifying road
surface condition in wintertime. They trained a segmentation CNN model with
images of different road conditions, and during inference the road condition
with most segmented pixels was chosen as the classification. Vosahlik et al.
[16] trained a ResNet50 CNN regression model to predict the vehicle tyre-road
friction in winter conditions. They trained their model based on ground truth
values from a slip-based friction estimation method. Ojala and Seppänen [17]
proposed a custom CNN regression model, SIWNet, for predicting road surface
friction in winter conditions with built-in uncertainty evaluation. Their
model was trained based on ground truths from an optical friction sensor.

On-board methods developed for summer conditions have featured the work of
Šabanovič et al. [18], who developed a custom CNN model capable of classifying
the road surface condition. Similar work has been carried out by Cordes et al.
[19], who also published an extensive open dataset called RoadSaw, which
featured ground truth values from an optical water film height sensor. Du et
al. [20] have developed regression-based road surface friction estimation for
summer conditions. They tested several CNN architectures for the task, with
VGG19 [7] providing the most accurate results. They collected ground truth
road surface friction information with a commercial contact-based monitoring
system.

During winter, road areas typically exhibit different surface condition and
friction properties in different sections of an image, due to for example tyre
tracks in snow or differences in water film height due to the road elevation
profile. Some studies have addressed this problem by increasing the spatial
resolution of the predictions. To overcome this issue, Roychowdhury et al.
[21] proposed an on-board approach where they split the image of the road into
a grid, and road surface condition in each cell was classified separately.
Pesonen [22] utilised partially supervised training to develop a deep learning
segmentation model to evaluate the road surface friction for each pixel from
an on-board camera. An optical road friction sensor was used to acquire
partial labels for the images. Higher spatial resolution in the prediction is
useful in on-board use cases, since the information can be utilised in for
example lateral control. For roadside camera applications however, a single
friction value for the entire road area could be sufficient information. This
is due to the probable applications being mostly related to longitudinal
control, such as instructing passing vehicles to operate at lower speeds and
avoid sharp decelerations.

###  II-B General Visual Features

With the emerge of larger deep learning models for computer vision, there have
been notable efforts in extracting increasingly general visual features. This
task is motivated by the need for generally applicable computer vision models,
which would require minimal further training for adoption to different tasks.
Visual transformer (ViT) models [23] have proven effective for the task. When
trained with large quantities of data in an self-supervised manner, the models
are capable of learning rich feature representations for downstream tasks.
These types of models are commonly referred to as foundation models [24].

A wide variety of vision foundation models have been published. Some are fully
self-supervised [25, 26, 27, 28, 29, 30] while others use weak supervision
from image captions [31, 32, 33]. Currently, DINOv2 [30] provides the best
performance both in image-level downstream tasks like classifications and
dense downstream tasks like segmentation. Due to its capabilities, DINOv2 was
chosen as the backbone for downstream friction prediction in this paper.

The suitability of foundation model features for friction estimation has not
been explored in the previous works. However, these features are proven to
generalise across domains, like different lighting conditions, scales, and
environments while encoding high-level information like depth and object class
accurately [30]. Thus the foundation model features are likely suitable also
for estimating friction from road appearance in varying lighting and weather
conditions.

###  II-C Research Gap

Previous works on road surface condition monitoring have developed and
extensively evaluated the capabilities of different CNN models, as well as the
ViT model. However, no previous works have explored utilisation of the recent
foundation models for the task. We propose the novel WCamNet model for
estimating road surface friction from roadside camera images. The model
combines general visual features acquired with the DINOv2 foundation model, as
well as a CNN processing pipeline trained for extracting relevant features for
the friction prediction task. Our results demonstrate that the proposed
architecture achieves higher accuracy in the road surface friction prediction
task than the state-of-the-art CNN and ViT models or the pretrained DINOv2
foundation model.

##  III Methods

###  III-A Problem Formulation and Dataset

This paper focuses on the application of deep learning for analysing the road
surface condition from a roadside camera image. More specifically, the task is
formulated as a regression problem, where the deep learning model predicts the
road surface friction as a scalar value, with an image as the input. The aim
of this work was to develop a novel prediction model for this task, which
outperforms previous state of the art in terms of accuracy. To facilitate
development, a large dataset was gathered from the national Finnish road
infrastructure network consisting of roadside cameras and optical road
friction sensors [34, 35]. Open access to the data was provided by Fintraffic,
the operator of the equipment.

A total of 31 pairs of roadside camera stations and weather stations around
Finland were utilised for data collection. Data samples were gathered at
approximately 20 minute intervals for roughly two months in February-March of
2023. From each station, 1-2 cameras were recorded depending on the number of
camera installations in the station. If 2 cameras were used from the same
station, they were looking at opposite directions of the road. A total of 58
cameras were recorded for the dataset. Vast majority of the cameras had a
resolution of 1280x720 pixels, with some cameras recording at different
resolutions. Highest resolution of the cameras was 1920x1080 pixels, whereas
lowest resolution of a recorded camera was 800x450 pixels. Similarly to the
camera stations, each weather station contained 1-2 optical road friction
sensors. If a station contained 2 road friction sensors, their readings were
averaged to reach a single ground truth value for the road surface friction.
Out of the 31 weather stations, 17 stations were equipped with a single road
friction sensor, whereas 14 stations had two sensors. The road friction
sensors reported friction as a continuous value, grip factor, with a
measurement range of 0.09-0.82. Lower values corresponded to slippery
conditions, and higher values indicated better tyre grip. To enable more
convenient processing, the readings were scaled to values between 0-1, defined
as friction factor, $f$.

Optical road friction sensor information from weather stations was used to
label the images captured from the corresponding roadside camera station.
Samples of the gathered data presented in Figure 1. The corresponding weather
stations and roadside camera stations were located up to 1.5 km from one
another, with majority of corresponding stations in immediate vicinity of one
another. Distribution of distances between the corresponding stations are
visualised in Figure 2. The gathered data was initially heavily biased on
certain friction factor values, since for example dry asphalt conditions were
prevalent across the data collection period. Therefore, random weighted
sampling was carried out on the gathered data to improve the balance of the
dataset. Sampled from the initially gathered data of over 160 thousand images,
the final dataset contained a total of 48791 images with corresponding
friction factor labels. The distribution of the friction factor values in the
final dataset is presented in Figure 3.

|  |
---|---|---
$f=0.89$ | $f=0.32$ | $f=0.34$
|  |
$f=0.95$ | $f=0.95$ | $f=0.51$
|  |
$f=0.74$ | $f=0.45$ | $f=0.15$
|  |
$f=0.63$ | $f=0.77$ | $f=0.56$
Figure 1: Samples of image data with corresponding friction factor values.
Figure 2: Distribution of distances between the corresponding camera and
weather stations. Figure 3: Distribution of friction factor values in the
dataset.

###  III-B Model Architecture

Our proposed WCamNet model infuses general visual features from a foundation
model with a CNN processing pipeline. Major motivation for the model was the
fact that standalone CNN architectures lack attention mechanisms due to local
feature extraction, consequently being unable to properly focus on the
relevant sections of the images. This is detrimental for road surface friction
evaluation, since the models are incapable of focusing on the road areas.
Consequently, standalone CNNs are not optimally suited for the prediction
task. WCamNet applies a visual foundation model, DINOv2, to fuse scene
understanding and attention for the CNN processing. Figure 4 visualises DINOv2
patch tokens acquired for resized samples from the gathered data, reduced to
three dimensions with principal component analysis. The samples highlight the
effectiveness of the foundation model in segmenting varying areas of the
scenery, which offers a useful attention mechanism for futher processing.
Furthermore, foundation models have learned rich feature representations from
vast datasets, and can extract directly useful features for the prediction
task. Therefore, foundation models are a more convenient approach for the task
than a training a ViT for the task to overcome the limitations of CNNs.
Transformer models are notorious for requiring extensive datasets for optimal
performance.

|  |  |  |  |  |  |
---|---|---|---|---|---|---|---
Figure 4: Image pairs demonstrating resized image samples and their respective
DINOv2 patch tokens reduced to three feature dimensions with principal
component analysis.

The proposed WCamNet architecture is depicted in Figure 5. The moderately
sized DINOv2 model, DINOv2-B, was chosen for the architecture. During training
of the network, the DINOv2 model remains frozen, while the weights of other
layers are optimised. When performing inference with the network, each RGB
input image is resized to 602x602, and then fed to DINOv2 and a high-
definition (HD) branch for parallel processing. The HD branch consists of
three CNN layers, which extract useful low-level features from the input
image. The final patch tokens produced by DINOv2 and the features from the HD
branch are concatenated into a single tensor, which is fed through two
consecutive standard residual squeeze-and-excitation (SE) blocks [36] with
reduction ratios of 8. These SE blocks enable weighing the relevant extracted
features, along with providing additional convolutional feature extraction.
The output of the consecutive SE blocks is pooled with global average pooling,
and fed through a fully connected layer with sigmoid activation. This output
is the friction factor prediction, $\hat{f}$, of the network.

Figure 5: Proposed WCamNet architecture. Tensor sizes are reported for
processing a single input image (batch size of one).

###  III-C Training and Testing

For training and testing, the dataset was split with an approximate
50%-15%-35% train, validation, and test split, respectively. A high portion of
testing data was chosen to ensure that the models were well generalisable to
different camera views. In the split, each roadside camera station was fully
allocated to a specific split, ensuring that data from different cameras of a
single station was not present in different splits.

To benchmark the performance of WCamNet, several other commonly applied deep
learning models were trained and tested with the data as well. ResNet50,
ResNet152, VGG19, DINOv2-B with a linear head, as well as ViT-B/14 with DINOv2
pretraining were included to the benchmark for comparison. The DINOv2-B with a
linear head featured a fully connected layer which produced the prediction
based on all the patch tokens as well as the final class token. The linear
head was trained on the training split, while the DINOv2 remained frozen.
VIT-B/14 with DINOv2 pretraining was essentially the DINOv2 model which was
fully trained on the training split, using the DINOv2 weights as the starting
point. All models utilised a sigmoid activation at the end of the architecture
for producing the final friction factor prediction.

WCamNet was trained for 15 epochs, and a Cosine Annealing learning rate
scheduler with a warm restart every five epochs. The CNN models, ResNet50,
ResNet152 and VGG19, were trained for 30 epochs with a step-based learning
rate scheduler reducing the learning rate to one tenth every ten epochs. The
ViT-B/14 model was trained in an identical fashion. DINOv2 with a linear head
was trained for six epochs with the step-based learning rate scheduler
decreasing the learning rate every two epochs. All models were evaluated with
PyTorch [37] implementations, trained with mean squared error loss function,
applying stochastic gradient descent with a momentum of 0.9, and a batch size
of 16. The initial learning rate and weight decay of each model were optimised
via grid search. These hyperparameters were selected based on performance on
the validation split.

Images fed to the networks were resized to 602x602 pixels, and normalised
based on the training set statistics. During training, image augmentation was
applied via random horizontal flipping, color jitter with a scale of 0.05,
random rotations of -45 to 45 degrees, as well as random cropping with a
padding of 64 pixels.

##  IV Results

###  IV-A Accuracy Results

In order to test the performance of the proposed WCamNet architecture, the
accuracy of the model was evaluated on the testing split of the utilised
dataset. Several other models commonly utilised for the road surface condition
prediction task were also evaluated for comparison. Resulting accuracies of
the models are presented in Table I with the mean absolute error (MAE) and
root-mean-square error (RMSE) metrics.

TABLE I: Accuracies of the models on the test set. Bolded values indicate the
best result in each column.
---
Method |  MAE |  RMSE
WCamNet (ours) |  0.150 |  0.195
ResNet50 |  0.184 |  0.233
ResNet152 |  0.201 |  0.260
VGG19 |  0.166 |  0.210
DINOv2-B w/ linear head |  0.172 |  0.219
ViT-B/14 (DINOv2 pretrained) |  0.183 |  0.232

The acquired results demonstrate that WCamNet outperformed the other models in
the accuracy benchmark. VGG19 was able to reach the second lowest errors in
the benchmark. Compared to VGG19, WCamNet reached a 10% lower MAE and a 7%
lower RMSE.

###  IV-B Ablation Study

To analyse the rationality of the design choices in the WCamNet architecture,
as well as the contribution of different parts of the network to the overall
performance, an ablation study was carried out. The training procedure was
repeated with different configurations of the WCamNet architecture, and the
accuracy results were evaluated on the testing split. The ablations included
changing the DINOv2-B model to the larger DINOv2-L model, removing the SE
blocks from the architecture, as well as removing the HD branch from the
architecture. Acquired results for these ablations are presented in Table II.

TABLE II: Ablation study of the proposed WCamNet. Bolded values indicate the
best result in each column.
---
Method |  MAE |  RMSE
WCamNet w/ DINOv2-L backbone |  0.155 |  0.197
WCamNet w/o SE blocks |  0.167 |  0.213
WCamNet w/o HD branch |  0.170 |  0.217

The results demonstrate that the design choices behind WCamNet are reasonable
and improve performance. Accuracy with the larger DINOv2-L model in the
architecture was highly similar as with the original architecture, yet the
larger model results in increased computational cost. Removal of SE blocks or
the HD branch resulted in notable increase in errors, indicating that these
elements are crucial for the performance of the model.

##  V Discussion

The presented WCamNet model was shown to improve the state of the art of
camera-based road surface friction estimation in winter conditions. The hybrid
architecture combining general visual features from DINOv2 and a CNN
processing pipeline was shown to affect performance favourably. Results
demonstrated that the model achieved higher accuracy than models generally
applied for the prediction task. The performance gains demonstrated by the
WCamNet model enable more reliable future utilisation of roadside camera-based
road surface friction monitoring applications in intelligent transportation
systems.

Although WCamNet was the most accurate computer vision model in the benchmark,
the error values were still quite notable on an absolute scale. This indicates
that even computer vision cannot perfectly mimic the performance of the
optical road friction sensor utilised as the ground truth. Future research
efforts are needed to more systematically map the reliability of computer
vision-based road surface friction estimation, and its short-comings.

Another factor affecting the absolute error is the spatial ambiguity in the
image data. A single ground truth reading was used to label each image.
However, this is sub-optimal, since the road surface condition can vary within
different parts of a single image. For example, some sections of the road
might be covered in snow, while other areas are not. This causes ambiguity in
the data, and has led to some authors proposing approaches with higher spatial
resolution in the data [21, 22]. Nevertheless, this type of data is difficult
to obtain, and therefore commonly a single label is used for the entire image.
Overall, the large quantity of the data used in this study should compensate
for ambiguities in the data.

To further enhance camera-based road surface friction estimation in winter
conditions, other weather data could be included as input in the prediction
models. Information such as temperature, precipitation, and humidity are
generally available for most areas online, and these data could provide
valuable contextual information for determining the road surface friction.
Fusion of camera images and weather information has been proposed in the past
[5], yet modern deep learning methods could likely demonstrate notable
performance gains. This type of data fusion could enable even higher accuracy
in computer vision-based approaches, facilitating adoption of camera-based
road monitoring methods in intelligent transportation applications.

## ACKNOWLEDGMENT

We acknowledge the funding provided by Henry Ford Foundation Finland, the
computational resources provided by the Aalto Science-IT project, and the data
provided by Fintraffic.

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# Monte Carlo Study of Agent-Based Blume-Capel Model for Political
Depolarization

Hung T<EMAIL_ADDRESS>Miron<EMAIL_ADDRESS>Sanda
<EMAIL_ADDRESS>1 Laboratoire de Physique Théorique et
Modélisation, CY Cergy Paris Université, CNRS, UMR 8089
2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.
2 Department of Physics, Cleveland State University, USA.
3 Levin School of Urban Affairs, Cleveland State University, USA.

###### Abstract

In this paper, using Monte Carlo simulations we show that the Blume-Capel
model gives rise to the social depolarization. This model borrowed from
statistical physics uses the continuous Ising spin varying from -1 to 1
passing by zero to express the political stance of an individual going from
ultra-left (-1) to ultra-right (+1). The particularity of the Blume-Capel
model is the existence of a $D$-term which favors the state of spin zero which
is a neutral stance. We consider the political system of the USA where voters
affiliate with two political groups: Democrats or Republicans, or are
independent. Each group is composed of a large number of interacting members
of the same stance. We represent the general political ambiance (or degree of
social turmoil) with a temperature $T$ similar to thermal agitation in
statistical physics. When three groups interact with each other, their stances
can get closer or further from each other, depending on the nature of their
inter-group interactions. We study the dynamics of such variations as
functions of the value of the $D$-term of each group. We show that the
polarization decreases with incresasing $D$. We outline the important role of
$T$ in these dynamics. These MC results are in excellent agreement with the
mean-field treatment of the same model.

††preprint: APS/123-QED

##  I Introduction

Social polarization has been investigated by numerous scholars using many
methods [1-6]. In democracies, each person is free to choose his/her stance
with respect to a social issue such as politics, immigration, race, economics,
… according to his/her preferences. People with the similar beliefs regarding
specific sets of social issues form groups that can be very large, such as
political parties [2]. Policies proposed by the left are very different of
those proposed by the right. Polarization derives from sharp differences
between individuals affiliated with the ”left” orientation and those with a
”right” orientation [3,4]. Depending on the social context and events, the
political polarization at any moment can be very sharp: people belonging to
different political parties may not agree on almost any subject. The cases of
France and of the USA in recent times are striking. For example, since 2022 in
France, no political party holds a majority in the parliament. Therefore, in
the absence of compromise between at least two parties, the majority required
to pass a law does not materialize. Instead, most of the time the government
uses article 49.3 of the Constitution to pass laws without a parliament vote
[6,7]. In the USA, the political polarization began to rise in the 70’s
[8-13]. We see this in periodic polls [11]. Other european democracies have
seen the same tendency of strong political polarization [16,17].

In general, each party proposed policies to solve societal problems ranging
from government aid to the needy, to race, immigration, national security, and
the environment [6]. However, in democracies, protracted conflict over
solutions to societal problems leads to more problems: political polarization
has serious deleterious societal [1,16,17,18] and economic [3] consequences.
One of these is that people gradually lose the ability to work together, to
compromise, make, and implement deals. In time, this can lead to societal
breakdown [17,19].

One notable problem caused by polarization is that individuals increasingly
tend to believe in information which agrees with, or justifies their political
perspective [17,20,21]. In time, the information sets from which they draw
support diverge to the point where polarized groups hold entirely
contradictory images of the shared reality. Therefore, strong polarization
prevents constructive debates. This may lead to political instability [17]–the
governing party changes often, leading to collective and individual
uncertainty. Changing the governing party at each round of elections prevents
implementation of long-term programs which are necessary in realms such as
economics and education. Since polities are complex systems [22] within which
interactions change with time, according to [23] and [24] empirical studies do
not suffice to help us understand political polarization dynamics. We also
need theoretical modeling to help explore the conditions under which specific
events can happen. Agent-based modeling has great potential in this regard
([25,26]). Modeling can help prepare information that might be useful in
reducing the impacts of polarization [27,28].

Together with the recognition of increasing polarization, there has been a
rise in the number of investigations of its causes and dynamics.
Sociophysics–namely applying physics tools to the study of social
phenomena–has been a very effective approach in this respect. It can handle
complexity in various domains, including politics, and provide insights
complementing those gleaned from other disciplines [28-33]. Sociophysics has
already been used in studies of polarization (e.g., [10,20,34-37]). Network
models can be used to explore polarization trends and to find avenues for
intervention, generating qualitative anticipatory scenarios which can be
queried (e.g., [38-43]). For instance, using network models we have
anticipated election outcomes in the US and in Bosnia-Hercegovina [28,42]; and
we have examined various outcomes of labor-management contract negotiations in
France [43]. As Ref. [44] has argued, anticipatory scenarios are useful in
supporting the development of robust strategies of action in the face of the
high levels of uncertainty characterizing complex systems.

Within Western democracies political polarization is on the rise, undermining
collective decision making abiity (e.g., [17,45,46]). We have examined
polarization dynamics in the USA between Democratic- and Republican-affiliated
individuals, using an agent-based model borrowed from statistical physics
using mean-field theory and Monte Carlo (MC) simulations [47,48].

We note that very recently Galam [49] has studied the political polarization
using his model of opiniion dynamics which consists in supposing there are
several categories of indiividuals in a communinity, each with a probability:
the contrarians, the floaters and the stubborn agents. He found several
scenarii of polarization depending on the case and the probability: from the
unanimity to the rigidity passing by the coexistence. His method is
probabilistic while ours uses the spin model with microscopic interaction
between individuals of the same group at time $t$ and interactiion of
individuals with the average stance of the other groups at the earlier time
$t-1$. We believe however that qualitatively we should find the same kinds of
polarization if we modify our model to match with his assumptions.

We note also that there exist several other physical models that can be mapped
into the social language to describe social phenomena. Let us mention that the
polarization between charged particles [50] can be seen as a social
polarization, or the mean-field approach treating the separation of ionic
liquids described via the Cahn-Hilliard term in a regular solution can be also
seen as a collective polarization [51].

Once strong polarizaton is present, we need to search for ways to reduce it,
namely to depolarize the society. We have used the Blume-Capel model from
statistical physics [52,53] to study depolarization using the mean-field
approximation [54]. As descibed below, this model has a term (called $D$-term)
which favors the neutral position in each individual, which may collectively
reduce polarization.

In [54], agents’ interactions had an infinite range (mean-field), meaning that
each individual interacting with each of the others in a political system with
three groups. As in [48], here we extend our work by assuming, instead, that
individuals interact only with their “neighbors”. We explore the insights to
be gained with the short-range interactions assuming a Bravais lattice, which
may be more realistic in terms of how individuals communicate and try to
persuade others to their political stance. Moreover, this kind of short-range
interaction matches a “massively parallel” approach proposed by [55] as a
practical means of reducing polarization. Our model may help assess the extent
to which the massively parallel approach can be effective in reducing
polarization. Note that agent-based modeling has been used to study attitude
change in societies [56].

In section II we describe the initial Blume-Capel model [54] and its
counterpart short-range model we use for Monte Carlo simulations in this
paper. In section III we present the simulation results and discuss their
meaning in terms of scenarios of depolarization trends. We conclude in section
IV with a summary.

##  II Model and Method

###  II.1 Model

Let us briefly explain the origin of depolarization in our recent paper
([54]), where we have used agent-based modeling to extend a sociophysics
2-group network model of conflict dynamics [38] to three political groups in
the US: Democrats (group 1), Republicans (group 2), and Independents (group
3).

To describe the political stance of an individual in a society, we use a spin
$S$ model, where the attitude $S$ ranges between -1 (extreme left) and 1
(extreme right). An individual’s stance can take any value on the continuum
within this range. Such a model is called a ”continuous Ising model” ([57],
[58]), as opposed to the discrete Ising model, where $S$ could only take two
values, -1 and 1. We use this continuous $S$ model here in the Blume-Capel
model described below. To compare with the mean-field approximation [54], we
use in this paper MC simulations with short-range interactions between
individuals, with real-time fluctuations.

Each individual in group $i$ ($i=1,2,3)$ has a stance compatible with the
group’s attitude $S_{i}$ regarding a specific issue under debate– economics,
social issues, defense, etc.—or (here) a package of such issues (in the [1]
and [6] sense). The individual stances have values between -1 and +1, where -1
corresponds to the democrats/progressive/left position ($i=1$), while +1
corresponds to the republicans/conservative/right position $(i=2)$.
Individuals thus align with the group whose average stance is compatible and
closest to their own [1].

Inside groups 1 and 2, individuals are homophilic [5]: they tend to prefer to
communicate with each other, rather than with individuals from a different
group. We denote $J_{i}$ the link between members of group $i$. It quantifies
the cohesiveness of group $i$. Through $J_{i}$, members inside each group
attempt to persuade each other to their own stance, effectively diminishing
intra-group differences and causing stances to converge.

Individuals in each group also keep an eye on the other groups’ average
attitudes, which in turn influence their own, either nudging the group average
to a more extreme value or to a more moderate one. These inter-group
interactions are described by parameters $K_{ij}$. For group 1, the inter-
group interaction terms, -$K_{12}S_{1}<S_{2}>$ and -$K_{13}S_{1}<S_{3}>$,
represent the influence of the mean stances of groups 2 and 3, $<S_{2}>$ and
$<S_{3}>$ respectively, on an individual in group 1. The inter-group
interactions $K_{12}$ and $K_{21}$ are not necessarily equal. At times,
members of one group may feel cooperative toward the other, who might not
reciprocate. Therefore, in general, $K_{ij}\neq K_{ji}$ because of human
agency. While physics phenomena obey Newton’s third law, the magnitudes of
human action and reaction do not have to be equal. Rather, the effect of group
$i$ on group $j$ can be different in magnitude and sign from the effect group
$j$ has on group $i$. Hence our model is not described by a single Hamiltonian
and its dynamics is not the Glauber dynamics (our spin is not Ising +/-1 but
is continuous). A temperature $T$, reflecting contextual factors, acts on each
individual indepently, the way the thermodynamic temperature acts on
particles.

The intra-group cohesion parameters $J$ and the inter-group influence
parameters $K$ affect the average group attitudes in time. For instance,
according to the recent Gallup polls [45], in early 2023 40% of adults
declared themselves independent—with zero internal cohesion $J_{3}$, since
they are not organized or formally linked, like Democrats or Republicans, but
rather a bin for the non-affiliated. However, in February 2023 all but 7% of
them leaned either Democrats or Republicans, at least partly in response to
persuasion efforts by the other two groups.

We also use a magnetic field $h_{i}$ to represents the effect of group i’s
leadership on group’s members. When $h_{i}>0$, group i’s mean stance is nudged
toward positive values; when $h_{i}<0$ the mean stance is nudged to negative
values.

The model’s Hamiltonian of group $i$ is inspired from the time-independent
Blume-Capel model. It is given by:

$H_{i}(t)=-J_{i}\sum_{m,n}S_{i}(m,t)\cdot
S_{i}(n,t)+D_{i}\sum_{i}S_{i}^{2}(m,t)-h_{i}\sum_{m}S_{i}(m,t)$ (1)

where $i$ indexes group $i$, and $S_{i}(m,t)$ is the stance of an individual
$m$ in group $i$ at time $t$. The sum is performed over the nearest neighbors
(NN) $m$ and $n$ belonging to group $i$. Note that for group 3 (Independents),
the first sum is zero because $J_{3}=0$. Also, a group at $t$ interacts with
the average stances of the other groups at $t-1$.

Note the that the positive sign of the $D$ term favors the small value of
$S_{i}(m,t)$ when $D_{i}$ is positive. Smaller $S_{i}$ causes smaller
polarization. In orther words, positive $D$ is at the origin of the
depolarization, as will be shown below.

When the three groups interact, the Hamiltonians of each group is as follows:

$\displaystyle{\cal H}_{1}(t)$ $\displaystyle=$ $\displaystyle
H_{1}(t)-K_{12}\sum_{m}S_{1}(m,t)<S_{2}(t-1)>$ (2) $\displaystyle-
K_{13}\sum_{m}S_{1}(m,t)<S_{3}(t-1)>$ $\displaystyle{\cal H}_{2}(t)$
$\displaystyle=$ $\displaystyle H_{2}(t)-K_{21}\sum_{m}S_{2}(m,t)<S_{1}(t-1)>$
(3) $\displaystyle-K_{23}\sum_{m}S_{2}(m,t)<S_{3}(t-1)>$ $\displaystyle{\cal
H}_{3}(t)$ $\displaystyle=$ $\displaystyle
H_{3}(t)-K_{31}\sum_{m}S_{3}(m,t)<S_{1}(t-1)>$ (4) $\displaystyle-
K_{32}\sum_{m}S_{3}(m,t)<S_{2}(t-1)>$

The model has the following parameters: three $J_{i}$, three $D_{i}$ for the
three groups’ respective internal cohesiveness and ”anisotropy”, three
$K_{ij}$ and three $K_{ji}$ describing the inter-group interactions (note that
$K_{ij}$ and $K_{ji}$ are not necessarily symmetric), and three $h_{i}$ to
describe leadership effects, if any. The $J$ and $K$ parameters can be
selected qualitatively, as we have done below, using publicly available poll
data (see also [6]; [39]; [33]).

We have solved this model using the mean-field approximation [54]. We define a
polarization measure $P$ as the distance between the mean stances of groups 1
and 2 at a given time $t$:

$P=(<S_{2}>-<S_{1}>)/2$ (5)

such that $-1\leq P\leq 1$. $<S_{i}>$ is the average individual stance of
group $i$ calculated at a time $t$. When $P=0$, there is no polarization. It
occurs when groups 1 and 2 have equal average stances $<S_{1}>=<S_{2}>$. When
$P=1$, polarization is extreme (also called hyperpolarization (e.g., Burgess
et al [17]). This can occur when Democrats’ stance $<S_{1}>=-1$ (most
progressive/left) and Republicans’ stance $<S_{2}>=1$ (most
conservative/right).$P$ can be negative if $<S_{1}>$ and $<S_{2}>$ change
their signs (it can be in politics).

Here, we use a similar model, but with short-range intra-group interactions
and perform MC simulations.

Before proceeding to the simulation method, let us discuss the role of the
“political” temperature $T$ introduced below, and borrowed from statistical
physics. The temperature in statistical physics represents thermal agitations
of the particles (spins, for example). Thus $T$ acts as a disordering factor:
at low $T$, particles stay in the lowest energy state (or very close to it),
while at high $T$, they vigorously change their state in an independent
manner, causing disorder in the system in spite of the inter-particle
interaction which favors order. The well-known example is the ferromagnets:
spins in ferromagnets are parallel at low $T$ but become disordered at high
$T$. In the context of political groups considered here, $T$ represents the
political ambiance of the society. When an election is not imminent, or the
society is calm, $T$ is low. Each group is relatively stable, with no
significant effect of inter-party interaction. Close to an election or during
politically fraught times with important issues at stake such as strained
economies or international tensions, intra-party cohesiveness may wane, due to
the fluctuation of individual stances of its members, equivalent to high
“political” temperature $T$. Then each group might attempt to take advantage
of the weakened cohesiveness of the other groups to enhance its influence in
the competition. As we shall see below, $T$ plays an important role in
outcomes of politic contests.

###  II.2 Simulation Method

We take the case of the US political system: there are three groups: Group 1
(Democrats), Group 2 (Republicans) and Group 3 (Independents). We assume Group
1 to have a stronger cohesiveness (largest $J$), and to be governing. Group 2
has weaker cohesiveness and is in opposition. Group 3 is composed of
individuals having no unified political framework, and no formal intra-group
communication links ($J_{3}=0$). The Independents are often attracted to the
stance of the opposition party, they play a contrarian role (see [59-61] for
other examples of contrarian used in a model).

Figure 1: Interaction parameters. Note that $J_{3}$, intra-group interaction
among the Independents, is zero. See text for comments.

For the MC simulations, we represent each of the three groups with a
triangular lattice of size $N\times N$, where each site is occupied by a
member. Each member interacts with its six nearest neighbors (NN) at time $t$,
and considers the average stances of the other groups calculated at $t$ – 1 (a
realistic lag). The choice of this lattice allows for a maximum number of NN
in 2D. Of course, we can use a 3D lattice to have more NN such as a FCC or a
HCP with 12 NN. However we believe it will not give new phenomena. For each
group, we use the periodic boundary conditions to reduce the size effects. In
general, we take the size of 100 × 100 lattice sites for each group. See
Figure 1 for the interaction parameters described in the previous subsection.
Note that the notion of NN interactions in politics does not necessarily mean
that people are geometrically close to each other. Rather, it refers to the
number of people generally in contact with an individual.

To simulate the three groups’ interactions at a given $T$, we carry out the
simulation as follows: for a group we generate initial individual stances -1
(Democrats) for group 1, 1 for group 2 (Replublicans), and 0 (Independents)
for group 3. We can also use the initial stances all equal to zero for three
groups. We use next the Metropolis algorithm to find the collective state of
each group at the time $t$, taking into account the average stances of the
other groups at $t-1$, as described in Eqs. (2)-(4). We follow the evolution
of each group with time $t$ (MC time).

The Metropolis algorithm used for updating the individual stance of a member
is as follows: at a time $t$, we calculate the interaction energy $E_{old}$ of
a member with its NN and with an effective field resulting from the two other
groups at time $t–1$. We make a trial change of its state by choosing a random
stance between -1 and 1. We calculate the member’s trial new energy $E_{new}$.
If $E_{new}<E_{old}$, the trial state is accepted. If $E_{new}>E_{old}$, it is
accepted with the probability $\exp[-(E_{new}-E_{old})/(k_{B}T)]$. We repeat
this updating procedure for all individuals in each of the three groups. Note
that the Metropolis algorithm obeys the detailed balance only when the system
is at equilibrium, namely when there is a probability conservation: state A to
state B has the same probability with that from B to A. Our purpose is to
study the time dependence of the polarization, so there is no such probability
conservation. Note that there are several popular dynamics such as Glauber
dynamics and Kawasaki dynamics, but to our knowledge all of them have been
devised for discrete spins, not for continuous spins used in this paper. The
advantage of the Metropolis algorithm is that it does not depend on the nature
of spin, it can be used for any kind of spin such as continuous spins used
here, XY spins or Heisenberg spins.

##  III Results and Discussion

As seen, our model has 9 principal interaction parameters $J_{i}(i=1,2,3)$ and
$K_{ij}(i\neq j,i=1,2,3,j=1,2,3$ in addition to $D_{i}(i=1,2,3)$ and
$h_{i}(i=1,2,3)$. However, in applications the choice of the parameters is
limited. As discussed in [47,48,54], this choice is guided by polls [45,46]
and by political common attitudes of the people: to produce anticipatory
scenarios of polarization, we made the following assumptions:
-The Democrats (group 1) are more cohesive than Republicans (group 2), i.e., $J_{1}>J_{2}$;
-Independents (group 3) have no cohesion ($J_{3}=0$) because they have no structure or means of identifying with each other, do not communicate, and do not recruit; therefore, they exert no influence on the other two groups and, as such,$K_{13}=K_{23}=0$;
-Independents tend to be contrarian to the party in power (here, group 1), thus $K_{31}<0$, and are not influenced by the opposition party, thus $K_{32}=0$.

With respect to parameter value selection, guided by media and professional,
frequent polling reports, we assigned parameter values such that they
qualitatively mimic general polls results [45,46]. To enable a comparison of
MC results with those obtained with the MFT model [54], we selected the same
values for parameters $J$ and $K$, as follows:

\- Intra-group interactions: $\displaystyle J_{1}=5,J_{2}=3,$ (6)
$\displaystyle J_{3}=0$ \- Inter-group interactions: $\displaystyle
K_{12}=-4,K_{21}=-5,$ $\displaystyle K_{13}=0,K_{31}=-3,$ $\displaystyle
K_{23}=0,K_{32}=0.$ (7)

\- Depolarization parameters $D_{i}$: we will choose several cases presented
below.

Note that a negative $K_{ij}$ indicates hostility (or resistance) of group $i$
toward group $j$, while a positive sign indicates attraction or potential
agreement between two groups. A variation of the above values keeping their
signs will not alter qualitatively the results shown in the following.

Each group is thermalized at temperature $T$ in interaction with the other
groups. We have calculated the following quantities:

\- Cohesive energy per individual $E_{i}(T)=<{\cal H}_{i}(T)>/N^{2}$ where
$<{\cal H}_{i}(T)>$ is the thermal average at $T$ given by

$<{\cal H}_{i}(T)>=\sum_{t=t1}^{t2}{\cal H}_{i}(t)/(t2-t1)$ (8)

where $t1$ is the starting averaging time and $t2$ the averaging end time. The
total cohesive energy $E(T)\sum_{i}E_{i}(T)$ is also calculated.

\- Stance of each group (sublattice magnetization) as a function of $T$:

$M_{i}(T)=<S_{i}(T)>=\sum_{t=t1}^{t2}\sum_{n}S_{i}(n,t)/(t2-t1)/N^{2}$ (9)

where $n$ belongs to group $i$. Within the assumption of the parameters given
above one has $M_{1}<0$, $M_{2}>0$. In the absence of $D$, $M_{3}>0$. We
define the strength of group $i$ by $Q_{i}(T)=|<S_{i}>|,$

\- Susceptibility or fluctuations of the stance of group $i$ at $T$:

$\chi(T)=[<M_{i}(T)^{2}>-<M_{i}(T)>^{2}]/(k_{B}T)$ (10)

####  III.0.1 The case $D_{1}=D_{2}=0$ and $D_{3}=5$

With $K_{31}=-3$ (Group 3 is contrarian to Group 1), if $D_{3}=0$ the stance
of Group 3 should be positive, though small but visible as seen in Fig. 2a.
But when $D_{3}=5$ as considered here, the stance of Group 3 is neutralized by
$D_{3}$. It is zero as shown in Fig. 2b.

Figure 2: Stances of the three groups with $D_{1}=D_{2}=0$ and (a) $D_{3}=0$;
(b) $D_{3}=5$ as function with MC time. Other interaction parameters are given
in Eqs. (6)-(7). Curves 1, 2 and 3 correspond respectively to $<S_{1}>$,
$<S_{2}>$ and $<S_{3}>$. Note that $J_{3}$, intra-group interaction among the
Independents, is zero. See text for comments.

####  III.0.2 The case $D_{1}=D_{2}=3$ and $D_{3}=5$

Figure 3: (a) Internal energy of three groups, curves 1, 2 and 3 correspond
respectively to Group 1, Group 2 and Group 3; (b) Total energy per individual
; (c) Total specific heat per individual, as functions of political
temperature $T$. See text for comments.

The equilibrium cohesive energy of each group is shown in Figure 3a, the total
cohesive energy of three groups is displayed in Figure 3b as a function of
temperature $T$. Since the three groups interact with each orther, there is
only a transition temperature $T_{C}\simeq 6.93$ where all of them become
disordered (the energy changes its curvature at this point). The specific heat
per individual is shown in Figure 3c where the peak temperature corresponds to
$T_{C}$. In terms of sociophysics, above $T_{C}$ there is no cohesiveness
between individuals. The society is in a turmoil state. Note that the energy
of Group 3 is due to its interaction with group 1. This is confirmed in Figure
4, showing the absolute values of stances $Q_{1}=|<S_{1}>|$, $Q_{2}=|<S_{2}>|$
and $Q_{3}=|<S_{3}>|$: we see that $Q$ is higher for larger $J$, and all drop
to zero (i.e., no cohesiveness) at $T_{C}$. In statistical physics, $T_{C}$ is
called transition temperatures [58] above which the systems become disordered.
At and close to $T_{C}$, the stances of the groups strongly fluctuate as shown
in Fig. 5. These fluctuations of the order parameter in statistical physics
correspond to the so-called susceptibilities which are the fluctuations of
$M_{i}$, namely $(<M_{i}^{2}>-<M_{i}>^{2})/(k_{B}T)$.

Figure 4: Strengths $Q$ of three groups as functions of political temperature
$T$. Curves 1, 2 and 3 correspond respectively to Group 1, Group 2 and Group
3. Figure 5: Fluctuations of the three groups’ stances as functions of
political temperature $T$. Curves 1, 2 and 3 correspond respectively to Group
1, Group 2 and Group 3.

Let us show now the polarization with evolving time $t$. Starting from the
highest state of polarization $P=1$, namely $<S_{1}>=-1$, $<S_{2}>=+1$, we see
that $P$ diminishes to 0.89 (see Fig. 6a) calculated at a low temperature
$T=2.254$, and to 0.58 at $T=5.762$. These values are much lower than the case
where there is no depolarization $D$ (see [48]). The role of $T$ is very
important since $P$ strongly depends on $T$.

Figure 6: Political polarization as a function of time $t$ at (a)
$T=2.254\simeq T_{C}/3$; (b) $T=5.762$ close to $T_{C}$. See text for
comments.

####  III.0.3 The case of higher $D_{1}$, $D_{2}$ and $D_{3}$

Let us take the cases of stronger depolarization $D_{1}=D_{2}=D_{3}=5$
($T_{C}=5.84$) and $D_{1}=D_{2}=D_{3}=10$ ($T_{C}=3.29$) . We show in Fig. 7
the polarization versus time $t$ .

Figure 7: Political polarization as a function of time $t$ at (a)
$D_{1}=D_{2}=D_{3}=5$ at $T=1.90<<T_{C}$; (b) for $D_{1}=D_{2}=D_{3}=10$ at
$T=3.2$ slightly above $T_{C}$. See text for comments.

If we go close to $T_{C}$ in all cases, we have a strong damping of the group
stances and of the polarization versus $t$. We show in Fig. 8 one example in
the case $D_{1}=D_{2}=D_{3}=5$ at $T=6$ slightly above $T_{C}$.

Figure 8: The case $D_{1}=D_{2}=D_{3}=5$: (a) Stances of three groups versus
$t$ at a political temperature $T$ higher than $T_{C}$, curves 1, 2 and 3
correspond respectively to $<S_{1}>$, $<S_{2}>$ and $<S_{3}>$. (b)
Polarization.

.

####  III.0.4 Strong oscillation of the polarization

One of the interesting cases is when the interactions between Group 1 and
Group 2 have the opposite signs. Let us take the case $D_{1}=D_{2}=3$ and
$D_{3}=5$ and $K_{21}=6$ and $K_{12}=-5$. We show in Fig. 9 the oscillating
stances of the three groups and the oscillation of the polarization. This
phenomenon occurs in a large temperature region below $T_{C}$. The amplitude
of the oscillation depends on the values of $K_{21}$ and $K_{12}$.

Figure 9: (a) Political stances as a function of time $t$, curves 1, 2 and 3
correspond respectively to $<S_{1}>$, $<S_{2}>$ and $<S_{3}>$; (b)
Polarization versus $t$. The parameters are $D_{1}=D_{2}=3$, $D_{3}=5$ with
$K_{21}=6$ and $K_{12}=-5$. See text for comments.

The attraction of the Republicans to the Democrats (positive $K_{21}$) is the
origin of the oscillation of $P$ in time. We recall that $P$ measures the
distance between the political stances.

At this stage, it is worth to emphasize that oscillation phenomena are found
in several systems due to the tendency of the matter to self-organize: the
phenomenon was first discovered by Boris Belousov in 1951, while he was trying
to find the non-organic analog to the Krebs cycle. When mixing potassium
bromate, cerium(IV) sulfate, malonic acid, and citric acid in dilute sulfuric
acid, he observed that the ratio of concentration of the cerium(IV) and
cerium(III) ions oscillated, causing the colour of the solution to oscillate
between a yellow solution and a colorless solution. This is due to the change
of cerium(IV) ions by malonic acid into cerium(III) ions, which are then
oxidized back to cerium(IV) ions by bromate(V) ions. The reader is referred to
Ref. [62] for a review. In our model, the oscillation occurs when a party is
attracted so far from its equilibrium by the other party, its intra-cohesive
energy gets it back. It is some kind of a restoring force analog to the
oscillatory motion of a pendulum.

To conclude this section, we have performed MC simulations on the same
statistical physics model as the one where we used the mean-field
approximation (see [54]). Despite the fact that the mean-field model neglects
fluctuations while MC simulations take into account space and time
fluctuations, the two methods yield qualitatively the same patterns of
political polarization.

##  IV Conclusion

We have proposed the Blume-Capel model to explore, using MC simulations,
whether depolarization is possible between Democrats and Republicans in the
USA as a function of time. The same model has been studied by mean-field
theory by our group [54]. We have considered three groups with initial
different political stances: Democrats, Republicans, and Independents. An
individual within any of these groups interacts with a limited number of
people sharing the same political viewpoint. At any time, individuals also
consider the average stance of other groups in the previous time period,
causing them to either become firmer or soften their stance. Although the
model represents the political structure in the USA, it can be adapted to
other three-group dynamics.

We find that MC results for short-range intra-group interactions agree well
with those obtained by the mean-field which assumed long-range interactions.
The Blume-Capel model assumes that each individual is represented by a
continuous Ising spin taking its values from -1 (left, liberal orientation) to
+1 (right, conservative orientation). What is important in our model is the
fact that there is the $D$ term in the Hamiltonian, which can soften the
opposite stances, which we called depolarization.The model shows that the
depolarization stems from the effect of the $D$ term on individual
orientations. It may be more efficient/durable than collective measures taken
by party leaders.

The MC simulation results show that polarization depends on $D$ and on the
nature of the inter-group interactions. It may advantage the party in
opposition and help it win an election. It may also give rise to an
oscillation of the polarization (whose sign changes in time). Therefore, the
outcome of an election depends on the moment in time when it occurs. It is
interesting to note that the MC and mean-field models yield qualitatively very
similar results with the same depolarization dynamics. Both approaches can be
used to generate scenarios that include various interventions to reduce
polarization. The MC near-neighbor approach lends itself to generating
scenarios for another kind of intervention proposed by [55] and [17], who have
called it “massively parallel.” It consists of independent individuals and
groups operating locally to reach out and initiate dialogues with people
holding opposite stances, thereby reducing the current acute homophily. Such
initiatives are already taking place around the USA (see [55]).

The two versions of the Blume-Capel models–mean-field, with infinite-range
interactions and Monte Carlo method, with near-neighbor interactions, show
ways to depolarize society and find practical actions which might correspond
to various $D$ values. Some on-going empirical studies and opinion polls will
provide data which we plan to include qualitatively in our models to refine
the scenarios we can generate, and make them relevant to spceific contexts
such as the United States and France.

In conclusion, we have illustrated how agent-based models from statistical
physics which contain sufficient ingredients can concisely describe complex
situations in social sciences which may appear intractable (for example, in
terms of number of variables and data) when studied with traditional, non-
dynamic methods.

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# A note on non-inclusion “interval” root solvers

Miodrag S. Petković 1,, Ljiljana D. Petković 2,111Corresponding author
1Faculty of Electronic Engineering, University of Niš,
A. Medvedeva 14, 18000 Niš, Serbia
2Faculty of Mechanical Engineering, University of Niš,
A. Medvedeva 14, 18000 Niš, Serbia

###### Abstract

In this note we give comments on recent interval extensions of the King-
Ostrowski multipoint methods for solving nonlinear equations published in at
least five scientific journals offering faulty “self-validated” methods. We
show that they do not possess essential feature of genuine interval methods –
inclusion property.
AMS Mathematical Subject Classification (2010): 65G20, 65H05, 30C15
Key words and phrases: Interval methods; Nonlinear equations; King-Ostrowski
method; Zero-enclosure; Convergence order

††E-mail address<EMAIL_ADDRESS>
##  1 Introduction

In this note we wish to draw attention to erroneous treatment of interval
methods for the inclusion of an isolated zero of a given real function,
published in the recent papers [2]–[6] and maybe in some other journals or
proceedings, unavailable to the author of this note. Using a simple derivation
of relations necessary for the development of genuine inclusion methods (also
called interval or self-validated methods), examples and counterexamples, we
point to the key mistakes made in the cited papers. Beside these comments, we
also wish to prevent possible construction of new methods that suffer from the
lack of the most important feature of interval methods – inclusion property.
The mentioned mistakes are mainly made due to the careless and incorrect
application of properties of interval arithmetics and the mean value theorem.
The basic operations and properties of real interval arithmetic can be found
in the books [7]–[10] and many papers cited therein so that we do not list
them in this note.

As well known, the main advantage of interval methods for the enclosure of an
isolated zero $x^{*}$ of a given function is the ability to produce a sequence
of nested intervals $X_{0}\subset X_{1}\subset X_{2}\subset\cdots$ such that
$x^{*}\in X_{k}$ for all $k=0,1,\ldots\;.$ Methods start from an initial
interval $X_{0}\owns x^{*}$ and terminate when the given stoping criterion is
satisfied.

Let
$X=[\underline{x},\overline{x}]\;(\underline{x},\overline{x}\in{\mathbb{R}},\;\underline{x}\leq\overline{x}])$
be a closed real interval. The set of all closed real intervals will be
denoted by $I({\mathbb{R}})$ and the members of this set by capital letters
$A,B,C,\ldots\ .$ For a real function $f,$ define its interval extension
$f(X)=\\{f(x)\;|\;x\in X\\}$ that satisfies the conditions

$\displaystyle f([x,x])=f(x)\quad(\mbox{\rm restriction}),$ $\displaystyle
f(x)\in f(X)\quad(\mbox{\rm inclusion property}).$ (1)

The goal of the papers [2]–[6] was to extend Ostrowski’s and King’s two-point
methods (applicable in ${\mathbb{R}}$) to the set $I({\mathbb{R}}).$ Both
methods can be expressed by the unique iterative formula

$\left\\{\begin{array}[]{l}$$y_{k}=x_{k}-\dfrac{f(x_{k})}{f^{\prime}(x_{k})},\\\\[10.0pt]
x_{k+1}=y_{k}-\dfrac{f(x_{k})+\beta
f(y_{k})}{f(x_{k})+(\beta-2)f(y_{k})}\cdot\dfrac{f(y_{k})}{f^{\prime}(x_{k})},\end{array}\quad(k=0,1,2,\ldots),\right.$
(2)

where $\beta$ is a real parameter. Formula (2) defines two-point King’s method
[11] of order four. Ostrowski’s method [12] is obtained from (2) as a special
case for $\beta=0.$

Let $X=[\underline{x},\overline{x}],$ then we define the midpoint and semi-
width of $X$ as

$\mbox{\rm mid\,}(X)=\frac{\underline{x}+\overline{x}}{2},\ \ \mbox{\rm
rad\,}(X)=\frac{\overline{x}-\underline{x}}{2},$

respectively. For two intervals $X_{k}$ and $Y_{k}$ let us introduce the
abbreviation

$t_{k}=\frac{f\bigl{(}\mbox{\rm mid\,}(Y_{k})\bigr{)}}{f\bigl{(}\mbox{\rm
mid\,}(X_{k})\bigr{)}}.$

The following two-point method was constructed in [2], [4], [6] as the
interval extension of (2)

Given $X_{0}\owns x^{*},$ for $k=0,1,2,\ldots$ calculate

$\left\\{\begin{array}[]{l}Y_{k}=N(X_{k})\cap X_{k},\\\
K(X_{k},Y_{k})=\mbox{\rm mid\,}(Y_{k})-\dfrac{1+\beta
t_{k}}{1+(\beta-2)t_{k}}\cdot\dfrac{f\bigl{(}\mbox{\rm
mid\,}(Y_{k})\bigr{)}}{F^{\prime}(X_{k})},\\\\[7.0pt]
X_{k+1}=K(X_{k},Y_{k})\cap X_{k},\end{array}\right.$ (3)

referred as Interval Ostrowski’s method for $\beta=0,$ and Interval King’s
method for $\beta\in{\mathbb{R}}$, see [3], [5]. By the same approach the
authors constructed the three-point method

Given $X_{0}\owns x^{*},$ for $k=0,1,2,\ldots$ calculate

$\left\\{\begin{array}[]{l}Y_{k}=N(X_{k})\cap X_{k},\\\
K(X_{k},Y_{k})=\mbox{\rm mid\,}(Y_{k})-\dfrac{1+\beta
t_{k}}{1+(\beta-2)t_{k}}\cdot\frac{f\bigl{(}\mbox{\rm
mid\,}(Y_{k})\bigr{)}}{F^{\prime}(X_{k})},\\\\[7.0pt]
Z(X_{k})=K(X_{k},Y_{k})\cap X_{k},\\\ M(X_{k},Y_{k},Z_{k})=\mbox{\rm
mid\,}(Z_{k})-\dfrac{1+\beta
t_{k}}{1+(\beta-2)t_{k}}\cdot\dfrac{f\bigl{(}\mbox{\rm
mid\,}(Z_{k})\bigr{)}}{F^{\prime}(X_{k})},\\\ X_{k+1}=M(X_{k},Y_{k},Z_{k})\cap
X_{k},\\\ \end{array}\right.$ (4)

referred as Interval modified Ostrowski method for $\beta=0$ in [4], [6].

##  2 Construction of self-validated methods

Mistakes given in the cited papers [2]–[6] can be divided into two groups:

(I) The loss of inclusion property due to the wrong interpretation of the mean
value theorem.

(II) The wrong estimation of the convergence order of the proposed interval
methods as a consequence of incorrect application of properties of interval
arithmetics and negligent estimation procedure.

First, we analyze the group (I) of mistakes that are of decisive importance.
As in the papers [2]–[6], in this note we also consider only real-valued
functions $f$ differentiable on a specific interval $X=[a,b]\in
I({\mathbb{R}})$ such that $f(a)f(b)<0.$ This means that there is at least one
zero $x^{*}$ in $X.$ Furthermore, assume that $f$ is monotonically increasing
(decreasing) on $X$, that is, $f^{\prime}(x)>0$ ($f^{\prime}(x)<0$) for all
$x\in X$. According to the previous conditions it follows that there exists
only one zero $x^{*}$ in $X$ and $0\notin f^{\prime}(X).$

For two numbers $a,b\in{\mathbb{R}}$ define the interval
$I(a,b):=[\min\\{a,b\\},\max\\{a,b\\}]$ and let $I(x,x^{*})\subseteq X\
(x,x^{*}\in X).$ According to the mean value theorem (shorter MVT) we have

$f(x)=f(x^{*})+(x-x^{*})f^{\prime}(\xi),\ \ \xi\in I(x,x^{*}),$ (5)

or (since $f(x^{*})=0$)

$f^{\prime}(\xi)=\frac{f(x)}{x-x^{*}}.$ (6)

Simple geometrical interpretation of MVT shows that given any chord of a
smooth curve, we can find a point (denoted here with $\xi$) lying between the
end-points of the chord such that the tangent at that point is parallel to the
chord. From (6), by using inclusion property (1), we obtain

$x^{*}=x-\frac{f(x)}{f^{\prime}(\xi)}\in
x-\frac{f(x)}{f^{\prime}(I(x,x^{*}))}\subseteq
x-\frac{f(x)}{F^{\prime}(X)}=:N(X),$ (7)

where $N$ is the Newton operator. $F^{\prime}(X)$ is an interval extension of
$f^{\prime}(I(x,x^{*}))$, that is, $F^{\prime}(X)\supseteq
f^{\prime}(I(x,x^{*})).$ Clearly, we are forced to use $F^{\prime}(X)$ since
$f^{\prime}(I(x,x^{*}))$ is unknown interval since $x^{*}$ is unknown. Note
that
$F^{\prime}(X)=I\bigl{(}f^{\prime}(\underline{x}),f^{\prime}(\overline{x})\bigr{)}$
since $f$ is monotone on $X=[\underline{x},\overline{x}].$

Taking $x=\mbox{\rm mid\,}(X)$ in (7) R. E. Moore constructed in [9] (see,
also, [7]–[10]) the following inclusion method of Newton’s type:

Given $X_{0}\owns x^{*},$ for $k=0,1,2,\ldots$ calculate

$\displaystyle\left\\{\begin{array}[]{l}N(X_{k})=\mbox{\rm
mid\,}(X_{k})-\dfrac{f\bigl{(}\mbox{\rm
mid\,}(X_{k})\bigr{)}}{F^{\prime}(X_{k})}\\\\[7.0pt] X_{k+1}=N(X_{k})\cap
X_{k}.\end{array}\right.$ (10)

The Moore-Newton interval method (10) is presented here only to demonstrate
essential characteristics of inclusion methods. Geometrical interpretation of
the interval method (10) is displayed on Fig. 1 (shaded area and the interval
$Y$), where the tangents at the endpoints are translated to the midpoint
$m(X).$ For clear demonstration, the simple function $f(x)=x^{2}-4$ and the
initial interval $X=[1,4]\owns x^{*}=2$ have been taken.

Fig. 1 The Moore-Newton interval method: geometrical interpretation

##  3 Lost inclusion property and incorrect convergence order

The key mistake in the construction of the methods (3) and (4), which causes
the lack of the inclusion property, is the use a “variation” of MVT (5) (given
by (11)) instead of MVT (5). In fact, the authors of the cited papers started
from the relation

$f(x)=f(x^{*})+(x-x^{*})cf^{\prime}(\xi),\ \ \xi\in I(x,x^{*}),$ (11)

where $c\neq 1$ is a real number that depends on the values of the function
$f$ evaluated at two points belonging to two successive intervals from the
sequence $\\{X^{(k)}\\}.$ Since $f(x^{*})=0,$ from (11) we obtain

$f(\xi^{\prime})=\frac{f(x)}{c(x-x^{*})}.$ (12)

The three-point method (4) has a similar form as the two-point method (3) so
that we will analyze only the method (3). For simplicity, we will omit the
iteration index $k.$ Without loss of generality, we will deal with arbitrary
points $x\in X=[\underline{x},\overline{x}]$ and $y\in
Y=(\underline{y},\overline{y}),$ including $x=\mbox{\rm mid\,}(X)$ and
$y=\mbox{\rm mid\,}(Y)$ (standing in (3)). The value $c$ appearing in (3) and
(4) is given by

$c=\frac{1+(\beta-2)t}{1+\beta t},\ \ t=\frac{f(y)}{f(x)}\ \ \
(\beta\in{\mathbb{R}}).$

Figure 2 The failure of Ostrowski-like “interval” method

Let us consider the second step of (3) (suppressing the index $k$) in the form

$K(X,Y)=x-\frac{1+\beta
t}{1+(\beta-2)t}\cdot\frac{f(y)}{f^{\prime}(I(\underline{x},\overline{x}))}=x-\frac{f(y)}{cf^{\prime}(I(\underline{x},\overline{x}))}=x-\frac{f(y)}{I\bigl{(}cf^{\prime}(\underline{x}\bigr{)},cf^{\prime}(\overline{x})\bigr{)}}.$
(13)

Since $t=f(y)/f(x)\neq 0$ there follows $c\neq 1.$ Hence we observe that the
slopes $cf^{\prime}(\underline{x})$ and $cf^{\prime}(\overline{x})$ do not
coincide with the tangents at the end-points $\underline{x}$ and
$\overline{x}$, used in (10). The new interval $K(X,Y)$ is now shifted and it
does not have to contain the zero $x^{*}$, as displayed in Fig. 2b. In
essence, MVT (5), which preserves the inclusion property, differs from the
relation (11) which leads to the described unfavorable situation. As a
consequence, the inclusion property of the method (3) is not guaranteed.

Geometrical interpretation (taking the function $f(x)=x^{3}+x^{2}-12$ and the
interval $[0.5,2.1]$ for demonstration) is displayed in Fig. 2. Observe that
the new interval $K(X,Y),$ constructed by green and red lines, does not
contain the zero $x^{*}=2.$ Numerical data are given in Example 1. For the
presented example the method (3) fails. The same would happen with the method
(4).

Remark 1. According to the results presented in the papers [2]–[6], we
conclude that the proofs of inclusion property are either wrong (see [2], [6],
[4]) or omitted (see [3], [5]). The wrong proofs are the consequence of
careless and incorrect manipulations with the basic arithmetic operations,
incorrect relations and inequalities, as well as the use of “variation” of
MVT. These mistakes are elementary so that we skip details.

Remark 2. The basic (wrong) idea presented in [5] (published 2015) is the same
as in [3] and [4]. Furthermore, a similar (wrong) idea exposed in [6] (2015)
appeared previously in [4] (2013). It is hard to believe that the same methods
are discovered independently since the latter authors cited the former
authors. However, in this note we do not intend to discuss ethical principles.
By the way, such kind of job is not necessary since the results of all papers
[2]–[6] are wrong.

Remark 3. To provide inclusion intervals in each iteration, in [2] and [4],
the authors need to check the conditions (i) $K(X_{k},Y_{k})\subset X_{k}\
(\owns x^{*})$ for (3) and (ii) $\\{K(X_{k},Y_{k})\subset X_{k}\ (\owns
x^{*})\ \wedge\ M(X_{k},Y_{k},Z_{k})\subset X_{k}\ (\owns x^{*})\\}$ for (4)
in each iteration. This is unacceptable iteration model, not only very
expensive but also contrary to standard iterative processes. What will happen
if (i) and (ii) do not hold for some $k$?

Remark 4. As numerical examples presented in [2]–[5] have shown, it is
possible to run the methods (3) and (4) in such a way that they produce
intervals containing the sought zero. Such situations exist in the case when
the factor $c$ in (3) and (4) is close to 1, which will happen when $\mbox{\rm
rad\,}(X)$ is sufficiently small so that the value $t=f(y)/f(x)=O(\mbox{\rm
rad\,}(X))$ is also small in magnitude. Then the dislocation of the shifted
interval
$cF^{\prime}(X)=I(cf^{\prime}(\underline{x}),cf^{\prime}(\overline{x})$ may be
small enough so that the intervals $K(X,Y)$ and $M(X,Y,Z)$ enclose the zero
$x^{*}$. However, since $c\neq 1$ (although $c$ can be very close to 1), using
multiple-precision arithmetic in order to obtain inclusion intervals of very
small widths it is possible that the zero $x^{*}$ comes out from some interval
in later iterations, see Example 3. Such unfavorable case could not have
happened if a proper interval method had been applied (e.g., the Moore-Newton
method (10)). Simply, as mentioned in Remark 1, the methods (3) and (4) cannot
guarantee the enclosure of a zero in each iteration.

Among many counterexamples we present two of them which, together with the
above discussion and Fig. 2, confirm that the methods published in [2]–[6] do
not guarantee the enclosure of the sought zero by produced intervals.

Example 1. We take $\beta=0$ in (3), that is, we wish to iterate “interval”
Ostrowski-like method (3). Let us find the enclosure for the zero $x^{*}=2$ of
the function $f(x)=x^{3}+x^{2}-12$ on the interval $X_{0}=[0.5,2.1].$ Since
$f^{\prime}(x)=3x^{2}+2x>0$ for all $x\in X_{0},$ $f$ is monotonically
increasing on this interval and $0\notin F^{\prime}(X_{0}).$ This means that
the division by zero-interval does not appear in (3) and (4). First, we find

$N(X_{0})=\mbox{\rm mid\,}(X_{0})-\dfrac{f(\mbox{\rm
mid\,}(X_{0}))}{F^{\prime}(X_{0})}=1.3-\frac{f(1.3)}{[f^{\prime}(0.5),f^{\prime}(2.1)]}=1.3-\frac{-8.113}{[1.75,17.43]}=[1.76546,5.936].$

Hence $x^{*}\in N(X_{0})$ (the inclusion holds, see Fig. 2a) and the process
can be continued. First, we find

$Y_{0}=N(X_{0})\cap X_{0}=[1.76546,5.936]\cap[0.5,2.1]=[1.76546,2.1]$

and calculate in interval arithmetic

$K(X_{0},Y_{0})=\mbox{\rm mid\,}(Y_{0})-\frac{1}{1-2f(\mbox{\rm
mid\,}(Y_{0}))/f(\mbox{\rm mid\,}(X_{0}))}\cdot\frac{f(\mbox{\rm
mid\,}(Y_{0}))}{F^{\prime}(X_{0})}=[2.01348,2.73702]$

(see Fig. 2b). We observe that $x^{*}=2\notin K(X_{0},Y_{0}))$ and the
Ostrowski-like method fails at the beginning of iterative process.

Example 2. King-like method (4) breaks down for the function $f(x)=x^{3}-8,$
the initial interval $X_{0}=[1.5,2.3]$ and $\beta=5.$ One obtains
$K(X_{0},Y_{0})=[2.024393,2.029699]\not\owns x^{*}=2.$

Example 3. To examine the extent of validity of the King-like method (3), we
have tested the monic polynomials of the 7th degree of the form

$P(x)=(x-1)(x^{6}+a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}),$

where the coefficients $a_{0},\ldots,a_{5}$ have been chosen randomly from the
interval (0,1) using the statement Random[ ] in computer algebra system
Mathematica. The initial interval $X_{0}=[0.6,3]$ containing the zero
$x^{*}=1$ has been chosen. We have tested 100 random polynomials with the
values of the parameter $\beta=-2+0.5m\ (m=0,1,\ldots,9).$ In this way, we
have tested 1000 examples in one experiment and found that the King-like
method (3) fell in 643 examples. Clearly, the number of failures depends on
the choice of the set $\\{a_{0},a_{1},a_{2},a_{3},a_{4},a_{5}\\}$ of random
coefficients. For this reason we have performed 20 experiments with 1000
examples (in total, 20 000 examples) and found that the percentage of failures
ranged between 55% and 65%.

Now we return to the group of mistakes (II). In essence, this group of errors
is irrelevant since the presented interval extensions do not guarantee the
enclosure of the sought zero and we could omit any analysis of further
mistakes (II). However, we will discuss in short the convergence order given
by theorems from the papers [2]–[6], as if the methods (3) and (4) had been
properly constructed (see Remark 4). The following academic question arises:
If the methods (3) and (4) could preserve the inclusion property during the
iterative process, what would be the order of convergence of the semi-widths
of produced inclusion intervals?

Let $X=[\underline{x},\overline{x}],$ and let $\mbox{\rm
rad\,}(X)=(\overline{x}-\underline{x})/2$ be the semi-width of the interval
$X.$ Similar denotation holds for the intervals appearing in (3) and (4). The
determination of convergence order of the Ostrowski-like method (equal to 4)
and the modified Ostrowski-like method (equal to 6) has been presented in [4]
and cited in [6] for the same methods. These iterative methods are given by
(3) and (4) (respectively) for $\beta=0.$ A very similar approach for finding
the order of the King-like method (3) ($\beta\in{\mathbb{R}}$) (equal to 4)
has been given in [3] and cited in [5]. All of these results are wrong! The
main reason for mistakes is careless and incorrect application of properties
of interval arithmetics, as mentioned at the beginning of this note. Although
the mistakes of type (I), the lack of inclusion property, make the “interval”
methods considered in [2]–[6] pointless, we give anyway the results of our
numerical experiments concerned with the convergence order.

Take for the moment $c=1$ in the method (2) realized in real arithmetic. Then
the order of the iterative method is 3 (see [13, Example 2.2]). The use of
corrective factor $c=(1+(\beta-2)t)/(1+\beta t)$ in (2) leads to the King
method (2) of order 4. Numerical experiments have confirmed that the
corrective factor $c$ also accelerates the King-like method (3) if this method
runs correctly. However, the application of $c=(1+(\beta-2)t)/(1+\beta t)$ in
King-like formula (3) (defined in $I({\mathbb{R}})$) shifts the interval
$F^{\prime}(X)$ which can cause the loss of inclusion property.

To estimate the increase of the order of convergence of King-like method (3)
that produces inclusion intervals in each iteration, we have tested 20
functions for the parameter $\beta=-2+0.5m\ (m=0,1,\ldots,9),$ furnishing in
this way 200 examples. To measure convergence rate of the sequence of
intervals $\\{X_{k}\\}$, we have applied the so-called computational order of
convergence (see [13])

$r_{c}=\frac{\log|\mbox{\rm rad\,}(X_{k+1})/\mbox{\rm
rad\,}(X_{k})|}{\log|\mbox{\rm rad\,}(X_{k})/\mbox{\rm rad\,}(X_{k-1})|},$
(14)

which gives very good results (quite close to the theoretical value of order
of convergence) if the radii involved in (14) are very small.

To obtain considerably narrow intervals $X_{k}$ we have run 5 iterations and
multi-precision arithmetic in computer algebra system Mathematica with 1000
significant digits, which simulates the limit process (appearing in finding
theoretical convergence order) in a quite satisfactory manner. In the test
examples when the King-like method (3) has worked well, we have been able to
apply the formula (14) and calculated $r_{c}=3.5+\varepsilon,$ with
$\varepsilon=O(10^{-2})$ (disagreements on the second decimal digit). We have
previously stressed that the determination of theoretical order of the
“interval” methods (3) and (4) is of negligible interest since these methods
are not defined well so that we have not analyzed the values of $r_{c}.$ In
every case, the order cannot be four, as asserted in the mentioned papers. We
end this note with the observation that numerical results presented in [2],
[4] and [5] show that the order of convergence is really slower than the order
given by theorems in the cited papers, which means that theoretical results
remain unconfirmed.

Acknowledgement. This work was supported by the Serbian Ministry of Education
and Science under Grant 174022.

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