# Nine Challenges in Artificial Intelligence and Wireless Communications for
6G

Wen Tong and Geoffrey Ye Li Dr. Wen Tong<EMAIL_ADDRESS>is with Wireless
Technology Labs, Huawei Technologies, Ottawa, Canada.Prof. Geoffrey Ye Li
<EMAIL_ADDRESS>is with Intelligent Transmission and Processing
(ITP) Lab., Dept. Electrical and Electronic Engineering, Imperial College
London, United Kingdom.
The main content of this article was presented at talks from the authors.

###### Abstract

In recent years, techniques developed in artificial intelligence (AI),
especially those in machine learning (ML), have been successfully applied in
various areas, leading to a widespread belief that AI will collectively play
an important role in future wireless communications. To accomplish the
aspiration, we present nine challenges to be addressed by the
interdisciplinary areas of AI/ML and wireless communications, with particular
focus towards the sixth generation (6G) wireless networks. Specifically, this
article classifies the nine challenges into computation in AI, distributed
neural networks and learning, and ML enabled semantic communications.

###### Index Terms:

computing crisis, distributed learning, and semantic communications, 6G
wireless networks.

##  I Introduction

By late 1980s, the basic ensemble of algorithms for computational neural
networks has been nearly completed even though the size of the neural networks
nowadays is much larger. Back to 30 years ago, a neural network under
exploratory study typically consists of only a handful of neurons configured
over a few layers. It can only handle a limited amount of training data due to
the constraints of computing resources and complexity, which limits its
capability, performance, and applications. A few scholars persisted through
early 2000s. They were able to produce new thinking, which resulted in the
huge success of deep learning (DL) and led them wining the prestigious Turing
Award in 2018.

Computing power plays a critical role in the development of artificial
intelligence (AI). Following the Moors’ Law, the computing power has been
increased by around 200,000 times compared with 30 years ago, which has
improved the capability of machine learning (ML) by 100,000 times. For a deep
neural network, each layer consists of millions or even tens of millions of
neurons with hundreds of millions of parameters. Thanks to the efforts of
those few dedicated scholars, brute force computation, and big data, DL has
surpassed the human level in many applications.

By the way, when we were PhD students in electrical engineering and computer
science around 30 years ago, we took some courses on neural networks. However,
the seniors also told us it was hard to find jobs if studying ML. Therefore,
we shifted our interest into wireless communications and participated in and
led the investigation from the first generation (1G) to the fifth (5G)
generation wireless networks.

In the next 10 to 20 years, our general mission is to facilitate various
applications of ML technologies through intelligent communications. We would
like to clarify a couple of concepts between AI and wireless networks,
including AI for communication network (AI4NeT) and communication network for
AI (Net4AI). In Figure 1, we demonstrate their correlation and difference at
the link level and the network level, respectively. Particularly, AI4Net
mainly belongs to 5G and 5.5G while Net4AI is the mainstay for the sixth
generation (6G) and future wireless networks. Nowadays, we are applying ML
techniques to improve the intelligence and transmission performance of
communication networks. For 6G and beyond, AI will be used everywhere in our
daily life and communication networks will transmit and collect massive amount
data required by ML. Therefore, AI will revolutionize 6G and will also be the
fundamental business for 6G. AI in 6G is far more than just over-the-top (OTT)
applications. For 6G and beyond, we would put particular focus on Net4AI.

Figure 1: Comparison of AI4Net or Net4AI at link level and network level,
respectively.

The brute-force computing will be a stimulant for the development of AI for a
while. The new understanding, new discoveries, and new innovations are largely
dependent on the computing power of AI. Moreover, the success and
popularization of AI require endless big data, which relays on future wireless
networks for data transmission and collection. Unfortunately, neither brute-
force computing nor endless big data is sustainable. Therefore, we identify
nine challenges that should be addressed for 6G, which are very critical and
to be detailed in the following. If there is no theoretical breakthrough in
the related areas, we still have to grope in the dark. Hopefully, the
interleaving of AI and 6G wireless networks will shed a light on producing new
theories for future wireless networks.

In this article, we will discuss nine fundamental scientific challenges in
communications and AI with focus on the problems and challenges in the
interdisciplinary areas of DL and wireless communications, especially those
key challenges for ensuring successful 6G wireless networks. Particularly,
they include general challenges in deep neural networks for wireless
communications, distributed DL and distributed neural networks, and DL enabled
semantic communications.

##  II General Challenges in Deep Neural Networks for Wireless Communications

In the following, we will discuss five fundamental challenges in ML with close
relation to wireless communications for 6G.

### Challenge 1: Computing Crisis of DL

Based on the research from the MIT team in [1], Figure 2 compares the
complexity of ML with the Moore’s Law in different eras. During the Dennard
growing era, the computing capability is improved by increasing the clock
speed, the power consumption of AI computation almost follows the Moore’s Law.
It starts to grow faster than the Moore’s Law during the multi-core era. Today
we have entered the era of DL. Due to the demand on brute-force computing in
AI, the power consumption increases far exceeding the growth rate of the
Moore’s Law, at least with $10^{5}$. Therefore, it is necessary to take energy
consumption into consideration to ensure sustainable development. Otherwise,
our efforts in the area may be in vain.

Moreover, to reduce the error rate of ML to as low as 5%, the computation cost
is extremely expensive, which requires 100 billion US dollars and 10
quintillion program instructions [1]. Therefore, reducing the computational
complexity becomes the primary task for the future AI. Even if some
techniques, such as pruning, low-dimensional compression, less quantization
levels, or smaller DNNs, can be applied to mitigate the issue, the AI
computing crisis is still a long-term engineering problem in the following
decades.

To address the above challenges, we should first establish a set of evaluation
methodologies for power consumption in computation and communication to
facilitate the selection of solutions.

Figure 2: Computing complexity at different eras.

### Challenge 2: Gradient Disappearance in DL

The backward-propagation (back-propagation) gradient iteration in DL is a very
headache issue for hardware acceleration architecture, which affects wireless
transmission latency and is of great importance for the end-to-end
communications[2].

In the past, many neural network patching methods have been developed to
address the issues on the back-propagation of iterative gradient descent
algorithms. Some of them are listed as following:

  * •

Saturation of sigmoid function, such as rectified linear unit (ReLU) function,
causes gradient disappearance. The negative semi-axis of the ReLU function is
a dead zone, where the gradient turns into zero. Therefore, leaky version of
ReLU (LeakyReLU) and parametric ReLU (PReLU) have been proposed to substitute
the ReLU.

  * •

To strengthen the stability of the gradient and weight distribution,
exponential linear unit (ELU) and scaled ELU (SELU) are adopted in deep neural
networks.

  * •

It is difficult to pass gradients if a neural network is too deep. Therefore,
the highway network [3] has been invented, in which the parameters are even
omitted and residual networks (ResNets) are used instead.

  * •

Batch normalization (BatchNorm) can stabilize the mean and the variance of
neural network parameters.

  * •

DropOut, which ignores some units/neurons during the training, can increase
noise to the gradient flow, therefore, it can mitigate overfitting and lower
the generalization error.

  * •

The gradient in recurrent neural networks (RNN) is sometimes unstable. So long
short-term memory (LSTM) is widely used, which can be further improved by
adding some gated recurrent units (GRUs).

  * •

The Jensen–Shannon (JS) divergence in the generative adversarial network (GAN)
causes gradients to disappear or become invalid, which can be solved by the
Wasserstein GAN (WGAN).

The above are just some patching methods, but they are unable to fully address
the issue .

By re-examining the Arnold-Kolmogorov’s arbitrary function approximation
theorem, it may be possible to develop a one-step or an iterative algorithm to
address the issue on iterative gradient descent in DL. Through in-depth
research on the nature and the behavior of the back-propagation in DL, perhaps
with some theory developments, we may be able to mitigate or address the issue
completely. Of course, this is a long-term goal and we are looking forward to
a major breakthrough one day in the near future.

### Challenge 3: Memory Capacity of Deep Neural Networks

It is predicted that the memory function of neural networks will be fully
exploited in the future communications. For example, semantic transmitter and
receiver require a huge memory capacity to store the background knowledge[13].
In general, the larger the memory capacity of a neural network, the higher
potential to improve communication efficiency, especially the efficiency of
semantic communications to be introduced in Section IV. A natural question
turns out to be how many types of objects or events a neural network can
remember or what is the memory capacity of a neural network.

It has been shown in [4] that the memory capacity of a Hopfield network with
$n$ neurons is

$C\approx\frac{n}{2\log_{2}n}.$ (1)

However, the memory capacity is still unknown for a general deep neural
network.

Another correlated question is the relationship between the memory capacity
and the computational complexity. Is it linear, exponential, or some other
relations? Such a relationship is of great importance for inference and
estimation at the receiver in communication systems.

### Challenge 4: Dependence of DL on Big Data (I still think is very
confusing)

The big data required by training deep neural networks is often collected by
wireless communications. Therefore, it becomes critical how to train neural
networks efficiently and quickly and consequently to lower the requirement on
big data and wireless communications.

Even if the relationship between the memory capacity and the size of deep
neural networks is still not clear as indicated in Challenge 3, the neural
network with a large memory capacity is usually with a massive number of
parameters and requires a huge amount of data to train the model.

There have been some initial works on exploiting the domain knowledge in the
area of communications to reduce the requirements on training data, such as
model-driven DL for communication systems [5]. It is a tricky issue to trade
off the domain knowledge usage, big data requirements, communication
performance, and system complexity. It is also important to know the minimum
amount of data required for a given communication applications in general.

### Challenge 5: Dynamic, Accretionary, and Meta Learning

Currently, AI models are trained under special learning hypothesis premise
that the environment (the statistics of the training data) is static, at least
during the training period. However, real communication scenarios, especially
for mobile communications, are constantly changing. The basic features of
dynamic DL for wireless communications require extensive investigations.

The challenge can be potentially addressed with the developments of
accretionary learning and meta learning. The theory of accretionary learning
for human beings was proposed by cognitive psychologists over 40 years ago.
Recently, the concept of accretionry learning[6] has been developed for
classification. Particularly, traditional ML trains a classification model
with fixed recognition sets, which fails for the data set with new features.
The accretionary learning[6] can address the issue by extending deep neural
networks through interaction of accretion, fine-tuning, and restructuring.
Moreover, meta learning[7], also known as learning-to-learn, becomes an active
research area recently. With its development, dynamic learning for
communication systems and networks could be achieved one day.

##  III Distributed Neural Networks and Distributed Learning

Nowadays, there are around 7 millions base stations and 6 billions mobile
terminals in the world and these numbers keep on growing. It is impossible to
collect all data from different base stations and terminals into a super
fusion server for centralized processing due to huge storage requirement,
heavy computing complexity, demanding energy consumption, and privacy issues.
Therefore, distributed neural networks and distributed learning become
extremely important in the future wireless communications.

### Challenge 6: Wireless Data Aided Collective Learning

Collective Learning is referred to that multiple servers collect data and
train ML models collectively. It belongs to a general category of distributed
learning, but with special relation to communications.

The architecture and performance of collective learning depend on both AI
itself and communications. In the integrated architecture of AI and wireless
communications, the primary problems are how to segment data and how to
separate communication and computing, where the data is mainly used for
training ML model while the communication model is used for data/parameter
transmission. Here, we pay close attention to communication issues.

Specifically, how to segment the neural network and the tasks for collective
learning is an unsolved problem, which is also a profound theoretical issue.
Specifically, the related research problems include:

  * •

How to collectively learn multiple small and identical neural networks?

  * •

How to segment a large network into multiple small networks?

  * •

What are the impacts of the size and the number of networks on the efficiency
and optimization of learning?

Figure 3: Distributed learning.

### Challenge 7: Wireless Communications Enabled Federated Learning

Federated learning [8, 9, 10], a type of distributed learning, is enabled by
wireless communications. Different from the collective learning in Challenge
6, federated learning relies more on communications. As shown in Fig. 3,
multiple clients/terminals jointly train/learn the same model in federated
learning and there are only parameter exchanges among multiple terminals,
rather than segmenting data, tasks, and neural networks as that in the
collective learning. The information exchange between the terminal and edge
neural networks depend on wireless communications. The same for that between
the edge neural networks and the central networks. Therefore, the transmission
error affects distributed/federated learning significantly.

In the past, the fields of wireless communications and AI have been developed
independently with different optimization goals. Particularly, communication
systems are usually designed to maximize data transmission rate and minimize
bit-error rate while the performance of AI is typically measured by the
recognition accuracy and the quality of the reproduced data. In general, the
interaction and connection between AI and wireless communications are new
trend that has never been jointly investigated before.

Let us take the learning process of a deep neural network as an example. Even
if the impact of the coefficient error in the first few iterations is trivial,
it becomes very critical and sensitive in the last few iterations of the last
few layers. Therefore, at different iterations and different layers of a
neural network, the demands for the quality of wireless transmission are
various. The similar issue shows in inference models.

In addition to the aforementioned error sensitivity issue, there are many
others. For example, how to compress model parameters to save communication
resources and how to perform robust parameter aggregation at the central
server to address communication errors and stragglers, as indicated in [10].
These issues pose innovation opportunities for 6G. It is more than desired to
find out the relationship between ML model errors and various wireless
transmission errors/imperfections from a theoretical basis to facilitate their
joint optimization[11].

##  IV Semantic Communications

Semantics communications is a problem in the post-Shannon era. Around 70 years
ago, Weaver [12] classified communications into three levels in his
masterpiece co-authored with Shannon, which include transmission of symbols,
semantic exchange, and effect of semantic exchange. When Shannon established
information theory, he only focused on the mathematical theory of symbol
transmission. Particularly, the amount of information in a source is indicated
by its entropy and the capacity of a communication channel is proved to be the
maximum mutual information between the channel input and output, both are
measured by bits. Due to the lack of general mathematical tools to model and
analyze semantics or meaning of the transmitted contents, limited progress has
been made in the past even if many outstanding researchers have tackled the
area.

Figure 4: Shannon channels and Weaver channels for typical communications and
semantic communications.

Recent success in DL and its application makes it possible to re-examine
Weaver’s second-level and even third-level models, where the goal is not to
transmit bit sequence, but the meaning of the content. We then colloquially
call them semantic communications. The preliminary communication architecture
is shown in Figure 4, where the Shannon channel is embedded inside and the
Weaver semantic channel is outside. Under this architecture, the scientific
questions include:

  * •

How do we measure semantics?

  * •

What is the limit of semantic communication systems?

  * •

Is there any theory analogical to Shannon information theory?

  * •

How to implement and perform semantic communications efficiently?

It is noticed that semantic communications have caught extensive attentions in
the past several years and there have been some initial research in this
area[13, 14, 15]. We believe that there will be more works on theory and
implementation of semantic communications in the near future.

### Challenge 8: Mathematical Foundation of Semantic Communications

Kolmogorov, a great mathematician in the last century, came into contact with
Shannon information theory from the other side of the Iron Curtain of the Cold
War in the Soviet Union in 1958 and further expanded the mathematical
foundation of information theory. Based on his investigation, the sufficient
and necessary condition for the mathematical equivalence of a large class of
automorphism systems is that they are with the same entropy. Kolmogorov also
extends the mathematical foundation of Shannon entropy into a new
interdisciplinary area of topology and algebra.

The new mathematical foundation for post-Shannon communications or semantic
communications is also particularly important. The entropy and capacity in
Shannon information theory are defined based on statistical probability, which
may not be applicable to semantic communications.

To develop mathematical foundation of semantic communications, the statistical
probability is substituted by logical probability of a message in [13], where
semantic source, semantic noise, semantic entropy, semantic capacity, and
semantic coding are defined accordingly. Moreover, semantic source theorem and
semantic channel coding theorem, analogical to their counterparts in Shannon
information theory, have been proved in [13]. However, it is more than desired
to build a comprehensive mathematical framework to identify the optimal
structure for semantic communication systems.

### Challenge 9: Structure of Semantic Communication Systems

Perhaps, many of us have ignored Pages 5 - 7 of the Shannon’s masterpiece
[12], which are the earliest statement on natural language AI. Based on
Dewey’s statistical results on the English language, Shannon attempted to
reconstruct English sentences from random English letters by exploiting the
probability and statistical methods in order to investigate semantic
communications. At that time, Shannon partially retreated to the fundamental
probability to support the concept of entropy in information theory, which
actually ignored the important elements in semantic communications.

In [14], the classic Huffman coding with the deep neural network method have
been compared for transmission of text in 2018. It has been found that the
improvement through using a deep neural network is pretty limited, which again
confirmed Shannon information theory.

It is a critical issue whether the information restoration in semantic
communications should be through a faithful recovery of the corresponding bits
or direct semantic restoration. It can be regarded as a structural problem in
semantic communications as indicated by Shannon and Weave. The question here
turns into whether to use a general DL neural network for training and
learning for semantic communications or to further explore different
structural levels of communications.

Recently, a DL-based semantic communication system for text transmission,
named DeepSC, has been developed in [15]. With the aid of Transformer and
novel loss function design, DeepSC aims at maximizing the system capacity and
minimizing the semantic errors by recovering the meaning of sentences, rather
than bit- or symbol-errors as in the traditional communications. To measure
the performance of semantic communications properly, a new metric, named
sentence similarity, has been proposed. Compared with the traditional
communication system without considering semantic information exchange, DeepSC
is more robust to channel variations and is able to achieve better
performance, especially in the low signal-to-noise (SNR) regime, as shown in
Figure 5. The initial results in [15] demonstrate the power of semantic
communications. More research in this area, such as semantic communications
for video, image, and speech transmission, is expected.

Figure 5: Robustness of the semantic communication system, DeepSC [15], in
comparison to benchmarks.

##  V Conclusion Remarks

In this article, we discussed nine challenges to ensure successful 6G,
including

  * •

the computing challenge in deep learning (DL) (Challenge 1)

  * •

The communication related challenges in deep DL (Challenges 2 - 5)

  * •

The challenges on the architecture of distributed learning enabled by
communications (Challenges 6 and 7)

  * •

The challenges in DL enabled semantic communications (Challenges 8 and 9)

We are certain that, in this rich research field, more promising solutions
will be developed to address the above challenges, which will generate huge
impacts on the evolution or even revolution of future communications.

##  VI Acknowledgement

We would like to acknowledge Professor Biing-Hwang Juang of Georgia Institute
of Technology, Dr. Yiqun Ge and Mr. Rong Li of Huawei Technologies for their
inspiring discussions and Dr. Zhijin Qin of Queen Mary University London for
useful comments, which have helped improve the quality of the article.

## References

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I_{k}}{R_{k}}\right.\right.\\\ \left.\left.+\frac{\eta
N_{k}}{S_{k}}+\kappa_{0}\left(e_{k}+\frac{1}{z_{k}}\right)+\frac{S_{k}\beta(e_{k},z_{k})}{\left[1+rI_{k}\right]+\eta
N_{k}}(1+I_{k})\right]\right\\}ds\\\
+\left\\{\sigma_{0}^{k}\left(1-\frac{1}{z_{k}}\right)(z_{k}-z_{k}^{*})+\sigma_{2}^{k}\left(1-\frac{1}{S_{k}}\right)(S_{k}-S_{k}^{*})\right.\\\
\left.+\sigma_{3}^{k}\left(1-\frac{1}{I_{k}}\right)(I_{k}-I_{k}^{*})+\sigma_{4}^{k}\left(1-\frac{1}{R_{k}}\right)(R_{k}-R_{k}^{*})\right\\}dB^{k}\\\
\leq\left\\{\zeta R_{k}+\eta
N_{k}+\mu(1+I_{k})+\tau(1+R_{k})+\kappa_{1}p(\eta_{k_{i}})+\kappa_{0}\left(1+\frac{e_{k}}{z_{k}}\right)\right.\\\
\left.+I_{k}\left(\frac{\beta^{k}(e_{k},z_{k})}{\left[1+rI_{k}\right]+\eta
N_{k}}+\frac{\mu+\tau}{S_{k}}\right)+\beta^{k}(e_{k},z_{k})\frac{S_{k}I_{k}}{\left[1+rI_{k}\right]+\eta
N_{k}}\right.\\\
\left.+\frac{(\sigma_{0}^{k})^{2}}{2}\left(1-\frac{z_{k}^{*}}{z_{k}}\right)^{2}+\frac{(\sigma_{2}^{k})^{2}}{2}\left(1-\frac{S_{k}^{*}}{S_{k}}\right)^{2}+\frac{(\sigma_{3}^{k})^{2}}{2}\left(1-\frac{I_{k}^{*}}{I_{k}}\right)^{2}\right.\\\
\left.+\frac{(\sigma_{4}^{k})^{2}}{2}\left(1-\frac{R_{k}^{*}}{R_{k}}\right)^{2}\right\\}ds+\left\\{\sigma_{0}^{k}\left(1-\frac{1}{z_{k}}\right)(z_{k}-z_{k}^{*})+\sigma_{2}^{k}\left(1-\frac{1}{S_{k}}\right)\right.\\\
\left.\times(S_{k}-S_{k}^{*})+\sigma_{3}^{k}\left(1-\frac{1}{I_{k}}\right)(I_{k}-I_{k}^{*})+\sigma_{4}^{k}\left(1-\frac{1}{R_{k}}\right)(R_{k}-R_{k}^{*})\right\\}dB^{k}\\\
\leq\mathfrak{M}\
ds+\left\\{\sigma_{0}^{k}\left(1-\frac{1}{z_{k}}\right)(z_{k}-z_{k}^{*})+\sigma_{2}^{k}\left(1-\frac{1}{S_{k}}\right)(S_{k}-S_{k}^{*})\right.\\\
\left.+\sigma_{3}^{k}\left(1-\frac{1}{I_{k}}\right)(I_{k}-I_{k}^{*})+\sigma_{4}^{k}\left(1-\frac{1}{R_{k}}\right)(R_{k}-R_{k}^{*})\right\\}dB^{k},$
(23)

where $\mathfrak{M}$ is a positive constant. Integration of both sides of the
Inequality (23) from $0$ to $\hat{s}_{m}\wedge t$ yield

$\displaystyle\int_{0}^{\hat{s}_{m}\wedge
t}d\mathfrak{W}[z_{k}(s),S_{k}(s),I_{k}(s),R_{k}(s)]$ $\displaystyle\hskip
28.45274pt\leq\int_{0}^{\hat{s}_{m}\wedge
t}\mathfrak{M}ds+\left\\{\sigma_{0}^{k}\left(1-\frac{1}{z_{k}}\right)(z_{k}-z_{k}^{*})+\sigma_{2}^{k}\left(1-\frac{1}{S_{k}}\right)(S_{k}-S_{k}^{*})\right.$
$\displaystyle\left.\hskip
56.9055pt+\sigma_{3}^{k}\left(1-\frac{1}{I_{k}}\right)(I_{k}-I_{k}^{*})+\sigma_{4}^{k}\left(1-\frac{1}{R_{k}}\right)(R_{k}-R_{k}^{*})\right\\}dB^{k},$

where $\hat{s}_{m}\wedge t=\min\\{\hat{s}_{m},t\\}$. After taking expectations
on both sides lead to

$\mathbb{E}\mathfrak{W}[z_{k}(\hat{s}_{m}\wedge t),S_{k}(\hat{s}_{m}\wedge
t),I_{k}(\hat{s}_{m}\wedge t),R_{k}(\hat{s}_{m}\wedge
t)]\leq\mathfrak{M}t+\mathfrak{W}[z_{k}(0),S_{k}(0),I_{k}(0),R_{k}(0)].$

Define $\aleph_{m}=\\{\hat{s}_{m}\leq t\\},\ \forall m\geq m_{1}$. Previous
discussion implies, for any $\varepsilon>0$ there exists an integer $m_{1}\geq
m_{0}$ such that, $Pr[\hat{s}_{m}\leq t]\geq\varepsilon$ therefore,
$Pr(\aleph_{m})\geq\varepsilon$. For each $\wp\in\aleph_{m}$, $\exists$ an $i$
such that $\hbar_{i}(\hat{s}_{m},\wp)=m\ \text{or}\ 1/m$ for $i=1,...,4$.
Therefore,
$\mathfrak{W}\left[z_{k}(\hat{s}_{m},\wp),S_{k}(\hat{s}_{m},\wp),I_{k}(\hat{s}_{m},\wp),R_{k}(\hat{s}_{m},\wp)\right]$
has the lower bound $\min\\{m-1-\ln m,1/m-1-\ln(1/m)\\}$. This yields,

$\displaystyle\mathfrak{M}t+\mathfrak{W}[z_{k}(0),S_{k}(0),I_{k}(0),R_{k}(0)]$
$\displaystyle\geq\mathbb{E}\left\\{\mathbbm{1}_{\aleph_{m}(\wp)}\mathfrak{W}[z_{k}(\hat{s}_{m}),S_{k}(\hat{s}_{m}),I_{k}(\hat{s}_{m}),R_{k}(\hat{s}_{m})]\right\\}$
$\displaystyle\geq\varepsilon\min\left\\{m-1-\ln(m),\frac{1}{m}-1-\ln\left(\frac{1}{m}\right)\right\\},$

where $\mathbbm{1}_{\aleph_{m}(\wp)}$ is a simple function on $\aleph_{m}$.
Letting $m\downarrow\infty$ leads to
$\infty=\mathfrak{M}t+\mathfrak{W}[z_{k}(0),S_{k}(0),I_{k}(0),R_{k}(0)]<\infty$,
which is a contradiction. Q.E.D.

### Proof of Lemma 2

As stochastic opinion dynamics is on $F$, this surface is oscillatory in
nature. Total social interaction variation between two probabilistic
interactions
$W_{k,\chi_{-k},\omega_{-k}}(s,h)$ and $W_{l,\chi_{-l},\omega_{-l}}(s,h)$ can
be defined in terms of a Hahn-Jordon orthogonal decomposition

$W=W_{k,\chi_{-k},\omega_{-k}}-W_{l,\chi_{-l},\omega_{-l}}=W_{k,\chi_{-k},\omega_{-k}}^{+}-W_{l,\chi_{-l},\omega_{-l}}^{-},$

such that

$\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}-W_{l,\chi_{-l},\omega_{-l}}\right)\right|\right|=W_{k,\chi_{-k},\omega_{-k}}^{+}(F)=W_{l,\chi_{-l},\omega_{-l}}^{-}(F).$

Therefore, for $h\in F$,

$\left|W_{k,\chi_{-k},\omega_{-k}}(s,h)-W_{l,\chi_{-l},\omega_{-l}}(s,h)\right|\\\
=\left|\int_{\mathbb{R}}h(s,\omega_{k})W_{k,\chi_{-k},\omega_{-k}}^{+}(F)(d\omega_{k})-\int_{\mathbb{R}}h(s,\omega_{k})W_{l,\chi_{-l},\omega_{-l}}^{-}(F)(d\omega_{l})\right|\\\
=\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|\left|\int_{\mathbb{R}}\left[h(s,\omega_{k})-h(s,\omega_{l})\right]\right.\\\
\left.\times\frac{W_{k,\chi_{-k},\omega_{-k}}^{+}(d\omega_{k})}{W_{k,\chi_{-k},\omega_{-k}}^{+}(F)}\times\frac{W_{l,\chi_{-l},\omega_{-l}}^{-}(d\omega_{l})}{W_{l,\chi_{-l},\omega_{-l}}^{-}(F)}\right|.$

Therefore,

$\left|W_{k,\chi_{-k},\omega_{-k}}(s,h)-W_{l,\chi_{-l},\omega_{-l}}(s,h)\right|\leq\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}-W_{l,\chi_{-l},\omega_{-l}}\right)\right|\right|.$

Supremum over $h\in F$ yields,

$\sup\left\\{\left|\left(W_{k,\chi_{-k},\omega_{-k}}(s,h)-W_{l,\chi_{-l},\omega_{-l}}(s,h)\right)\right|\right\\}\leq\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}-W_{l,\chi_{-l},\omega_{-l}}\right)\right|\right|.$

The reverse inequality can be checked trivially by introducing a simple
function $\mathbbm{1}_{\mathcal{G}}$, with $\mathcal{G}\in\mathcal{E}$, belong
to F. Therefore, we are able to show that

$\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|=\sup\left\\{\left|\left(W_{k,\chi_{-k},\omega_{-k}}(s,h)-W_{l,\chi_{-l},\omega_{-l}}(s,h)\right)\right|\right\\}.$

Now, by construction , there exists two disjoint subsets $F_{+}$ and $F_{-}$
such that, $W^{+}(F_{-})=0=W^{-}(F_{+})$ (Moral, 2004). For any graph
$\mathcal{G}\in\mathcal{E}$, $W^{+}(\mathcal{G})=W(\mathcal{G}\cap F_{+})\geq
0$ and, $W^{-}(\mathcal{G})=-W(\mathcal{G}\cap F_{+})\geq 0.$ Hence,

$W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G}\cap F_{+})\geq
W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}\cap F_{+}),$

and,

$W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}\cap F_{-})\geq
W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G}\cap F_{-}).$

Consider $\hat{h}$ be another probability measure for any
$\mathcal{G}\in\mathcal{E}$ by,

$\hat{h}(\mathcal{G})=W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G}\cap
F_{-})+W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}\cap F_{+}).$

By construction,

$\hat{h}(\mathcal{G})\leq W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G})\wedge
W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}),$ (24)

and,

$\hat{h}(F)=W_{k,\chi_{-k},\omega_{-k}}(F_{-})+W_{l,\chi_{-l},\omega_{-l}}(F_{+}).$
(25)

As

$\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|=W^{+}(F)=W(F_{+})\\\
=W_{k,\chi_{-k},\omega_{-k}}(F_{+})-W_{l,\chi_{-l},\omega_{-l}}(F_{-})=1-\left[W_{k,\chi_{-k},\omega_{-k}}(F_{+})+W_{l,\chi_{-l},\omega_{-l}}(F_{-})\right],$

by Equation (25) one obtains

$1-\sup_{\hat{h}\in\left(W_{k,\chi_{-k},\omega_{-k}}(s),W_{l,\chi_{-l},\omega_{-l}}(s)\right)}\tilde{h}(F)\leq
1-\hat{h}(F)=\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|.$

The reverse inequality is proved as follows. Suppose, $\tilde{h}$ be a non-
negative measure such that for any graph $\mathcal{G}\in\mathcal{E}$ we have

$\tilde{h}(\mathcal{G})\leq W_{k,\chi_{-k}\omega_{-k}}(\mathcal{G})\wedge
W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}).$

Assuming $\mathcal{G}=F_{+}$ and $\mathcal{G}=F_{-}$ give us

$\tilde{h}(F_{+})\leq W_{k,\chi_{-k},\omega_{-k}}(F_{+})\ \text{and,}\
\tilde{h}(F_{-})\leq W_{l,\chi_{-l},\omega_{-l}}(F_{-}).$

Therefore,

$\tilde{h}(F)\leq
W_{k,\chi_{-k},\omega_{-k}}(F_{+})+W_{l,\chi_{-l},\omega_{-l}}(F_{-})=1-\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|,$

which implies

$1-\tilde{h}(F)\geq\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|.$

Taking the infimum over all the distributions $\tilde{h}\leq
W_{k,\chi_{-k}\omega_{-k}}(s)$ and $W_{l,\chi_{-l},\omega_{-l}}(s)$, we get

$\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|=1-\sup_{\hat{h}\in\left(W_{k,\chi_{-k},\omega_{-k}}(s),W_{l,\chi_{-l},\omega_{-l}}(s)\right)}\hat{h}(F).$

To prove the final part of the lemma note that,

$W_{l,\chi_{-l},\omega_{-l}}(F_{+})=W_{k,\chi_{-k},\omega_{-k}}(F_{+})\wedge
W_{l,\chi_{-l},\omega_{-l}}(F_{+}),$

and

$W_{k,\chi_{-k},\omega_{-k}}(F_{-})=W_{k,\chi_{-k},\omega_{-k}}(F_{-})\wedge
W_{l,\chi_{-l},\omega_{-l}}(F_{-}).$

Hence,

$\hat{h}(F)=W_{k,\chi_{-k},\omega_{-k}}(F_{-})+W_{l,\chi_{-l},\omega_{-l}}(F_{+})\\\
=\left[W_{k,\chi_{-k},\omega_{-k}}(F_{-})\wedge
W_{l,\chi_{-l},\omega_{-l}}(F_{-})\right]+\left[W_{k,\chi_{-k},\omega_{-k}}(F_{+})\wedge
W_{l,\chi_{-l},\omega_{-l}}(F_{+})\right].$

As $F_{+}$ and $F_{-}$ are mutually exclusive, therefore,

$\hat{h}(F)\geq\inf\sum_{i=1}^{I}\left[W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G}_{i})\wedge
W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}_{i})\right],$

where the infimum is taken over all resolutions of $F$ into pairs of
nonintersecting subgraphs $\mathcal{G}_{i}$, $1\leq i\leq I$, $I\geq 1$.
Reverse inequality can be shown by using the definition of $\hat{h}$. By
Equation (25) for any finite subgraph $\mathcal{G}_{i}\in\mathcal{E}$, we have

$\hat{h}(\mathcal{G}_{i})\leq
W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G}_{i})\wedge
W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}_{i}).$

Therefore,

$\hat{h}(F)=\sum_{i=1}^{I}\hat{h}(\mathcal{G}_{i})\leq\sum_{i=1}^{I}\left[W_{k,\chi_{-k},\omega_{-k}}(\mathcal{G}_{i})\wedge
W_{l,\chi_{-l},\omega_{-l}}(\mathcal{G}_{i})\right]$

By taking the infimum over all subgraphs yields

$\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|=1-\inf\sum_{i=1}^{I}\left(W_{k,\chi_{-k},\omega_{-k}}(s,\mathcal{G}_{i})\wedge
W_{l,\chi_{-l},\omega_{-l}}(s,\mathcal{G}_{i})\right),$

since

$\hat{h}(F)=1-\left|\left|\left(W_{k,\chi_{-k}\omega_{-k}}(s)-W_{l,\chi_{-l},\omega_{-l}}(s)\right)\right|\right|.$

This completes the proof. Q.E.D.

### Proof of Proposition 2

Consider $\mho:\Omega_{E}\rightarrow\mathbb{E}$ is an increasing function
which represents the influence of risk-group $k$ in the network which is a
convex function of the odds of themselves to get the signals from the
neighbors about their social interactions and is defined by
$\hat{\pi}=\log[\varrho/(1-\varrho)]$. Assume for $\mathfrak{q}_{3}$, the
signal profile $\varrho$ of risk-group $k$ is in $\mathbb{D}\subseteq(0,1)$.
Now suppose, $\mathcal{J}$ is the total number of interactions of risk-group
$k$ with open edges with the set of edges with boxes $\Lambda$ as
$\mathbb{E}_{\Lambda}$. Then by Theorem $4.2$ of Grimmett (1995) and by
Picard-Lindelof theorem there exists a unique random opinion in $G$ (Board and
Meyer-ter Vehn, 2021). Q.E.D.

### Proof of Proposition 3

I have divided the proof into two cases.

$\mathbf{Case\ I}$: There are total $K$-risk-groups with an individual risk-
group $k$ such that $k=1,2,...,K$. I assume that $m\subset\mathbb{N}$, a set
$\beth$ with condition $|\beth|=m+1$, and affinely independent state variables
and lock-down intensity
$\\{Z_{k}(s)\\}_{k\in\beth}\subset\mathbb{R}^{6K}\times G$ such that
$\widetilde{\Xi}$ coincides with the simplex convex set of
$\\{Z_{k}(s)\\}_{k\in\beth}$. For each $Z(s)\subset\Xi$, there is a unique way
in which the vector $Z(s)$ can be written as a convex combination of the
extreme valued state variables and lock-down intensity , namely,
$Z(s)=\sum_{k\in\beth}\alpha_{k}(s,Z)Z_{k}(s)$ such that
$\sum_{k\in\beth}\alpha_{k}(s,Z)=1$ and $\alpha_{k}(s,Z)\geq 0,\ \forall
k\in\beth$ and $s\in[0,t]$. For each risk-group $k$, define a set

$\widetilde{\Xi}_{k}:=\left\\{Z\in\widetilde{\Xi}:\alpha_{k}[\mathcal{L}_{k}(s,Z)]\leq\alpha_{k}(s,Z)\right\\}.$

By the continuity of the quantum Lagrangian of $k^{th}$ risk-group
$\\{\mathcal{L}_{k}\\}_{k\in\beth}$, $\widetilde{\Xi}_{k}$ is closed. Now we
claim that, for every $\tilde{\beth}\subset\beth$, the convex set consists of
$\\{Z_{k}\\}_{k\in\tilde{\beth}}$ is proper subset of
$\bigcup_{k\in\tilde{\beth}}\widetilde{\Xi}_{k}$. Suppose
$\tilde{\beth}\subset\beth$ and $Z(s)$ is also in the non-empty, convex set
consists of the state variables and the lock-down intensity
$\\{Z_{k}(s)\\}_{k\in\tilde{\beth}}$. Therefore, there exists
$k\in\tilde{\beth}$ such that
$\alpha_{k}(s,Z)\geq\alpha_{k}\left[\mathcal{L}_{k}(s,Z)\right]$ which implies
$Z(s)\in\tilde{\Xi}\subset\bigcup_{l\in\tilde{\beth}}\tilde{\Xi}_{l}$. By
_Knaster-Kuratowski-Mazurkiewicz Theorem_ , there is
$\bar{Z}_{k}^{*}\in\bigcap_{k\in\beth}\tilde{\Xi}_{k}$, in other words, the
condition
$\alpha_{k}\left[\mathcal{L}_{k}(s,\bar{Z}_{k}^{*})\right]\leq\alpha_{k}(s,\bar{Z}_{k}^{*})$
for all $k\in\beth$ and for each $s\in[0,t]$ (González-Dıaz, Garcıa-Jurado and
Fiestras-Janeiro, 2010). Hence,
$\mathcal{L}_{k}(s,\bar{Z}_{k}^{*})=\bar{Z}_{k}^{*}$ or $\mathcal{L}_{k}$ has
a fixed-point.

$\mathbf{Case\ II}$: Again consider
$\widetilde{\Xi}\subset\mathbb{R}^{6K}\times G$ is a non-empty, convex and
compact set. Then for $m\subset\mathbb{N}$, a set $\beth$ with condition
$|\beth|=m+1$, and affinely independent state variables and lock-down
intensity $\\{Z_{k}(s)\\}_{k\in\beth}\subset\mathbb{R}^{6K}\times G$ such that
$\widetilde{\Xi}$ is a proper subset of the convex set based on
$\\{Z_{k}(s)\\}_{k\in\beth}$ for all $s\in[0,t]$. Among all the simplices,
suppose $\hat{\aleph}$ is the set with smallest $m$. Let $\tilde{Z}(s)$ be a
dynamic point in the $m$-dimensional interior of $\hat{\aleph}$. Define
${\hat{\mathcal{L}}}_{k}$, an extension of $\mathcal{L}_{k}$ to the whole
simplex $\hat{\aleph}$, as follows. For every $Z(s)\in\hat{\aleph}$, let

$\bar{\zeta}(s,Z):\max\left\\{\bar{\zeta}\in[0,1]:(1-\bar{\zeta})\tilde{Z}(s)+\bar{\zeta}Z(s)\in\widetilde{\Xi}\right\\},\
\forall s\in[0,1],$

and,

${\hat{\mathcal{L}}}_{k}(s,Z):\mathcal{L}_{k}\left\\{\left[1-\bar{\zeta}(s,Z)\right]\tilde{Z}(s)+\bar{\zeta}(s,Z)Z(s)\right\\}.$

Therefore, $\bar{\zeta}$ is continuous which implies
${\hat{\mathcal{L}}}_{k}(s,Z)$ is continuous. Since the codomain of
${\hat{\mathcal{L}}}_{k}(s,Z)$ is in $\tilde{\Xi}$, every fixed-point of
${\hat{\mathcal{L}}}_{k}(s,Z)$ is also a fixed-point of $\mathcal{L}_{k}$. Now
by _$\mathbf{Case\ I}$_ , ${\hat{\mathcal{L}}}_{k}(s,Z)$ has a fixed-point and
therefore, $\mathcal{L}_{k}$ also does. Q.E.D.

## Funding declaration

No funding was used to write this paper.

## Conflict of interest

The author declares that he has no conflicts of interest.

## Author contribution

The author declares that he has solely contributed the whole paper.

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