# Adversarial Robustness in Cognitive Radio Networks

Makan Zamanipour Makan Zamanipour is with Lahijan University, Shaghayegh
Street, Po. Box 1616, Lahijan, 44131, Iran<EMAIL_ADDRESS>Copyright (c) 2015 IEEE. Personal use of this material is permitted. However,
permission to use this material for any other purposes must be obtained from
the IEEE by sending a request to<EMAIL_ADDRESS>
###### Abstract

When an adversary gets access to the data sample in the adversarial robustness
models and can make data-dependent changes, how has the decision maker
consequently, relying deeply upon the adversarially-modified data, to make
statistical inference? How can the resilience and elasticity of the network be
literally justified $-$ if there exists a tool to measure the aforementioned
elasticity? The principle of byzantine resilience distributed hypothesis
testing (BRDHT) is considered in this paper for cognitive radio networks
(CRNs) $-$ without-loss-of-generality, something that can be extended to any
type of homogeneous or heterogeneous networks $-$ while the byzantine primary
user (PU) has a signal-to-noise-ratio (SNR) including the coefficient of
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
which is in relation to the temporal rate of the $\alpha-$leakage as the
appropriate tool to measure the aforementioned resilience. Our novel online
algorithm $-$ which is named $\mathbb{OBRDHT}$ $-$ and solution are both
unique and generic over which an evaluation is finally performed by
simulations $-$ e.g. an evaluation of the total error as the false alarm
probability in addition to the miss detection probability versus the sensing
time.

###### Index Terms:

Age-of-information, asynchronous, belief, byzantine attacks, distributed
inference, falsification, first-eigenvalue, multi-agent system.

##  I Introduction

Accurate and efficient procedures are required to find some unique ans
specific information sets e.g. distributions of data sets or the number of
libraries in the controlled network. However, the controlled network may be of
a technically adversarial situation in which the users are able to be
potentially subjected to some knock-on effects $-$ resulting in an a priori
unknown subset of Byzantines111Those users who do data falsification $-$
something that happens in the physical-layer. who purposefully follow totally
different trajectories.

Aim at classifying the Byzantines, each good user should be collaboratively
required to repeatedly share their beliefs, i.e., observations with others $-$
since in reality, only a single good agent cannot be useful for the
aforementioned classification, due to practical limitations such as a noisy
plant. However, there arises an important question that how much reliable the
aforementioned collaborative sharing is, as Byzantines undoubtedly have a
potential to arbitrarily share over-written information aimed at preventing of
being infered.

Since every possible subset relating to Byzantines is considered a hypothesis,
the classification issue expressed above is called distributed hypothesis
testing $-$ where a subset of users cooperatively identifies the unknown true
hypothesis according to their local beliefs.

Distributed networks have mainly two valid beneficiaries compared with
centralised systems while the former ones: (i) can technically guarantee a
resilience to fraud; and (ii) can tolerate failures. Meanwhile, distributed
systems have three main features: (i) Agreement, that is, an overall decision
on the same value by all true processes; (ii) termination, that is, to make a
decision in an expected finite time-zone by all true processes; and (iii)
validity, that is, the agents who are in consensus never leave their
agreement.

Inversely, aimed at preventing the consensus to occur, all Byzantine processes
are controlled by an Adversary. In fact, while binary byzantine consensus
degrades the overall performance, distributed solutions where each agent
communicates to its neighbors without a fusion center can enrich the
resilience.

Distributed algorithms rely mostly upon beliefs which are probability
distributions over the hypotheses. Now, each agent’s actual belief should
theoretically converge to the true hypothesis in order to guarantee the
resilient against the bad agents who arbitrarily share modified information.

(a)  Figure 1: Flow of the problem-and-solution and our contribution in this
paper.

###  I-A Related work

In this part, we explore the related work recently appeared in the literature.

####  I-A1 Byzantine resilience distributed hypothesis testing (BRDHT)
[1]-[19]

In [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], some novelties from either practical or
theoretical or both points of view were proposed for real world applications
such as: the Internet of Things, the multi-sinks, the one-bit double
quantisation algorithms, multi-cast routing in multi-hop wireless networks,
decentralized learning, vehicular social networks or even cognitive radio
networks. In [11], an unknown subset of auctioned agents was considered that
may consciously share modified information to diminish the global
optimisation. While each agent try to maintain and update two sets of beliefs,
namely actual and local and actual ones, two distributed algorithms were
consequently proposed in [11]. A sparse encoding framework was proposed in
[12], while a trade-off was defined between the resource requirements and the
corruption threshold. in [13], the jamming behaviour of Byzantines were
considered for the first time. The work in [14] as the first research to
byzantine-resilient optimisation in which no leader player existed for
coordination, the authors found the global optima in the context of a convex
fashion. In [15] as the first single-server byzantine-resilient scheme for
secure federated learning based on an integrated stochastic quantisation, some
important trade-offs were examined for the network size, privacy protection
and user dropouts. In [16], the asymptotically learning issue of the true
hypothesis in an almost sure sense was explored. In [17], it was perfectly
proven that the probability of sample paths through which the rejection rate
of false observations tangentially passes from the mean-rate, would
theoretically experience an exponential plummet. Under the assumptions that:
(i) the number of Byzantines is locally upper-bounded, and (ii) each good
agent neglects the most extreme values on his local vicinity consequently
altering his observations, some novel and secure ideas were well-developed in
[18] for both types of first-order and second-order systems. Instead of the
case that agents update their observations according to a min-rule, a novel
distributed learning rule was proposed in [19], which does not concentrate on
“belief-averaging”. In fact, the research in [19] works in the sense that each
agent is able to learn the true hypothesis asymptotically almost surely. It
was also proven that each falsification is ruled out by every agent
exponentially fast: (i) with probability 1, and (ii) at an enhanced network-
independent rate.

####  I-A2 Byzantine attacks in CRNs [20]-[24]

In [20], the issue of optimal sensing disruption for a cognitive radio
adversary was technically explored. In [21], a secure and energy-efficient
cooperative spectrum sensing scheme was newly proposed. In [22], a generalized
framework for a power-limited adversary in terms of the optimal sensing
disruption was theoretically proposed. In [23] and for the process of spectrum
sensing, secure routing scenarios were deeply taken into account which was
based mainly upon the nodes’ behaviour. In [24] and with the aid of the
Thompson sampling technique, an online primary user emulation attack was
investigated.

####  I-A3 Asynchronous Byzantine consensus [25]-[30]

The issue of asynchronous byzantine agreement was taken theoretically into
account for incomplete networks in [25]. In [26], through a weak round
installer, a highly effective binary byzantine faulty tolerant consensus
algorithm was proposed. In the aforementioned algorithm proposed in [26], an
initial broadcast was fulfilled by non-faulty agents. In [27], through
asynchronous message-passing networks which may include the issue of byzantine
failure, distributed causal shared memory in the client-server model was
considered. The issue of lattice agreement was investigated in asynchronous
systems in [28] where each process would make a decision over an metric from a
per-defined lattice under the assumption of the problem of byzantine failure.
In [29], the problem of online payments was explored, where resources were
transferable among agents. Towards such end, Astro, a system was highly
effectively proposed to make a solution in a completely asynchronous manner.
It was also proven that Astro can enable an adequately more effective
implementation than consensus-based frameworks. In [30], it was proposed a
novel model by which one can use those extra founds being able to be created
in byzantine fault-tolerant protocols in asynchronous networks. The
aforementioned model aimed at partitioning the nodes into clusters and
concurrently and independently processing transactions on different
partitions.

####  I-A4 Information theory & game-theory [31]-[33]

From an information theoretic game-theoretical point of view, although some
rare work has been done e.g. [31, 32, 33], this dominant area of research is
still open. In [31], and for general alphabets, the saddle-point of the
conditional Kullback-Leibler-Divergence to the conditional Re’nyi divergence
was strongly evaluated. In [32], the information pattern of a mean-field-
theoretical scheme was evaluated. In [33], a trade-off was physically realised
in the context of ”how much fast and how much secure” w.r.t. two essential
limits in mean-field-games (MFGs).

(a)  Figure 2: Distributed hypothesis testing model.

###  I-B Motivations and contributions

In this paper, we are interested in responding to the following questions:
What if a network generally is under a potentially hidden risk? How can we
explore it from a game-theoretical point of view, so as to control of it? Is
stabilisability explorable for our game theoretical paradigm? How can deal
with the asynchronous behaviour of the Byzantines in order to further enhance
the synchronisation of the overall system? With regard to the incomplete
version of the literature, the questions expressed here strongly motivate us
to find an interesting solution, according to which our contributions are
described in the following.

While the data set is distributively compressed and transfered to an estimator
that aims to decide between two possible distributions for the data, a
hypothesis testing issue is investigated in this paper from a game theoretical
point of view. More specifically: (i) To the best of our knowledge and from an
information theoretic perspective, a MFG theoretical point of view has never
ever been explored for the issues of hypothesis testing or/and maximum a
posteriori before. Therefore, it is revealed that the literature technically
lacks in the theoretical context of the aforementioned issue $-$ something
that is actualised in this paper. (ii) The Newton’s second law, that is,
$\mathbb{F}=ma$ is evaluated in relation to the inertia forces which
characterise the information leakage to the network. (iii) In addition, some
further discussions are gone over such as the rotation density in a given
volume by calculating the curl of the leakage, or defining the suspension
dynamics, stabilizability of the system is explored $-$ as well as the fact
that the synchronisation of the system according to the asynchronous behaviour
of the Byzantines are evaluated.

###  I-C General notation & preliminaries

Notations used throughout the paper are mainly given in Table I. The term
$i\in\\{1,\cdots,\mathscr{u}\\}$ stands for the users where
$\mathscr{U}\supseteq\mathscr{u}$.

TABLE I: List of useful notations. Notation |  Definition |  Notation |  Definition
---|---|---|---
$\mathcal{W}$ |  Winner Walk |  $t$ |  Time
$(\cdot)^{(in)}$ |  Inner Game |  $\theta$ |  Belief
$(\cdot)^{(out)}$ |  Outer Game |  $(\cdot)^{\star}$ |  Optimum
$\ell(\cdot|\cdot)$ |  Conditional Likelihood |  $\mathscr{L}_{\alpha}$ |  $\alpha-$Leakage
$\mathbb{E}$ |  Expected-value |  $\mathcal{X}$ |  State
$\mathcal{M}$ |  PDF of MFG |  $\vartheta$ |  Cost
$\mathbb{N}ull$ |  Null |  $\nabla$ |  Nabla
$Vol$ |  Volume |  $\mathbb{B}\mathscr{all}$ |  Balloon

###  I-D Organisation

The rest of the paper is organised as follows. The system set-up and our main
results are given in Sections II and III. Subsequently, the evaluation of the
framework and conclusions are given in Sections IV and V. The proofs are also
presented in the appendices. The flow of the problem-and-solution in this
paper is also shown in Fig. 1.

##  II System model and preliminaries

In this section, we go over the opening topics. We initially discuss about the
system model from both information theoretic and control theoretic standpoints
and we also talk about some useful and fundamental preliminaries.

Consider Fig. 2 $-$ temporarily considering only one Bob. Assume that a sender
named Alice has some private data denoted by the random variable
$\mathcal{S}\in S$ which is correlated with some non-private data
$\mathcal{X}\in X$. Alice is supposed to share $\mathcal{X}$ with an analyst
named Bob. However, due to the correlation between $\mathcal{X}$ and
$\mathcal{S}$ which is captured by the joint distribution
$\mathcal{P}_{\mathcal{S},\mathcal{X}}$, Bob may be able to draw some
inference on the private data $\mathcal{S}$. Alice consequently decides to,
instead of $\mathcal{X}$, release a distorted version of $\mathcal{X}$ defined
by $\mathcal{Y}\in Y$ in order to alleviate the inference threat over
$\mathcal{S}$ logically acquirable from the observation of $\mathcal{X}$. The
distorted data $\mathcal{Y}$ is generated by passing through the following
privacy mapping, i.e., the conditional distribution
$\mathcal{P}_{\mathcal{Y}|\mathcal{X}}$. It should be noted that, in fact, Bob
may also be able to act as an adversary by using $\mathcal{Y}$ to
illegitimately infer the private data set $\mathcal{S}$, even though he is a
legitimate recipient of the data set $\mathcal{Y}$. Therefore, the privacy
mapping should be designed in the sense that we can be assured about a
reduction to the inference threat on the private set $\mathcal{S}$ as follows:
while preserving the utility of $\mathcal{Y}$ by maintaining the correlation,
i.e., dependency between $\mathcal{Y}$ and $\mathcal{X}$, we aim at
alleviating the dependency between $\mathcal{S}$ and $\mathcal{Y}$. This kind
of two-fold information-theoretic goal balances a trade-off between utility
and privacy. As also obvious, the Markov chain
$\mathcal{S}\rightarrow\mathcal{X}\rightarrow\mathcal{Y}$ holds.

Let us go in datails from an information-theory point of view. Alice observes
the source-symbol sets $-$ sequences $-$
$\mathcal{S}^{(\ell)}=\\{s^{(\ell)}\\}^{\mathcal{L}_{1}}_{\ell=1}$ and
$\mathcal{X}^{(\ell,q)}=\\{x^{(\ell,q)}\\}^{\mathcal{L}_{2}}_{\ell=1}$ where
$q-$th Bob observes
$\mathcal{Y}^{(\ell,q)}=\\{y^{(\ell,q)}\\}^{\mathcal{L}_{3}}_{\ell=1}$ which
are i.i.d discrete memoryless variables respectively with the probability mass
functions $\mathcal{P}_{\mathcal{S}}$, $\mathcal{P}_{\mathcal{X}}$ and
$\mathcal{P}_{\mathcal{Y}}$ where
$\\{\cdot\\}^{\big{(}\ell\in\left\\{1,\cdots,\mathcal{L}_{i},i\in\\{1,2,3\\}\right\\}\big{)}}$
stands literally for the $\ell$-th source-symbol per block belonging to the
source-symbol set of size $1$-by-$\mathcal{L}_{i},i\in\\{1,2,3\\}$. In fact,
we principally see a simple scheme where: (i) the Alice-encoder follows the
Borel measurable map
$f^{(\ell,q)}_{x}:\;\mathcal{X}^{(\ell,q)}\longmapsto\mathcal{M}^{(x,q)}=\\{1,2,\cdots,2^{\ell\mathcal{R}^{(\ell,q)}_{x}}\\}$,
while $\sum{q}\mathcal{R}^{(\ell,q)}_{x}$ is the rate of Alice; moreover, (ii)
the Bob-decoder legitimately follows the Borel measurable map
$g^{(\ell,q)}:\;\\{1,\cdots,2^{\ell\mathcal{R}^{(\ell,q)}_{x}}\\}\longmapsto\mathcal{X}^{(\ell,q)}$
after reception via the channel. The hypothesis is consequently defined as
$\hat{\mathscr{H}}\triangleq
g^{(\ell,q)}\big{(}\mathcal{M}^{(x,q)},\mathcal{Y}^{(\ell,q)}\big{)}$ [34].

Now, assume a finite set $\Theta\supseteq\theta$ of possible hypotheses where
its cardinality declares the total number of hypotheses. In relation to the
observation $\mathscr{s}\in\mathscr{S}$, the observation probability is given
by a conditional likelihood function
$\ell\big{(}\mathscr{s}|\theta^{\star}\big{)}\in[0;1]$, and
$\sum\limits_{\mathscr{s}\in\mathscr{S}}\ell\big{(}\mathscr{s}|\theta^{\star}\big{)}=1$.
Meanwhile, the term $\theta^{\star}$ traditionally indicates the unknown but
fixed true hypothesis to be learnt.

Additionally, for the $i\in\\{1,\cdots,\mathscr{u}\\}$ over the time horizon
$t$, let us have the following essential definitions [17, 35].

Definition 1. Our basic definitions are as follows:

  * •

$\alpha-$leakage [35]: The $\alpha-$leakage222Relating to Sibson’s mutual
information: see e.g. [36]. is defined as

$\begin{split}\;\mathscr{L}_{\alpha}\triangleq\frac{\alpha}{\alpha-1}log\frac{\mathop{{\rm
max}}\limits_{\ell{\big{(}\theta^{\star}|\mathscr{s}}\big{)}}\mathbb{E}\left\\{\Big{(}\ell{\big{(}\theta^{\star}=\theta|\theta,\mathscr{s}}\big{)}\Big{)}^{\frac{\alpha}{\alpha-1}}\right\\}}{\mathop{{\rm
max}}\limits_{\ell{\big{(}\theta^{\star}}\big{)}}\mathbb{E}\left\\{\Big{(}\ell{\big{(}\theta^{\star}=\theta|\theta}\big{)}\Big{)}^{\frac{\alpha}{\alpha-1}}\right\\}},\end{split}$

for $\alpha\in(1,\infty)$ and by the continuous extension for
$\alpha=1,\infty$, without loss of generality;

  * •

Rate of convergence to true hypothesis [17]: Rate of convergence to true
hypothesis $\theta^{\star}$ and over the time horizon is theoretically written
by

$\begin{split}\;\omega^{(i)}_{1}\triangleq\lim\limits_{t\rightarrow\infty}inf\bigg{(}-\frac{1}{t}log\Big{(}\ell(\theta_{i}^{\star};i;t)-1\Big{)}\bigg{)};\end{split}$

  * •

Rate of convergence to false hypothesis [17]: Rate of convergence to true
hypothesis $\theta^{\star}$ and over the time horizon is theoretically written
by

$\begin{split}\;\omega^{(i)}_{2}\triangleq\lim\limits_{t\rightarrow\infty}inf\bigg{(}-\frac{1}{t}log\Big{(}\ell(\theta_{i}^{\star};i;t)-0\Big{)}\bigg{)},\end{split}$

  * •

Rate of social learning [17]: Rate of social learning and with regard to the
true hypothesis $\theta^{\star}$ is presented as

$\begin{split}\;\omega_{3}\triangleq\lim\limits_{t\rightarrow\infty}inf\bigg{(}-\frac{1}{t}log\Big{(}\mathop{{\rm
max}}\limits_{i}\sum\limits_{i=1}^{\mathscr{u}}\sum\limits_{j\neq
i}\ell(\theta_{j};t)\Big{)}\bigg{)}.\end{split}$

##  III Main results

Our main achievements are given in this section providing some novel
propositions where the proofs are given in the appendices.

###  III-A Objective $-$ ”The Byzantine attack and defense are mutually
exclusive but indivisible the same as spear and shield”, declared in [10]

Defining the hypothesis pair $\big{(}\mathscr{H}_{0},\mathscr{H}_{1}\big{)}$
as the following

$\begin{split}\begin{cases}\;\mathscr{H}_{1}:\ell\big{(}\theta^{\star}\big{)}\geq\nu_{1},\\\
\mathscr{H}_{0}:\ell\big{(}\theta^{\star}\big{)}<\nu_{1},\end{cases}\end{split}$

w.r.t. the threshold $\nu_{1}$, the following error probability
$\mathbb{P}_{\mathscr{e}}$ should be minimised

$\begin{split}\;\mathbb{P}_{\mathscr{e}}\triangleq\mathbb{P}\Big{(}\mathscr{H}_{1}\big{|}\mathscr{H}_{0}\Big{)}+\mathbb{P}\Big{(}\mathscr{H}_{0}\big{|}\mathscr{H}_{1}\Big{)},\end{split}$

where the first term is called the false alarm probability, and the second one
stands for the miss detection probability. Indeed, Byzantines aim to the
cooperation in the network to fail via an increment in both the false alarm
probability and the miss detection probability. They do this in the sense that
they can conveniently do their own communications in a more efficient manner
at the cost of other users’ loss333Byzantine naturally arise when the
resources in the overall network is scarce.. The Byzantine attack issue
principally is in fact a zero-sum game between the detection process and
malicious users and that is additionally a pursuit-evation game for
Byzantines.

Remark 0. We use the temporal rate of the $\alpha-$leakage as the tool which
measures the resilience.

Now, let us go in details, where in Fig. 1 the flow of the problem-and-
solution is briefly shown.

###  III-B $\mathbb{F}=ma$

###### Proposition 1

The inertia forces which actualise the information-leakage are attainable.

Proof: See Appendix A.$\;\;\;\blacksquare$

###### Corollary 1

According to the third law of thermodynamics, when the temporal rate of the
$\alpha-$leakage goes to be constant if and only if the temperature set, that
is, the term $\theta$ thechnically goes to zero.

Proof: The proof is easy to follow, thus, it is neglected
here.$\;\;\;\blacksquare$

###  III-C MFG

###### Proposition 2

In the consequence of Proposition 1 since one consider it as the control law
of an MFG, there exists a couple of HJB and FPK equations, which can
thoroughly solve the game.

Proof: See Appendix B $-$ Similar to [39].$\;\;\;\blacksquare$

###  III-D Stabilizability

###### Proposition 3

The overall system may experience an absolute stabilizability.

Proof: See Appendix C.$\;\;\;\blacksquare$

###  III-E Nash-equilibrium

###### Proposition 4

Our bi-level MFG even experiences a Nash-equilibrium where its probability is
bounded.

Proof: See Appendix D.$\;\;\;\blacksquare$

Algorithm 1 A Dog-Leg-method based ADMM-oriented iterative online algorithm
for computing
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
named $\mathbb{F}$unction $dog-leg$.

$\mathbb{F}$unction $dog-leg$

Input: $\theta$, $\Theta\supseteq\theta$,
$\mathscr{u},\;\mathscr{U}\supseteq\mathscr{u}$,
$d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$, $d\ell\big{(}\theta\big{)}$,
$\mathscr{s}_{0}$.

Initialisation: Set
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
and the input parameters as $\emptyset$. for $i\in\mathscr{U}$ do for $k-$th
iteration do while
$\bigg{|}\frac{d\ell_{k,i}\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell_{k,i}\big{(}\theta\big{)}}-\frac{d\ell_{k,i}\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell_{k-1,i}\big{(}\theta\big{)}}\bigg{|}>\phi_{k,i}$
do (a.i.) Set $d\ell\big{(}\theta\big{)}$ to be fixed as
$d\ell^{{f}}\big{(}\theta\big{)}$; (a.ii.) Solve $\mathop{{\rm
argmin}}\limits_{\theta}\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell^{{f}}\big{(}\theta\big{)}}$;
and (a.iii.) Update:
$\Big{(}\theta^{\star};d\ell^{\star}\big{(}\theta^{\star}|\mathscr{s}_{0}\big{)}\Big{)}\leftarrow\Big{(}\theta_{k,i};d\ell_{k,i}\big{(}\theta_{k,i}|\mathscr{s}_{0}\big{)}\Big{)}$.
(b.i.) Set $d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$ to be fixed as
$d\ell^{{f}}\big{(}\theta|\mathscr{s}_{0}\big{)}$; (b.ii.) Solve $\mathop{{\rm
argmin}}\limits_{\theta}\frac{d\ell^{{f}}\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$;
and (b.iii.) Update:
$\Big{(}\theta^{\star};d\ell^{\star}\big{(}\theta^{\star}\big{)}\Big{)}\leftarrow\Big{(}\theta_{k,i};d\ell_{k,i}\big{(}\theta_{k,i}\big{)}\Big{)}$.
endwhile endswitch $\;\;\;\;\;k\leftarrow k+1$ endfor $\;\;\;i\leftarrow i+1$
endfor Return
$\frac{d\ell^{\star}\big{(}\theta^{\star}|\mathscr{s}_{0}\big{)}}{d\ell^{\star}\big{(}\theta^{\star}\big{)}}$

END-function

###  III-F Vector field

###### Proposition 5

$\alpha-$leakage $\mathscr{L}_{\alpha}$ is a vector field through which the
rotation density in each point over the given surfaces and enclosed loops
1-dimensionally or/and 2-dimensionally created by the likelihood functions
$\ell(\cdot)$ is analysable $-$ whether or not this density is equated with
zero.

Proof: See Appendix E.$\;\;\;\blacksquare$

###### Corollary 2

According to the third law of thermodynamics and Corollary 1, the
$\alpha-$leakage is not curl-free, although according to Proposition 4 it can
be, relating to the Nash-equilibrium.

Proof: The proof is easy to follow, thus, it is neglected here $-$ the
divergence is non-conservative over the closed-loops enclosed to the change in
the likelihood functions, i.e., $d\ell(\cdot)$.$\;\;\;\blacksquare$

###  III-G Ascending/descending nature

###### Proposition 6

So far, it has been proven that the $\ell(\cdot)$ terms are linear or non-
linear functions of each other $-$ in the appendices. Now, one can assert that
for each group of Byzantines and good agents separately, all the $\ell(\cdot)$
are of either a purely ascending nature or a descending one.

Proof: See Appendix F.$\;\;\;\blacksquare$

###  III-H Probabilistic formalisation of the true hypothesis
$\mathscr{H}_{1}$

###### Proposition 7

The true hypothesis $\mathscr{H}_{1}$ can be mapped to a probabilistic one
where it is both attainable and analysable.

Proof: See Appendix G.$\;\;\;\blacksquare$

Algorithm 2 An iterative online algorithm for byzantine-resilient distributed
hypothesis testing named $\mathbb{OBRDHT}$.

$\mathbb{P}$rocedure $\mathbb{OBRDHT}$

Input:
$\;\;\;\;\mathbb{I}\text{nput}\triangleq\bigg{\\{}\theta,\Theta\supseteq\theta,\mathscr{u},\mathscr{U}\supseteq\mathscr{u},\mathcal{T}^{(case)}_{start}$,
$\mathcal{T}^{(case)}_{end}$, $\mathcal{X}^{(case)}_{i}$,
$\mathcal{X}^{(case)}_{-i}$, $c^{(case)}_{1}$, $c^{(case)}_{2}$,
$c^{(case)}_{3}$, $\zeta^{(case)}_{local}(\cdot)$, $\mathcal{M}^{(case)}$,
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}\bigg{\\}}$

Initialisation: (i) Invoke
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
from $\mathbb{F}$unction $dog-leg$. ${\;\%\;Algorithm\;\ref{ffff}}\cdots$ (ii)
Set $\ell\big{(}\theta\big{)}$ and
$\bigg{\\{}\mathbb{I}\text{nput}\setminus\bigg{\\{}\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}\bigg{\\}}\bigg{\\}}$
as $\emptyset$. for $i\in\mathscr{U}$ do for $k-$th iteration do switch $case$
$\;\;\;\;\;\;\;\%\;(\cdot)^{(case)}\in\\{(\cdot)^{(in)},(\cdot)^{(out)}\\}$
case 1 $case=in$ $\;\;\;\;\;\;\;\%\;(\cdot)^{(case)}=(\cdot)^{(in)}$ case 2
$case=out$ $\;\;\;\;\;\;\;\%\;(\cdot)^{(case)}=(\cdot)^{(out)}$ if
$\theta\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\Theta\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathscr{u}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathscr{U}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathcal{T}^{(case)}_{start}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathcal{T}^{(case)}_{end}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathcal{X}^{(case)}_{i}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathcal{X}^{(case)}_{-i}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;c^{(case)}_{1}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;c^{(case)}_{2}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;c^{(case)}_{3}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\zeta^{(case)}_{local}(\cdot)\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\mathcal{M}^{(case)}\neq\emptyset\;{\color[rgb]{1,0,0}\&}\;\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
then while
$\Big{|}\mathcal{X}^{(case)}_{k,i}-\mathcal{X}^{(case)}_{k-1,i}\Big{|}>\gamma_{k,i}^{(case)}$
do Phase 0: at Server. Server sends a predefined global belief to Users. Phase
1: at Users. Users send their local beliefs to Server. Phase 2: at Server.
Server receives the local observations from Users and compares the gradient of
the local beliefs of Users with the global belief according to the relative
control laws: (i) Find $\ell_{k,i}\big{(}\theta\big{)}$ w.r.t. HJB-FPK;
${\;\%\;Record\;the\;optimum}\cdots$ (ii) Update:
$\ell\big{(}\theta^{\star}\big{)}\leftarrow\ell_{k,i}\big{(}\theta\big{)}$;
and (iii) Server sends the global belief to Users. endwhile else then Break;
endif endswitch $\;\;\;\;\;k\leftarrow k+1$ endfor $\;\;\;i\leftarrow i+1$
endfor Return $\ell\big{(}\theta^{\star}\big{)}$

END-procedure

###  III-I How to deal with the asynchronous behaviour of the Byzantines

There arises a tremendously crucial query that what if the Byzantines
asynchronise themselves and if there exists a solution to this type of issue.
The greedy Algorithm 3 as well as Algorithm 4 indeed provide an opportunity
for us to tackle this kind of issue.

###### Proposition 8

One can probabilistically go towards a solution to the asynchronous behaviour
of the Byzantines.

Proof: See Appendix H.$\;\;\;\blacksquare$

###### Proposition 9

One can go towards a solution to the sasynchronous behaviour of the Byzantines
from a non-convex optimisation point of view.

Proof: See Appendix I.$\;\;\;\blacksquare$

###### Proposition 10

The criterion in relation to the freshness of the information, that is, the
age-of-information (AoI) is analysable in a given domain.

Proof: See Appendix J.$\;\;\;\blacksquare$

###  III-J Complexity

The online444An online algorithm is the one where the data set is only
available in terms of a real-time fashion, in contrast to offline ones where
the whole data set is completely available at the initial point. Algorithm 2
named $\mathbb{OBRDHT}$ has the following complexity. In each iteration and
for every user, our MFG principally creates a complexity order of bound
$NlogN$, where the function $\mathbb{F}\textit{unction}\;dog-leg$ being
invoked from Algorithm 1 including the alternating direction method of
multipliers (ADMM) creates the one as
$\mathcal{O}\big{(}\rho^{-1}\big{)}+\mathcal{O}\big{(}\rho^{-2}\big{)}$ w.r.t.
the given accuracy coefficient $\rho$. Therefore, the total mount of the
complexity has the order of bound
$\mathcal{T}\times\mathscr{u}\times\Big{(}\mathcal{O}\big{(}\rho^{-1}\big{)}+\mathcal{O}\big{(}\rho^{-2}\big{)}+NlogN\Big{)}$
where $\mathcal{T}$ stands literally for the overall running time.

##  IV Numerical results

We have done our simulations w.r.t. the Bernoulli-distributed data-sets using
GNU Octave of version $4.2.2$ on Ubuntu $16.04$. The parameters used in the
simulations are listed in Table II [37, 38]. Recall
$\mathbb{P}_{\mathscr{e}}\triangleq\mathbb{P}\Big{(}\mathscr{H}_{1}\big{|}\mathscr{H}_{0}\Big{)}+\mathbb{P}\Big{(}\mathscr{H}_{0}\big{|}\mathscr{H}_{1}\Big{)}$
from the previous parts while the first term is the false alarm probability as
$\frac{\kappa_{2}}{\kappa_{1}+\kappa_{2}}\mathscr{Q}\Big{(}\frac{\partial-2\tau_{s}B_{a}\partial_{n}^{2}}{\sqrt{4\tau_{s}B_{a}\partial_{n}^{4}}}\Big{)}$,
and the second one stands for the miss detection probability as
$\frac{\kappa_{1}}{\kappa_{1}+\kappa_{2}}\bigg{(}1-\mathscr{Q}\Big{(}\frac{\partial-2\tau_{s}B_{a}(\partial_{p}^{2}+\partial_{n}^{2})}{\sqrt{4\tau_{s}B_{a}(\partial_{p}^{2}+\partial_{n}^{2})^{2}}}\Big{)}\bigg{)}$
while $\partial$ is the detection threshold, $\mathscr{Q}(\cdot)$ is
Q-function and $\tau_{s}$ is the sensing time while
$\tau_{s}=\tau_{tot}-\tau_{t}$ holds w.r.t. the transmission time $\tau_{t}$
calculable as
$\tau_{t}=\frac{E_{t}}{\frac{\kappa_{4}N_{0}N_{rx}B_{a}(\frac{4\pi}{\kappa_{5}})^{\kappa_{3}}10^{\kappa_{3}}}{G_{a}\kappa_{6}}r_{s}^{\kappa_{3}}+P_{elec}}$
w.r.t. the energy consumed for transmission $E_{t}$ and the transmission range
$r_{s}$. It is noteworthy to add that we get access to the energy from the MFG
theoretical discussion and the Hamiltonian, and the time frame w.r.t. AoI.

TABLE II: Simulation parameters $-$ similar to [37, 38]. Parameter |  Symbol |  Value
---|---|---
PU birth rate |  $\kappa_{1}$ |  $3/s$
PU death rate |  $\kappa_{2}$ |  $3/s$
Path loss exponent |  $\kappa_{3}$ |  $3$
Reference SU SNR (dB) |  $\kappa_{4}$ |  $20$
Byzantine-PU SNR (dB) |  $\frac{\partial^{2}_{p}}{\partial^{2}_{n}}$ |  $10\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
Non-Byzantine-PU SNR (dB) |  $\frac{\partial^{2}_{p}}{\partial^{2}_{n}}$ |  $10$
Receiver noise figure |  $N_{rx}$ |  $12.589$
Thermal noise ($\frac{W}{Hz}$) |  $N_{0}$ |  $417\times 10^{-23}$
Bandwidth |  $B_{a}$ |  $10KHz$
Signal wavelength |  $\kappa_{5}$ |  $0.125m$
Amplifier efficiency |  $\kappa_{6}$ |  $0.2$
Antenna gain |  $G_{a}$ |  $0.01$
TX circuit power cons. |  $P_{elec}$ |  $3.63mW$

In Fig. 3, the total error
$\mathbb{P}_{\mathscr{e}}\triangleq\mathbb{P}\Big{(}\mathscr{H}_{1}\big{|}\mathscr{H}_{0}\Big{)}+\mathbb{P}\Big{(}\mathscr{H}_{0}\big{|}\mathscr{H}_{1}\Big{)}$
as discussed in the previous paragraph is depicted versus the $\tau_{s}$
regime while changing $r_{s}$. The more $r_{s}$ we choose, the less
probability we can assure.

In Fig. 4, the $\alpha-$included rate versus the iteration set is depicted. As
obvious, the higher $\alpha$ we have, the worse overall system performance we
experience.

In Fig. 5, the cumulative distribution function (CDF) of the iterations
required for the convergence of Algorithm 2 named $\mathbb{OBRDHT}$ $-$ which
invokes Algorithm 1 named $dog-leg$ $-$ is presented while changing the
normalised version of
$\beta\triangleq\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$.
It is revealed that our hidden control law which will be discussed in Appendix
B $-$ Remark 4 $-$ is a highly important key in our scheme $-$ something that
emphatically proves our contribution in this context.

In Fig. 6, in terms of the Jain’s fairness index, the fairness

$\begin{split}\;\frac{\bigg{(}\sum\limits_{i=1}^{u}\mathscr{h}\Big{(}\big{(}\mathscr{L}^{(i)}_{\alpha}\big{)}^{-1}\Big{)}\bigg{)}^{2}}{u\sum\limits_{i=1}^{u}\mathscr{h}\Big{(}\big{(}\mathscr{L}^{(i)}_{\alpha}\big{)}^{-1}\Big{)}^{2}},\end{split}$

is shown while changing $\alpha$ where $\mathscr{h}\big{(}\cdot\big{)}$ is a
reverse function of $\mathscr{L}^{(i)}_{\alpha}$. As absolutely obvious, the
same as Fig. 4, the higher $\alpha$ we have, the worse overall system
performance we literally experience.

In Fig. 7, the true hypothesis rate (%), that is, $\omega_{1}$ from Definition
1 is shown versus the response time vs. the $1-\beta$ regime, while changing
$\alpha$. This figure also emphasises on the results achieved from the
previous figures $-$ something that recoreds different maximum values.

In Fig. 8, pen-ultimately speaking, Algorithm 3 and Algorithm 4 are compared
with each other. The first sub-figure shows the amount of $\alpha-$leakage
obtained from Algorithm 3 divided to the one obtained by Algorithm 4 is shown.
As revealed, Algorithm 4 out-performs Algorithm 3 in the context of
synchronisation. Meanwhile, the second and the third sub-figures also show a
significantly more adequate and acceptable CDF of synchronisation for
Algorithm 4 compared to Algorithm 3 $-$ for the various values of $\alpha$ and
$\beta$.

In Fig. 9, finally speaking, the average normalised AoI versus the iteration
regime is finally depicted while changing the values for $\alpha$ and $\beta$.
This figure also, as obvious, proves the effect of $\alpha$ and $\beta$ the
same as the previous figures as well.

##  V conclusion

Novel insights to byzantine-resilient distributed hypothesis testing were
provided in this paper. We proposed a new iterative algorithm and examined its
performance by simulations and interpreted it from a theoretical point of view
as well.

##  Appendix A Proof of Proposition 1:
$\mathbb{F}=ma$

For the proof, we need to go over the Newton’s second law as $\mathbb{F}={ma}$
as below.

Initially speaking, the temporal rate of the $\alpha-$leakage can be
principally calculated as

$\begin{split}\;\frac{d\mathscr{L}_{\alpha}}{dt}\triangleq\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\\
\underbrace{\frac{d\mathscr{L}_{\alpha}}{d\ell\big{(}\theta^{\star}|\mathscr{s}\big{)}}\times\cancelto{\ell\big{(}\theta^{\star}|\theta\big{)}}{\frac{d\ell\big{(}\theta^{\star}|\mathscr{s}\big{)}}{d\ell\big{(}\theta|\mathscr{s}\big{)}}}\times\frac{d\ell\big{(}\theta|\mathscr{s}\big{)}}{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}\times\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}}_{\text{Outer
MFG}}\times\underbrace{\frac{d\ell\big{(}\theta\big{)}}{dt}}_{\text{Inner
MFG}},\end{split}$

according to the chain rule in derivatives, where:

  * •

(i) the first term comes from the fact that $\mathscr{L}_{\alpha}$ is mainly a
function of $\ell\big{(}\theta^{\star}|\mathscr{s}\big{)}$;

  * •

(ii) for the second term, that is,
$\frac{d\ell\big{(}\theta^{\star}|\mathscr{s}\big{)}}{d\ell\big{(}\theta|\mathscr{s}\big{)}}$
which is equal to $\ell\big{(}\theta^{\star}|\theta\big{)}$, we need555See
e.g. [40]. $\ell\big{(}{\mathscr{s}}|\theta^{\star}\big{)}$ and
$\ell\big{(}{\mathscr{s}}|\theta\big{)}$ which are both available;

  * •

(iii) the third term theoretically originates from the fact that we need to
define a constraint according to which the divergence between the current
state with the major player $-$ in terms of a hidden game where the
observations are the agents $-$ , that is, the initial condition can be less
than a threshold666See e.g. [33].;

  * •

(iv) the penultimate term comes from the issue that we are interesting in
doing a distillation of $\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$ in finding
$\ell\big{(}\theta\big{)}$ $-$ as it is possible since
$\ell\big{(}\theta\big{)}=\sum\limits_{\mathscr{s}_{0}}\ell\big{(}\mathscr{s}_{0}\big{)}\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$
holds; and

  * •

(v) the last term can be justified by the Kushner-Zakai principle777See e.g.
[41]..

Regarding the issue that all the four terms represented above are achievable,
therefore, we can say that the temporal rate of the $\alpha-$leakage is
attainable as well.

Remark 1. The iterative Dog-leg-method based ADMM oriented online algorithm 1
is presented for the issue expressed here. This algorithm can find the optimum
value for the pair
$\Big{(}d\ell^{\star}\big{(}\theta^{\star}|\mathscr{s}_{0}\big{)};d\ell^{\star}\big{(}\theta^{\star}\big{)}\Big{)}$
as the Leg-Knee pair.

The proof is now completed.$\;\;\;\blacksquare$

##  Appendix B Proof of Proposition 2:
MFG realisation $-$ Similar to [39]

The sketch of the proof is easy to follow. First, we consider a nested bi-
level MFG for which we define $(\cdot)^{(in)}$ and $(\cdot)^{(out)}$,
respectively. Indeed, there exists888See e.g. [42]. a bi-level online
stochastic optimisation as $\;\mathop{{\rm
min}}\limits_{\ell\big{(}\theta\big{)}}\mathbb{E}_{\theta}\bigg{(}\Omega_{1}\Big{(}\ell\big{(}\theta\big{)},\mathscr{L}_{\alpha}\Big{)}\Big{|}\mathscr{H}_{1}\bigg{)}$$s.t.\;\
\mathop{{\rm
min}}\limits_{\theta}\mathbb{E}_{t}\Big{(}\Omega_{2}\big{(}\theta,t\big{)}\Big{|}\mathscr{H}_{1}\Big{)},$
w.r.t. the given $\Omega_{1}(\cdot)$ and $\Omega_{2}(\cdot)$.999Step 1:
Control law. • (i) Inner MFG. Defining the state
$\mathcal{X}^{(in)}_{i}(t)\triangleq\ell^{(i,t)}\big{(}\theta\big{)}$ in
relation to the $i-$th agent, we have the control law as the game dynamics as
follows
$d\mathcal{X}^{(in)}_{i}(t)=\mathscr{f}^{(in)}\big{(}\mathcal{X}^{(in)}_{i}(t)\big{)}dt+\mathcal{W}^{(in)}_{i,0}$
$-$ as the Kushner-Zakai equation as discussed above $-$ with regard to the
arbitrary function $\mathscr{f}^{(in)}(\cdot)$ and the Winner random walk
process $\mathcal{W}^{(in)}_{i,0}$. • (ii) Outer MFG. Defining the state
$\mathcal{X}^{(out)}_{i}(t)\triangleq\mathscr{L}_{\alpha}^{(i)}(t)$ in
relation to the $i-$th agent, we have the control law as the game dynamics as
follows
$d\mathcal{X}^{(out)}_{i}(t)=\mathscr{f}^{(out)}\big{(}\mathcal{X}^{(out)}_{i}(t)\big{)}dt+\mathcal{W}^{(out)}_{i,0}$
with regard to the arbitrary function $\mathscr{f}^{(out)}(\cdot)$ and the
Winner random walk process $\mathcal{W}^{(out)}_{i,0}$. Remark 2. Hereinafter,
let us the following holds:
$(\cdot)^{(case)}\in\left\\{(\cdot)^{(in)},(\cdot)^{(out)}\right\\}$. Remark
3. The terms $\mathscr{f}^{(in)}\big{(}\mathcal{X}^{(in)}_{i}(t)\big{)}$ and
$\mathscr{f}^{(out)}\big{(}\mathcal{X}^{(out)}_{i}(t)\big{)}$ physically
consist of the evolutions to respectively $\mathcal{X}^{(in)}_{i}(t)$ and
$\mathcal{X}^{(out)}_{i}(t)$ with regard to the relative gradients. In
addition, the amount of the aforementioned divergences and evolutions are
upper-bounded, according to the concentration-measure as
$\;\mathbb{P}\bigg{\\{}\Big{|}\mathcal{X}^{(case)}_{i}(t){\displaystyle\pm}\mathscr{f}^{(case)}\big{(}\mathcal{X}^{(case)}_{i}(t)\big{)}\Big{|}\leq\varkappa_{1}^{(i,t)}\bigg{\\}}\leq\varkappa_{2}^{(i,t)},$
w.r.t. concentration-measure thresholds $\varkappa_{1}^{(i,t)}$ and
$\varkappa_{2}^{(i,t)}$, correspondingly. Step 2: Cost. For the trajectory
$\left\\{\mathcal{X}^{(case)}_{i}(\tau)\right\\}_{[\mathcal{T}^{(case)}_{start},\mathcal{T}^{(case)}_{end}]}$,
the cost
$\vartheta^{(case)}\big{(}\mathcal{X}^{(case)}_{i};\mathcal{X}^{(case)}_{-i};t\big{)}$
consists of the term
$\mathscr{g}^{(case)}\big{(}\mathcal{X}^{(case)}_{i}(\tau);\mathcal{X}^{(case)}_{-i}(\tau)\big{)}$
with given states of other agents, i.e., $\mathcal{X}^{(case)}_{-i}$, and with
given initial and final time instants $\mathcal{T}^{(case)}_{start}$ and
$\mathcal{T}^{(case)}_{end}$, where we have
$\;\vartheta^{(case)}\big{(}\mathcal{X}^{(case)}_{i};\mathcal{X}^{(case)}_{-i};t\big{)}\triangleq\mathbb{E}\Bigg{(}\int_{\mathcal{T}^{(case)}_{start}}^{\mathcal{T}^{(case)}_{end}}\cdots$${\mathscr{g}^{(case)}\big{(}\mathcal{X}^{(case)}_{i}(\tau);\mathcal{X}_{-i}^{(case)}\big{)}}d\tau\Bigg{)}$$=\mathbb{E}\Bigg{(}\int_{\mathcal{T}^{(case)}_{start}}$${}^{\mathcal{T}^{(case)}_{end}}\cdots$$\bigg{(}c^{(case)}_{1}\zeta^{(case)}_{local}$$\big{(}\mathcal{X}^{(case)}_{i}(\tau)\big{)}$$+$$c^{(case)}_{2}\zeta^{(case)}_{global}$$\big{(}\mathcal{X}^{(case)}_{i}(\tau)$$;\mathcal{X}^{(case)}_{-i}(\tau)\big{)}+c^{(case)}_{3}\bigg{)}d\tau\Bigg{)},$
where $c^{(case)}_{i},i\in\\{1,2,3\\}$ are positive constants, and the terms
$\zeta^{(case)}_{local}$ and $\zeta^{(case)}_{global}$ stand respectively for
the local and global costs. Step 3: HJB & FPK. Regarding the fact that we
concentrate on the following problem $\;\mathop{{\rm
min}}\limits_{\ell{\big{(}\theta^{\star}}\big{)}}\mathbb{E}\Bigg{(}\int_{\mathcal{T}^{(case)}_{start}}^{\mathcal{T}^{(case)}_{end}}\bigg{(}c^{(case)}_{1}\zeta^{(case)}_{local}\Big{(}\ell{\big{(}\theta^{\star}}\big{)};\mathcal{X}^{(case)}_{i}(\tau)\Big{)}+$$c^{(case)}_{2}\zeta^{(case)}_{global}\Big{(}\ell{\big{(}\theta^{\star}}\big{)};\mathcal{X}^{(case)}_{i}(\tau);\mathcal{X}^{(case)}_{-i}(\tau)\Big{)}+c^{(case)}_{3}\bigg{)}d\tau\Bigg{)},$$s.t.$$d\mathcal{X}^{(case)}_{i}(t)=\mathscr{f}^{(case)}\big{(}\mathcal{X}^{(case)}_{i}(t)\big{)}dt+\mathcal{W}^{(case)}_{i,0},$
one can integrate the both phrases given above in terms of $\;\mathop{{\rm
min}}\limits_{\ell{\big{(}\theta^{\star}}\big{)}}c^{(case)}_{1}\zeta^{(case)}_{local}\Big{(}\ell{\big{(}\theta^{\star}}\big{)};\mathcal{X}^{(case)}_{i}(t)\Big{)}$$+c^{(case)}_{2}\zeta_{global}\Big{(}\ell{\big{(}\theta^{\star}}\big{)};\mathcal{X}^{(case)}_{i}(t);\mathcal{X}^{(case)}_{-i}(t)\Big{)}+c^{(case)}_{3}\;\\\
+\mathbb{E}\;\Big{(}\vartheta\big{(}\mathcal{X}^{(case)}_{i}+d\mathcal{X}^{(case)}_{i};\mathcal{X}^{(case)}_{-i}+d\mathcal{X}^{(case)}_{-i};t+dt\big{)}\Big{)},$
while using the second order Taylor expansion for the expected-value term
achieved above $-$ since a non-linear function can be considered a linear one
in the context of its projection over a specific horizon. $-$ one can
mathematically write HJB as $\;\partial_{t}\vartheta^{(case)}+$$\mathop{{\rm
min}}\limits_{\ell{\big{(}\theta^{\star}}\big{)}}\Big{(}\mathcal{X}^{(case)}_{i}+\ell{\big{(}\theta^{\star}}\big{)}\partial_{\mathcal{X}^{(case)}_{i}}\vartheta^{(case)}\Big{)}+\mathcal{W}^{(case)}_{i,1},$
with the given Winner random walk process $\mathcal{W}^{(case)}_{i,1}$, while
FPK is written as
$\;\partial_{t}\mathcal{M}^{(case)}=\partial_{\mathcal{X}^{(case)}_{i}}\Big{(}\ell{\big{(}\theta^{\star}}\big{)}\mathcal{M}^{(case)}\Big{)}+\mathcal{W}^{(case)}_{i,2},$
with the given Winner random walk process $\mathcal{W}^{(case)}_{2}$, while
$\mathcal{M}^{(case)}$ is the probability distribution function (PDF) of the
population. Step 4: Convergence rule. The whole updating process will be
automatically terminated when the following condition is satisfied
$\;\sum\limits_{i}\big{|}d\mathcal{X}^{(case)}_{i}\big{|}\leq\gamma^{(case)},$
where the term $\gamma^{(case)}$ is a predefined threshold as well, by which
the convergence of the relative algorithm is principally guaranteed. Remark 4.
In parallel with the inner game, since we have the term
$\frac{d\ell\big{(}\theta|\mathscr{s}\big{)}}{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}$
as discussed in the previous proof, we can conclude that we can even interpret
our main problem in terms of a nested 3-level MFG as follows. In fact, the
aforementioned term indicates that, in the context of a hidden game, we have
some agents as the observations while the initial condition
$\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$ is the major-player. Therefore, one
should take into account a threshold for the maximum divergence between every
user with the major-player as
$\frac{d\ell\big{(}\theta|\mathscr{s}\big{)}}{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}\leq\eta$.
The issue expressed here also implicitly emphatically proves that
$\mathcal{M}^{(in)}$ is 2-dimensional. Indeed, there exists a 3-level online
stochastic optimisation as $\;\mathop{{\rm
min}}\limits_{\ell\big{(}\theta\big{)}}\mathbb{E}_{\theta}$$\bigg{(}\Omega_{1}\Big{(}\ell\big{(}\theta\big{)},\mathscr{L}_{\alpha}\Big{)}\Big{|}\mathscr{H}_{1}\bigg{)}$$s.t.$$\mathop{{\rm
min}}\limits_{\theta}$$\mathbb{E}_{t}\Big{(}$$\Omega_{2}\big{(}$$\theta,t\big{)}\Big{|}\mathscr{H}_{1}\Big{)}$$\mathop{{\rm
min}}\limits_{\ell\big{(}\theta|\mathscr{s}_{0}\big{)},\ell\big{(}\theta\big{)}}$$\mathbb{E}_{\mathscr{S}_{0}}$$\Big{(}\Omega_{3}$$\big{(}\ell\big{(}\theta|\mathscr{s}_{0}\big{)},$$\ell\big{(}\theta\big{)},t\big{)}$$\Big{|}\mathscr{H}_{1}\Big{)},$
w.r.t. the given $\Omega_{1}(\cdot)$, $\Omega_{2}(\cdot)$ and
$\Omega_{3}(\cdot)$, and conditioned on the occurrence of the true hypothesis
$\mathscr{H}_{1}$.$\;\;\;\blacksquare$

##  Appendix C Proof of Proposition 3:
Stabilizability

Solution 1.

The information cannot be completely and throughly leaked. This is because of
the fact that although the data processing inequality says the information
leakage is non-negative, when the system experiences a critical point, of
course it can be non-impossibly automatically shut down $-$ according to
Corollary 1 and the thrid law of thermodynamics.

Now, one can define the suspension dynamics of the overall system. Indeed, it
has been theoretically shown that101010See e.g. [43]. the function
$\mathscr{Sus}(\tau,\mathscr{w})$ mathematically expresses the suspension
dynamics w.r.t. the following partial derivate equation

$\begin{split}\;\xi\frac{\partial}{\partial\tau}\Bigg{(}f(\mathscr{Sus})+\mathscr{E}\frac{\partial\mathscr{Sus}}{\partial\tau}\Bigg{)}=\frac{\partial^{2}\mathscr{Sus}}{\partial\mathscr{w}^{2}},\end{split}$

w.r.t. the boundary conditions
$\frac{\partial\mathscr{Sus}}{\partial\mathscr{w}}(\tau,0)=0$ and
$\mathscr{Sus}(\tau,\mathscr{w}_{0})=\sigma$ where $\xi$ is the density
spectra of the $\alpha-$leakage in the system, $\frac{1}{\mathscr{E}}$ stands
for the medium amount of the elasticity111111Resilience: As the degree to
which a cost is updated in response to price or income changes. or in other
words, $\mathscr{E}$ stands for the viscosity, and $f(\mathscr{Sus})$
expresses the correlation of the shear-rate on the shear stress $-$ such as in
the Couette flow $-$ where the relevance belongs to the gradient of the
estimated version of the uncertainties about Byzantines.

Remark 5. Please note that the term $\ell(\cdot)$ actualises the shear-stress
and the relative reformation of the statistical geometry.

Now, according to the zeroth law of thermodynamics we continue. This law says
that when two isolated sub-system separately experience an equilibrium with a
given system, consequently, one can say that the two aforementioned sub-
systems are in an equilibrium with each other as well. Indeed, according to
the zeroth law of thermodynamics, we see that in the suspension case, the
inertia forces discussed in Proposition 1 and Appendix A, are in equilibrium
with each other where these isolated sub-systems can be interpreted according
to the Divergence theorem where the total gradient of each sub-part or sub-
surface plays the key role for the total system.

The proof is now completed $-$ although it can be evaluated with the following
solution as well.

Solution 2.

From Definition 1, one can see that $\mathop{{\rm
min}}\limits_{\ell{\big{(}\theta^{\star}}\big{)}}\mathop{{\rm
max}}\limits_{\ell{\big{(}\theta^{\star}|\mathscr{s}}\big{)}}\\{\cdots\\}$ is
a non-increasing operand since $\ell{\big{(}\theta^{\star}}\big{)}$ which
experiences a minimisation, is a function of
$\ell{\big{(}\theta^{\star}|\mathscr{s}}\big{)}$. Therefore, the temporal rate
of the $\alpha-$leakage is non-positive, specifically regarding the fact that
the total rate of the system should be non-negative $-$ as discussed above.

The proof is now completed.$\;\;\;\blacksquare$

##  Appendix D Proof of Proposition 4:
Nash-equilibrium

In consequence of the stabilizability, the issue of Nash-equilibrium is also
strongly provable. In other words, even this issue is provable since there
exists some specific cases between two groups of the Byzantines and the fusion
centre where they experience a zero-sum game the final result of which is:
either (i) lose-win, either (ii) win-lose, or (iii) half-half, where the later
forms an equilibrium. In addition, the Nash-equilibrium has the probability
order of bound
$\mathbb{P}\Big{\\{}\big{|}\nu_{2}\leq\theta^{\star}\leq\nu_{3}\big{|}\leq\lambda_{1}\Big{\\}}\leq\lambda_{2}$,
according to the concentrayio-measure principle and w.r.t. the concentration-
measure thresholds $\lambda_{1}$ and $\lambda_{2}$ and w.r.t the thresholds
$\nu_{2}$ and $\nu_{3}$ in relation to the true-and-false hypothesis.

The proof is now completed.$\;\;\;\blacksquare$

##  Appendix E Proof of Proposition 5:
Curl for the vector field $\mathscr{L}_{\alpha}$

The $\alpha-$leakage $\mathscr{L}_{\alpha}$ is a vector field theoretically
consisting of $\ell\big{(}\theta\big{)}$, $\ell\big{(}\mathscr{s}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}\big{)}$, $\ell\big{(}\mathscr{s}|\theta\big{)}$,
$\ell\big{(}\theta|\mathscr{s}\big{)}$,
$\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}|\theta\big{)}$,
$\ell\big{(}\mathscr{s}|\mathscr{s}_{0}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}|\mathscr{s}\big{)}$, and also
$\ell\big{(}\theta|\mathscr{Q}\big{)}$,
$\ell\big{(}\mathscr{s}|\mathscr{Q}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}|\mathscr{Q}\big{)}$,
$\ell\big{(}\mathscr{Q}|\theta\big{)}$,
$\ell\big{(}\mathscr{Q}|\mathscr{s}\big{)}$,
$\ell\big{(}\mathscr{Q}|\mathscr{s}_{0}\big{)}$ w.r.t. the arbitary variable
$\mathscr{Q}$.

Now, the Stoke’s theorem in relation to the curl principle indiates that for
the given surface $\psi\supseteq\partial\psi$ and the given closed-loop
$\partial\psi\subseteq\psi$ over this surface121212That is, the enclosed
boundary curve., one can write the following for the line
$\partial\psi_{0}\subseteq\partial\psi$ and the surface
$\psi_{0}\subseteq\psi$

$\begin{split}\;\iint\limits_{\psi}\big{(}\nabla\times\mathscr{L}_{\alpha}).d\big{(}\psi_{0}\big{)}=\int\limits_{\partial\psi}\mathscr{L}_{\alpha}.d\big{(}\partial\psi_{0}\big{)},\end{split}$

where

$\begin{split}\;\iint\limits_{\psi\triangleq
h\big{(}\ell(\theta^{\star}|\mathscr{s}),\ell(\theta|\mathscr{s})\big{)}}\big{(}\nabla\times\mathscr{L}_{\alpha}).d\Big{(}\psi_{0}\triangleq
h_{0}\big{(}\ell(\theta|\mathscr{s}),\ell(\theta|\mathscr{s}_{0})\big{)}\Big{)}=\\\
\int\limits_{\partial\psi\triangleq\ell(\theta|\mathscr{s}_{0})}\mathscr{L}_{\alpha}.d\big{(}\partial\psi_{0}\triangleq\ell(\theta)\big{)},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{split}$

holds, as a case in point, for the surface
$h\big{(}\ell(\theta^{\star}|\mathscr{s}),\ell(\theta|\mathscr{s})\big{)}\supseteq\ell(\theta|\mathscr{s}_{0})$
and the enclosed closed-loop $\ell(\theta|\mathscr{s}_{0})\subseteq
h\big{(}\ell(\theta^{\star}|\mathscr{s}),\ell(\theta|\mathscr{s})\big{)}$ over
this surface, and for the sub-loop
$\ell(\theta)\subseteq\ell(\theta|\mathscr{s}_{0})$ and the enclosed sub-
surface
$h_{0}\big{(}\ell(\theta|\mathscr{s}),\ell(\theta|\mathscr{s}_{0})\big{)}\subseteq
h\big{(}\ell(\theta^{\star}|\mathscr{s}),\ell(\theta|\mathscr{s})\big{)}$ for
the given functions $f(\cdot)$, $h(\cdot)$ and $h_{0}(\cdot)$.

(a)  Figure 3: total error
$\mathbb{P}_{\mathscr{e}}\triangleq\mathbb{P}\Big{(}\mathscr{H}_{1}\big{|}\mathscr{H}_{0}\Big{)}+\mathbb{P}\Big{(}\mathscr{H}_{0}\big{|}\mathscr{H}_{1}\Big{)}$
as discussed in the previous paragraph is depicted versus the $\tau_{s}$
regime while changing $r_{s}$.  (a)  Figure 4: Rate including $\alpha-$leakage
against the iteration regime, changing $\alpha$.  (a)  Figure 5: CDF of the
iterations required for the convergence of online Algorithm 2 $-$ named
$\mathbb{OBRDHT}$ $-$ changing the normalised version of
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$,
i.e., $\beta$.  (a)  Figure 6: Fairness index (%) versus the number of the
iterations changing $\alpha$.

Remark 6. Regarding the fact that the likelihood functions have the polar
structure of $re^{i\phi}$ for the given $r\in[0,1]$ and $\phi\in[0,2\pi]$
where $i\triangleq\sqrt{-1}$, the divergence of $\mathscr{L}_{\alpha}$ in a
given point $\theta_{\beta}$ w.r.t. $\ell\big{(}\theta\big{)}$,
$\ell\big{(}\mathscr{s}\big{)}$, $\ell\big{(}\mathscr{s}_{0}\big{)}$,
$\ell\big{(}\mathscr{s}|\theta\big{)}$,
$\ell\big{(}\theta|\mathscr{s}\big{)}$,
$\ell\big{(}\theta|\mathscr{s}_{0}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}|\theta\big{)}$,
$\ell\big{(}\mathscr{s}|\mathscr{s}_{0}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}|\mathscr{s}\big{)}$,
$\ell\big{(}\theta|\mathscr{Q}\big{)}$,
$\ell\big{(}\mathscr{s}|\mathscr{Q}\big{)}$,
$\ell\big{(}\mathscr{s}_{0}|\mathscr{Q}\big{)}$,
$\ell\big{(}\mathscr{Q}|\theta\big{)}$,
$\ell\big{(}\mathscr{Q}|\mathscr{s}\big{)}$,
$\ell\big{(}\mathscr{Q}|\mathscr{s}_{0}\big{)}$ is given as

$\begin{split}\;\oiint\limits_{\Gamma}\bigg{\\{}\frac{\partial\mathscr{L}^{(\epsilon)}_{\alpha}}{\partial\epsilon}+\frac{1}{\epsilon}\partial\mathscr{L}^{(\epsilon)}_{\alpha}+\frac{1}{\epsilon}\frac{\partial\mathscr{L}^{(\epsilon)}_{\alpha}}{\partial\phi}\bigg{\\}}d\Gamma_{0},\end{split}$

where the key terms $\epsilon$ and $\phi$ are respectively the diameter
function and the angle function relating to the diameters and angles of all
the likelihood functions defined above, physically including the test point
$\theta_{\beta}$, while $\Gamma\supseteq\Gamma_{0}$ is the test surface
consisting of every 2 curves being defined from the likelihood functions over
which the normal vectors are correspondingly defined.

Remark 7. It should also be careful that the divergence of the density of a
fluid is zero iff (if and only if) there exists a source or sink $-$ as more-
or-less discussed before in the context of the third law of thermodynamics.

The proof is now completed.$\;\;\;\blacksquare$

(a) $\alpha=8$

(b) $\alpha=18$

Figure 7: Tru hypothesis rate(%) versus the response time (second) vs. the
$1-\beta$ regime, changing $\alpha$ $-$ something that recoreds different
maximum values.

##  Appendix F Proof of Proposition 6:
Ascending/descending nature

First, we know we have to classify the agents into two groups of Byzantines
and good agents since Byzantines aim at maximising the error
$\mathbb{P}_{\mathbb{e}}$ or equivalently

$\begin{split}\;\mathcal{D}_{\mathscr{kl}}\Big{\\{}\mathscr{H}_{0}\big{|}\big{|}\mathscr{H}1\Big{\\}}\triangleq\ell\Big{(}\theta\big{|}\mathscr{H}_{1}\Big{)}\frac{log\ell\Big{(}\theta\big{|}\mathscr{H}_{1}\Big{)}}{log\ell\Big{(}\theta\big{|}\mathscr{H}_{0}\Big{)}},\end{split}$

whereas good agents follow the reverse, while the term
$\mathcal{D}_{\mathscr{kl}}\big{\\{}\cdot\big{|}\big{|}\cdot\big{\\}}$ stands
for the information theoretic principle of the Kullback-Leibler Divergence.

On the other hand, we see that for every pair
$\big{(}\ell(\cdot),\ell(\cdot)\big{)}$ which mathematically creates a
surface, the change in the threshold $\nu_{1}$ relating to the hypothesis pair
$\big{(}\mathscr{H}_{1},\mathscr{H}_{0}\big{)}$ results in a change in the
size of the relative enclosed surface, and consequently, in the size of the
closed-loop over which and according the Stokes’ theorem we apply an integral.
Therefore, we can see that for each group of Byzantines and good agents
separately, the likelihood functions are either ascending or descending which
actualise respectively a maximisation or a minimisation.

Remark 8. In relation to the changes of the relative closed-loops $-$
discussed above $-$ enclosed to the likelihood functions, the following
analysis can be applied. Through every given Riemannian manifold, one can
define a Wilson-loop131313See e.g. [44], in order to understand what it is.
for which it is satisfied a differential equation of the Schrodinger type. The
aforementioned differential equation can justify the positive/negative
evolution of the relative closed-loop.

The proof is now completed.$\;\;\;\blacksquare$

##  Appendix G Proof of Proposition 7:
Probabilistic formalisation of the true hypothesis $\mathscr{H}_{1}$

The proof is provided in the context of the following multi-step solution.

Step 1.

As proven e.g. in [45], the following holds

$\begin{split}\;\mathscr{f}\big{(}\mu_{1}\big{)}=\mathop{{\rm
inf}}\limits_{\Omega_{1}\triangleq\big{\\{}\Omega:\mathscr{L}_{\alpha}\setminus\\{0\\}\big{\\}},\;\mathbb{N}ull\big{\\{}\Omega\setminus
0\big{\\}}\neq\emptyset}\frac{\int\limits_{\Omega}curl\big{(}\mathscr{L}_{\alpha}\big{)}d\Omega}{\mathscr{L}_{\alpha}},\end{split}$

where $\mathscr{f}\big{(}\mu_{1}\big{)}$ is technically a function of the
first eigenvalue $\mu_{1}$. It should also be noticed that the above equation
is valid when $\mathscr{L}_{\alpha}\neq 0$, since as declared in the equation,
the search region emphasis this issue.

(a) $\alpha-$leakage obtained from Algorithm 3 divided to the one obtained by
Algorithm 4.

(b) CDF of synchronisation vs. the iterations while changing $\alpha$ and
$\beta$: Algorithm 4.

(c) CDF of synchronisation vs. the iterations while changing $\alpha$ and
$\beta$: Algorithm 3.

Figure 8: An overall comparison between Algorithm 3 and Algorithm 4. (a)
Average optimal policy $\pi_{t}(a|s)$ Figure 9: Average normalised AoI versus
the iterations while changing $\alpha$ and $\beta$..

Remark 9. As perfectly shown in the literature e.g. in [46], the first-
eigenvalue is the central one that plays a totally important role in the
creation of a Riemannian manifold. Meanwhile, as proven e.g. in [47], the
facts $\mu_{1}\big{(}\mathbb{B}\mathscr{all}^{(0)}\big{)}\leq
f\Big{(}\big{(}\mu_{1}(\Upsilon)\big{)}\Big{)}$ and
$\mathbb{B}\mathscr{all}^{(0)}=\mu_{1}\big{(}\mathbb{B}\mathscr{all}^{(0)}\big{)}\partial\mathbb{B}\mathscr{all}^{(0)},\Upsilon=\mu_{1}(\Upsilon)\partial\Upsilon$
hold since we theoretically have $\partial\mathbb{B}\mathscr{all}^{(0)}\leq
f\Big{(}\partial\Upsilon\frac{\mathbb{V}ol\;\big{(}\mathbb{B}\mathscr{all}^{(0)}\big{)}}{\mathbb{V}ol\;\big{(}\Upsilon\big{)}}\Big{)}$
and w.r.t. the sample ball $\mathbb{B}\mathscr{all}^{(0)}\in\mathbb{R}^{3}$
out of the total manifold $\Upsilon\in\mathbb{R}^{3}$ where
$\partial\Upsilon\in\mathbb{R}^{2}$ and
$\partial\mathbb{B}\mathscr{all}^{(0)}\in\mathbb{R}^{2}$ stand respectively
for the surface areas enclosed with the relative volumes $\Upsilon$ and
$\mathbb{B}\mathscr{all}^{(0)}$. This kind of interpretation can also be
originated from the principle ”hearing the shape of a drum” invented by M. Kac
1966 [48] where one can infer the shape of the manifold w.r.t. the frequencies
of the enclosed susrface area, that is, the eigenvalues of the Laplacian over
the manifold. Finally speaking, it should also be emphatically taken into
account that the amount of the orientability named above is upper-bounded.
This is because of the fact that the spectral radius, that is, the first
eigenvalue theoretically experiences141414See e.g. [49]
$\mu_{1}\leq\mathop{max}\limits_{v_{1},v_{2}}\Big{\\{}\mathscr{d}_{v_{2}}+\mathscr{d}_{v_{2}}\Big{\\}}$,
with equality if and only if the relative graph relating to the manifold is a
regular or semi-regular bipartite one, where the pair of
$\big{(}v_{1},v_{2}\big{)}$ stands physically for two vertices with the
debsity $\mathscr{d}_{v_{1}}$ and $\mathscr{d}_{v_{2}}$, respectively. The
discussion given here is shown in Fig. 10.

(a)  Figure 10: How to hear the orientability of the enclosed surfaces.

Step 2.

Additionally, as discussed in Appendix F, the flexibility of the controller-
threshold $\nu_{1}$ may result in a change in the size of the relative
surfaces and closed-loops created by the likelihood functions as well. In
other words, the resultant vector born from the curl can theoretically change
w.r.t. the threshold $\nu_{1}$. Therefore, one can map the true hypothesis
$\mathscr{H}_{1}:\theta\geq\nu_{1}$ to a probabilistic one as
$\mathscr{H}_{1}:\mathbb{P}\big{\\{}curl\big{(}\mathscr{q}\big{)}>\nu_{4}\big{\\}}=1-\mathbb{P}\big{\\{}curl\big{(}\mathscr{q}\big{)}\leq\nu_{4}\big{\\}}$
where
$\mathbb{P}\big{\\{}curl\big{(}\mathscr{q}\big{)}\leq\nu_{4}\big{\\}}\triangleq\mathbb{F}_{curl\big{(}\mathscr{q}\big{)}}\Big{(}{\nu_{4}}\Big{)}$
w.r.t. the CDF
$\mathbb{F}_{curl\big{(}\mathscr{q}\big{)}}\triangleq\frac{1}{2\pi
i}\oint_{c-i\infty}^{c+i\infty}\frac{-1}{s}x^{-s}\Delta_{1}(s+1)ds$ according
to the Mellin transform [50] where
$\Delta_{1}(s)=\mathbb{E}_{curl\big{(}\mathscr{q}\big{)}}\bigg{\\{}\Big{(}curl\big{(}\mathscr{q}\big{)}\Big{)}^{s-1}\bigg{\\}}$,
while $\mathscr{q}$ may be every possible vector field relating to the
surfaces enclosed to the relative likelihood functions.

###### Corollary 3

151515See e.g. [45, 46, 47, 48].

For the evolution of the first-eigenvalue, we have the control-law
$\frac{\partial\mu^{\prime}_{1,r}}{\partial
t}+\mathscr{A}_{0}\mu^{\prime}_{1,r}(t)+\mathscr{A}_{1}\mu^{\prime}_{1,r}(t-1)=0$
where $\mu^{\prime}_{1,r}\triangleq\mu_{1,r}e^{\mu_{1}t}$ and $\mu_{1,r}$ is
the right-hand eigenvector in relation to the first eigenvalue $\mu_{1}$.

Remark 10. The corollary defined above also proves a hidden control-law
according to which one can extend our 3-level nested game to a 4-level one.

The proof is now completed.$\;\;\;\blacksquare$

##  Appendix H Proof of Proposition 8:
Probabilistic solution to the asynchronous behaviour of the Byzantines

The proof is totally easy to follows which is given in this part, prior to
which we go as follows. The pioneer research [51] indicates that: ‘’In a
synchronous system, processes run in lock step, and messages sent in one step
are received in the next.” In other words, the term of asynchronous indicates
that each process may be unknown to the other processes since it can go on at
its own speed. In fact, one can say that there exists a bounded time delay for
message transmission and one can technically guarantee the information as no
message is received.

Now, the rate of the $\alpha-$leakage is a perfect tool to probabilistically
cope with the asynchronous behaviour of the Byzantines where this rate should
be instantaneously negative as much as possible. Indeed, we follow
$\mathop{{\rm
argmin}}\limits_{\theta}\mathbb{P}r\Big{(}\frac{d\mathscr{L}_{\alpha}}{dt}\geq
0\Big{)}$, or equivalently,
$\mathbb{P}r\Big{(}\frac{d\mathscr{L}_{\alpha}}{dt}\geq
0\Big{)}\leq\gamma_{th}$ w.r.t. the threshold $\gamma_{th}$, something that
can be theoretically actualised by the greedy Algorithm 3 as well.

Algorithm 3 A Greedy algorithm to cope probabilistically with the asynchronous
behaviour of the Byzantines.

Initialisation: (i) Invoke
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
from $\mathbb{F}$unction $dog-leg$. ${\;\%\;Algorithm\;\ref{ffff}}\cdots$ (ii)
Initialise the ground set
$\Xi\leftarrow\\{0,1,2,\cdots,\frac{1}{\varsigma}-1\\}$, the feasible set
$\Theta\leftarrow\emptyset$, and the objective function
$f(\Theta)\triangleq\mathbb{P}r\Big{(}\frac{d\mathscr{L}_{\alpha}}{dt}\geq
0\Big{)}$. while $|f(\Theta)|<f_{max}$ do
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Theta_{new}\leftarrow\mathop{{\rm
max}}\limits_{{\theta}\;\in\;\Xi\;\setminus\;\Theta}{\rm\;}f(\Theta\;\cup\;\\{\theta\\})-f(\Theta),$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Theta\leftarrow\Theta\cup\\{\Theta_{new}\\}$
endwhile Output: $\Theta$ end

Remark 11. The total mount of the complexity has the order of bound
$\mathcal{T}\times\Big{(}\mathcal{O}\big{(}\rho^{-1}\big{)}+\mathcal{O}\big{(}\rho^{-2}\big{)}+(NlogN)^{2}\Big{)}$
where $\mathcal{T}$ stands literally for the overall running time.

The proof is now completed.$\;\;\;\blacksquare$

Algorithm 4 An SPSA method based algorithm to cope with the asynchronous
behaviour of the Byzantines.

Initialisation. Invoke
$\frac{d\ell\big{(}\theta|\mathscr{s}_{0}\big{)}}{d\ell\big{(}\theta\big{)}}$
from $\mathbb{F}$unction $dog-leg$. ${\;\%\;Algorithm\;\ref{ffff}}\cdots$
while $\mathbb{TRUE}$ do Compute
$f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}$, Compute $\nabla
f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}$, Update
$f\Big{(}\ell^{(n+1)}\big{(}\theta\big{)}\Big{)}\leftarrow
f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}+\nabla
f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}$. endwhile Output:
$\ell\big{(}\theta\big{)}$ end

##  Appendix I Proof of Proposition 9:
Solution to the asynchronous behaviour of the Byzantines

In addition to the first solution given above, there exists another one which
is given in the following.

As e.g. declared in [52] and shown in Fig. 11, a synchronous process is the
one where
$\mathop{max}\limits_{\mathscr{Q}}\mathbb{P}r\big{(}\mathscr{Q}\big{)}\rightarrow
1$ holds w.r.t. the arbitary $\mathscr{Q}$. This means that, after a careful
look at the denominator of the equation in relation to the $\alpha-$leakage,
it is revealed that a minimisation over the $\alpha-$leakage relies upon a
maximisation of the synchronisation of the overall system. Now, the
asynchronous behaviour of the Byzantines can result a degradation in the total
synchronisability of the system. Therefore, one can assign an unknown random
variable $\xi_{b}\in[0,1)$ coordinated by the Byzantines161616As a strictly
positive weighting scalar. where we aim at maximising the synchronisability.
This maximisation can be fully performed according to non-convex optimisation
algorithms such as the Simultaneous Perturbation Stochastic Approximation
(SPSA) method171717See e.g. [40, 53] to understand what it is. as follows.
With the goal of approximately estimating the gradient of the relative
objective function, the SPSA method theoretically utilizes finite differences
between two randomly perturbed inputs in terms of the following term

(a)  Figure 11: How the asynchronous behaviour of the Byzantines creates an
adversarial effect on the algorithm: where the asynchronisability creates a
degradation in the gradients.

$\begin{split}\;\nabla
f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}\triangleq\frac{f\Big{(}\ell^{(n)}\big{(}\theta\big{)}+\mathscr{C}_{n}\Delta_{n}\Big{)}-f\Big{(}\ell^{(n)}\big{(}\theta\big{)}-\mathscr{C}_{n}\Delta_{n}\Big{)}}{2\mathscr{C}_{n}\Delta_{n}},\end{split}$

where

$\begin{split}\;f\Big{(}\ell\big{(}\theta^{\star}\big{)}\Big{)}\triangleq\mathop{\rm
max}\limits_{\ell\big{(}\theta^{\star}\big{)}}\xi_{b}\mathbb{E}\left\\{\Big{(}\ell\big{(}\theta^{\star}=\theta|\theta\big{)}\Big{)}^{\frac{\alpha}{\alpha-1}}\right\\},\xi_{b}\in[0,1),\end{split}$

where $\Delta_{n}$ is a random perturbation vector whose elements are
Bernoulli disributed181818See e.g. [40, 53]., and w.r.t. the term
$\mathscr{C}_{n}$ as the hyper-parameter for the SPSA191919See e.g. [40, 53].,
and the term $n$ is the index of the iteration.

Finally, the algorithm updates as
$f\Big{(}\ell^{(n+1)}\big{(}\theta\big{)}\Big{)}\leftarrow
f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}+\nabla
f\Big{(}\ell^{(n)}\big{(}\theta\big{)}\Big{)}$.

The discussion given theoretically above is presented in Algorithm 4.

Remark 12. The total mount of the complexity has the order of bound
$\mathcal{T}\times\Big{(}\mathcal{O}\big{(}\rho^{-1}\big{)}+\mathcal{O}\big{(}\rho^{-2}\big{)}+0.33N\Big{)}$
where $\mathcal{T}$ stands literally for the overall running time. The term
$0.33N$ also arises from SPSA202020See e.g. [40, 53]..

The proof is now completed.$\;\;\;\blacksquare$

##  Appendix J Proof of Proposition 10:
AoI

In e.g. [54]212121Page $3$, Eq. $3$., it was proven that the probability of
transmission at the time instant $t+1$ at the $i$ node is given by
$\mathscr{P}_{i}(t+1)=max\bigg{\\{}\mathscr{P}^{(min)}_{i},\mathscr{f}\Big{(}\mathscr{P}_{i}(t);-\frac{1}{\vartheta^{(aoi)}_{i}(t)}\Big{)}\bigg{\\}}$,
where $\mathscr{P}^{(min)}_{i}$ is a constant independently and individually
pre-adjusted at each node $i$ and $\vartheta^{(aoi)}$ is AoI which we are
interested in making a reduction in.

After a careful look at the equation
$max\bigg{\\{}\mathscr{P}^{(min)}_{i},\mathscr{f}\Big{(}\mathscr{P}_{i}(t);-\frac{1}{\vartheta^{(aoi)}_{i}(t)}\Big{)}\bigg{\\}}$,
we see that this can be re-presented as
$\mathscr{P}^{(min)}_{i}u(t)+\big{(}t-\mathscr{P}^{(min)}_{i}\big{)}u\big{(}t-\mathscr{P}^{(min)}_{i}\big{)}$
where $u(\cdot)$ is the step function, i.e., $u(t)=1,t\geq 0$ and $u(t)=0,t<0$
hold. Now, we see that for
$\mathscr{P}^{(min)}_{i}\leq\mathscr{f}\Big{(}\mathscr{P}_{i}(t);-\frac{1}{\vartheta^{(aoi)}_{i}(t)}\Big{)}$
one can see a direct way to reach out AoI $\vartheta^{(aoi)}$ according to
$\mathscr{P}_{i}(t+1)$ and $\mathscr{P}_{i}(t)$ $-$ that is, the two terms
that can be found by the temporal rate of the $\alpha-$leakage according to
$\mathbb{F}=ma$.

On the other hand, a maximisation over $-\frac{1}{\vartheta^{(aoi)}_{i}(t)}$
means that we should take into account the worst-case of it, i.e., the the
value we look for AoI should be the least upper-bound of it $-$ or vice versa.

The proof is now completed.$\;\;\;\blacksquare$

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