# MeritRank: Sybil Tolerant Reputation for Merit-based Tokenomics*††thanks:
*pre-print BRAINS confrence, Paris, September 27-30, 2022

Bulat Nasrulin, Georgy Ishmaev, Johan Pouwelse  Delft University of Technology
{b.nasrulin, m.a.devos-1, g.ishmaev<EMAIL_ADDRESS>
###### Abstract

Decentralized reputation schemes present a promising area of experimentation
in blockchain applications. These solutions aim to overcome the shortcomings
of simple monetary incentive mechanisms of naive tokenomics. However, there is
a significant research gap regarding the limitations and benefits of such
solutions. We formulate these trade-offs as a conjecture on the
irreconcilability of three desirable properties of the reputation system in
this context. Such a system can not be simultaneously generalizable,
trustless, and Sybil resistant. To handle the limitations of this trilemma, we
propose MeritRank: Sybil tolerant feedback aggregation mechanism for
reputation. Instead of preventing Sybil attacks, our approach successfully
bounds the benefits of these attacks. Using a dataset of participants’
interactions in MakerDAO, we run experiments to demonstrate Sybil tolerance of
MeritRank. Decay parameters of reputation in MeritRank: transitivity decay and
connectivity decay, allow for a fine-tuning of desirable levels of reputation
utility and Sybil tolerance in different use contexts.

###### Index Terms:

Reputation, Sybil attack, Tokenomics, Feedback Aggregation

##  I Introduction

Reputation mechanisms in blockchain applications can provide many desirable
properties as system components. In general, reputation mechanisms can be
employed at different layers of blockchain systems: protocols layer e.g
Delegated Proof-of-Stake [1], middleware e.g. MEV-Boost [2], and most
prominently at the application layer of Decentralized Autonomous Organizations
(DAOs) [3, 4, 5, 6]. However, the implementation of reputation-based
incentives is still not well researched and rife with challenges. There is a
deficit of systematic understanding regarding general trade-offs of different
approaches to reputation. The most prominent research gap in this context is
the issue of Sybil attacks, a problem well known in peer-to-peer protocols
[7]. This problem presents a significant barrier to the implementation of
reputation systems proposed as a way to address the limitations of simple
tokenomics [de_filippi_reputation_2021, 9].

Token-based incentives became one of the most prominent mechanisms in
blockchain protocols, solving a number of open incentivization problems in
peer-to-peer networks, such as network liveness, network security, and open-
sources software maintenance. Experimentation with these mechanisms has
contributed to the emergence of a subfield in the blockchain design space,
sometimes labelled ‘tokenomics’. In fact, all prominent application cases for
blockchain protocols, including cryptocurrencies, Decentralized Finance
(DeFi), DAOs, and Non-Fungible Tokens (NFTs), employ these incentives
mechanisms at different levels of abstraction [11].

Limits of such incentives, however, are only slowly becoming apparent with
empirical observations. One is a misalignment of incentives that monetary
incentives create between different participants in more complex systems [13,
14]. The second limitation is the vulnerability of governance schemes built on
naive monetary incentives [15]. Finally, token-based incentives in blockchain
protocols disproportionately rewarding large holders tend to encourage re-
centralization in decentralized networks [16]. Allocation of rewards for
participants111We use terms: participant, node and peer interchangeably. of
decentralized protocols based on merits or contributions reflected in their
non-transferable ’reputation’ offers an appealing approach to overcome these
limitations. Merit-based rewards can provide alignment of incentives for
participants in complex communities and allow for a sustainable distribution
of resources [17].

This paper aims to contribute to the theoretical research gap on trade-offs of
different reputation solutions and propose a practical, scalable Sybil
tolerant reputation mechanism scheme in decentralized networks. We propose and
demonstrate feasibility on the experimental basis of MeritRank: Sybil-tolerant
Aggregated Feedback Reputation Mechanism, compatible with token-based
incentive mechanisms. Our contributions are the following:

  * •

We analyze and formulate general trade-offs between desirable properties of
reputation in decentralized settings as decentralized reputation trilemma in
section II.

  * •

We describe the system model for logic and key components for reputation-based
distribution of rewards in a generic DAO in section III.

  * •

We present the formalization of MeritRank and three types of reputation decay
as a method to achieve tolerance against Sybil attacks to manipulate feedback
aggregation reputation in section IV.

  * •

Using experiments on data set from more than 150 weeks of user interactions in
MakerDAO, we demonstrate that transitivity decay and connectivity decay
provide high tolerance against Sybil attacks on reputation. Another
experimental finding is that epoch decay, implemented in other reputation
mechanisms [6] worsens Sybil tolerance (section V).

##  II Background and Related Work

Reputation systems in decentralized environments have been proposed for
various application-specific cases, and as general models in peer-to-peer
systems [18, 19, 20, 21, 22, 23]. Accordingly, the limitations of these
solutions are relatively well-understood. Some of these limitations include
scalability [21], contextual accuracy [20], partial reliance on trusted-setups
[20, 21], vulnerability to certain types of attacks [24, 25, 26, 27, 28], and
significant privacy trade-offs [22]. However, there is also an identifiable
research gap regarding general trade-offs inherent to any reputation systems
implemented in decentralized environments. We identify only a small number of
surveys on reputation solutions in decentralized settings [20, 21], with only
limited comparative trade-off analysis [22]. The absence of engineering
research on reputation in the context of tokenomics and DAOs is characterized
by the general deficit of academic research in this area [29].

###  II-A Decentralized Reputation Trilemma

Figure 1: Reputation mechanism trade-off triangle. A reputation mechanism can
be only one of the two: Generalizable, Sybil resistant, Trustless.

We formulate these trade-offs as a conjecture on the irreconcilability of
three desirable properties of the reputation system in this context. Such a
system can not be simultaneously generalizable, trustless, or Sybil resistant.
The trustless property means that reputation accounting and evaluation does
not rely on specific trusted entities. In a fully Sybil-resistant reputation
system, the attacker cannot unfairly manipulate the reputation system by
controlling multiple identities [35]. Generalizable reputation system allows
to account and evaluate any type of participants’ contribution.

All proposed reputation mechanisms inevitably sacrifice one of these
properties to achieve two others, thus forming a Decentralized Reputation
Trilemma as illustrated in figure 1. To illustrate this problem, we look into
three conceptually different methods to implement reputation.

Trusted Oracles. This reputation method is based on some trusted oracle
keeping track of reputation scores. In the context of a decentralized system,
such an oracle can take input from the actions of participants to calculate
their reputation scores based on a pre-determined reputation function. This is
a generalizable approach as oracles can take any input for reputation function
[21]. The problem of Sybil resistance can be addressed with a task of Sybil
detection outsourced to an oracle verifying the identities of participants
[30].

Participants have to trust that oracles use correct input and provide honest
outputs on reputation function, which undermines the trustless properties of a
decentralized system. Furthermore, dependency on oracles introduces points of
failure in a system that can be exploited, as demonstrated by price oracles in
DeFi [31]. Attempts to provide trustless properties to oracles essentially
create a peer prediction market. However, this reduces their applicability to
a set of cases where agreement on reputation can be achieved through social
consensus voting, thus sacrificing generalizability [32].

Cryptographic Proofs. Another popular tool in reputation accounting is a
method of cryptographic proofs, that can be employed to address the problem of
Sybil attacks. Each peer performs some contribution or work verifiable by any
other peer in the network. Peers keep track of their and others’ reputation
scores, generating proofs of contributions and sending them to all other peers
in the network. After validating the proofs, clients update their reputation
scores. A Sybil attack is undermined by the requirement to provide verifiable
proof of contribution. Such a solution can be trustless and decentralized
since no single party is responsible for reputation accounting.

However, this approach also has significant limitations regarding
generalizability of application. Certain types of contributions like
computation work, proof of bandwidth, proof of transaction, or proof of
storage allow for accounting mechanisms based on cryptographic proofs, where
contribution accounting can be trustless and universally verifiable. However,
this type of accounting does not grasp semantics for all types of
collaborative work and human contributions Furthermore, even in those
application cases where such type of accounting is conceptually meaningful,
practical scalability can be hampered by the high overhead [wust2018you].

Feedback Aggregation. A peer-to-peer reputation is an approach where
participants interact with each other, each giving feedback on the
contribution or work provided by others. The resulting reputation is computed
by aggregating feedback from participants.

The main benefits of this approach are generalizability and the absence of
trusted third parties. Peer-to-peer feedback makes such an approach feasible
for various application-specific contexts, as peers can provide context-
specific feedback on any interaction or work. Unlike the cryptographic proofs
approach, feedback aggregation does not introduce high costs from overhead.
However, the big disadvantage of this approach is that it is susceptible to
manipulation, especially through Sybil attacks [35].

###  II-B Sybil tolerant feedback aggregation

The observations based on the reputation trilemma necessitate a solution for
reputation systems in decentralized environments that do not sacrifice
completely one of the corners of this triangle. Hard scalability limitations
for Cryptographic Proofs, and fundamental nature of limitations presented by
Trusted Oracles suggest that the Feedback Aggregation side presents the most
feasible direction to achieve acceptable trade-offs.

In the context of open, permissionless systems, strict Sybil resistance is not
achievable entirely [36]. Furthermore, attempts to emulate Sybil resistant
properties of closed systems undermine the privacy of participants and other
desirable properties of decentralized solutions. Different approaches can be
used to minimize the impact of Sybil attacks on peer feedback-based
reputation: Sybil detection and Sybil tolerance [37, 38]. The efficiency of
Sybil detection, in general, is limited by the assumptions on the behavior of
Sybil nodes that is distinctively different from the behavior of honest nodes
in the network, which is not necessarily always the case [36].

Sybil tolerance for aggregated feedback reputation mechanisms is a system
property evaluated in the context of specific Sybil attacks on reputation.
Sybil tolerance does not try to identify and filter Sybil identities but
instead limits their impact on the system. This approach allows for
straightforward application in those cases where impact can be evaluated in
consumed system resources, e.g., rewards pool in a DAO treasury. We also
demonstrate in Section V that two properties of this approach suggest its
feasibility. (1) Tolerance threshold remains constant with the increase in the
size of Sybil participants. (2) Threshold can be adjusted for different
fairness and accuracy levels depending on reputation’s contextual semantics.

##  III Merit-based Tokenomics

In this section, we present a system model for merit-based tokenomics. This
model is generic and does not make assumptions about implementation details.
It can describe a reputation system for the participants of a generic DAO,
where peers provide feedback to each other resulting in a reputation ranking
that can be used to distribute rewards from DAO treasury proportionally to
accrued reputation. Figure 2 represents the logic and key components of the
system model.

Figure 2: Merit-Based Tokenomics system model

###  III-A System Model

Accounting mechanism. We assume that peers observe and evaluate each others’
contributions. The evaluation is recorded in a personal ledger. We model the
interaction between peers as a directed graph $G=\left(V,E,w\right)$, called
_the feedback graph_. An edge $\left(i,j\right)\in E$ is directed from $i$ to
$j$ and total feedback given to $j$ by $i$. We model the feedback as a special
token value, which accumulates with time. Peers perform some work contributing
to the community; as a result, they receive feedback from other peers. The
function $w:V\times{}V\rightarrow{}\mathbb{R}_{\geq{}0}$ denotes the weight of
the edges and shows the total feedback received.

An example of a feedback graph is presented in Figure 3. The weights of the
edges are cumulative weighted feedback from participants. Our model is
agnostic to the way the feedback graph is formed or how the graph weights are
chosen. Specific semantics of feedback are left out of the scope of the model.
Some examples may include code contributions, governance proposals, or a
rating assigned to $j$ based on the reactions of $i$ on Github [39]. An
accounting mechanism can be implemented with middleware that defines feedback
semantics and records evaluations in a personal ledger, similar to SourceCred
[5].

Gossip mechanism. We assume that there is a way for the peer to discover the
feedback graph, for example, through a gossip protocol. While we acknowledge
that reputation mechanisms in distributed systems face problems, such as the
incompleteness of information about peer interactions, peer discovery, etc.,
we disregard these issues assuming that global information is available
through a fault-free gossip protocol. Practical implementation may use overlay
for peer-to-peer gossip similar to [40].

Reputation mechanism. A reputation mechanism assigns some reputation scores
for every known node in the feedback graph. In principle, a reputation score
reflects the level of contributions made by the node compared with others.

Figure 3: Example of feedback graph $G$. Edge represent the sum of evaluations
made by the participant about another participant.

######  Definition III.1 (Reputation score).

A reputation score $R(G,j)$ is a value assigned to node $j$ given known
feedback graph $G$:

$R(G,j)\in\mathbb{R}\quad\forall j\in V$

We refer to $R(G)$ as a set of reputation scores for all nodes in the graph
$G$. The reputation score is computed by aggregating the feedback from all
nodes in the graph.

Allocation mechanism. Reputation mechanisms are used as an input for some
allocation policy that decides, given a set of nodes to whom distribute the
reward. For model simplicity, it is assumed that allocation happens
automatically per each epoch without participants’ actions. Participants may
claim tokens from the reward pool for each epoch in practical implementation,
corresponding to their reputation at the end of each epoch.

######  Definition III.2 (Allocation policy).

An allocation policy $A$ returns rewards given the feedback graph $G$ and
reputation scores $R_{i}(G)$:

$A_{i}(R_{i}(G_{i}),j)\in\mathbb{R}$

We do not make any assumptions about the allocation policies but assume that
nodes holding a high reputation score are more likely to receive rewards or
receive higher rewards. One example of an allocation is winners-take-all, in
which the top ten nodes with the highest reputation receive all the rewards.
Another example of allocation policy is the quadratic distribution [9], in
which nodes receive proportionally less with the increase in reputation.
Different policies will have different application fit, but discussion on
specific properties is out-of-scope.

Dynamic model. To capture the diachronic nature of contributions and rewards
we consider epochs. This reflects the semantics of reputation which
accumulates over time. One epoch represents one full cycle of system model
shown in Figure 2. We use superscript to refer to the epoch. For example,
$G^{(\tau)}$, for the feedback graph at epoch $\tau\geq 0$.

###  III-B Sybil Tolerance Model

We model a Sybil attack as a strategic, in a sense that the attacker is
interested in receiving as many rewards with only a small amount of work. It
does this by creating fake identities and fake edges connecting identities it
controls with one another. The weights of the edges in the Sybil region can be
chosen arbitrarily by the attacker. We assume that the attacker knows which
reputation algorithm is used and can execute an optimal attack on the given
reputation mechanism.

######  Definition III.3 (Sybil Attack).

Given the feedback graph $G=(V,E,w)$, an attacker $j$ performs a Sybil attack
$\sigma_{S}$ by introducing the following elements to the graph:

  * •

A set of Sybil identities $S=\left\\{{}j,s_{1},\ldots{},s_{m}\right\\}$, each
of which is called a Sybil and is indistinguishable from an honest node by
other nodes.

  * •

A set of Sybil edges $E_{S}\subset{}S\times{}S$ with edge weights
$w_{S}:S\times S\rightarrow{}\mathbb{R}_{\geq{}0}$.

  * •

A set of attack edges $E_{a}$ with weights
$w_{a}:V\times{}S\rightarrow\mathbb{R}_{\geq{}0}$.

After the attack has been carried out, we obtain a modified feedback graph,
denoted $G^{\prime}:=G\downarrow\sigma_{S}$.

Sybil attack benefit. The attacker is allowed to create arbitrarily many Sybil
nodes and fake edges between Sybils. However, we require any edge between
honest nodes $V$ and Sybils $S$ to be a real transaction. Therefore, we assume
that the attacker also makes contributions to receive feedback from honest
nodes in order to create an edge connecting the attacker’s identities to the
honest part of the network. To maximize the benefit of such, the attacker
might decide to get feedback from a highly reputable node via its attack edge.

Due to dynamic nature of allocation we use reputation scores as proxies for
cost and profit of the Sybil attack. The profit for the Sybil attack
$\sigma_{S}$ given the modified feedback graph $G^{\prime}$ is defined as:

$\omega^{+}(\sigma_{S})=\sum\limits_{s\in S\setminus\\{j\\}}R(G^{\prime},s)$

The cost for the Sybil attack is defined as reputation gained through honest
work through attack edges $E_{a}$. Given an modified graph
$G^{\prime\prime}=(V\cup S,E\cup E_{a},w)$ the cost is defined of the Sybil
attack is defined as:

$\omega^{-}(\sigma_{S})=\sum\limits_{s\in S}R(G^{\prime\prime},s)$

We model the Sybil-Tolerance as a bound on the benefit that the attacker can
gain through a Sybil attack $\sigma_{S}$ on feedback graph $G$. A reputation
mechanism $R$ is Sybil tolerant if the gain after performing a Sybil attack is
limited by some constant $c\geq{}0$:

$\lim\limits_{|S|\to\infty}\frac{\omega^{+}(\sigma_{S})}{\omega^{-}(\sigma_{S})}\leq
c$

##  IV MeritRank: Sybil Tolerant Reputation Mechanism

This section introduces a generic Sybil tolerant modification to the flow and
walk-based reputation mechanisms. We briefly demonstrate the limits to Sybil
tolerance for these types of reputation mechanisms. We propose three
modifications that limit the gain from a Sybil attack.

###  IV-A Sybil Tolerance of Existing Reputations

(a) Cycle attack

(b) Serial attack

(c) Parallel attack

Figure 4: Sybil attack strategies. A beneficial Sybil attack is a combination
of these three strategies.

We consider reputation mechanisms that satisfy the following requirements:
$1.)$ The reputation mechanism should be a graph-based measure of centrality,
$2.)$ It should work without requiring any central third party, and lastly,
$3.)$ It should reward contributors.

Naive approaches that are based on simple global centralities, such as vertex
degrees, are easily manipulable. In fact, even complex global reputation
mechanisms, such as PageRank, are not Sybil tolerant [24]. Such vulnerability
trivially follows from the fact that any node in the network can gain a
reputation by adding any new edge. Thus, the attacker can unboundedly gain
benefits from the Sybil attack without the need to create any attack edge.

To address this issue, personalized reputation mechanisms were proposed [42].
A personalized reputation mechanism assigns a positive reputation to the node
$j$ only if there is a path from a seed node $i$ to the node $j$. This, in
turn, provides inherent tolerance to a simple Sybil attacks, as participants
need to perform some work to establish a path.

We analyze three generic reputation mechanisms previously reported to be Sybil
tolerant. Specifically, we consider MaxFlow [44], personalized PageRank [43],
personalized Hitting Time [42]. We, however, show that these reputation
mechanisms are vulnerable to the Sybil attack as defined in III.3. To
illustrate their vulnerabilities, we present three Sybil strategies in Figure
4. These strategies, when combined, cover all Sybil beneficial strategies as
shown in [23].

Personalized PageRank: The personalized PageRank algorithm (PPR) with source
node $i$ and reset probability $\alpha\in{}[0,1]$ on feedback graph $G_{i}$ is
given by the steady-state probability that an $\alpha$-terminating random walk
initiated at $i$ arrives at $j$. In practice, personalized PageRank is
implemented as a random walk-based reputation, counting the number of
encounters in a random walk.

The Sybil-Tolerance of PPR follows from its damping factor $\alpha$, which
comes at the cost of reputation’s accuracy. The most effective way to attack
PageRank is to use cycles with big weights. The attack is shown in the Figure
4(a). The cycle attack goes as follows: The attacker inserts one Sybil
identity and adds two edges forming a cycle between the Sybil through its
Sybils and, as a result, increasing the total number of encounters in a random
walk.

Personalized Hitting Time: Given a feedback graph $G$, the personalized
Hitting Time (PHT) algorithm is given by the probability of an
$\alpha$-terminating random walk that starts at $i$ visiting node $j$ before
it terminates. As with the PageRank, PHTs are implemented as random-walk
reputation. The difference is that it accounts only for facts of node
encounters in a random walk.

PHT bounds the gains from both cycle attacks and parallel attacks. However is
still vulnerable to the serial attack (Figure 4(b)). The attacker can create a
long path connecting all Sybil nodes. As a result, all Sybil nodes in the path
can have a positive reputation.

MaxFlow: Given a seed node $i$, MaxFlow (MFW) reputation score of node
$j\in{}V_{i}$ is given by the maximum flow of contributions from $j$ to $i$ in
the feedback graph $G$. MaxFlow takes into account all possible flows of work
from the source node to the target node. The amount a node can gain is bounded
by its aggregated contributions.

MaxFlow is vulnerable to a parallel attack shown in Figure 4(c). The attacker
can create multiple Sybil identities $s_{1},...,s_{m}$, with each identity
gaining the same amount of reputation, i.e., for all
$k=1..m|R(G,s_{k})=R(G,s_{1})$.

###  IV-B Bounding the Attacks

We propose three modifications to the above-mentioned reputation mechanisms to
improve their Sybil tolerance. To address the vulnerabilities, we require to
provide bounds on parallel reports and serial reports and provide bounds on
transitivity. Formally these bounds are defined as:

  * •

Parallel report bound. Let the attacker execute a parallel attack or cycle
attack, as shown in Figure 4, adding at each time one Sybil identity. A
reputation $R$ is parallel report bound if
$\sum\limits_{l=1}^{m}R(G^{(m)},s_{l})\leq R(G^{(1)},s_{1})$.

  * •

Serial report bound. Reputation $R$ is serial-bound if there exists a bound on
the gain after a serial attack (as shown in Figure 4(b)):
$\sum\limits_{l=1}^{m}R(G^{(m)},s_{l})<\infty$.

  * •

Bounded Transitivity. Let node $j$ received some positive reputation $R(G,j)$
because of the existence of the path $P(i,j)$. The reputation $R$ is
transitive-bound if $R(G,j)\leq\min\limits_{k\in P(i,j)}(R(G,k))$.

Relative feedback. To achieve parallel report bound reputation each edge
weight must be considered with respect to the degree of the node. This can be
achieved by modifying each weight in the graph $G$, such that,
$\bar{w}(i,j)=w(i,j)/\sum\limits_{k\in\mathcal{N}(i)}(w(i,k))$, where
$\mathcal{N}(i)$ is the set of neighbours of node $i$ in the graph $G$.

Transitivity $\alpha$ decay. To achieve serial report bounds, we introduce
transitivity decays. For random walk-based reputation, we terminate the random
walks at each new step with probability $\alpha$. The value of $\alpha$ limits
the length of random walks, limiting the effects of serial attacks.

Connectivity $\beta$ decay. We increase Sybil tolerance even further by
introducing punishment for being in a separate connected component. The
intuition behind this decay is that the attack edges, as defined in definition
III.3 are often bridges, i.e., their cut creates two separate components.

We punish the nodes for forming a separate component with a decay multiplier
$\beta$. Given the feedback graph $G$ and seed node $i$, the modified
reputation for node $j$ is defined as follows:

$R_{\beta}(G,j)=\begin{cases}(1-\beta)*R(G,j),&\text{if }I_{i}(j)\\\
R(G,j),&\text{otherwise}\end{cases}$ (1)

, where $I(i,j)$ is a index indicating that nodes $i$ and $j$ are connected
through a bridge.

In practice, the index $I(i,j)$ is calculated based on the proportions of
random walks $T_{ij}$ starting from a seed node $i$ that reach node $j$
through some node $k$ in a path $P(i,j)$. Specifically, $k$ is a cut vertex if
the proportion of random walks is higher that some given threshold $t>0$:

(a) Gain for given reputation algorithms with $\alpha=0.4$

(b) Attacker’s gain with given $\alpha$ decay with $|S|=50$.

Figure 5: The total reputation gain by adding Sybil nodes with transitivity
$\alpha$ decay.

$I_{i}(j)=\frac{\max\limits_{k\in P(i,j)}|T_{ij}(k)|}{|T_{ij}|}\geq t$

If the threshold is equal to 1, this would indicate that vertex $k$ is the
only one connecting $i$ and $j$. The higher the connectivity decay $\beta$,
the more important the diversity of feedback received.

Epoch $\gamma$ decay. Finally, we consider an epoch decay $\gamma$. The
attacker can gain an additional advantage by reusing an old connection. For
example, the attacker can create a few attack edges and then, as a result,
gain a positive reputation for a prolonged period of time.

We provide two implementations of this decay: (1) when the multiplier is
applied to reputation values, for i.e.,
$R_{\gamma}(G^{(\tau+1)},i)=(1-\gamma)*R_{\gamma}(G^{(\tau)},i)+R(G^{(\Delta(\tau+1,\tau))},i)$,
where
$G^{(\Delta(\tau+1,\tau))}=(V^{(\tau+1)},E^{(\tau+1)}\setminus)E^{(\tau))},w^{(\tau+1)})$,
and (2) when the decay is applied to graph weights, i.e.,
$w_{\gamma}^{(\tau+1)}(i,j)=w^{(\tau+1)}(i,j)-(2-\gamma)*w^{(\tau)}(i,j)$.

##  V Experiments

In this section, we show the result of a quantitative study of the attacker
manipulation of reputation by simulating multiple Sybil attacks using real-
world dataset. We show effects of different decay values on different
reputation algorithms and resulting thresholds for Sybil tolerance.

###  V-A Experimental Set-Up

We create a dataset from the forum of one of the biggest DAOs to this date -
MakerDAO. We parse all user interactions, replies, likes, created posts, and
votes on the proposals. The users’ actions are recorded as contributions to
the DAO weighted in work units. Each participant is rewarded with DAI token
proportional to the reputation at each epoch. Thus the attacker gain is
directly expressed in DAI tokens.

We use weights as defined in SourceCred MakerDAO
project222http://makerdao.sourcecred.io. For instance, a post created by a
user is evaluated by the number of likes it receives. One like of the post is
equal to 4 work units. The contribution graph is a user-to-user evaluation,
where the weight of an edge $(i,j)$ is the total sum of work done by $j$ as
evaluated by $i$. Our dataset 333We release source code and the dataset at
https://github.com/grimadas/meritrank contains activity starting from Jun 24,
2019, to May 26, 2022. We use the epoch of one week. The resulting graph
$G^{(153)}$ contains 2057 nodes and 35853 edges. The peak week 149 has data of
291 users and 1528 edges.

For our computations, we use a special virtual seed node inserted to the
graph. At each epoch, we connect the seed node with the top 10 most reputable
nodes with equal weights.

We perform the Sybil attack as described in definition III.3. We implement all
attack strategies as shown in Figure 4. We pick the attacker node $j$ randomly
from the set of 10 most reputable nodes. The attack is performed repeatedly
every epoch after a period of 20 epochs. We argue that this attack have the
most prominent effects on reputation.

Figure 6:  Attacker’s gain with $\beta$ connectivity decay. We fix
$\alpha=0.4$.

###  V-B Transitivity $\alpha$ Decay

We apply the transitivity decay to personalized PageRank (PPR), Hitting Time
(PHT), and MaxFlow (MFW). We show that it successfully limits the attacker
gain in Figure 5(a). Note that random walk-based reputation achieves higher
tolerance than flow-based reputation.

We report the attack again with the increase of Sybils in Figure 5(b).
Transitivity decay can successfully bound the number of the gain of the
attacker. The effect is most visible with higher numbers of $\alpha$. A
downside, however, is the potential loss of reputation utility, as shown in
section V-E.

###  V-C Connectivity $\beta$ Decay

We report the effects of connectivity decay on the attacker’s gain in Figure
6. We fix the transitivity decay $\alpha=0.4$. The connectivity decay
mechanism can further improve the Sybil tolerance of transitive decay. Note
that with $\beta=0.0$, we are able to eliminate the gains of the attacker
fully.

We show in Figure 7 that connectivity and transitivity decay combined provided
a sufficient level of Sybil tolerance. The figure shows the effect of decays
on the attacker’s gain. Two decay mechanisms complement each other, providing
higher level of Sybil tolerance.

Figure 7: Cumulative effect of both $\alpha$ and $\beta$ decay mechanisms with
$|S|=50$.

###  V-D Epoch $\gamma$ Decay

(a) $\gamma$ decay applied to reputation values.

(b) $\gamma$ decay applied to graph weights.

Figure 8: The total reputation gain of the attacker with $\gamma$-epoch decay.
We fix $\alpha=0.4$ and $\beta=0.5$.

We run the experiments by applying the $\gamma$ epoch decay to reputation and
the weights. Surprisingly, epoch decay worsens Sybil-Tolerance of the
reputation mechanism, as can be seen from Figure 8(a) and Figure 8(b).
Attacker gains the minimum when there are no epoch decays, i.e., $\gamma=0.0$.
With the increase of $\gamma$, the attacker is able to gain more from the
Sybil attack. This effect is explained by the fact that reputation weights are
higher for new feedback. However, as the attacker can always insert new Sybil
identities at each epoch, eventually, these identities receive more walks than
honest nodes.

###  V-E Effect of Decays on Informativeness

Figure 9: Mean absolute percentage deviation (MAPD) compared with decay value
0 for transitivity decay ($\alpha$) and connectivity decay ($\beta$).

Informativeness of reputation is an application-specific metric that shows the
utility and goals of the reputation mechanism. As a reference, we propose to
measure the informativeness loss by comparing it with the reputation without
the decay. Specifically, we use Mean Absolute Percentage Deviation (MAPD) for
decay type and decay value $d$ defined as:

$MAPD(d)=\frac{1}{|V|}\sum\limits_{j\in
V}\lvert\frac{R_{decay=0}(G,j)-R_{decay=d}(G,j)}{R_{decay=0}(G,j)}\rvert$

We show the effects of decays on the informativeness of the reputation
mechanism in Figure 9. The less is the value of MAPD; the less is the effect
of decay on reputation informativeness. As expected, connectivity $\beta$
decay can achieve higher informativeness compared to transitivity decay
$\alpha$. Note that a small increase of MAPD is expected as the decay value
get’s closer to 1. This happens because first-hop neighbors get a higher
reputation than more distant nodes, which are not reachable by random walks.

###  V-F Discussion

Our experiments show that flow-based mechanisms can be translated from credit
networks [37] to feedback graphs to achieve Sybil tolerance. However, random
walk-based mechanisms PPR and PHT provide better tolerance than flow-based MFW
in feedback-based reputation mechanisms. Connectivity decay is preferable as
it allows for minimal informativeness loss while providing the highest
tolerance than transitivity decay. In practice, both of these parametrizations
can be combined for a desirable threshold tolerance.

Our findings also suggest that the intuitively appealing concept of time-based
epoch-based decay [6] should be implemented with caution. When applied
naively, Reputation mechanism with this decays favor new feedback and feedback
from new users assigning higher weights improving gains from dynamic Sybil
attack.

##  VI Conclusion

The advances in complex blockchain applications reveal limits of naive
tokenomics based on simple models of monetary incentives for token holders.
Merit-based non-transferable reputation schemes that reward active
contributors present a promising direction of engineering for novel blockchain
applications such as DAOs. However, application of reputation in decentralized
environments is limited by the reputation trilemma. We argue that the feedback
aggregation mechanism is the most feasible approach for such solutions as it
does not sacrifice the trustless properties of decentralization. While some
existing solutions, such as SourceCred address the problem of feedback
accounting, no solutions are providing Sybil tolerance for reputation
mechanisms based on feedback aggregation.

We propose MeritRank: a Sybil tolerant reputation mechanism based on feedback
aggregation from the participants of decentralized applications. We use data
set from interactions of MakerDAO participants to show experimentally that
MeritRank reputation mechanism bounds the profit of attacker using Sybils. The
novelty of this approach lies with modifications for reputation decay:
transitivity decay, epoch decay, and connectivity decay. Experimental results
show that a combination of transitivity decay and connectivity decay can
provide a desirable level of Sybil tolerance. Another advantage of our
approach is the ability to fine-tune the parameters in order to achieve the
context-specific balance between the utility of reputation and Sybil
tolerance. These properties suggest that MeritRank provides practical solution
to the reputation trilemma.

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# Differential Chaos Shift Keying-based Wireless Power Transfer over a
Frequency Selective Channel

Priyadarshi Mukherjee, Constantinos Psomas, and Ioannis Krikidis  IRIDA
Research Centre for Communication Technologies
Department of Electrical and Computer Engineering, University of Cyprus
Emails: {mukherjee.priyadarshi, psomas<EMAIL_ADDRESS>
###### Abstract

This paper studies the performance of a differential chaos shift keying
(DCSK)-based wireless power transfer (WPT) setup in a frequency selective
scenario. Particularly, by taking into account the nonlinearities of the
energy harvesting (EH) process and a generalized frequency selective
Nakagami-$m$ fading channel, we derive closed-form analytical expressions for
the harvested energy in terms of the transmitted waveform and channel
parameters. A simplified closed-form expression for the harvested energy is
also obtained for a scenario, where the delay spread is negligible in
comparison to the transmit symbol duration. Nontrivial design insights are
provided, where it is shown how the power delay profile of the channel as well
as the parameters of the transmitted waveform affect the EH performance. Our
results show that a frequency selective channel is comparatively more
beneficial for WPT compared to a flat fading scenario. However, a significant
delay spread negatively impacts the energy transfer.

###### Index Terms:

Differential chaos shift keying, wireless power transfer, frequency selective
channel, nonlinear energy harvesting.

##  I Introduction

With the rapid evolution of the Internet of Things (IoT) in the recent years,
the wireless traffic is expected to significantly increase between $2019$ and
$2025$ [1]. For applications, where a large number of devices are deployed,
the network lifetime is significantly affected due to their limited battery
life. As a result, charging these devices becomes critical and hence, low-
powered wireless communication networks is a relevant and important topic of
research. In this context, based on the advances made in recent years,
wireless power transfer (WPT) is emerging as a promising solution. WPT is
especially useful for applications, where the devices are wirelessly powered
by harvesting energy from ambient/dedicated radio-frequency (RF) signals [2].
This is achieved by employing a rectifying antenna (rectenna) at the receiver
that converts the received RF signals to direct current (DC).

Accurate mathematical modeling of the energy harvesting (EH) circuit is
extremely crucial for designing efficient WPT architectures. In this context,
some works propose simplified models of the EH circuit, namely, linear [3],
piece-wise linear [4], and the nonlinear logistic saturation-based [5].
However, unlike these works, the authors in [2] propose a realistic circuit-
based model of the harvester circuit that also enables the design of waveforms
that maximize the WPT efficiency. Based on this model, the aspect of designing
waveforms that result in an enhanced harvested energy gained importance. The
authors in [6] show that the nonlinearity of the rectification process at the
EH circuit causes certain waveforms, with high peak-to-average-power-ratio
(PAPR) to provide higher output DC power, compared to conventional constant-
envelop sinusoidal signals. Based on this observation, there are some works,
which investigate the effect of the transmitted symbols and modulation
techniques on WPT. By considering the nonlinear EH model proposed in [2], the
authors investigate the use of multisine waveforms for WPT due to their high
PAPR. The work in [7] proposes a simultaneous wireless information and power
transfer (SWIPT) architecture based on the superposition of multi-carrier
unmodulated and modulated waveforms at the transmitter. Apart from the
multisine waveforms, experimental studies demonstrate that due to their high
PAPR, chaotic waveforms outperform conventional single-tone signals in terms
of the WPT efficiency [8].

Due to its properties such as sensitivity to initial data and aperiodicity,
chaotic waveforms have been extensively used in the past to improve the
performance of wireless communication systems. In this context, the non-
coherent modulation technique of differential chaos shift keying (DCSK) is one
of the most widely studied chaotic signal-based communication system [9]. The
majority of the related works focus on the error performance of such systems
for various scenarios. To exploit the benefits of both DCSK and WPT, there are
few works in the literature that investigate DCSK-based SWIPT, e.g. [10, 11,
12, 13]. The work in [10] proposes a short-reference DCSK-based SWIPT
architecture to achieve higher data rate than the conventional system. The
authors in [11] investigate adaptive link selection for buffer-aided relaying
in a DCSK-SWIPT architecture, where two link-selection schemes based on
harvested energy are proposed. The work in [12] investigates a chaotic multi-
carrier system in a SWIPT framework via the sub-carrier index to reduce the
energy consumption. However, the aforementioned studies consider a simplified
linear model for the EH, and as a result, they are independent of the circuit
characteristics as well as the design of excitation waveforms [2]. In this
context, the authors in [13] propose a novel DCSK-based WPT architecture based
on the nonlinearities of the EH process. Furthermore, they propose a WPT-
optimal DCSK-based waveform that results in enhanced energy transfer. The
harvested energy is analytically characterized in terms of the transmitted
waveform parameters, by considering an ideal flat fading scenario. However, it
is worth noting that DCSK-based signals are essentially wideband signals [14].

Motivated by this, in this paper, we investigate a point-to-point DCSK-based
WPT topology by considering a frequency selective fading scenario. Closed-form
analytical expressions of the harvested DC are obtained in terms of the
transmitted waveform parameters and the power delay profile of a generalized
Nakagami-$m$ multipath channel. By considering practical channel conditions,
an approximation of the harvested energy is also provided, when the delay
spread is negligible compared to the transmitted symbol duration. Our results
demonstrate that while a frequency selective channel results in enhanced WPT
when compared against its flat fading counterpart, a significant delay spread
is detrimental for energy transfer.

##  II A Chaotic Signal-based WPT System Architecture

###  II-A System model

Figure 1: Architecture for DCSK-based WPT.

We consider a simple point-to-point WPT topology, where the transmitter
employs a DCSK generator and the receiver consists of an analog correlator,
followed by an EH circuit [13], as shown in Fig. 1. Note that, DCSK signals,
until now, have been mainly considered for information transfer. However, here
we focus on the WPT aspect and ignore the information side of the signal. We
assume that the wireless link suffers from both large-scale path-loss effects
and small-scale frequency-selective fading. Specifically, the received power
is proportional to $r^{-a}$, where $r$ is the transmitter-receiver distance
and $a>0$ denotes the pathloss exponent. Moreover, unlike [13], a generalized
frequency selective fading channel is considered here. In wideband wireless
communication systems, a commonly used channel model is the two-ray fading
channel model [14]. Using this generalized channel model, the output of the
channel is

$y_{k}=\alpha_{1}s_{k}+\alpha_{2}s_{k-\tau},$ (1)

where $s_{k}$ is the $k$-th transmitted chaotic chip, $\tau$ is the time delay
between two rays, and $\alpha_{1},\alpha_{2}$ are independent Nakagami-$m$
distributed random variables. Generally, the two paths will have unequal
average power gains and hence, we assume $\alpha_{1},\alpha_{2}$ to have power
gains $\Omega_{1},\Omega_{2}$, such that $\Omega_{1}+\Omega_{2}=1$ [14]. In
general, a frequency-selective channel adopts an $n$-ray $(n\geq 2)$ fading
model. However, in this work, we consider a two-ray model for analytical
simplicity. Furthermore, the aim of this work is to demonstrate how the
chaotic signal-aided energy transfer is affected by the frequency-selective
nature of the channel. Hence, the analysis can be extended to any arbitrary
number of multipaths.

###  II-B Chaotic signals

Assume a DCSK signal, where the current symbol is dependent on the previous
symbol [9] and different sets of chaotic sequences can be generated by using
different initial conditions. Each transmitted bit is represented by two sets
of chaotic signal samples, with the first set representing the reference, and
the other conveying information. If $+1$ is to be transmitted, the data sample
will be identical to the reference sample. Otherwise, an inverted version of
the reference sample will be used as the data sample [9]. Let $\beta$ be a
non-negative integer, defined as the spreading factor. Then, $2\beta$ chaotic
samples are used to spread each information bit. During the $l$-th
transmission interval, the output of the transmitter is

$\displaystyle
s_{l,k}=\begin{cases}x_{l,k},&k=2(l-1)\beta+1,\dots,(2l-1)\beta,\\\
d_{l}x_{l,k-\beta},&k=(2l-1)\beta+1,\dots,2l\beta,\end{cases}$ (2)

where $d_{l}=\pm 1$ is the information bit, $x_{l,k}$ is the chaotic sequence
used as the reference signal, and $x_{l,k-\beta}$ is its delayed version.
Furthermore, $x_{l,k}$ can be generated according to various existing chaotic
maps. Due to its good correlation properties, we consider the Chebyshev map of
degree $\xi$ for chaotic signal generation, which is defined as [9]

$\displaystyle x_{k+1}=\cos(\xi\cos^{-1}(x_{k})),\forall\>\>|x_{k}|\leq 1.$
(3)

###  II-C Analog correlator

The proposed WPT architecture employs an analog correlator, followed by an EH
rectifier circuit. An analog correlator essentially consists of a series of
$(\psi-1)$ delay blocks, where $\psi$ is a positive integer; the rationale
behind this application is that, the signal can be effectively integrated over
a certain time interval [15]. In what follows, for the sake of simplicity, we
will consider $\psi$ equal to the transmitted symbol length, i.e.
$\psi=2\beta$. As a result, the correlator output $\eta_{l}$ for the $l$-th
received symbol is

$\displaystyle\eta_{l}$
$\displaystyle=\sqrt{P_{t}}\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}y_{l,k}$
$\displaystyle=\sqrt{P_{t}}\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}\alpha_{1}s_{k}+\alpha_{2}s_{k-\tau},$
(4)

which follows from (1) and $P_{t}$ is the transmission power. It is worthy to
note that $\psi=1$ corresponds to the conventional case without a
correlator111An analog correlator is placed at the receiver, prior to the
harvester, because the work in [13] analytically proves and demonstrates why
the correlator is beneficial for energy transfer [13, Proposition $1$]..

###  II-D Energy transfer model

The WPT receiver is equipped with an antenna followed by a rectifier. The
rectifier, which generally consists of a diode (e.g., a Schottky diode) and a
passive low pass filter, acts as an envelope detector [16] and therefore
neglects the phase of the received signal $y$. Based on the nonlinearity of
this circuit, the harvested power in terms of $y$ is [2]

$P_{\rm
harv}=k_{2}R_{ant}\mathbb{E}\\{|y|^{2}\\}+k_{4}R_{ant}^{2}\mathbb{E}\\{|y|^{4}\\},$
(5)

where the parameters $k_{2},k_{4},$ and $R_{ant}$ are constants determined by
the characteristics of the circuit. The conventional linear model is a special
case of this nonlinear model and can be obtained by considering only the first
term in (5). Moreover, the energy transfer model considered in this work is
based on the assumption that the harvester operates in the nonlinear region
[2]. If the power of the harvester input signal becomes too large, the diode
inside the harvester will be forced into the saturation region of operation,
making the derived analytical results inapplicable. For the sake of
presentation, we will use $\varepsilon_{1}=r^{-a}k_{2}R_{ant}P_{t}$ and
$\varepsilon_{2}=r^{-2a}k_{4}R_{ant}^{2}P_{t}^{2}$. Hence, by incorporating a
$\psi$-bit analog correlator at the receiver prior to the harvester, the
average harvested power is [13]

$\displaystyle P_{{\rm harv}}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\psi}y_{k}\right)^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\psi}y_{k}\right)^{4}\right\\}.$
(6)

##  III Analysis of harvested energy

In this section, we analyse the harvested energy in terms of the transmitted
chaotic waveform parameters and also the frequency-selective channel
parameters.

The authors in [13], proposed a WPT-optimal signal design of length $\beta+1$,
where during the $l$-th symbol duration, the transmitter output is

$\displaystyle s_{k}$
$\displaystyle=\\!\begin{cases}\\!x_{k},&\\!\\!k=(\beta+1)(l-1)+1,\\\
\\!d_{l}x_{(\beta+1)(l-1)+1},&\\!\\!k=(\beta+1)(l-1)+2,\cdots,(\beta+1)l.\end{cases}$
(7)

If this waveform is transmitted by considering a Nakagami-$m$ block fading
scenario and a $\beta+1$ bit analog correlator is implemented at the receiver
prior to the harvester, the obtained harvested power is [13]

$P_{\rm
harv}=\frac{1}{2}\varepsilon_{1}(1+\beta^{2})+\frac{3(1+m)}{8m}\varepsilon_{2}(1+6\beta^{2}+\beta^{4}).$
(8)

However, this is an ideal scenario and in practice, chaotic signals are
wideband signals [14]. As a result, to obtain a realistic insight into the
harvested power of this WPT-optimal waveform, we consider here a frequency
selective scenario as defined in (1), where we have the delayed copy of the
$l$-th transmitted symbol, i.e. $s_{k-\tau}$ as

$s_{k-\tau}=\begin{cases}d_{l-1}x_{(\beta+1)(l-2)+1},&\begin{aligned}
k&=(\beta+1)(l-1)+1,\\\ &\quad\cdots,(\beta+1)(l-1)+\tau,\end{aligned}\\\
x_{(\beta+1)(l-1)+1},&k=(\beta+1)(l-1)+\tau+1,\\\
d_{l}x_{(\beta+1)(l-1)+1},&\begin{aligned} k&=(\beta+1)(l-1)+\tau+2,\\\
&\quad\cdots,(\beta+1)l.\end{aligned}\end{cases}$ (9)

Hence, by using (1), (II-C), (III), and (9) in (6), we obtain

$\displaystyle P_{\rm harv}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(\sum\limits_{k=(\beta+1)(l-1)+1}^{(1+\beta)l}y_{k}\right)^{2}\right\\}$
$\displaystyle\quad+\varepsilon_{2}\mathbb{E}\left\\{\left(\sum\limits_{k=(\beta+1)(l-1)+1}^{(1+\beta)l}y_{k}\right)^{4}\right\\},$
(10)

where the expectation is taken over $\alpha_{1},\alpha_{2},$ and $s_{k}$.
Towards this direction, we provide the following theorem.

######  Theorem 1.

The harvested power for the WPT-optimal signal in a frequency selective
Nakagami-$m$ fading scenario is given by (1).

$\displaystyle P_{\rm harv}$
$\displaystyle=\frac{\varepsilon_{1}}{2}\left(\tau^{2}\Omega_{2}+\Omega_{1}\left(1+\beta^{2}\right)+\Omega_{2}\left(1+(\beta-\tau)^{2}\right)+2\left(1+\beta^{2}-\beta\tau\right)\left(\frac{\Gamma(m+0.5)}{\Gamma(m)}\right)^{2}\frac{\sqrt{\Omega_{1}\Omega_{2}}}{m}\right)$
$\displaystyle\quad+\frac{\varepsilon_{2}}{2}\left(\frac{3}{4}\left(\tau^{4}\Omega_{2}^{2}\left(\frac{m+1}{m}\right)+\Omega_{1}^{2}\left(1+6\beta^{2}+\beta^{4}\right)\left(\frac{m+1}{m}\right)+4\frac{\Gamma(1.5+m)\Gamma(0.5+m)}{m^{2}\Gamma^{2}(m)}\Omega_{1}^{1.5}\Omega_{2}^{0.5}\right.\right.$
$\displaystyle\quad\left.\left.\times\left(\left(1+3\beta^{2}\right)+\left(\beta-\tau\right)\left(3\beta+\beta^{3}\right)\right)+6\Omega_{1}\Omega_{2}\left(\left(\beta-\tau\right)^{2}\left(1+\beta^{2}\right)+4\beta\left(\beta-\tau\right)+1+\beta^{2}\right)\right.\right.$
$\displaystyle\quad\left.\left.+4\frac{\Gamma(1.5+m)\Gamma(0.5+m)}{m^{2}\Gamma(m)^{2}}\Omega_{1}^{0.5}\Omega_{2}^{1.5}\left(1+3\left(\beta-\tau\right)^{2}+\beta\left(\beta-\tau\right)^{3}+3\beta\left(\beta-\tau\right)\right)+\Omega_{2}^{2}\left(1+6(\beta-\tau)^{2}\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.+(\beta-\tau)^{4}\right)\left(\frac{m+1}{m}\right)\right)+3\tau^{2}\Omega_{2}\left(\Omega_{1}\left(1+\beta^{2}\right)+\Omega_{2}\left(1+(\beta-\tau)^{2}\right)+2\left(1+\beta^{2}-\beta\tau\right)\left(\frac{\Gamma(m+0.5)}{\Gamma(m)}\right)^{2}\right.\right.$
$\displaystyle\quad\left.\left.\frac{\sqrt{\Omega_{1}\Omega_{2}}}{m}\right)\right).$
(11)

###### Proof.

See Appendix A. ∎

We observe from (1) how the power distribution over the two paths, i.e.
$\Omega_{1},\Omega_{2}$ and the delay spread $\tau$ affect the WPT
performance. The theorem demonstrates that a considerable $\tau$ is
detrimental for energy transfer. Note that the flat fading channel is a
special case of its frequency selective counterpart. By replacing
$\Omega_{1}=1,\Omega_{2}=0,$ and considering delay spread $\tau=0$ in (1),
$P_{\rm harv}$ reduces to (8). Furthermore, in a frequency selective scenario,
for an $N$-tone multisine signal, the linear term of the harvested power is
independent of $N$ and the nonlinear term is linearly dependent on $N$ [2]. On
the contrary, with a DCSK-based waveform, the linear and nonlinear terms of
$P_{\rm harv}$ in (1) are proportional to $\beta^{2}$ and $\beta^{4}$,
respectively. As a result, this waveform significantly outperforms multisine
waveforms, in terms of WPT.

To the best of our knowledge, this is the first analytical closed-form
expression obtained for DCSK-based WPT, which characterizes the harvested
energy in terms of the power delay profile of a frequency selective wireless
channel and also the nonlinearities of the EH process at the harvester.
Finally, note that (1) is an extremely complex and lengthy equation. However,
the work in [14] demonstrates that practically we have $0<\tau\ll\beta$. As a
result, an approximation for $P_{\rm harv}$ can be obtained corresponding to
the $\tau\rightarrow 0$ scenario as an upper bound, given in the following
corollary.

######  Corollary 1.

By considering the limiting case of negligible delay spread, the harvested
power is given by

$\displaystyle\lim\limits_{\tau\rightarrow 0}P_{\rm harv}\\!$
$\displaystyle=\\!\frac{\varepsilon_{1}}{2}\left(1+\beta^{2}\right)\\!\left(\Omega_{1}+\Omega_{2}+2\left(\frac{\Gamma(m+0.5)}{\Gamma(m)}\right)^{2}\right.$
$\displaystyle\quad\left.\times\frac{\sqrt{\Omega_{1}\Omega_{2}}}{m}\right)+\frac{3\varepsilon_{2}}{8}\left(1+6\beta^{2}+\beta^{4}\right)$
$\displaystyle\quad\times\left(\left(\Omega_{1}^{2}+\Omega_{2}^{2}\right)\left(\frac{m+1}{m}\right)+6\Omega_{1}\Omega_{2}\right.$
$\displaystyle\quad\left.4\frac{\Gamma(1.5+m)\Gamma(0.5+m)}{m^{2}\Gamma(m)^{2}}\\!\\!\left(\Omega_{1}^{0.5}\Omega_{2}^{1.5}+\Omega_{1}^{1.5}\Omega_{2}^{0.5}\right)\\!\\!\right)\\!\\!.$
(12)

The above corollary follows directly from Theorem 1 by replacing
$\tau\rightarrow 0$ in (1). Note that $m=1$ leads to an enhanced harvested
energy compared to the $m\rightarrow\infty$ scenario, i.e. wireless fading
enhances WPT. This observation corroborates the claims made in [13] regarding
the beneficial role of fading in DCSK-based WPT systems.

##  IV Numerical Results

We validate our theoretical analysis through extensive Monte-Carlo
simulations. Unless otherwise stated, lines correspond to analysis whereas
markers correspond to simulation results. Without any loss of generality, we
consider a transmission power of $P_{t}=30$ dBm, a Tx-Rx distance $r=20$ m,
and a pathloss exponent $\alpha=4$. The parameters considered for the EH model
are taken as $k_{2}=0.0034,k_{4}=0.3829,$ and $R_{ant}=50$ $\Omega$ [7].

Figure 2: WPT performance over a two-ray Nakagami-$m$ fading channel ($z_{\rm
MC}$ versus $\beta$) with $m=4,\tau=3$. Figure 3: WPT performance over a two-
ray Nakagami-$m$ fading channel ($z_{\rm MC}$ versus $\tau$) with
$m=4,\Omega_{1}=0.75,\Omega_{2}=0.25$.

Fig. 2 depicts the WPT performance of the proposed DCSK-based energy waveform
with respect to the spreading factor, over a two ray Nakagami-$m$ frequency
selective channel. Firstly, we observe the theoretical results (lines) match
very closely with the simulation results (markers); this verifies our proposed
analytical framework. Moreover, the figure also demonstrates that the
frequency selective nature of the channel, i.e. $\Omega_{2}\neq 0$ is
beneficial for WPT and the worst performance is obtained as the channel moves
towards the limiting case of a flat fading scenario, i.e.
$\Omega_{1}=1,\Omega_{2}=0.$ While [13] proved that fading is beneficial for
DCSK-based WPT, we extend the claim in this work by demonstrating that
frequency selective fading is more beneficial for DCSK-based WPT, when
compared against its flat fading counterpart.

Fig. 3 illustrates the impact of the delay spread $\tau$ on the WPT
performance of the proposed energy waveform for various values of the
spreading factor $\beta$. The figure demonstrates that the harvested energy
decreases with $\tau$, which is inline with (1) in Theorem 1. Moreover, for a
particular $\tau$, greater $\beta$ results in higher harvested DC, which
corroborates the claim made in Fig. 2. Finally, we observe that the effect of
$\tau$ can be neglected when $\tau\ll\beta$. However, for a considerable
$\tau$, the approximation does not agree with the exact harvested energy.
Fortunately, in most practical applications, the condition $\tau\ll\beta$
holds [14] and as a result, the degradation in WPT performance due to delay
spread of the channel is negligible. Finally, we observe that the gap between
the actual and the approximated harvested energy decreases with increasing
$\beta$; for example, observe the performance gap at $\tau=10$ between
$\beta=20$, $\beta=30$, and $\beta=40$.

Fig. 4 demonstrates the combined effect of the frequency-selectivity and delay
spread of the channel. In this figure we obtain the harvested DC by jointly
varying the power ratio $\frac{\Omega_{2}}{\Omega_{1}}$ and delay spread
$\tau$. It is worthy to note that in this figure,
$\frac{\Omega_{2}}{\Omega_{1}}\in\left(0,1\right)$; while
$\frac{\Omega_{2}}{\Omega_{1}}=0$ denotes a flat fading channel,
$\frac{\Omega_{2}}{\Omega_{1}}=1$ implies equal power distribution between the
two paths of the considered two-ray path model. We do not consider
$\Omega_{2}>\Omega_{1}$ because for all practical scenarios, we always have
$\Omega_{2}\leq\Omega_{1}$ [14]. An interesting observation obtained from Fig.
2 and Fig. 3 is that while frequency selectivity of the channel enhances WPT
performance, increasing delay spread $\tau$ has a negative impact on the same.
As $\tau\neq 0$ is the signature of a frequency selective fading scenario, the
best performance is observed at $\frac{\Omega_{2}}{\Omega_{1}}=1$, i.e.
$\Omega_{1}=\Omega_{2}=0.5$ and $\tau=1$.

Figure 4: Joint effect of the multipath components and the delay spread on the
harvested energy with $m=4,\beta=30$.

##  V Conclusion

In this paper, we investigated the framework of DCSK-based WPT by considering
a generalized frequency selective Nakagami-$m$ fading scenario and also taking
into account the nonlinearities of the EH process. To the best of our
knowledge, this is the first work where a closed-form expression of harvested
energy in a frequency-selective fading scenario is obtained as a function of
the multipath components and the delay spread. An approximation of the
harvested DC is also provided for the scenario, where the delay spread is
negligible compared to the transmitted symbol duration. It was shown that
while a frequency selective channel enhances WPT compared to its flat fading
counterpart, an increasing delay spread is detrimental to wireless energy
transfer. While the analytical expressions are derived by considering a
simplified two-ray fading channel model, the analysis can be generalized for
any arbitrary number of multipaths.

##  Appendix A Proof of Theorem 1

From (1), we have the channel output

$y_{k}=\alpha_{1}s_{k}+\alpha_{2}s_{k-\tau},$ (13)

where $\alpha_{1},\alpha_{2}$ are independent Nakagami-$m$ distributed random
variables with power gains $\Omega_{1},\Omega_{2}$, respectively such that
$\Omega_{1}+\Omega_{2}=1$. Therefore, the probability density function (PDF)
of $\alpha_{i},$ $\forall$ $i\in\\{1,2\\}$ is

$f_{\alpha_{i}}(z)=\frac{2m^{m}z^{2m-1}e^{-\frac{mz^{2}}{\Omega_{i}}}}{\Gamma(m)\Omega_{i}^{m}},\>\>\forall\>\>z\geq
0,$ (14)

where $\Gamma(\cdot)$ denotes the complete Gamma function, $m\geq 1$ controls
the severity of the amplitude fading, and
$\mathbb{E}\\{\alpha_{i}^{2}\\}=\Omega_{i}$. Moreover, the chaotic sequences
generated from the considered Chebyshev map have the following invariant PDF
[9]

$\displaystyle f_{X}(x)=\begin{cases}\frac{1}{\pi\sqrt{1-x^{2}}},&|x|<1,\\\
0,&\text{otherwise}.\end{cases}$ (15)

Here, we aim to obtain a closed-form expression for

$\displaystyle P_{\rm harv}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(\sum\limits_{k=(\beta+1)(l-1)+1}^{(1+\beta)l}y_{k}\right)^{2}\right\\}$
$\displaystyle\quad+\varepsilon_{2}\mathbb{E}\left\\{\left(\sum\limits_{k=(\beta+1)(l-1)+1}^{(1+\beta)l}y_{k}\right)^{4}\right\\},$
(16)

where the expectation is taken over $\alpha_{1},\alpha_{2},$ and $s_{k}$. In
this context, from (III) and (9), we obtain

$\displaystyle\sum\limits_{k=(\beta+1)(l-1)+1}^{(1+\beta)l}y_{k}=\sum\limits_{k=(\beta+1)(l-1)+1}^{(1+\beta)l}\alpha_{1}s_{k}+\alpha_{2}s_{k-\tau}$
$\displaystyle=\tau\alpha_{2}d_{l-1}x_{(\beta+1)(l-2)+1}$
$\displaystyle\quad+\left(\alpha_{1}\left(1+\beta
d_{l}\right)+\alpha_{2}\left(1+(\beta-\tau)d_{l}\right)\right)x_{(\beta+1)(l-1)+1}$
$\displaystyle=\delta_{1}x_{(\beta+1)(l-2)+1}+\delta_{2}x_{(\beta+1)(l-1)+1},$
(17)

where $\delta_{1}=\tau\alpha_{2}d_{l-1}$ and $\delta_{2}=\alpha_{1}(1+\beta
d_{l})+\alpha_{2}(1+(\beta-\tau)d_{l})$. Accordingly, (A) is rewritten as

$\displaystyle P_{\rm harv}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(\delta_{1}x_{(\beta+1)(l-2)+1}+\delta_{2}x_{(\beta+1)(l-1)+1}\right)^{2}\right\\}$
$\displaystyle\quad+\varepsilon_{2}\mathbb{E}\left\\{\left(\delta_{1}x_{(\beta+1)(l-2)+1}+\delta_{2}x_{(\beta+1)(l-1)+1}\right)^{4}\right\\}.$
(18)

Then, the first term of (A) can be evaluated as

$\displaystyle\mathbb{E}\left\\{\left(\delta_{1}x_{(\beta+1)(l-2)+1}+\delta_{2}x_{(\beta+1)(l-1)+1}\right)^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\delta_{1}^{2}x_{(\beta+1)(l-2)+1}^{2}+\delta_{2}^{2}x_{(\beta+1)(l-1)+1}^{2}\right.$
$\displaystyle\quad\left.+2\delta_{1}\delta_{2}x_{(\beta+1)(l-2)+1}x_{(\beta+1)(l-1)+1}\right\\}$
$\displaystyle\overset{(a)}{=}\frac{1}{2}\left(\mathbb{E}\left\\{\delta_{1}^{2}\right\\}+\mathbb{E}\left\\{\delta_{2}^{2}\right\\}\right)$
$\displaystyle\overset{(b)}{=}\frac{1}{2}\left(\tau^{2}\Omega_{2}+\Omega_{1}\left(1+\beta^{2}\right)+\Omega_{2}\left(1+(\beta-\tau)^{2}\right)\right.$
$\displaystyle\quad\left.+2\left(1+\beta^{2}-\beta\tau\right)\left(\frac{\Gamma(m+0.5)}{\Gamma(m)}\right)^{2}\frac{\sqrt{\Omega_{1}\Omega_{2}}}{m}\right),$
(19)

where $(a)$ follows from (15) as $\mathbb{E}\\{x_{i}\\}=0$ and
$\mathbb{E}\\{x_{i}^{2}\\}=\frac{1}{2}$ $\forall$ $i$. Moreover, by assuming
equally likely transmissions of $d_{l},d_{l-1}=\pm 1$, $(b)$ follows from

$\displaystyle\mathbb{E}\left\\{\delta_{1}^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{(\tau\alpha_{2}d_{l-1})^{2}\right\\}=\tau^{2}\mathbb{E}\left\\{\alpha_{2}^{2}\right\\}\mathbb{E}\left\\{d_{l-1}^{2}\right\\}=\tau^{2}\Omega_{2},$
$\displaystyle\mathbb{E}\left\\{\delta_{2}^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\left(\alpha_{1}\left(1+\beta
d_{l}\right)+\alpha_{2}\left(1+(\beta-\tau)d_{l}\right)\right)^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\alpha_{1}^{2}\left(1+\beta
d_{l}\right)^{2}+\alpha_{2}^{2}\left(1+(\beta-\tau)d_{l}\right)^{2}\right.$
$\displaystyle\quad\left.+2\alpha_{1}\alpha_{2}\left(1+\beta
d_{l}\right)\left(1+(\beta-\tau)d_{l}\right)\right\\}$
$\displaystyle=\Omega_{1}\left(1+\beta^{2}\right)+\Omega_{2}\left(1+(\beta-\tau)^{2}\right)$
$\displaystyle\quad+2\left(1+\beta^{2}-\beta\tau\right)\left(\frac{\Gamma(m+0.5)}{\Gamma(m)}\right)^{2}\frac{\sqrt{\Omega_{1}\Omega_{2}}}{m},$
(20)

where

$\displaystyle\mathbb{E}\left\\{\\!\left(1+\beta d_{l}\right)^{2}\\!\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\left(1+\beta\right)^{2}+\left(1-\beta\right)^{2}\right\\}=1+\beta^{2},$
$\displaystyle\mathbb{E}\left\\{\\!\left(1+(\beta-\tau)d_{l}\right)^{2}\\!\right\\}$
$\displaystyle=\\!\frac{1}{2}\\!\left\\{\\!\left(1+(\beta-\tau)\right)^{2}\\!\\!+\\!\\!\left(1-(\beta-\tau)\right)^{2}\\!\right\\}$
$\displaystyle=1+(\beta-\tau)^{2}.$ (21)

Furthermore, we also use

$\displaystyle\mathbb{E}\\{\alpha_{i}\\}$
$\displaystyle=\frac{2m^{m}}{\Gamma(m)\Omega_{i}^{m}}\int_{0}^{\infty}z^{2m}e^{-\frac{mz^{2}}{\Omega_{i}}}dz$
$\displaystyle=\frac{1}{\Gamma(m)}\sqrt{\frac{\Omega_{i}}{m}}\int_{0}^{\infty}v^{m-0.5}e^{-v}dv\\!=\\!\frac{\Gamma(m+0.5)}{\Gamma(m)}\sqrt{\frac{\Omega_{i}}{m}},$
(22)

which follows from the transformation $\frac{mz^{2}}{\Omega_{i}}\rightarrow
v$.

Similarly, by using the multinomial theorem, the second term of (A) can be
expanded as

$\displaystyle\mathbb{E}\left\\{\left(\delta_{1}x_{(\beta+1)(l-2)+1}+\delta_{2}x_{(\beta+1)(l-1)+1}\right)^{4}\right\\}$
$\displaystyle\overset{(a)}{=}\mathbb{E}\left\\{\delta_{1}^{4}x_{(\beta+1)(l-2)+1}^{4}+6\delta_{1}^{2}\delta_{2}^{2}x_{(\beta+1)(l-2)+1}^{2}x_{(\beta+1)(l-1)+1}^{2}\right.$
$\displaystyle\quad\left.+\delta_{2}^{4}x_{(\beta+1)(l-1)+1}^{4}\right\\}$
$\displaystyle\overset{(b)}{=}\frac{3}{8}\mathbb{E}\left\\{\delta_{1}^{4}\right\\}+\frac{3}{2}\mathbb{E}\left\\{\delta_{1}^{2}\right\\}\mathbb{E}\left\\{\delta_{2}^{2}\right\\}+\frac{3}{8}\mathbb{E}\left\\{\delta_{2}^{4}\right\\},$
(23)

where $(a)$ follows from $\mathbb{E}\left\\{x_{i}^{s}\right\\}=0$ $\forall$
$i$ when $s$ is odd and $(b)$ follows from
$\mathbb{E}\left\\{x_{i}^{2}\right\\}=\frac{1}{2}$ $\forall$ $i$. Furthermore,
$\mathbb{E}\left\\{\delta_{1}^{2}\right\\},\mathbb{E}\left\\{\delta_{2}^{2}\right\\}$
are obtained from (A) and

$\displaystyle\mathbb{E}\left\\{\delta_{1}^{4}\right\\}=\mathbb{E}\left\\{(\tau\alpha_{2}d_{l-1})^{4}\right\\}$
$\displaystyle=\tau^{4}\mathbb{E}\left\\{\alpha_{2}^{4}\right\\}\mathbb{E}\left\\{d_{l-1}^{4}\right\\}=\tau^{4}\Omega_{2}^{2}\left(\frac{m+1}{m}\right),$
$\displaystyle\mathbb{E}\left\\{\delta_{2}^{4}\right\\}=\mathbb{E}\left\\{\left(\alpha_{1}\left(1+\beta
d_{l}\right)+\alpha_{2}\left(1+(\beta-\tau)d_{l}\right)\right)^{4}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\alpha_{1}^{4}\left(1+\beta
d_{l}\right)^{4}+4\alpha_{1}^{3}\alpha_{2}\left(1+\beta
d_{l}\right)^{3}\left(1+(\beta-\tau)d_{l}\right)\right.$
$\displaystyle\quad\left.+6\alpha_{1}^{2}\alpha_{2}^{2}\left(1+\beta
d_{l}\right)^{2}\left(1+(\beta-\tau)d_{l}\right)^{2}\right.$
$\displaystyle\quad+4\alpha_{1}\alpha_{2}^{3}\left(1+\beta
d_{l}\right)\left(1+(\beta-\tau)d_{l}\right)^{3}$
$\displaystyle\quad+\left.\alpha_{2}^{4}\left(1+(\beta-\tau)d_{l}\right)^{4}\right\\}$
$\displaystyle=\Omega_{1}^{2}\left(\\!1+\\!6\beta^{2}+\\!\beta^{4}\right)\\!\\!\left(\\!\frac{m+1}{m}\\!\right)\\!+\\!4\frac{\Gamma(1.5+m)\Gamma(0.5+m)}{m^{2}\Gamma^{2}(m)}$
$\displaystyle\quad\times\Omega_{1}^{1.5}\Omega_{2}^{0.5}\left(\left(1+3\beta^{2}\right)+\left(\beta-\tau\right)\left(3\beta+\beta^{3}\right)\right)$
$\displaystyle\quad+6\Omega_{1}\Omega_{2}\left(\left(\beta-\tau\right)^{2}\left(1+\beta^{2}\right)+4\beta\left(\beta-\tau\right)+1+\beta^{2}\right)$
$\displaystyle\quad+4\frac{\Gamma(1.5+m)\Gamma(0.5+m)}{m^{2}\Gamma(m)^{2}}\Omega_{1}^{0.5}\Omega_{2}^{1.5}\left(1+3\left(\beta-\tau\right)^{2}\right.$
$\displaystyle\quad\left.+\beta\left(\beta-\tau\right)^{3}+3\beta\left(\beta-\tau\right)\right)+\Omega_{2}^{2}\left(1+6(\beta-\tau)^{2}\right.$
$\displaystyle\quad\left.+(\beta-\tau)^{4}\right)\left(\frac{m+1}{m}\right).$
(24)

Here, we use the $n$-th order moment of $\alpha_{i}$, i.e.
$\displaystyle\mathbb{E}\\{\alpha_{i}^{n}\\}=\frac{2m^{m}}{\Gamma(m)\Omega_{i}^{m}}\int_{0}^{\infty}\\!\\!z^{2m+n-1}e^{-\frac{mz^{2}}{\Omega_{i}}}dz\\!=\\!\frac{\Gamma\left(m+\frac{n}{2}\right)}{\Gamma(m)}\left(\frac{\Omega_{i}}{m}\right)^{\frac{n}{2}}$,
which can be obtained in a way similar to (A).

Finally, by substituting (A) and (A) in (A), we obtain $P_{\rm harv}$ as
stated in (1).

## Acknowledgment

This work was co-funded by the European Regional Development Fund and the
Republic of Cyprus through the Research and Innovation Foundation, under the
project INFRASTRUCTURES/1216/0017 (IRIDA). It has also received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 819819).

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