Fast Probabilistic Consensus with Weighted Votes

Abstract

The fast probabilistic consensus (FPC) is a voting consensus protocol that is robust and efficient in Byzantine infrastructure. We propose an adaption of the FPC to a setting where the voting power is proportional to the nodes reputations. We model the reputation using a Zipf law and show using simulations that the performance of the protocol in Byzantine infrastructure increases with the Zipf exponent. Moreover, we propose several improvements of the FPC that decrease the failure rates significantly and allow the protocol to withstand adversaries with higher weight. We distinguish between cautious and berserk strategies of the adversaries and propose an efficient method to detect the more harmful berserk strategies. Our study refers at several points to a specific implementation of the IOTA protocol, but the principal results hold for general implementations of reputation models.

Full text

Fast Probabilistic Consensus with Weighted
Votes
Sebastian M¨uller∗,1, Andreas Penzkofer∗,2, Bartosz Ku´smierz2, Darcy
Camargo2,3, and William J. Buchanan4
1Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M - UMR 7373, 13453
Marseille, France,
sebastian.muller@univ-amu.fr
2IOTA Foundation, 10405 Berlin, Germany
{andreas.penzkofer, bartosz.kusmierz, darcy.camargo}@iota.org
3Department of Mathematics, Weizmann Institute, POB 26, Rehovot 7610001, Israel
4Blockpass ID Lab, Edinburgh Napier University, Edinburgh, UK
b.buchanan@napier.ac.uk
Abstract. The fast probabilistic consensus (FPC) is a voting consen-
sus protocol that is robust and efficient in Byzantine infrastructure. We
propose an adaption of the FPC to a setting where the voting power is
proportional to the nodes reputations. We model the reputation using a
Zipf law and show using simulations that the performance of the protocol
in Byzantine infrastructure increases with the Zipf exponent. Moreover,
we propose several improvements of the FPC that decrease the failure
rates significantly and allow the protocol to withstand adversaries with
higher weight. We distinguish between cautious and berserk strategies
of the adversaries and propose an efficient method to detect the more
harmful berserk strategies. Our study refers at several points to a specific
implementation of the IOTA protocol, but the principal results hold for
general implementations of reputation models.
Keywords: Distributed systems, consensus protocols, fairness, Sybil at-
tack, Byzantine infrastructures, simulation studies
1 Introduction
Distributed consensus algorithms allow networked systems to agree on a required
state or opinion in situations where centralized decision making is difficult or even
impossible. As distributed computing is inherently unreliable, it is necessary to
reach consensus in faulty or Byzantine infrastructure. The importance of this
problem stems from its omnipresence and fault tolerance is one of the most
fundamental aspects of distributed computing, e.g., [1].
This article focuses on a consensus protocol that falls into the class of binary
majority voting consensus protocols. The basic idea is that nodes query other
∗These authors contributed equally.

2 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
nodes about their current opinion, and adjust their own opinion over the course
of several rounds based on the proportion of other opinions they have observed.
The functional principle of this protocol, already observed by the Marquis de
Condorcet in 1785 [4], relies on the law of large numbers; suppose there is a
large population of voters, and each one independently votes ”correctly” with
probability p > 1/2. Then as the population size grows, the probability that the
outcome of a majority vote is ”correct” converges to one.
While voting consensus protocols have their limitations, they have been suc-
cessfully applied not only in decision making but also in a wide range of engi-
neering and economical applications, and lead to the emerging science of socio-
physics [3].
We continue the works of [11] and [2] and propose several adaptions, Section
8, of the fast probabilistic consensus protocol (FPC) that decreases the failure
rate of at least one order of magnitude, e.g., see Fig. 6. The main contribution is
the adaption of the protocol to a setting allowing defense against Sybil attacks.
In FPC nodes need to be able to query a sufficiently large proportion of the
network directly, which requires that nodes have global identities (node IDs)
with which they can be addressed. In a decentralized and permissionless setting
a malicious actor may gain a disproportionately large influence on the voting by
creating a large number of pseudonymous identities. In the blockchain environ-
ment, mechanisms such as proof-of-work and (delegated) proof-of-stake can act
as a Sybil mitigation mechanism in the sense that the voting power is propor-
tional to the work invested or the value staked [14].
For the IOTA protocol [12] introduces mana as a Sybil defense, where mana
is delegated to nodes and proportional to the active amount of IOTA in the
network. While in the remainder of the paper we will always refer to mana, the
protocol can be implemented using any good or resources that can be verified
via resource testing or recurring costs and fee, e.g., [10]. In Section 3 we propose
a weighted voting consensus protocol that is fair in the sense that the voting
power is proportional to the nodes’ reputation.
In general, values in (crypto-)currency systems are not distributed equally; [8]
investigates the heterogeneous distribution of the wealth across Bitcoin addresses
and finds that it follows certain power laws. Power laws satisfy a universality
phenomenon; they appear in numerous different fields of applications and have,
in particular, also been utilised to model wealth in economic models [7]. In this
paper we consider a Zipf law to model the proportional wealth of nodes in the
IOTA network: the nth largest value y(n) satisfies
y(n) = Cn−s,(1)
where C−1=PN
n=1 n−s,Nis the number of nodes, and sis the Zipf parameter.
Fig. 1 shows the distribution of IOTA for the top 100 richest addresses1together
with a fitted Zipf distribution. Since (1) only depends on two parameters, sand
N, this provides a convenient model to investigate the performance of FPC in a
1https://thetangle.org

Fast Probabilistic Consensus with Weighted Votes 3
wide range of network situations. For instance, networks where nodes are equal
may be modelled by choosing s= 0, while more centralized networks can be
considered for s > 1. We refer to Section 4 for more details on the Zipf law.
Fig. 1. Distribution of relative IOTA value on the top 100 addresses with a fitted Zipf
distribution with s= 0.9.
Outline
The rest of the paper organizes as follows. After giving an introduction to the
original version of FPC in Section 2, we summarize results on the fairness of
this protocol in Section 3. In Section 4 we propose modelling of the weight
distribution using a Zipf law, we highlight the skewness of this distribution in
Section 5, and in Section 6 we discuss how the properties of the Zipf law influence
the message complexity of the protocol.
After defining the threat model in Section 7 we propose several improvements
of the Vanilla FPC in Section 8. In Section 9, we outline a protection mechanism
against the most severe attack strategies. The quorum size is an important pa-
rameter of FPC that dominates its performance; we give in Section 10 a heuristic
to choose a quorum size for a given security level.
Section 11 presents simulation results that show the performance of the pro-
tocol in Byzantine infrastructure for different degrees of centralization of the
weights. We conclude in Section 12 with a discussion.

4 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
2 Vanilla FPC
We present here only the key elements of the proposed protocol and refer the
interested reader to [11] and [2] for more details. In order to define FPC we have
to introduce some notation. We assume the network to have Nnodes indexed by
1,2, . . . , N and that every node is able to query any other nodes.2Every node
ihas an opinion or state. We note si(t) for the opinion of the node iat time t.
Opinions take values in {0,1}. Every node ihas an initial opinion si(0).
At each (discrete) time step each node chooses krandom nodes Ci=Ci(t),
queries their opinions and calculates
ηi(t+ 1) = 1
ki(t)X
j∈Ci
sj(t),
where ki(t)≤kis the number of replies received by node iat time tand sj(t)=0
if the reply from jis not received in due time. Note that the neighbors Ciof
a node iare chosen using sampling with replacement and hence repetitions are
possible.
As in [2] we consider a basic version of the FPC introduced in [11] in choosing
some parameters by default. Specifically, we remove the cooling phase of FPC
and the randomness of the initial threshold τ. Let Ut,t= 1,2, . . . be i.i.d. random
variables with law Unif ([β , 1−β]) for some parameter β∈[0,1/2]. The update
rules for the opinion of a node iis then given by
si(1) = 1,if ηi(1) ≥τ,
0,otherwise,
and for t≥1:
si(t+ 1) = 


1,if ηi(t+ 1) > Ut,
0,if ηi(t+ 1) < Ut,
si(t),otherwise.
Note that if β= 0.5, FPC reduces to a standard majority consensus. The above
sequence of random variables Utare the same for all nodes; we refer to [2] for a
more detailed discussion on the use of decentralized random number generators.
We introduce a local termination rule to reduce the communication complex-
ity of the protocols. Every node keeps a counter variable cnt that is incremented
by 1 if there is no change in its opinion and that is set to 0 if there is a change
of opinion. Once the counter reaches a certain threshold l, i.e., cnt ≥l, the
node considers the current state as final. The node will therefore no longer send
any queries but will still answer incoming queries. In the absence of autonomous
termination the algorithm is halted after maxIt iterations.
2This assumption is only made for sake of a better presentation; a node does not need
to know every other node in the network. While the theoretical results in [11] are
proven under this assumption, simulation studies [2] indicate that it is sufficient if
every node knows about half of the other nodes. Moreover, it seems to be a reasonable
assumption that large mana nodes are known to every participant in the network.

Fast Probabilistic Consensus with Weighted Votes 5
3 Fairness
Introducing mana as a weighting factor may naturally have an influence on the
mana distribution and may lead to degenerated cases. In order to avoid this
phenomenon we want to ensure that no node can increase its importance in
splitting up into several nodes, nor can achieve better performance in pooling
together with other nodes.
We consider a network of Nnodes whose mana is described by {m1, .., mN}
with PN
i=1 mi= 1. In the sampling of the queries a node jis chosen now with
probability
pj=f(mj)
PN
i=1 f(mi).
Each opinion is weighted by gj=g(mj), resulting in the value
ηi(t+ 1) = 1
Pj∈CigjX
j∈Ci
gjsj(t).
The other parts of the protocol remain unchanged.
We denote by yithe number of times a node iis chosen. As the sampling is
described by a multinomial distribution we can calculate the expected value of
a query as
Eη(t+ 1) =
N
X
i=1
si(t)vi,
where
vi=X
y∈NN:Pyi=k
k!
y1!···yN!
yigi
PN
n=1 yngn
N
Y
j=1
pyj
j
is called the voting power of node i. The voting power measures the influence of
the node i. We would like the voting power to be proportional to the mana.
Definition 1. A voting scheme (f, g )is fair if the voting power is not sensitive
to splitting/merging of mana, i.e., if a node isplits into nodes i1and i2with a
mana splitting ratio x∈(0,1), then
vi(mi) = vi1(xmi) + vi2((1 −x)mi) (2)
In the case where g≡1, i.e., the ηis an unweighted mean, the existence of a
voting scheme that is fair for all possible choices of kand mana distributions is
shown in [9]:
Lemma 1. For g≡1the voting scheme (f, g)is fair if and only if fis the
identity function f=id.
For this reason we fix from now on g≡1 and f=id.

6 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
4 Zipf’s law and mana distribution
One of the most intriguing phenomenon in probability theory is that of univer-
sality; many seemingly unrelated probability distributions, which may involve
large numbers of unknown parameters, can end up converging to a universal law
that only depends on few parameters. Probably the most famous example of this
universality phenomenon is the central limit theorem.
Analogous universality phenomena also show up in empirical distributions,
i.e., distributions of statistics from a large population of real-world objects. Ex-
amples include Benford’s law, Zipf’s law, and the Pareto distribution3; we refer
to [15] for more details. These laws govern the asymptotic distribution of many
statistics which
1. take values as positive numbers;
2. range over many different orders of magnitude;
3. arise from a complicated combination of largely independent factors; and
4. have not been artificially rounded, truncated, or otherwise constrained in
size.
Out of the three above laws, the Zipf law is the appropriate variant for modelling
the mana distribution. The Zipf law is defined as follows: The nth largest value
of the statistic Xshould obey an approximate power law, i.e., it should be
approximately Cn−sfor the first few n= 1,2,3, . . . and some parameters C, s >
0.
The Zipf law is used in various applications. For instance, Zipf’s law and the
closely related Pareto distribution can be used to mathematically test various
models of real-world systems (e.g., formation of astronomical objects, accumula-
tion of wealth and population growth of countries). An important point is that
Zipf’s law does in general not apply on the entire range of X, but only on the
upper tail region when Xis significantly higher than the median; in other words,
it is a law for the (upper) outliers of X.
The Zipf law tends to break down if one of the hypotheses 1) - 4) is dropped.
For instance, if the statistic Xconcentrates around its mean and does not range
over many orders of magnitude, then the normal distribution tends to be a much
better model. If instead the samples of the statistics are highly correlated with
each other, then other laws can arise, as for example, the Tracy-Widom law.
Zipf’s law is most easily observed by plotting the data on a log-log graph,
with the axes being log(rank order) and log(value). The data conforms to a Zipf
law to the extent that the plot is linear and the value of scan be found using
linear regression. For instance, Fig. 1 shows the distribution of IOTA for the top
100 richest addresses.
Due to universality phenomemon, the plausibility of hypotheses 1) - 4) above
and Fig. 1 we assume a Zipf law for the mana distribution. In Section 12 we give
more details on the validity of the model.
3Interesting to note here that these three distributions are highly compatible with
each other.

Fast Probabilistic Consensus with Weighted Votes 7
5 Skewness of mana distribution
For s > 0 the majority of the nodes would have a mana value less than the
average and hence, in the case of an increasing function f, these nodes would
be queried less than in a homogeneous distribution. As a consequence the initial
opinion of small mana nodes may become negligible.
We define the γ-effective number of nodes Nγ-eff as the number of nodes
whose proportional mana is more than or equal to γ/N:
Nγ-eff =
N
X
i=1
1{mi≥γ/N}
where 1is the standard indicator function. Fig. 2 shows the relative proportion
of effective nodes nγ-eff =Nγ-eff/N with s. We show the figure for N= 1000,
although the distribution hardly changes when changing N. Note that for γ= 1
and s→0 a large proportion of the nodes would have less than a proportion
1/N of the mana and hence nγ-eff approaches, as s→0, to a value strictly less
than 1. Note that for values of s'1 the effective number of nodes can be very
small. This is also reflected in the distribution of IOTA. The top 100 addresses
shown in Fig. 1 own 60% of the total funds, albeit there are more than 100.000
addresses in total1.
Fig. 2. Proportion of effective number of nodes.

8 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
6 Message complexity
Let us start with the following back-of-the-envelope calculation. Denote by h(N)
the mana rank of a given node. At every round this node is queried on average
N·h(N)−s
PN
n=1 n−s(3)
times. Now, if s < 1 this becomes asymptotically Θ(Nsh(N)−s), if s= 1 we
obtain Θ(N
log Nh(N)−1), and if s > 1 this is Θ(Nh(N)−s). In particular, the
highest mana node, i.e., h(N) = 1, is queried Θ(Ns), Θ(N
log N), or Θ(N) times,
and might eventually be overrun by queries. Nodes whose rank is Θ(N) have to
answer only Θ(1) queries. This is in contrast to the case s= 0 where every node
has the same mana and every node is queried in average a constant number of
times.
The high mana nodes are therefore incentivized to gossip their opinions and
not to answer each query separately. Since not all nodes can gossip their opinions
(in this case every node would have to send Ω(N) messages) we have to find a
threshold when nodes gossip their opinions or not. If we assume that high mana
nodes have higher throughput than lower mana nodes a reasonable threshold is
log(N), i.e., only the Θ(log(N)) highest mana nodes do gossip their opinions,
leading to Θ(log N) messages for each node in the gossip layer. In this case the
expected number of queries the highest mana node, that is not allowed to gossip
its opinions, receives is Θ(( N
log N)s) if s < 1, Θ(N
(log N)2) if s= 1, and Θ(N
(log N)s)
if s > 1. In this case, nodes of rank between Θ(log N) and Θ(N) are the critical
nodes with respect to message complexity.
Another natural possibility would be to choose the threshold such that every
node has to send the same amount of messages. In other words, the maximal
number of queries a node has to answer should equal the number of messages
that are gossiped. For s < 1 this leads to the following equation
Nsh(N)−s=h(N) (4)
and hence we obtain that a threshold of order Ns
s+1 leads to Θ(Ns
s+1 ) messages
for every node to send. For s > 1 one obtains similarly a threshold of N1
1+slead-
ing to Θ(N1
1+s) messages. In the worst case, i.e., s= 1, the message complexity
for each node in the network is O(√N).
We want to close this section with the remark that, as mentioned in Section 4,
Zipf’s law does mostly not apply on the entire range of the observations, but only
on the upper tail regions of the observations. Adjustments of the above thresh-
old and more precise message complexity calculations have to be performed in
consideration of the real-world situation of the mana distribution. Moreover,
the optimal choice of this threshold has also to depend on the structure of the
network, and the performances of the different nodes.

Fast Probabilistic Consensus with Weighted Votes 9
7 Threat model
We consider the ”worst-case” scenario where adversarial nodes can exchange
information freely between themselves and can agree on a common strategy. In
fact, we assume that all Byzantine nodes are controlled by a single adversary.
We assume that such an adversary holds a proportion qof the mana and thus
has a voting power vq=q.
In order to make results more comparable we assume that the adversary
distributes the mana equally between its nodes such that each node holds 1/N
of the total mana. Fig. 3 shows an exemplary distribution of mana between all
nodes. Nodes are indexed such that the malicious nodes have the highest indexes,
while honest nodes are indexed by their mana rank.
Fig. 3. Mana distribution with s= 1, N= 100 and q= 0.2.
We assume an ”omniscient adversary”, who is aware of all opinions and
queries of the honest nodes. However, we assume that the adversary has no
influence nor prior knowledge on the random threshold.
The adversary can take several approaches in influencing the opinions in
the network. In a cautious strategy the adversary sends the same opinion to
all enquiring nodes, while in a berserk strategy, different opinions can be sent
to different nodes; we refer to [11, 2] for more details. While the latter is more
powerful it may also be easily detectable, e.g., see [12]. The adversary may also
behave semi-cautious by not responding to individual nodes.

10 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
7.1 Communication model
We have to make assumptions on the communication model of the FPC. We as-
sume the communication between two nodes to satisfy authentication, i.e., senders
and receivers are who they claim to be, and data integrity, i.e., data is not
changed from source to destination. Nodes can also send a message on a gossip
layer; these messages are then available to all participating nodes. All messages
are signed by a private key of the sending node.
As we consider omniscient adversaries we do not assume confidentiality. For
the communication of the opinions between nodes we assume a synchronous
model. However, we want to stress that similar performances are obtained in
a probabilistic synchronous model, in which for every ε > 0 and δ > 0.5, a
majority proportion δof the messages is delivered within a bounded (and known)
time, that depends on εand δ, with probability of at least 1 −ε. Due to its
random nature, FPC still shows good performances in situations where not all
queries are answered in due time. Moreover, the gossiping feature of high mana-
nodes allows to detect whether high mana nodes are eclipsed or are encountering
communication problems.
7.2 Failures
In the case of heterogeneous mana distributions there are different possibilities
to generalize the standard failures of consensus protocols: namely integration
failure, agreement failure and termination failure. In this paper we consider only
agreement failure since in the IOTA use case this failure turns out to be the
most severe. In the strictest sense an agreement failure occurs if not all nodes
decide on the same opinion. We will consider the α-agreement failure; such a
failure occurs if at least a proportion of αnodes differ in their final decision.
7.3 Adversary strategies
While [11] studies robustness of FPC against all kinds of adversary strategies, [2]
proposes several concrete strategies in order to perform numerical simulations.
In particular, [2] introduced the cautious inverse voting strategy (IVS) and the
berserk maximal variance strategy (MVS). It was shown that, as analytically
predicted in [11], the efficacy of the attacks is reduced when a random threshold
is applied. The studies also show that the berserk attack is more severe, however
in the presence of the random threshold the difference to IVS is not significant.
Moreover, in Section 9 we propose efficient ways to detect berserk behavior. The
simpler dynamic of the IVS may also allow to approach the protocol more easily
from an analytical viewpoint. For these reasons, we consider in this paper only
a cautious strategy that is an adaption of the IVS to the setting of mana.
manaIVS We consider the cautious strategy where the adversary transmits at
time t+1 the opinion of the mana-weighted minority of the honest nodes of step

Fast Probabilistic Consensus with Weighted Votes 11
t. More formally, the adversary chooses
arg min
j∈{0,1}
N
X
i=1
mi1{si(t) = j}(5)
as its opinion at time t+ 1. We call this strategy the mana weighted inverse vote
strategy (manaIVS).
8 Improvements of FPC
We suggest several improvements of the Vanilla FPC described in [11].
Fixed threshold for last rounds In the original version of FPC nodes query
at random including itself and finalize after having the same opinion for lcon-
secutive rounds [11]. We analyzed various situations when the Vanilla FPC en-
countered failures. One key finding was that the randomness of the threshold
has sometimes a negative side effect. In fact, due to its random nature it will
from time to time show abnormal behavior.4In order to counteract this effect
we can fix the threshold to a given value, e.g., τ= 0.5, for the last l2rounds.
The initial l−l2rounds enable the original task of FPC to create an honest
super majority even in the presence of an adversary. Once a super majority is
formed a simple majority rule is sufficient for the network to finalize on the same
opinion, while the likelihood of nodes switching due to unusual behavior of the
threshold is decreased significantly.
Bias towards own opinion In Section 3 we showed that with the introduc-
tion of mana as a Sybil protection we can adopt the FPC protocol in a fair
manner by querying nodes with probability proportional to their mana. How-
ever, this can lead to agreement failures if a mana high node over-queries the
adversary in round l. Part of the network would then finalize the opinion, while
the mana-weighted majority of nodes could still switch their opinion. In an ex-
treme situation it is possible that a node that holds the majority of the funds
adjusts its opinion according to a minority of the funds, which is undesirable.
In order to prevent this we propose the following adaption. Each node biases
the received mean opinion ηto its current own opinion. More specifically, a node
jcan calculate its η-value of the current round iby
ηi(t+ 1) = mjsi(t) + (1 −mj)η∗
i(t+ 1),
where mjis j’s proportion of mana and η∗
i(t+ 1) is the mean opinion from
querying nodes without self-query.
4This is a common phenomenon for stochastic processes in random media; e.g., see
[6].

12 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
Fixed number of effective queries As discussed in Section 3 in order to
facilitate a fair quorum (thereby preventing game-ability) we select for a given
vote a node at random with a probability proportional to the mana. If a node is
selected mtimes it is given mvotes (of which all would have the same opinion).
However this can lead to a quorum with a population of nodes kdiff < k, in par-
ticular in scenarios where Nis low or sis large. Furthermore, if there is a fixed
bandwidth reserved to ensure the correct functioning of the voting layer, individ-
ual nodes could regularly under-utilize this bandwidth since the communication
overhead is proportional to kdiff. We can alleviate this deficit by increasing kdy-
namically to keep kdiff constant, and thereby improve the protocol by increasing
the effective quorum size kautomatically.
Through this approach the protocol can adopt dynamically to a network with
fewer nodes or different mana distributions.
9 Berserk detection
Since berserk strategies are the most severe attacks, e.g., [11, 2], the security of
the protocol can be improved if berserk nodes can be identified and removed from
the network. We, therefore, propose in this section a mechanism that allows to
detect berserk behavior. This mechanism is based on a ”justification of opinion”
where nodes exchange information about the opinions received in the previous
rounds. As the set of queried nodes changes from round to round this information
does not necessarily allow a direct direction of a berserk behavior but berserk
behavior is detectable indirectly with a certain probability. Upon discovering
malicious behavior, nodes can gossip the proofs of this behavior, such that all
other honest nodes can ignore the berserk node afterwards.
9.1 The berserk detection protocol
We allow that a node can ask a queried node for a list of opinions received during
the previous round of FPC voting. We call such a list a vote list and write v-list.
A node may request for it in several ways. For example, the full response message
to the request of a v-list and the opinions could be comprised of the opinion in
the current round and the received opinions from the previous round. We do not
require nodes to apply this procedure for every member of the quorum or every
round. For instance, each node could request the list with a certain probability
or if it has the necessary bandwidth capacity available. Furthermore, we can set
an upper bound on this probability on the protocol level so that spamming of
requests for v-lists can be detected. We denote this probability that an arbitrary
query request includes a request for a v-list by pB.
A more formal understanding of the approach is the following: assume that
in the last round a node yreceived kvotes, submitted by nodes z1, ..., zk. If
a node xasks yfor a v-list, then ysends votes submitted by z1, ..., zkalong
with the identities of z1, ..., zkbut without their signatures. This reduces the
message size. Node xcompares the opinions in the v-list submitted by ywith

Fast Probabilistic Consensus with Weighted Votes 13
other received v-lists. If xdetects a node that did send different opinions it will
ask the corresponding nodes for the associated signatures in order to construct
a proof of the malicious behaviour. Having collected the proof the honest node
gossips the evidence to the network and the adversary node will be dropped by
all honest nodes after they have verified the proof.
Note that a single evidence for berserk behaviour is sufficient and that further
evidence does not yield any additional benefit.
9.2 Expected number of rounds before detection
To test how reliable this detection method is and what the communication over-
head would be, we carry out the following back-of-the-envelope calculations for
s= 0 and s > 0. We are interested in the probability of detecting a berserk
adversary since the inverse of this probability equals the estimated number of
rounds that are required to detect malicious behaviour of a given node.
Let us start with s= 0 and consider the following scenario. Among Nnodes
there is a single berserk node B. In the previous round, the adversarial node is
(in expectation) queried ktimes. To see this note that in the case of s= 0, nodes
are queried with uniform probability and every node has to receive on average
the same number of queries. Furthermore, the berserk node sends freplies with
opinion 0 to the group of nodes G0and (k−f) replies with opinion 1 to the
group of nodes G1.
The probability that a node xreceives v-lists that allow for the detection of
the berserk node is in this case bounded below by
P(xreceives v-list from G0and G1)
≥2k
2p2
B
f
N·k−f
N−1·N−k
N−2··· N−2k+ 3
N−k+ 1 =γ0.
The probability that some node detects the berserk behaviour satisfies
P(some node detects malicious node) ≥1−(1 −γ0)N−1.
For example, in a system with N= 1000, k= 20, pB= 0.1 and f=k−f= 10
the detection probability is bounded below by 0.23. Assuming that the full FPC
voting (i.e., a voting cycle) for a conflict takes about 15 rounds, berserk nodes
can be detected within one FPC voting cycle with high probability.
Precise calculations are more difficult to obtain for s > 0 and we give rough
bounds instead. Let us assume that Bholds the mana proportion mB. In the
case of mana, i.e., s > 0, it is not the number of nodes, that are querying the
berserk node, that is essential, but their mana. The probability that any given
honest node queries the berserk node is at least mB, which implies that the
average sum of mana of honest nodes that query the berserk node is at least
mQ=mB(1 −mB). We assume that we can split up these nodes into two
groups G0and G1of equal mana weight, i.e., mG1=mG2. The berserk node
answers 0 to the nodes in G0and 1 to the ones in G1. Then the probability that

14 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
an honest node xqueries and requests a v-list from a node from the group Gi
(i= 0,1) is at least pBmQ/2. Moreover,
P(xreceives v-list from G0and G1)
≥2pBmQ
22=γ1.
Similarly to above,
P(some node detects malicious node) ≥1−(1 −γ1)N−1.
For instance, if N= 1000, pB= 0.1 and mB= 0.2 the detection probability
is greater than 0.12. Note that the above bound holds already for k= 2. Hence,
higher values of kwill lead to detection probabilities close to 1.
10 Heuristic for choosing the quorum size
An important parameter that dominates the performance is the quorum size k.
It may be chosen as large as the network capacity allows, in a dynamic fashion
or as small as security allows to be sustainable. Previous results, e.g. [11] and
[5], show that an increase of kdecreases the failures rates exponentially. Let
us give here some heuristic probabilistic bounds on what kind of values of k
may be reasonable. Here we consider only the Vanilla FPC but note that the
same behaviour occurs for the changed protocol. The case s= 0 can be treated
analytically as follows.
One disadvantage of the majority voting is that even if there is already a
predominant opinion present in the network, e.g., opinion 1 if p>τ, that a node
picks by bad chance too many nodes of the minority opinion.
Let pbe the average opinion in the network and τthe threshold with which
a node decides whether to choose the opinion 1 or 0 for the next round. More
specifically if more than τ k nodes respond with 1 the node selects 1, or 0 other-
wise. The number of received 1 opinions follows a Binomial distribution B(k, p).
Hence, the probability for a node to receive opinions that result in an η-value
leading to the opinion 0 is given by
P0,k(τ) = P(Y≤ bτ kc) =
bτ kc
X
m=0 k
mpm(1 −p)k−m,
where Y∼ B(k, p).As we are interested in the exponential decay of the latter
probability as k→ ∞ we use a standard large deviation estimate, e.g., [6], to
obtain for τ < p:
P0,k(τ)≈e−k I(τ),(6)
with rate function
I(τ) = τlog τ
p+ (1 −τ) log 1−τ
1−p.(7)

Fast Probabilistic Consensus with Weighted Votes 15
Fig. 4. Probability for a node to choose the opinion 0 for τ= 0.5 in the mana setting.
This shows an exponential decay of P0,k(τ) in kand that the rate of decay
depends on the ”distance” between pand τ.
An exact calculation in the mana setting of these probabilities is more difficult
to obtain. We consider the situation where the top mana holders have opinion
1 and the remaining nodes have opinion 0 such that a proportion pof the mana
has opinion 1. Fig. 4 shows estimates, obtained by Monte-Carlo simulations, of
the probability that the highest mana node will switch to opinion 0.
11 Simulation results
We perform simulation studies with the parameters given in Table 11 and study
the 1%-agreement failure. In order to make the study of the protocol numerically
feasible we choose the system parameters such that a high agreement failure is
allowed to occur. However as we will show the parameters can be adopted such
that a significantly lower failure rate can be achieved.
The source code of the simulations is made open source and available online.5
The initial opinion is assigned as follows. The highest mana nodes that hold
together more than p0of the mana are assigned opinion 1 and the remaining
opinion 0. More formally, let
J:= min{j:
j
X
i=1
mi> p0},
5https://github.com/IOTAledger/fpc-sim

16 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
Parameter Value
NNumber of nodes 1000
p0Initial average opinion 0.66
τThreshold in first round 0.66
βLower random threshold bound 0.3
kQuorum size 20
lFinal consecutive round 10
maxIt Max termination round 50
qProportion of adversarial mana 0.25
αMinimum proportion of mana 0.01
for agreement failure
Fig. 5. Default simulation parameters
then si(0) = 1 for all i≤Jand si(0) = 0 for j > J.
We investigate a network with a relatively small quorum size, k= 20 and
a homogeneous mana distribution (s= 0). The adversary is assumed to hold
a large proportion of the mana with q= 0.25. Fig. 6 shows the agreement
failure rate with N. We observe that the improvements from Section 8 increase
the protocol significantly for the lower range of N. For a large value of Nthe
improvements are still of the order of one magnitude.
Fig. 7 shows the agreement failure rate with the adversaries’ mana proportion
q. First, we can see that for the vanilla version the protocol performance remains
approximately the same for small values of s, however for s= 2 we can observe
a deterioration in performance. This effect may be explained by the skewness
of the Zipf law, leading to a more centralized situation where high mana nodes
opinion are susceptible to sampling effects described in Section 8.
We can also observe that the improvements enable the protocol to withstand
a higher amount qof adversarial mana and that for most values of qthe im-
provement is at least one order of magnitude. As we increase swe can observe
an agreement failure that is several orders of magnitudes smaller than without
the improvements.
Fig. 8 shows the failure rate with the quorum size k. As discussed in Section
10 the probability for a node to select the minority opinion in a given round
decreases exponentially with kand this trend is also well reflected in the agree-
ment failure rate, apart for small values of k. We show that the improvement of
the failure rate becomes increasingly pronounced as the quorum size is raised.
In Vanilla FPC the improvement decreases in the query size. Interesting to note
that for small query sizes (k≤60), the centralized situation, s > 1, is more stable
against attacks, but for larger kthe centralized situations become more vulner-
able than the less centralized ones. The improved FPC clearly performs better
and the improvement of the agreement rate is more important as sincreases.

Fast Probabilistic Consensus with Weighted Votes 17
Fig. 6. Agreement failure rates with N, for s= 0. The improvements from Section 8
are applied individually.
Fig. 7. Agreement failure rates with qfor three different mana distributions.

18 M¨uller, Penzkofer, Ku´smierz, Camargo, Buchanan
Finally, for s= 2 no failures are found in 106simulations for the improved
algorithm, i.e., the failure rate is less than 10−6. This is in agreement with the
performance increase observed in Fig. 7.
Fig. 8. Agreement failure rates with k.
We want to highlight that the experimental study above is only the first
step towards a precise understanding of the protocol. There are not only many
numerous parameters of the protocol itself, different ways to distribute the ini-
tial opinions, other types of failures to consider, but also many possible attack
strategies that were not studied in this paper. We refer to [2] for a more com-
plete simulation study on the Vanilla FPC and like to promote research in the
direction of [2] for the FPC with weighted votes.
12 Discussions
A main assumption in the paper is that every node has a complete list of all
other nodes. This assumption was made for the sake of simplicity. We want to
stress out that in [2] it was shown, for s= 0, that in general it is sufficient that
every node knows about 50% of the other nodes. These results transfer to the
setting s > 0 in the sense that a node should know about nodes that hold at
least 50% of the mana. In many applications it is reasonable that all large mana
nodes are publicly known and that this assumption is verified.
Another simplification that we applied in the presentation of our results is
that we assumed that the mana of every node is known and that every node has

Fast Probabilistic Consensus with Weighted Votes 19
the same perception of mana. However, such a consensus on mana is not nec-
essary. Generally, it is sufficient if different perceptions of mana are sufficiently
close. The influence of such differences on the consensus protocol clearly depends
on the choice of parameter sand may be controlled by adjusting the protocol
parameters. However, a detailed study of the above effects is beyond the scope
of the paper and should be pursued in future work.
For the implementation of FPC in the Coordicide version of IOTA, [12],
it is important to note that the protocol, due to its random nature, is likely to
perform well even in situations where the Zipf law is partially or even completely
violated.
The fairness results in Section 3 concern the Vanilla FPC. Similar calculations
for the adapted versions are more difficult to obtain and beyond the scope of this
paper. In particular, the sampling is no longer a sampling with replacement, but
the sampling is repeated until kdifferent nodes are sampled; we refer to [13] for a
first treatment of the difference of these two sampling methods. The introduced
bias towards its own opinion likely increases the voting power with respect to its
own opinion but does not influence the voting power towards other nodes. Due
to this fact and that linear weights are the most natural choice, we propose this
voting scheme also for the adapted version.
Acknowledgment
We are grateful to all members of the coordicide team for countless valuable
discussions and comments on earlier versions of the manuscript.
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