# Heterogeneity- and homophily-induced vulnerability of a P2P network
formation model: the IOTA auto-peering protocol

Yu Gao  Blockchain & Distributed Ledger Technologies Group, Department of
Informatics, University of Zürich, Switzerland  Carlo Campajola  UZH
Blockchain Center, University of Zürich, Switzerland  Institute of Finance and
Technology, University College London, United Kingdom  DLT Science Foundation,
London, United Kingdom  Nicolò Vallarano  Blockchain & Distributed Ledger
Technologies Group, Department of Informatics, University of Zürich,
Switzerland  UZH Blockchain Center, University of Zürich, Switzerland
Corresponding author<EMAIL_ADDRESS>Andreia Sofia Teixeira  LASIGE,
Departamento de Informática, Faculdade de Ciências, Universidade de Lisboa,
Portugal  Claudio J. Tessone  Blockchain & Distributed Ledger Technologies
Group, Department of Informatics, University of Zürich, Switzerland  UZH
Blockchain Center, University of Zürich, Switzerland

###### Abstract

IOTA is a distributed ledger technology that relies on a peer-to-peer (P2P)
network for communications. Recently an auto-peering algorithm was proposed to
build connections among IOTA peers according to their “Mana” endowment, which
is an IOTA internal reputation system. This paper’s goal is to detect
potential vulnerabilities and evaluate the resilience of the P2P network
generated using IOTA auto-peering algorithm against eclipse attacks. In order
to do so, we interpret IOTA’s auto-peering algorithm as a random network
formation model and employ different network metrics to identify cost-
efficient partitions of the network. As a result, we present a potential
strategy that an attacker can use to eclipse a significant part of the
network, providing estimates of costs and potential damage caused by the
attack. On the side, we provide an analysis of the properties of IOTA auto-
peering network ensemble, as an interesting class of homophile random networks
in between 1D lattices and regular Poisson graphs.

###### Index Terms:

IOTA, peer-to-peer, Mana, eclipse attack, networks

††publicationid: pubid:  This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

##  I Introduction

Over the last few decades peer-to-peer networks (P2P) have been widely
employed in a variety of settings as a base layer for distributed systems,
including the worldwide web [1], file sharing [2], instant messaging in social
networks [3, 4, 5], and distributed computing [6]. In contrast to the client-
server system, P2P networks stand out as more egalitarian and decentralised,
where peers share data resources with each other without resorting to a
central authoritative node [7]. Particularly, the emergence of Bitcoin [8] has
brought revolutionary technology in socio-technical and socio-economic
systems, and the decentralised P2P network layer is the backbone of its
Distributed Ledger Technology (DLT).

Blockchains and Distributed Ledger Technologies have gained tremendous
traction in the last decade, and advances pervade socio-technical and socio-
economic aspects surrounding them. A DLT is a consensus-driven system where
digital data is replicated, shared, and synchronised across multiple sites,
countries, or institutions in the absence of a central authority, ensuring
geographic decentralisation. A blockchain is a specific type of DLT, with
Bitcoin [8], the first type of a DLT to reach worldwide diffusion, using a
blockchain for transaction storing.

Generally, the P2P network in DLTs is the infrastructure that enables the
distribution of data among peers, and the establishment of consensus is
significantly influenced by the topology of connections within the P2P network
[9, 10]. The nodes in a generic P2P network are computers, and the information
they contain and exchange can be files, contracts, and transaction records, to
name a few [11]. To give an example, if we think of DLTs storing specifically
economic transactions like Bitcoin, the “miners” i.e. the peers driving
consensus play the role of clerks in a bank: they clear the systems
transactions in order for the everyday Bitcoin user to be able to execute
their transactions. Their incentive in doing so is the coinbase reward
associated with each block attached to the blockchain, which also serves as a
Bitcoin minting mechanism.

Due to the growing adoption of cryptocurrencies, numerous researchers have
extensively explored their P2P networks. One of the main challenges in
studying real-world P2P networks follows from their decentralised nature: they
rarely can be fully monitored by a single agent, as each node only knows about
its peers and has no reliable information beyond them, nor are they
incentivised to share this information as it may lead to malicious attacks. To
address this problem, several scholars have proposed methodologies for mapping
these networks and conducting subsequent analyses. To cite a few, Miller
introduced “AddressProbe” to uncover the topology of the Bitcoin P2P network
[12]; Deshpande et al. devised a framework to map the Bitcoin P2P network,
presenting an insightful snapshot of its topology [13]; Kim et al. developed
NodeFinder, a tool to measure the Ethereum P2P network’s characteristics and
dissect the intricate DEVp2p ecosystem [14]. From a different viewpoint,
Imtiaz et al. revealed the intermittent network connectivity displayed by
Bitcoin nodes [15].

Notably, there exists research concerning attacks on P2P networks: Heilman et
al. [16] proposed employing extremely low-rate TCP connections for network
attacks, while Nayak et al. [17] devised a combination of mining attacks and
network-level attacks to undermine the consensus in DLT protocols.
Additionally, other papers outline a variety of technical approaches to
network attacks [18, 19].

In P2P networks, nodes actively communicate with nearby nodes, facilitating
the exchange of information. This dynamics is crucial for maintaining
consensus among nodes and deterring potential efforts by individuals or
attackers to gain complete control of the network. Despite the original
intention of most DLT systems’ P2P networks to build random, uncontrollable
networks, vulnerabilities can arise when certain network information is
exploited [20, 21]. To prevent such attacks, it is essential to have a
comprehensive understanding of the inherent topology of the empirical P2P
network within a static network model, achieved by substituting technical
cryptographic elements with graph-based parameters [22, 23, 24, 25]. This
approach can uncover potential weaknesses in the design of P2P network
protocols.

Network Science offers a comprehensive tool-set to effectively model these
types of complex systems [26, 27]. Socio-technical systems have been largely
studied in the context of complex networks [28, 29]: translating such systems
into network representations allows us to exploit their topological properties
in order to identify vulnerabilities and nonlinear properties. P2P protocols
have a natural network representation, where peers can be represented as
nodes, while edges embody peer connections. An obstacle to a straightforward
application of network analysis to these systems is that by design P2P
protocols make the connection construction unknown globally, as peers only
know their local connectivity. On the contrary, peers generally have a huge
security incentive not to reveal their connections to possible malicious
adversaries [16, 30]. A possible approach to this challenge is to define
theoretical models that describe P2P networks, starting from the P2P protocol
rules, in order to gain further insight.

The prevailing research naturally focuses on Bitcoin and Ethereum, both
operating in a permissionless manner. Generally, these platforms incentivise
validators — i.e. the peers tasked with verifying transactions and upholding
network integrity — by issuing rewards for their honest contribution and
penalties if their actions contradict rules or network interests. However, for
platforms lacking such reward and penalty systems for peers like
IOTA111https://www.iota.org/, the design of the peer connection algorithms can
have a profound influence on the consensus integrity.

IOTA has been introduced to address one of the main limitations of mainstream
blockchains, namely their scalability. Blockchain addresses the challenge of
decentralising digital transaction systems while enhancing data transparency
and security. However, it may encounter scalability issues when managing a
large volume of transactions [31]. Each node is required to verify and store a
complete copy of the blockchain, leading to limited network capacity and
slower transaction processing; in addition to this, blockchains have an
intrinsic transaction volume limit, depending on how many transactions can fit
inside a block and the time that occurs between two blocks. To tackle this,
IOTA introduced the Tangle [32], a new ledger data structure based on a
directed acyclic graph (DAG) instead of the classic linear chain. By enabling
concurrent transaction processing, the Tangle can theoretically handle
increased transaction throughput as the network expands. Another peculiarity
of IOTA is “Mana”, a scarce quantity associated with a peer that would
tokenize its reputation, which has been proposed to be introduced in an
upcoming upgrade [33]. Mana is intended as a Sybil protection mechanism [34,
35], i.e. a mechanism that sets a cost (in this case, acquiring reputation)
for nodes to participate in the consensus formation. This prevents Sybil
attacks, where malicious agents are able to manipulate consensus by the
creation of a large number of nodes.

It was proposed to use Mana within a pseudo-random P2P network formation
model, the IOTA auto-peering module [20], which favours homophilic
connections, i.e. among nodes in a close Mana range. Homophily is known in
sociology as the tendency of individuals to bind by similarities. In network
theory, we observe homophily when nodes tend to connect preferentially with
nodes with similar characteristics [36, 37, 38, 39]. Many social networks
demonstrate connection mechanisms in which each node possesses a distinct
trait, such as wealth, age, or trustworthiness. Connections in these networks
arise from similarities, as observed in friendship networks [40],
collaboration networks [41] or actor networks [42].

The goal of our work is to identify potential weaknesses in the P2P network
formation model that may result in vulnerability to attacks. In particular, we
focus on eclipse and partitioning attacks, where malicious actors aim at
controlling the flow of information between portions of the network to an
extent where they gain an unfair amount of power in determining the consensus.
To investigate the security of the IOTA P2P network, we analyse the networks
generated by the IOTA auto-peering formation model
222https://github.com/iotaledger/autopeering-sim and develop two attack
strategies: the “Betweenness” strategy and the “Greedy” strategy. From an
attacker’s perspective, both strategies can be translated into network
partitioning algorithms: what the attacker is interested in is to find the
“control set”, i.e. the subset of nodes they need to take control of in order
to censor the flow of information between the two network partitions. We
initially employ the “Betweenness” strategy, which utilises the link
betweenness centrality metric [43, 44, 45] to split the P2P network by
removing the top betweenness links and by identifying the control set as the
nodes that are endpoints of the removed links. The second strategy we employ,
which we call “Greedy”, involves sorting all nodes in the network by Mana
endowment and defining a possible partition by splitting nodes according to
their ranking. To quantify the impact of the attack, we introduce the concepts
of “Damage” and “Cost”, whose ratio we employ as a measure of attack
efficiency.

Although both strategies rely on complete topological information about the
network, it is important to note that in real-world scenarios the specific P2P
network topology is not publicly accessible. Therefore, to address this
limitation, we also propose a “Blind” strategy. This strategy is inspired by
the results of the “Betweenness” and “Greedy” strategies, but it is executed
assuming the attacker has no knowledge of the actual topology of the P2P
network except for the knowledge of the Mana distribution and the IOTA auto-
peering network formation model parameters, which are public information
available to all IOTA nodes. The blind strategy identifies a “target” node,
i.e. the centre of the control set, which is then composed of a specific
number of nodes before and after the target node in terms of Mana ranking. Our
experiments show that blind strategies can be applied to obtain advantageous
control sets in terms of attack efficiency, leading to the P2P network
partition into two separate components that causes more damage than the attack
cost. We also test the performances of blind strategy attacks on networks
generated using the Watts-Strogatz (WS) model [42, 27], in order to have a
reference baseline. Indeed, as will be clear from the results, the WS networks
ranging from the 1D lattice to a fully randomised regular network act as the
ideal reference for the IOTA auto-peering networks. These are in fact
themselves random regular networks, with a preferential dimension for
connections given by the Mana endowment rather than by the position in the 1D
lattice.

The rest of the paper is organised as follows: Section II describes our
approach to the problem, setting the theoretical framework for the simulation
study. Section III presents the results of the simulations. Section IV
concludes the paper.

##  II Approach

###  II-A Mana

The auto-peering module, as described by IOTA3, generates a P2P network
composed of $N$ nodes. Each node has a given amount of Mana and, for
convenience, we will index them from the largest endowment of Mana (identified
as $1$) to the lowest (identified as $N$). These Mana values are publicly
known to all nodes and regularly broadcast over the P2P network itself. The
distribution of Mana is the result of a dynamical process, where nodes can
accrue their endowment through fair participation in the consensus mechanism,
and their chance to do so is itself a function of their Mana endowment; in our
setting we assume Mana to be an equilibrium value, not depending on time. As
is commonly found in similar systems like Proof-of-Work mining [46, 47],
Proof-of-Stake validation [48, 49] or any system where the growth of a
quantity is proportional to its current level, the so-called “Matthew effect”
enters the picture and a Zipf’s law often emerges after sufficient time. For
this reason, we assume the Mana distribution follows a Zipf’s Law [50] with
exponent $s$ so that the Mana value of node $i$ is

$m(i)=Ki^{-s},$ (1)

where $m(i)$ is the Mana endowment of the $i$-th ranked node and $K$ is an
arbitrary positive constant used for numeric stability, in our case
$K=10^{10}$.

###  II-B Peer discovery

The protocol prescribes that each node $i$ will add node $j>i$ to its list of
eligible neighbours, $\mathcal{N}_{i}$, if $m(i)<\rho m(j)$, where $\rho$ is a
given constant. If this condition is not met for at least $R$ other nodes, $i$
will add the $R$ nodes ranked after itself to the list. Any time a node $j$ is
added to the set $\mathcal{N}_{i}$, $i$ will be added to $\mathcal{N}_{j}$ to
reciprocate.

Therefore, the potential neighbours of any node $i$ are formally identified as

$\small\mathcal{N}_{i}=\left\\{j=1,\dots,N\,:\,\frac{1}{\rho}m(i)<m(j)<\rho
m(i)\,\vee\,\left|j-i\right|<R\right\\}.$ (2)

From this set, $i$ randomly picks $k$ elements to link to and receives $k$
links from other nodes in the set. The result is a random regular network with
coordination number $k$ (or in other words where each node has degree $2k$),
and links appear only between nodes with similar Mana endowment. From a
modelling perspective, $N$, $k$, $R$, $\rho$ and the Zipf’s exponent $s$ are
the parameters needed to generate a network.

###  II-C Network Formation

We base our simulation on the publicly available Go code by the IOTA research
team333https://github.com/iotaledger/autopeering-sim. A node will compile its
potential neighbours’ list and cease searching for neighbours when it has
established $k$ connections, while it accepts connection requests from other
nodes until it has received another $k$ connections. Communication is then
bidirectional along these links, so the resulting network is undirected. As a
result, each node ultimately has $2k$ neighbours.

Combining the aforementioned rules, we can deduce that an increase in the
Zipf’s exponent $s$ leads to the most affluent nodes having relatively short
potential neighbour lists. This happens because of the increased inequality in
the distribution of Mana among nodes, meaning a larger Mana gap between nodes.
The parameters $R$ and $\rho$, although not as influential, still impact the
IOTA P2P network and contribute to its structure by tuning the tolerance that
the nodes have in terms of the Mana endowment of their potential neighbours.
As can be seen by the definition of Eqs. 1 and 2, when $s$ is small the
distribution is relatively homogeneous and peers easily have many potential
neighbours through the range condition set by $\rho$, forming a network that
is closely resembling a regular (i.e. where all nodes have the same degree)
version of the Erdős-Rényi (ER) random network. However, as $s$ increases,
nodes that fail to find enough peers within the Mana ratio condition can
acquire sufficient potential neighbours through the alternative condition
based on $R$. In this situation, peers that have fewer potential neighbours
can solely establish connections with peers that are in their immediate
vicinity in terms of Mana ranking. Consequently, the network will tend towards
resembling a k-regular 1D lattice network.

Figure 1: Intuitive visualisation of the different topologies realised by the
IOTA auto-peering formation model as a function of $\rho$ and $s$. The graphs
are generated with fixed $N=100,R=10$, $k=4$. Colour represents the Mana
endowment of nodes, in a logarithmic scale.

To have a more intuitive understanding of the IOTA auto-peering model, we
provide a visualisation of networks generated under various $s$ and $\rho$
values in Figure 1. As it can be seen, when the Mana (shown in colour in log
scale) is relatively homogeneous and when the tolerance parameter $\rho$ is
high enough the network does not show any particular asymmetry, but in a low
$\rho$ scenario and whenever $s$ is increased there is a clear orientation of
the network, with the nodes with the largest Mana endowment being shifted
towards one end of the network and the shape becomes more linear, following
the nodes Mana hierarchy.

###  II-D Attack Damage and Cost

Figure 1 neatly highlights the intuition behind a potential attack strategy
for this communication network. When $s$ is large and $\rho$ is small, the
elongated shape taken by the network results in the creation of a choke point
separating the few high Mana nodes from the many low Mana ones.

In an eclipse attack, the attacker’s goal is to control all the victim’s
incoming and outgoing communications, isolating the target from the rest of
the peers and destroying the ability of all peers to reach a consensus[30]. In
our work, the goal of the attacker is to achieve control over all links
separating two sub-components of the network: this allows the attacker to
manipulate the flow of information between the two sub-components, gaining
advantage and twisting consensus towards their needs. Since the parameters of
the network formation protocol are known to the attacker, they only need to
find an allocation of their Mana endowment across multiple Sybil nodes in a
way that allows them to control the crucial choke points.

To measure the attack efficiency, we introduce the concept of damage $D$ and
cost $x$. The damage is measured in terms of the fraction of Mana the attacker
is able to disconnect from the network: if the total Mana in the system is
denoted as $M=\sum_{i}m(i)$ and the network nodes are split into two
components $A$ and $B$ where without loss of generality $\sum_{i\in
A}m(i)\leq\sum_{j\in B}m(j)$, then

$D(A)=\sum_{i\in A}m(i)/M.$ (3)

Clearly, the maximum attainable damage is then $D_{max}=1/2$.

To control a link, the attacker needs to control one of the endpoint nodes.
The cost of controlling a link $(i,j)$ is defined as the minimum Mana of its
endpoint nodes $x(i,j)=\min(m(i),m(j))$. Then, for a set of links $C(A,B)$
which separates partition $A$ from $B$ and the set of least Mana endpoints
(the “frontier”) $F=\\{\arg\min(m(i),m(j))\;\forall(i,j)\in C(A,B)\\}$, the
cost of attack reads

$\displaystyle x(C)=\sum_{i\in F}m(i).$ (4)

The goal of the attacker is, for a given Mana cost $x_{max}$ they are able to
commit, to find the optimal set $C$ such that $x(C)\leq x_{max}$ and
$x(C^{\prime})>x_{max}\forall C^{\prime}$ such that $D[x(C^{\prime})]\geq
D[x(C)]$.

This is a NP-hard problem, as there are exponentially many partitions and cuts
from which to choose. In the following, we propose two simplified strategies
for the attacker: a “Betweenness” strategy and a “Greedy” strategy, which both
require full information about the P2P structure, and a derived “Blind”
strategy that only requires the more realistic assumption of knowledge of the
parameters used to generate the network.

###  II-E Attack strategies

#### Betweenness strategy

The betweenness strategy, as the name suggests, relies on edge betweenness
centrality to identify the optimal cut. Similar to node betweenness, edge
betweenness serves as a metric that quantifies the fraction of shortest paths
passing through a specific edge between any two nodes within a network [51].
Inspired by the Girvan-Newman algorithm for community detection [52], the
strategy prescribes removing edges from the network starting from the ones
with the highest betweenness centrality. However, unlike the community
detection process, in this scenario, the removal of the top betweenness links
is halted once the network is divided into two parts. It is crucial to note
that after each removal of the top betweenness link, the link betweenness is
recalculated for the entire network. The attacker proceeds iteratively,
removing the top betweenness links until the original network is fragmented
into two distinct sub-networks, denoted as $A$ and $B$. The removed links
constitute the associated cut $C$, and the attack damage $D$ is the total Mana
associated to the nodes in the sub-network deriving from cut $C$, which has
the smaller total Mana share.

#### Greedy strategy

In the literature, we can find many different instances of network
partitioning algorithms, such as spectral partitioning [53], clustering
coefficient [54], community detection [55] or heuristic based partitioning
[56]. However, none of these algorithms takes advantage of node attributes,
such as the Mana endowment. Here, we introduce a simple method to split the
auto-peering network according to the Mana ranking information and the network
topological information, which we call the “Greedy” strategy.

The Greedy strategy is much more simplistic than the Betwenness one: the
attacker splits the nodes into two sets $A=\\{i\leq i^{*}\\}$, and
$B=\\{i>i^{*}\\}$ for each possible target node $i^{*}=1,\dots N-1$. It
follows that the cut is the set of links connecting $A$ and $B$. For each
$i^{*}$ we calculate the cost of the split and choose the $i^{*}$ that
maximises damage per unit cost to run the attack.

#### Blind strategy

In practical situations, the specific topology of the P2P network is not
available, as nodes do not have a full picture of all connections in the
network but only know their own peers. A blind strategy, one that does not
require full knowledge about the network topology, is therefore needed. We
design a blind strategy that is informed by the results obtained with the
Betweenness and Greedy strategies, namely measuring the frequency with which a
specific node, given the network formation parameters, ends up being in the
control set identified by the full-information strategy. We then take the most
frequently targeted node $i_{\sigma}$, where $\sigma\in\\{B,G\\}$ (for
Betweenness and Greedy, respectively) and define the control set for the
strategy $\mathcal{C}_{\text{blind}}$ as all nodes within a range $L$ in the
Mana ranking from node $i_{\sigma}$, i.e.

$\mathcal{C}_{\text{blind}}(\sigma)=\left\\{i\;:\;|i_{\sigma}-i|<L\right\\}$
(5)

The attacker is assumed to take control of this set of nodes and we verify
whether this was successful to split the network and quantify the attack
efficiency according to our Damage and Cost measures.

##  III Results

###  III-A Simulation results for full information attacks

We explore the parameter space and find that the resulting network structure
is most sensitive to the value of Zipf’s exponent $s$, which describes Mana
heterogeneity (see section III-C). We fix all other parameters to reasonable
values $N=100$, $R=10$, $\rho=4$, $k=4$, and focus on the attack vulnerability
varying $s$ from $0.5$ to $1.5$ at intervals of $0.1$. For each set of
parameters, we generate a sample of 1000 networks and apply our attack
strategies, measuring the damage over cost ratio $D/x$.

In Figure 2, we present the results for the average damage over cost
$\mathbb{E}[D/x]$ over 1000 simulations and across varying values of $s$ for
the full information attacks. We observe that both the Betweenness and Greedy
strategies, respectively represented by the blue and orange violins, attain
maximum efficiency when $s\approx 1$. Additionally, we show the result of
applying the Betweenness strategy to a random-regular network where Mana does
not have a role in the formation of connections, i.e. the fully randomised
Watts-Strogatz (WS) model of small-world networks [27] with coordination
number $k$. Compared to the auto-peering network formation model, the attack
efficiency in the WS model appears to be much closer to 0 in the vast majority
of cases. The result in Figure 2 shows that in specific Mana distribution
conditions, the attacker can achieve relatively large damage to the network
with a small cost and that this vulnerability is induced by the presence of
Mana in the network formation protocol. Both strategies achieve maximal
$D/x\approx 3.5$ for $s=1$: this is particularly relevant because Zipf laws
with exponent in the close range around $1$ have been observed to describe
wealth distribution in cryptocurrency token holdings in the literature [57].

Figure 2: The average damage over cost ratio by attack strategy and underlying
network formation model. Each auto-peering generated network has $N=100$,
$\rho=4,R=10$, $k=4$. WS generated networks have $N=100,k=4,p=1$, the data
size is 1000 graphs.

Next, we compare the frequency of endpoint removals between the betweenness
strategy and the greedy strategy under the conditions of $\rho=4$, $R=10$, and
$N=100$, based on a sample of $1000$ graphs. Our findings, which for the sake
of conciseness we report in Fig.A2 and Fig.A3 in the Appendix, reveal that the
greedy strategy, which aims to optimise the $D/x$ ratio, typically removes
node $i_{G}=14$ as the maximum frequency endpoint. On the other hand, for the
betweenness strategy, the maximum frequency endpoint that is removed is node
$i_{B}=12$. We find that in the case of $s=1$, the endpoints selected with
maximum frequency $F$ by both the “betweenness” strategy and the “greedy”
strategy are typically in the same range, albeit not precisely the same nodes.
We are going to use the maximum frequency endpoints hereby identified to
inform the blind attacks analysed in the next section.

###  III-B Simulation results for blind attacks

A pitfall of the two strategies above is that they both require the attacker
to have perfect knowledge of the network structure, which is unrealistic. A
more realistic relaxation of this assumption is that the attacker is only
aware of the network formation model parameters. Given such information, the
attacker can reproduce the above results and inform a “Blind” strategy, aiming
to control the nodes that are most often needed to perform the split. In
particular, for the specific case of $N=100$, we take the “target” nodes
$i_{B}=12$ and $i_{G}=14$ (for the Betweenness and Greedy strategy
respectively) and define the parameter $L$ such that the blind attacker
controls nodes $\mathcal{C}_{\mathrm{blind}}(\sigma)$ for
$\sigma\in\\{B,G\\}$. We then measure the $D/x$ ratio over $1000$ simulated
networks along with the success ratio $p$, i.e. the fraction of simulations
where the network is successfully split by the blind attacker, and show the
results in Figure 3 as a function of $L$.

Figure 3: Success ratio $p$ (on the right $y$-axis) and average damage
$\mathbb{E}[D/x]$ (on the left $y$-axis) per unit cost of a Blind strategy
informed by the Greedy (“BG”) and Betweenness (“BB”) strategies. The results
are averaged over a sample of $1000$ simulated networks with $N=100$,
$\rho=4$, $R=10$ and $s=1$ while the error bars are $95\%$ confidence
intervals. As comparisons, we also show the success ratio $p$ on an equivalent
1D lattice and a fully randomised regular WS-type network in the inset.

The red and yellow lines show $\mathbb{E}[D/x]$, the average damage over cost
attack efficiency associated with a Blind strategy informed either by the
Greedy (red, “BG”) or Betweenness (yellow, “BB”) results, averaged over a
sample of $1000$ networks. The blue and green lines track the attack success
rate $p$ using the target node from the “Greedy” strategy (blue line) and from
the “Betweenness” strategy (green line).

As should be expected, for low choices of $L$ the strategy is unsuccessful in
splitting the network. The Blind-Betweenness (BB) strategy reaches a peak
average efficiency of $\mathbb{E}[D/x]=1.75$ for $L^{*}_{b}=7$, corresponding
to $100\%$ attack success. On the other hand, the Blind-Greedy (BG) strategy
reaches peak efficiency for $L^{*}_{g}=8$: while requiring more nodes to be
controlled, the efficiency is larger on average ($\mathbb{E}[D/x]=2$).
Summarising, both strategies allow the attacker to successfully split the
network from a low information context. Based on our simulations the BG
strategy attains a better efficiency, but we observe that we did not assign an
additional cost to the number of controlled nodes, we only considered the Mana
required to control the node: the number of nodes to control may affect the
cost of an attack in a real-world situation. Interestingly, both strategies at
their peak efficiency $L$ command the same cost of around $24\%$ of all the
Mana in the system, as shown in Figure 5. This means that an attacker
controlling $24\%$ of the system’s Mana would be able to identify a
distribution strategy to control the choke points of the P2P network $100\%$
of the times.

To compare the severity of the attacks with some baseline scenarios, we also
run the same attacks on a 1D lattice with the same coordination number $k$ as
the auto-peering network and on a fully randomised Watts-Strogatz (WS) network
model, i.e. a random-regular network of degree $2k$. For the one-dimensional
lattice, for which we show the success ratio with the cyan triangles in Figure
3, it is straightforward to see that the network is always split whenever
$L\geq k$, where in this case $k=4$. Compared with a 1D k-regular lattice, the
auto-peering networks are more resilient. This makes perfect sense: 1D lattice
are completely predictable networks, where the attacker does not need to
acquire information to predict the topology of the network. On the other
extreme the WS networks, being fully randomised, are much more resistant to
attacks and the networks are almost never successfully split unless $L$ is
chosen to be of the same order of magnitude as $N$. We show this result in the
inset of Figure 3.

This result shows clearly that the IOTA auto-peering protocol produces
networks that are in between the fully deterministic 1D lattice and a fully
random regular network. Depending on the parameter values, the Mana-based
connection rules can severely restrict the options for nodes to connect
between themselves, resulting in the loss of the security provided by
randomness and consequently in predictable network structures that can be
exploited.

###  III-C Strategy Robustness

In order to verify the robustness of our results, we test the proposed
strategies by simulating the model on a realistic range in the parameters’
space. We choose $\rho\in[1.5,4]$ and $s\in[0.5,1.5]$ as they should provide a
comprehensive overview of realistic parameters choices.

We find that, as long as $N$ is large enough compared to $k$, varying the
number of nodes has no particular effect on the results. This is mostly due to
the fact that the Mana is distributed according to Zipf’s law, and so most of
the Mana is concentrated in the first few nodes in the ranking regardless.
Similarly, we do not find effects of particular interest when changing $R$
within reasonable ranges. For the sake of space, we report additional results
testing $N$ and $R$ dependence in Fig.A5 and Fig.A6 (see Appendix). For each
combination of parameters, the results were averaged over a sample of $1000$
graphs generated by the auto-peering network formation model. On the other
hand, we find an interesting interplay between the $\rho$ and $s$ parameters,
which we discuss in the present section.

In Figure 4 we show the expected damage over cost ratio across different
parameter combinations for the two informed strategies. While the performances
of the two strategies are slightly different, we do observe a similar trend:
both attain maximum success (in terms of damage over cost) for low $\rho$ and
for $s$ between $0.5$ and $1$, i.e. when the Mana distribution is sufficiently
heterogeneous compared to nodes’ “tolerance” $\rho$, but not so heterogeneous
that attacks can only be successful by controlling the top Mana nodes
themselves. At the same time, we observe that $\rho$ has a similar effect on
both strategies for all values of $s$, with a stronger effect when $s$ is low
(so when Mana distribution is homogeneous). This is explained by the role of
$\rho$, which is increasing (by a multiplicative factor) the tolerance for
each node in terms of Mana difference to its potential neighbours; this means
that when the network is homogeneous, a large $\rho$ implies longer range
connections, while when $s$ is very large, the randomness effect of $\rho$ is
diminished.

Figure 4: Heatmaps of the expected damage over cost, $\mathbb{E}[D/x]$,
plotted over $\rho$ and $s$, for the betweenness (up plot) and greedy (down
plot) strategy. $R=10$ is kept constant. The maximum for each fixed $s$ is
when $\rho$ is minimal, meaning that whenever the P2P model is more similar to
a chain, the network is maximally vulnerable.

In Figure 5 we present the Mana cost (as a percentage of the total Mana) that
is necessary to successfully split the network with $100\%$ success, following
a blind attack with the BB or BG strategy. We notice that the regions of the
parameters space which deliver the worst output in terms of attack efficiency
are also the ones with the largest cost of attack, where graphs generated form
the auto-peering model resemble random regular graphs. The cost is
monotonically decreasing in $s$ and increasing in $\rho$ for both the blind
strategies.

Figure 5: Cost in Mana (as a percentage of the total Mana in the system) of a
blind attack strategy inspired by betweenness strategy results (bottom plot)
and greedy strategy results (upper plot) necessary to obtain a successful
network split $100\%$ of the times, the results are averaged over $1000$
simulations for each parameters combination. $s$ and $\rho$ vary, while $R=10$
and $N=100$ are kept constant.

##  IV Conclusions

In conclusion, our results point out that a malicious agent controlling just
over $24\%$ of the total Mana can divide their Mana into a set of nodes that
will most likely act as choke points of the P2P network from the auto-peering
network formation model, leading to the attacker successfully splitting the
P2P network and achieving damage that is close to the maximum possible. Such
an attack would potentially end up compromising the operations of the IOTA
network, as the attacker can control the flow of information between the high-
Mana portion of the network and the vast majority of nodes on the other side.
For comparison, we also apply the strategies on 1D lattices and fully rewired
Watts-Strogatz (WS) networks, finding that the IOTA auto-peering protocol is
far less resistant than the latter while slightly more resistant than the 1D
lattice.

To conclude, we want to provide a practical comment to interpret our results:
the auto-peering network formation model is not currently implemented on the
IOTA system; as such there is no immediate risk to the IOTA network.

The results of the present work are to be interpreted as a policy suggestion,
presenting a potential attack vector. At the same time, from a network science
perspective, we believe that the auto-peering model provides an interesting
interpretation of random assortative formation models, and provides an
alternative bridge between k-regular Poisson graphs and 1D lattices,
complementing the popular Watts-Strogatz model.

## Acknowledgments

YG acknowledges financial support from the China Scholarship Council. AST
acknowledges support by FCT through the LASIGE Research Unit, ref.
UIDB/00408/2020 (https://doi.org/10.54499/UIDB/00408/2020) and ref.
UIDP/00408/2020 (https://doi.org/10.54499/UIDP/00408/2020).

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In this appendix, for a better understanding of the strategies employed in the
paper, we present additional details regarding the simulation results of the
two full information strategies (“Betweenness strategy” and “Greedy
strategy”), which in turn were used to develop the “Blind strategy.”

In Figure A1, we show the simulation results of $E[D/x]$ for $1000$ graphs,
where $N=100$, $R=10$, $\rho=1$, and $N=100$, when applying the Greedy
strategy. This figure explains the target selection mechanism of the Greedy
strategy, which selects the target node $i^{*}$ based on the highest $E[D/x]$
value among the $N-1$ potential splits.

We then compare the consistency of the Betweenness strategy and Greedy
strategy in terms of the nodes that are typically included in the control set.
Under the same parameters conditions used to produce Figure A1, our findings
in Figure A2 reveal that the greedy strategy most often targets node
$i_{G}=14$, whereas the betweenness strategy’s most frequently targeted node
is $i_{B}=12$.

To compare the attack efficiency in these two strategies, we also reframe the
result shown in Figure 4 of the main text by showing the ratio between the two
$E[D/x]$. The result is displayed in Figure A3, which shows that while for the
large majority of parameters the two strategies are similarly effective, in
the top-left corner (when $\rho$ is large and $s$ is small) the greedy
strategy significantly outperforms the betweenness strategy. This is the same
region of the parameters space where both strategies have the worst
efficiency, as reported in Figure 4 of the main text. It is also the range of
parameters where the networks produced by the auto-peering model are maximally
close to a random-regular topology. In Erdos-Renyi graphs the betweenness
information is homogeneous and the attacker gains no advantage from the
distribution of link betweenness.

Figure A4 provides an overview of the effect of BB (Blind-Betweenness) and BG
(Blind-Greedy) attacks. Here the heatmap indicates the minimum $L$, that is
the count of nodes removed before and after the target nodes $i_{\sigma}$
identified by the full-information strategies, such that the blind attack is
$100\%$ successful in splitting the simulated graphs. This measure can be
intended as a measure for the robustness of the generated networks. We see
that in BB attacks the value of $L$ is relatively stable until it explodes in
the region of the parameters’ space where the network is more akin to an
unstructured random regular network, i.e. low $s$ and high $\rho$. The $L$
values of BG attacks are more gradually increasing when moving towards the
same region of the parameters’ space. The overall take-home message about the
robustness of the different methodologies is that BG attacks are less
sensitive to parameter variation.

Finally, we provide a sensitivity analysis for our results with respect to the
parameters $R$ and $N$ in Figures A5 and A6. The analysis demonstrates that
both network size $N$ and parameter $R$ present minimal quantitative influence
on the results, and do not affect the overall qualitative behaviour. In Figure
A5, varying $N$ yields consistently marginal differences in $E[D/x]$ across
the subplots. Similarly, Figure A6 indicates that altering $R$ produces
negligible variations in $E[D/x]$ within each subplot grid. These observations
confirm the robustness of the outcomes to changes in both network size $N$ and
the parameter $R$ in the additional peer discovery condition.

Figure A1: The average Greedy damage over cost by targeted node $i^{*}=1,\dots
N-1$, where N=100, $\rho=4,R=10$, and sample size is $1000$ graphs. Figure A2:
Histogram of frequencies $F(i)$ with which node $i$ is in the control set
according to the Greedy and Betweenness strategies on a sample of $1000$
graphs, where $\rho=4,R=10,N=100$, $k=4$, $s=1$.  Figure A3: The heatmap shows
the ratio of $\mathbb{E}[D/x]$ for the Betweenness strategy over the
$\mathbb{E}(D/x)$ for the Greedy strategy, effectively representing the
relative efficiency of the betwenness attack with respect to the greedy
strategy. $s$ and $\rho$ varies, while $R=10$ is kept constant. The reader can
notice a minimum placed at $s=0.5$ and $\rho=3.5$. That is the region of the
parameters’ space where the auto-peering model networks most resemble regular
Erdos-Renyi random graphs, and the link betweenness becomes homogeneous and
highly uninformative. Figure A4: Each cell’s number indicates the minimum
value of $L$ necessary to obtain a successful network split $100\%$ of the
times in the betweenness and greedy strategies respectively. The results are
averaged over $1000$ simulations for each parameters combination. $s$ and
$\rho$ vary, while $R=10$ is kept constant. Figure A5: The heatmap shows the
ratio of $\mathbb{E}[D/x]$ for the Betweenness strategy $\mathbb{E}(D/x)$,
where $s$, $N$ and $\rho$ varies, while $R=10$ is kept constant. From the
three heat map plots, we can see that there is no big gap for the damage over
cost in different network sizes. Figure A6: The heatmap shows the ratio of
$\mathbb{E}[D/x]$ for the Betweenness strategy over the $\mathbb{E}(D/x)$ for
the Greedy strategy varying $s$, $R$ and $\rho$, where $N=100$.


# Recognizing Arrow Of Time In The Short Stories

Fahimeh Hosseini
Shenakht Pajouh / Sharif
University of Technology,
Tehran, Iran
fahim.hosseini.77@
gmail.com
Hosein Fooladi
Shenakht Pajouh / Sharif
University of Technology,
Tehran, Iran
fooladi.hosein@
gmail.com
Mohammad Reza Samsami
Shenakht Pajouh / Sharif
University of Technology,
Tehran, Iran
mohammadrezasamsami76@
gmail.com

###### Abstract

Recognizing arrow of time in short stories is a challenging task. i.e., given
only two paragraphs, determining which comes first and which comes next is a
difficult task even for humans. In this paper, we have collected and curated a
novel dataset for tackling this challenging task. We have shown that a pre-
trained BERT architecture achieves reasonable accuracy on the task, and
outperforms RNN-based architectures.

##  1 Introduction

Recurrent neural networks (RNN) and architectures based on RNNs like LSTM
[Hochreiter and Schmidhuber, 1997] has been used to process sequential data
more than a decade. Recently, alternative architectures such as convolutional
networks [Dauphin et al., 2017, Gehring et al., 2017] and transformer model
[Vaswani et al., 2017] have been used extensively and achieved the state of
the art result in diverse natural language processing (NLP) tasks.
Specifically, pre-trained models such as the OpenAI transformer [Radford et
al., 2018] and BERT [Devlin et al., 2018] which are based on transformer
architecture, have significantly improved accuracy on different benchmarks.

In this paper, we are introducing a new dataset which we call
ParagraphOrdering, and test the ability of the mentioned models on this newly
introduced dataset. We have got inspiration from ”Learning and Using the Arrow
of Time” paper [Wei et al., 2018] for defining our task. They sought to
understand the arrow of time in the videos; Given ordered frames from the
video, whether the video is playing backward or forward. They hypothesized
that the deep learning algorithm should have the good grasp of the physics
principle (e.g. water flows downward) to be able to predict the frame orders
in time.

Getting inspiration from this work, we have defined a similar task in the
domain of NLP. Given two paragraphs, whether the second paragraph comes really
after the first one or the order has been reversed. It is the way of learning
the arrow of times in the stories and can be very beneficial in neural story
generation tasks. Moreover, this is a self-supervised task, which means the
labels come from the text itself.

Label |  1
---|---
First Paragraph |  Now they were walking through the trees, one of them carrying him in its huge arms, quite gently. He was scarcely conscious of his surroundings. It was becoming more and more difficult to breathe.
Second Paragraph |  Then he felt himself laid down on something soft and dry. The water was not falling on him now. He opened his eyes.
Table 1:  A single example of ParagraphOrdering dataset.

##  2 Paragraph Ordering Dataset

We have prepared a dataset, ParagraphOrdreing, which consists of around
300,000 paragraph pairs. We collected our data from Project Gutenberg. We have
written an API for gathering and pre-processing in order to have the
appropriate format for the defined task.111API for downloading the dataset:
https://github.com/ShenakhtPajouh/transposition-data. The implementation of
different algorithms: https://github.com/ShenakhtPajouh/transposition-simple
Each example contains two paragraphs and a label which determines whether the
second paragraph comes really after the first paragraph (true order with label
1) or the order has been reversed (Table 1). The detailed statistics of the
data can be found in Table 2.

#Train Samples | 294265
---|---
#Test Samples | 32697
Unique Paragraphs | 239803
Average Number of Tokens | 160.39
Average Number of Sentences | 9.31
Table 2:  Statistics of ParagraphOrdering dataset.

##  3 Approach

Different approaches have been used to solve this task. The best result
belongs to classifying order of paragraphs using pre-trained BERT model. It
achieves around $84\%$ accuracy on test set which outperforms other models
significantly.

Model | Accuracy ($\pm 0.01$)
---|---
LSTM+Feed-Forward | 0.518
LSTM+Gated CNN+Feed-Forward | 0.524
BERT Features(512 tokens)+Feed-Forward | 0.639
BERT Classifier(30 tokens / 15 tokens from each paragraph) | 0.681
BERT Classifier(128 tokens / 64 tokens from each paragraph) | 0.717
BERT Classifier(256 tokens / 128 tokens from each paragraph) | 0.843
Table 3:  Accuracy on Test set.

###  3.1 Encoding with LSTM and Gated CNN

In this method, paragraphs are encoded separately, and the concatenation of
the resulted encoding is going through the classifier. First, each paragraph
is encoded with LSTM. The hidden state at the end of each sentence is
extracted, and the resulting matrix is going through gated CNN [Dauphin et
al., 2017] for extraction of single encoding for each paragraph. The accuracy
is barely above $50\%$, which depicts that this method is not very promising.

###  3.2 Fine-tuning BERT

We have used a pre-trained BERT in two different ways. First, as a feature
extractor without fine-tuning, and second, by fine-tuning the weights during
training. The classification is completely based on the BERT paper, i.e., we
represent the first and second paragraph as a single packed sequence, with the
first paragraph using the A embedding and the second paragraph using the B
embedding. In the case of feature extraction, the network weights freeze and
CLS token are fed to the classifier. In the case of fine-tuning, we have used
different numbers for maximum sequence length to test the capability of BERT
in this task. First, just the last sentence of the first paragraph and the
beginning sentence of the second paragraph has been used for classification.
We wanted to know whether two sentences are enough for ordering classification
or not. After that, we increased the number of tokens and accuracy
respectively increases. We found this method very promising and the accuracy
significantly increases with respect to previous methods (Table 3). This
result reveals fine-tuning pre-trained BERT can approximately learn the order
of the paragraphs and arrow of the time in the stories.

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# Shifting of Fermi Level and Realization of Topological Insulating Phase in
the Oxyfluoride BaBiO2F

Bramhachari Khamari and B. R. K Nanda1 1 Condensed Matter Theory and
Computational Lab, Department of Physics, Indian Institute of Technology
Madras, Chennai 600036, India<EMAIL_ADDRESS>
###### Abstract

The disadvantage of BaBiO3 of not being a topological insulator despite having
symmetry protected Dirac state is overcome by shifting the Fermi level (EF)
via fluorination. The DFT calculations reveal that the fluorination neither
affects the spin-orbit coupling nor the parity of the states, but it acts as a
perfect electron donor to shift the EF. We find that 33 % fluorination is
sufficient to shift the EF by $\sim$ 2 eV so that the invariant Dirac state
lies on it to make BaBiO2F a topological insulator. The fluorinated cubic
compound can be experimentally synthesized as the phonon studies predict
dynamical stability above $\sim$ 500 K. Furthermore, the Dirac states are
found to be invariant against the low-temperature phase lattice distortion
which makes the structure monoclinic. The results carry practical significance
as they open up the possibility of converting the family of superconducting
oxides, ABiO3 (A = Na, K, Cs, Ba, Sr, Ca), to real topological insulator
through appropriate fluorination.

##  1 Introduction

Band topology, one of the highly active research area in materials science and
physical chemistry for the last one decade, has given rise to the novel exotic
concept of topological insulator (TI) [1, 2, 3, 4, 5, 6, 7, 8, 9]. The TI
compounds with an insulating interior and conducting exterior [1, 4], and
determined through Z2 invariance[10], are promising for next generation spin
based device applications such as spintronics and quantum information and
quantum computing [11, 12]. The spin-orbit coupling (SOC) driven complex band
structure of these compounds, is a great recipe to realize high thermoelectric
efficiency[13]. The conducting exterior of the TIs exhibit topologically
invariant Dirac cones (TIDC), where linearly dispersing valence band (VB) and
conduction band (CB) touch each other at the Fermi level (EF) and are
protected by time reversal symmetry.

While the widely investigated topological insulators, namely selenides,
tellurides[14, 15, 16] and Heusler[7, 17] alloys, have demonstrated
interesting physical concepts, the narrow band gap ($\sim$ 0.3eV) hinders
their application prospects[18, 19] and hence search for wide band gap TIs is
highly desirable. In addition, from the research point of view, wide-band gap
3D systems exhibiting high resistance are useful to experimentally reveal the
TI properties, as the surface and bulk states can be easily distinguished[20].
In this context, covalent bonding driven large band gap oxides should be of
great interest. However, due to negligible contribution towards SOC by the
oxygen states, many of the oxides fail to exhibit the TI behavior.

There are very few oxides, such as Bi based double perovskites A2BiXO6, where
A is a group-II element and X is either Br and I and single perovskites ABiO3
with A being a group-I and II element, B being a group-V and VI heavy element
which exhibit topologically protected surface Dirac states in their
equilibrium structure [21, 18, 22, 23]. Since the compounds Ba1-xKxBiO3 are
superconductors (TC $\sim$ 30K[24]), it is suggested that both the
superconducting behavior and the topologically invariant surface states[22]can
together be exploited to realize Majorona Fermion in these matrials[25]. The
perovskites also carry significance, as their layered geometry facilitates to
construct heterostructures so that the competition between the topological
phase and various orders can be examined to explore novel interfacial
states[26, 27]. However, the major disadvantage of ABiO3 is that the bulk band
gap ($\sim$ 0.7 eV) within which the surface TIDC appears, lies away from the
Fermi level (EF). The separation between TIDC and EF is in the range 2 to 5
eV[23]. Hence, for all practical purposes, these compounds are not TI, unless
the EF is shifted to the TIDC. The shifting of EF is usually achieved through
certain mechanism like thermal excitation, electrical gating or chemical
doping. While thermal excitation can hardly shift the EF by few meV, the
electrical gating can push it to a maximum of 1 eV[28]. Therefore, chemical
substitution remains the only viable option though it can distort the lattice
to lower the symmetry and thereby to affect the parity of the states which
determine Z2 invariance. In ABiO3, while O and A have fixed charge states, 2-,
1+/2+ respectively, Bi is multivalent and show ambiguous (flexible) charge
states[23]. Hence, the chemical doping is preferable either at the A site or
at the anion site so that the charge neutrality is maintained.

In this work, with the objective of realizing the TIDC at EF through the
substitution, we have examined the electronic structure of BaBiO3 as a
prototype. In order to achieve the electron doping, we replace O by F.
Fluorine is a preferable substituent as studies on other oxides suggest that
it maintains the thermodynamical stability while replacing the oxygen. For
example, F is substituted for O to achieve electron doping in the iron based
superconductor, LaO0.5F0.5BiS2 (x= 0.5)[29, 30], LaO1-xFxFeAs (x= 0.05 $-$
0.12)[31]. Similarly, experimental reports suggest the formation of
La1-xSrxFeO3-xFx (x=1, 0.8, 0.5, 0.2)[32]. Furthermore, it has been found that
perovskites BaFeO2F [33, 34] and SrFeO2F [35, 36] can be experimentally
synthesized. In these perovskites, one out of three Oxygens are replaced with
Fluorine.

Figure 1: (a) The cubic perovskite structure of BaBiO3. (b - c) Representative
structures of the [001] BaBiO2F slabs with F being placed at different face
centered positions. The real slabs, on which the calculations were carried
out, consist of 15 units of BaBiO2F with 15 Å thick vacuum separating two
consecutive slabs. (d) and (e) depict the bulk and surface Brillouin zones
(BZ) respectively. For these systems, the time reversal invariant momenta
(TRIM) are point R and $\Gamma$ which are mapped to $\overline{M}$ and
$\overline{\Gamma}$ respectively in the surface BZ.

The compound BaBiO3 undergoes a structural transition from cubic to
rhombohedral to monoclinic phase with decrease in the temperature[37] and the
earlier studies suggest that the TIDC states are protected against these
structural transitions[18]. Therefore, as the high-symmetry configurations are
computationally less expensive, in this work we deal with the high
temperature(750 K-800 K) cubic phase of BaBiO3[38, 39]. The crystal structure,
the slab geometry and their corresponding Brillouin zones (BZ) are shown in
Fig. 1.

To examine the band topology of the pure and fluorinated compounds, density
functional calculations are carried out using the full potential linearized
plane wave (FP-LAPW) formalism [40] as implemented in WIEN2k simulation
package[41]. Augmented plane waves in the interstitial and localised orbitals
within the muffin-tin sphere are used to construct the basis sets. To account
for exchange and correlation effect generalized gradient approximation (GGA)
[42] along with the modified Becke-Johnson (mBJ) correction[43] is considered.
For the the BZ integration, $10\times{10}\times{10}$ $k$-mesh, is used to
study the bulk electronic properties. Proportionate $k$-mesh is considered for
slab calculation. The largest vector in the plane wave expansion is obtained
by setting R$K_{max}$ = 7.0. For the fluorinated compound BaBiO2F, the GGA
optimized lattice parameter of 4.60 Å, which is 0.25 Å more than that of the
pristine BaBiO3, is used for the calculations.

The slabs, considered for the calculations, are of 15 units thick and grown
along [001]. A vacuum of 15 Å thick separates the consecutive slabs. Both top
and bottom surfaces are taken to be Ba terminated (i.e. either BaF or BaO) as
earlier studies have established the formation of TIDC with the Ba
termination[18]. The schematic representative of the slabs are shown in Fig.
1(b), (c). The surface band structure is calculated along the high-symmetry
path of the surface BZ which is basically a projection of the bulk BZ in the
$k_{x}$-$k_{y}$ plane (see Fig. 1(e)).

The band structure and hence the transport properties of the pervoskites ABX3
with X being a halogen or oxygen can be determined through the valence
electron count (VEC)[44], i.e. the number of valence electrons per formula
unit. The molecular orbital picture, drawn based on the DFT obtained band
structure (Fig. 2a), suggests that if the VEC is 20, the bulk compound is a
semimetal or semiconductor[44]. Depending on the strength of spin-orbit
coupling (SOC) and thereby the s-p band inversion, these
semimetals/semiconductors give rise to the formation of TIDC at the surface
boundary[23]. If the VEC differs from 20, the system is a metal and ceases to
exhibit the TI phase.

Figure 2: The band structure of BaBiO3 (a and b) and BaBiO2F (c and d) without
and with spin-orbit coupling. The Fermi level (EF) is set to zero. The red and
blue circles represent proportionate contribution of Bi-s and Bi-p states
respectively towards forming the bands. The partial DOS, plotted on the right
of each band structure further illustrates the orbital composition of the
bands. The molecular orbital picture[23] responsible for the formation of
these bands is shown in the inset. For the pure compound, while the s-p band
inversion occurs without SOC, the gap appearing at $\sim$ 2 eV above EF is SOC
driven. For the fluorinated compound, both band inversion and band gap opening
are SOC driven. Here, F provides the extra electron to shift the band gap to
EF.

The contrasting transport properties of ABX3, i.e. semimetal/semiconductor
leading to TI behavior for VEC = 20 and metallic otherwise, is solely governed
by a single salient feature of the band structure. An universal band gap (zero
or finite) exists at the TRIM R ($\pi/a$, $\pi/a$, $\pi/a$) between the
downward dispersive lower lying B-s dominated singlet state and upward
dispersive B-p dominated triplet states which can be noticed from Fig. 2a.
These dispersive bands emerge out of the antibonding bands formed by the
Bi-$\\{$s,p$\\}$\- O-p chemical bonding, as schematically demonstrated by the
molecular-orbital picture (inset of Fig. 2a). With VEC equals 20, the singlet
state is completely occupied to form the band gap at EF[44]. For BaBiO3, the
VEC is 19 and hence the the singlet state is partially occupied to produce the
(zero) band gap approximately 2 eV above EF (see Fig. 2a). The spin-orbit
coupling amplifies the band gap (see Fig. 2b) within which time reversal
symmetry protected Dirac states appear[23]. Fluorine with one additional
valence electron compared to oxygen acts as a perfect electron donor. For
BaBiO2F, the VEC is 20 and therefore, the band gap appears at EF (Fig. 2d).
Furthermore, the F-p states dominate the valence band spectrum in the range -6
to -10 eV w.r.t. EF. We may note that the family of ABiO3 exhibits multiple
Dirac states[23], one in the valence band spectrum (around 7 eV below EF),
formed by the bonding Bi-$\\{$s,p$\\}$\- O-p states, and other in the
conduction band spectrum (around 2eV above EF) formed by the antibonding
states. With substitution of fluorine, the F-p states dominate the bonding
states (see Fig. 2c) instead of the Bi-p states. Therefore, the s-p band
inversion does not occur (see Fig. 2d) and hence the TIDC is not formed in the
valence band. On the other hand, the antibonding spectrum is unperturbed by
the F-p states and hence neither the s-p band inversion nor the SOC driven
band gap is affected by the fluorination as can be observed from Fig. 2d.
Hence, it is expected that the TIDC will form in the antibonding spectrum.

Figure 3: Comparison of the band structure obtained from DFT (red) and from
the TB model using MLWF basis(black).

To verify the formation of TIDC for the pristine and fluorinated compounds,
the surface states are calculated using Wannier formalism. First, the
maximally localized Wannier functions (MLWF) as well as the strength of the
hopping integral between these functions, are obtained from the bulk DFT
calculations using Wannier90[45]. Taking MLWF as the elements of the basis, a
tight-binding (TB) model was employed on a slab structure to calculate the
surface Green’s function through an iterative method[46, 47]as implemented in
WannierTools package[48]. The appropriateness of the basis can be confirmed
from the fact that the TB band structure has an excellent agreement with the
full band structure obtained with DFT as can observed from Fig. 3. The
imaginary part of the surface Green’s function is the local density of state
LDOS($k$, E).

Figure 4: The DFT+SOC simulated ARPES spectra (LDOS ($\kappa$, E)) of (001)
surface for both BaBiO3 and BaBiO2F. EF is set to zero. The surface states are
calculated using Wannier formalism for BaBiO3 (a, b), BaBiO2F with BaF
termination (c, d) and BaBiO2F with BaO termination (e, f). The box highlights
the formation of TIDC. With doping the EF shifts to the TIDC irrespective of
the surface termination.

For the pristine and oxyfluoride [001] slabs (Fig. 1), the LDOS($k$, E) are
plotted in Fig. 4. The colour gradient is a measure of LDOS. Deep red and
purple correspond to highest and lowest LDOS respectively. The faded green
color reflects the bulk band structure. As expected for the pristine system,
the TIDC appears at $\sim$ 2 eV above EF and $\sim$ -7 eV below EF (see Fig.
4(a) and (b)). With fluorination, the TIDC appears at EF irrespective of
whether F replaces O from the BaO plane (Fig. 4c) or from the BiO2 plane (Fig.
4e) with respect to the growth direction (see Fig. 1-b and c). However, unlike
the pristine compound, the TIDC does not form in the valence band spectrum for
the fluorinated systems as can be seen from the Fig. 4(d) and Fig. 4(f). It is
due to the absence of s-p band inversion as discussed in the bulk band
structure. For further confirmation, we calculated the full band structure of
a 15 units cell thick BaBiO2F slab using DFT, which is shown in Fig. 5, and
observed that the TIDC, which is marked by the red circle, is formed very
close to EF.

Figure 5: The DFT + SOC obtained band structure of 15 units cell thick BaBiO2F
slab with BaF termination (a, c) and BaO termination (b, d). The TIDC,
indicated by red circle, is observed near EF for both the terminations. Figure
6: Temperature dependent phonon spectra of BaBiO2F. The thermodynamical
stability of the high temperature cubic phase is ascertained as the negative
frequencies are absent for T $>$ 500 K.

While the electronic structure of BaBiO2F satisfies the necessary and
sufficient criteria to form the TIDC near EF, the structural stability of such
a system needs to be examined for its practical realization. The pristine
BaBiO3 exists in the cubic phase when the temperature is beyond $\sim$ 750 K.
[39] Keeping this in mind, we have examined the structural stability of the
doped system in the temperature range 500 to 700K using the following two step
process. In the first step, ab-initio molecular dynamics (AIMD) simulations
were performed on a $2\times 2\times 2$ supercell using VASP package[49, 50].
The PBE-GGA exchange-correlation functional was used and the plane wave cutoff
energy was set to 500 eV. The system was equilibriated at different
temperatures (NVT) using velocity scaling method for ∼1.0 ps with a time step
of 0.5 fs followed by ∼0.5 ps -long production run for collecting the atomic
displacement and force datasets. In the second step ALAMODE[51] package was
used to get the harmonic and interatomic force constants (IFCs). The phonon
band dispersion were obtained by solving the dynamical matrix for the given
wave vectors. The phonon dispersion curves for different temperatures are
shown Fig. 6. The soft modes (negative frequencies) are observed when the
temperature decreases below 500 K. Therefore, the cubic phase of these
compounds exhibit dynamical stability for temperature greater than 500 K. By
lowering the temperature, the soft modes introduce structural distortion.

The stability of the oxyfluoride is further quantified by calculating the
formation and cohesive energies. They are estimated using the following
expressions.

$\displaystyle E_{Form}$ $\displaystyle=$ $\displaystyle
E_{BaBiO_{3-x}F_{x}}-E_{BaBiO_{3}}-x\frac{1}{2}E_{F_{2}}+x\frac{1}{2}E_{O_{2}},$
(1) $\displaystyle E_{Coh}$ $\displaystyle=$ $\displaystyle
E_{BaBiO_{3-x}F_{x}}-E_{Ba}-E_{Bi}-(3-x)\frac{1}{2}E_{O_{2}}-x\frac{1}{2}E_{F_{2}},$
(2)

Where $E_{BaBiO_{3-x}F_{x}}$, $E_{BaBiO_{3}}$ are total energy in their ground
state. $E_{F_{2}}$ and $E_{O_{2}}$ are estimated by keeping respective
molecules in a very large cubic box (size = 15 Å). While the $E_{Form}$ is
found to be $-$ 0.69 eV, the $E_{Coh}$ is found to be -2.45 eV/atom. The
negative values indicates that the oxyfluoride can be synthesized
independently or via substitution of Fluorine in BaBiO3.

Figure 7: (a) The monoclinic (C2/m) crystal structure of BaBiO3. (b) Schematic
illustration of breathing and stretching mode of distortions, and tilting of
octahedra observed in the monoclinic structure. (c) The Brillouin zone for the
monoclinic lattice. (d) The bulk band structure and (e) the simulated (001)
surface ARPES spectra for BaBiO3. (f) and (g) are same as (d) and (e) but for
BaBiO2F. The proportionate contribution of Bi-s and Bi-p states in each band
are represented by red and blue respectively. The s-p band inversion is
highlighted by circles and the formation of TI states are highlighted by
rectangles. The Fermi level is set to zero.

Experimentally a series of structural transitions are observed in BaBiO3.[52]
The high temperature cubic phase yields to rhombohedral phase at 750K and at
405K the monoclinic structure (C2/m) forms the ground state . When the
temperature goes below 104K, the compound stabilizes with the primitive
monoclinic structure with (P21/c).

The room temperature monoclinic phase (see Fig. 7(a)) is obtained from the
simultaneous occurring of the breathing and tilting distortions of the
octahedra. Interestingly, the earlier reported band structure calculations
suggests that the TI Dirac state is protected against such distortions[18].
The present work reconfirms the protection of the TI states as the bulk band
structure for the monoclinic structure demonstrates s-p band inversion (see
Fig. 7(d)) and the surface ARPES spectra demonstrates the formation of TI
Dirac states (Fig. 7(e)). Anticipating a similar structural transition as in
the pure compound, it is prudent to examine robustness of the TI state in the
fluorinated system in the monoclinic (C2/m) phase.

The bulk band structure of monoclinic BaBiO2F is shown in Fig. 7(f). Like the
pure compound (see Fig. 7(d)), the s-p band inversion is observed at $\Gamma$,
but now at the Fermi level. As mentioned earlier, it is due to the fact that
the participatory bands are shifted below by approximately 2.5 eV due to extra
electron through F substitution. The ARPES spectra in Fig. 7(g) shows that the
s-p band inversion leads to the formation of surface TI Dirac state in the
monoclinic structure.

To summarize, our electronic structure calculations show that fluorine acts as
a perfect electron donor to shift the Fermi level of the oxide perovskite
BaBiO3 without affecting its symmetry protected surface Dirac states. For
33$\%$ fluorine substitution, the Dirac states lie exactly on the Fermi level
to make the doped perovskite as a topological insulator. The thermodynamical
stability of the fluorinated system beyond 500 K, as ascertained from phonon
studies, suggests that the experimental realization of BaBiO3 as topological
insulator is possible through doping. Universal nature of the band structure
of cubic oxide perovskites implies that the finding of this work can be easily
extended to other members of ABiO3 where A is a divalent group-II element.
Furthermore, we reveal that the TI state remain invariant with respect to the
structural distortion that leads to the low temperature phase monoclinic
structure.

##  2 Acknowledgement

The authors acknowledge HPCE, IIT Madras for providing the computational
resources. This work is supported by DST, India through grant no.
EMR/2016/003791. B. K. acknowledges K. Murali for computational help.

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