# Chainlet Orbits: Topological Address Embedding for the Bitcoin Blockchain

Poupak Azad
Department of Computer Science
University of Manitoba
Winnipeg, MB
<EMAIL_ADDRESS>
&Baris Coskunuzer
Department of Mathematical Sciences
University of Texas at Dallas
Richardson, TX, USA
<EMAIL_ADDRESS>
Murat Kantarcioglu
Department of Computer Science
University of Texas at Dallas
Richardson, TX, USA
<EMAIL_ADDRESS>
&Cuneyt Gurcan Akcora
Departments of Computer Science and Statistics
University of Manitoba
Winnipeg, MB, Canada
cuneyt.akcora@umanitoba

###### Abstract

The rise of cryptocurrencies like Bitcoin, which enable transactions with a
degree of pseudonymity, has led to a surge in various illicit activities,
including ransomware payments and transactions on darknet markets. These
illegal activities often utilize Bitcoin as the preferred payment method.
However, current tools for detecting illicit behavior either rely on a few
heuristics and laborious data collection processes or employ computationally
inefficient graph neural network (GNN) models that are challenging to
interpret.

To overcome the computational and interpretability limitations of existing
techniques, we introduce an effective solution called Chainlet Orbits. This
approach embeds Bitcoin addresses by leveraging their topological
characteristics in transactions. By employing our innovative address
embedding, we investigate e-crime in Bitcoin networks by focusing on
distinctive substructures that arise from illicit behavior.

The results of our node classification experiments demonstrate superior
performance compared to state-of-the-art methods, including both topological
and GNN-based approaches. Moreover, our approach enables the use of
interpretable and explainable machine learning models in as little as 15
minutes for most days on the Bitcoin transaction network.

##  1 Introduction

The success of Bitcoin [24] has also encouraged significant usage of
cryptocurrencies for illegal activities ranging from ransomware payments [2]
to darknet market transactions [13]. For instance, the CryptoLocker ransomware
spreads through spam emails, and once installed, it establishes communication
with a command-and-control center, encrypts the system resources, and requests
Bitcoin payments to unlock the encrypted resources.

Since the Bitcoin transaction network can be represented as a heterogeneous
directed graph, we can apply graph machine learning techniques to detect
illicit transactions in node or edge classification tasks [36, 18]. Among the
techniques used, graph neural networks (GNNs) (e.g., GCN [14], DGCNN [35], GIN
[34] and GraphSage [11]) allow powerful node embeddings by first summarizing
the substructure information of node neighborhoods for each node as vectors,
and then combining them to get a representation of the whole graph. While GNNs
are quite powerful methods, for very large graphs, they fail to embed the
crucial domain information in a computationally efficient manner. Furthermore,
in many cases, the results are not interpretable and explainable due to
complex aggregations and embeddings done by the GNNs.

Alternatively, orbit analysis can be employed to examine the graph geometry
surrounding a node, leading to the creation of interpretable behavioral
patterns [19]. In the realm of graph theory and network analysis, an orbit
denotes a unique configuration or arrangement of nodes and edges within a
graph that recurs throughout its structure. This notion captures the symmetry
and repetitive patterns present in the graph. Blockchain transaction networks
lend themselves well to orbit analysis due to their heterogeneous nature [4],
which organizes nodes into clearly defined substructures, namely transactions.

We define a notion of orbits to determine the common structural patterns
between nodes in Bitcoin transaction graphs (e.g., see Figure 1). Using orbit-
based node embedding, we study e-crime in Bitcoin networks by focusing on
special substructures induced by the illicit behavior and show that we can
efficiently match the existing GNN detection accuracy while allowing
interpretable and explainable machine learning models. To the best of our
knowledge, this is the first work to provide search and query capabilities
over UTXO graphs. Our framework is well-suited for human-in-the-loop
scenarios, particularly when there is a need for manual analysis of numerous
Bitcoin addresses due to compliance requirements.

Figure 1: By analyzing transaction patterns within the directed Bitcoin
network, we identify specific address (circular nodes) behaviors in
transactions (rectangular nodes) and assign them structural roles which we
refer to as orbits. In this example, a red (e-crime) address has been
identified and assigned orbits 9, 13, and 14, which are commonly used by
addresses associated with ransomware.

Our key contributions can be summarized as follows:

  * •

Efficient Orbit Discovery: We present a topological-equivalence-based node
embedding called "orbit" that can effectively handle very large directed
graphs, and give the mathematical foundations of the theory by applying group
actions on graphs.

  * •

E-Crime Address Discovery: We distinguish orbits that capture the transaction
behaviors of e-crime participants, enabling an analyst to focus on the actions
of specific groups of addresses such as those controlled by ransomware
operators.

  * •

Extensive Empirical Evaluation: We conduct node classification experiments on
the entire Bitcoin transaction network, and demonstrate that orbit-based
models outperform existing topological, and graph neural network-based
methods. 111Our implementation is available at
https://github.com/chainletRepo/chainlet.

  * •

Interpretable and Explainable Models in Graph ML: We use orbit patterns to
profile typical behavior and enable searches and similarity queries on all
blockchains similar to the Bitcoin transaction graph.

In section 2, we discuss the necessary background related to our research. In
section 3, we define the chainlet and orbit notations that are used for the
efficient embedding of the node information on Bitcoin graphs. In section 4,
we report extensive experimental evaluation results on real-life datasets.
Finally, in section 5, we conclude by discussing the implications of our
results and future work.

##  2 Background and Preliminaries

In this section, we introduce preliminaries on Blockchain, Bitcoin, and recent
Blockchain research. Afterward, we introduce graph machine learning on
blockchain networks and discuss node classification methods.

###  2.1 E-crime on Blockchain

The success of Bitcoin [24] has also encouraged significant usage of
cryptocurrencies for illegal activities. Bitcoin provides pseudo-anonymity;
although all transactions are public by nature, user identification is not
required to join the network.

The earliest results aimed at tracking the transaction network to locate
bitcoins used in illegal activities, such as money laundering and blackmailing
(e.g., [6]), by using heuristics. At the same time, mixing schemes (e.g., [21,
30]) have been developed to hide the flow of coins in the network. Earlier
research results have shown that some bitcoins can be traced [22] in the
network. As a result, obfuscation efforts [26] by malicious users have become
increasingly sophisticated. This in return increased the need for illegal
activity detection techniques that go beyond simple heuristics.

Ransomware. Ransomware is a type of malware that infects a victim’s computer
systems, IoT devices, and mobile devices and demands payment in exchange for
releasing control over the infected resources [20]. Ransomware can either lock
access to resources or encrypt their contents. It can be delivered through
various means, such as email attachments or web-based vulnerabilities, and
more recently through mass exploit campaigns. For instance, the CryptoLocker
ransomware used the Gameover ZeuS botnet to spread through spam emails. Once
the ransomware is installed, it establishes communication with a command-and-
control center. While earlier variants of ransomware used fixed IP addresses
or domain names, newer versions may utilize anonymity networks, such as TOR,
to connect to a hidden command-and-control server.

Montreal [27], Princeton [12] and Padua [9] studies have analyzed ransomware
networks on cryptocurrencies and found that hacker behavior can help us
identify undisclosed ransomware payments. The datasets of these three studies
are publicly available.

Early studies in ransomware detection have used decision rules on amounts and
times of known ransomware transactions to locate undisclosed ransomware
(CryptoLocker) payments [17]. More recent studies are joint efforts between
researchers and Blockchain analytics companies. For example, Huang et al. [12]
identify shared hacker behavior and use heuristics to identify ransomware
payments. The authors estimate that 20,000 victims have made ransomware
payments. Nevertheless, these studies utilize proprietary features for
identifying ransomware payments and families, and some studies do not disclose
their feature definitions [32].

Darknet Markets. Darknet markets are online platforms that can only be
accessed through the use of privacy-enhancing tools such as TOR and I2P. These
markets enable sellers to create online stores and connect with buyers. The
merchandise offered on these markets ranges from illicit goods such as drugs
and weapons to legal items such as fake passports. Darknet markets employ an
escrow service, where buyers are provided with a one-time use address to make
payments for goods. After payment, the market instructs the seller to
physically ship the item to the buyer. Once the item is delivered, the market
releases the coins to the seller’s account, who can then cash out the coins.
Our analysis in Section 4 corroborates the delay results; we found that most
market payments remain in an address for some days.

Cryptocurrencies are ideal for darknet markets as payments can be made pseudo-
anonymously and can be received anywhere in the world. Starting with the Silk
Road in 2011, darknet markets have been processing an ever-growing amount of
illicit goods. Law enforcement agencies have seized multiple darknet markets,
but the ease of setting up new markets has turned this into a cat-and-mouse
game. Typically, Bitcoin and Monero are used for Darknet payments. In multiple
cases, the seized markets’ data have been made public, where each sale is
recorded with information such as the buyer, seller, good, amount, and time.
The Darknet Market Archives, which we use in our experiments, is a collection
of such released sales [7].

###  2.2 Blockchain Transaction Network

A blockchain is an immutable public ledger that records transactions in
discrete data structures called blocks [24]. The earliest blockchains are
cryptocurrencies such as Bitcoin and Litecoin where a transaction is a
transfer of coins.

A UTXO (Unspent Transaction Output) network, such as Bitcoin, is a directed,
weighted multigraph where an edge denotes coin transfers.

We model the Bitcoin UTXO network with a weighted, directed heterogeneous
graph with address and transaction nodes. We create the graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{B})$ from a set of transactions
$TX$ and input and output addresses in $TX$. On $\mathcal{G}$, $\mathcal{V}$
is a set of nodes, and $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ is a
set of edges.
$\mathcal{B}=\left\\{\textbf{Address},\textbf{Transaction}\right\\}$
represents the set of node types. A node $u\in\mathcal{V}$ has a node type
$\phi(u)\in B$.

For each edge $e_{uv}\in\mathcal{E}$ between adjacent nodes $u$ and $v$, we
have $\phi(u)\neq\phi(v)$. In other words, there is no edge between the same
node types (Transaction$\to$Transaction or Address$\to$Address), and an edge
$e\in\mathcal{E}$ represents a coin transfer between an address node and a
transaction node. This heterogeneous graph model subsumes the homogeneous case
(i.e., $\left|B\right|=1$), where only transaction or address nodes are used,
and edges link nodes of the same type. Here, we focus on the case where each
address node is linked (i.e., input or output address of a transaction) via a
transaction node to another address node. We use $\Gamma_{a}^{i}$ and
$\Gamma_{a}^{o}$ to refer to predecessors (in-neighbors) and successors (out-
neighbors) of an address $a$, respectively. On the heterogeneous Bitcoin
network, the in-neighbors $\Gamma_{n}^{i}$ of a transaction $tx_{n}$ is
defined as the set of transactions (not addresses) whose one or more outputs
are input to transaction $tx_{n}$. The out-neighbors of $tx_{n}$ are denoted
as $\Gamma_{n}^{o}$.

A transaction has inputs and outputs; the sum of output amounts of a
transaction $tx_{n}$ is defined as
$\mathcal{A}^{o}(n)=\sum\limits_{a_{u}\in\Gamma_{n}^{o}}{{A}_{u}^{o}(n)}$,
where ${A}_{u}^{o}(n)$ is the amount of coins the output address $a_{u}$
receives from transaction $tx_{n}$. $\mathcal{A}^{i}(n)$ is defined similarly,
and $\mathcal{A}^{i}(n)=\mathcal{A}^{o}(n)$, i.e., the total input amount is
equal to the total output amount for any transaction $tx_{n}$.

On the Bitcoin network, an address may appear multiple times with different
inputs and outputs. An address $u$ that appears in a transaction at time $t$
can be denoted as $a_{u}^{t}$. To mine address behavior in time, we divide the
Bitcoin network into 24-hour long windows by using the UTC-6 timezone as a
reference. This window approach serves two purposes. First, the induced
24-hour network allows us to capture how fast a coin moves in the network. The
speed is measured by the number of blocks in the 24-hour window that contains
a transaction involving the coin. Second, temporal information of
transactions, such as the local time, has been found useful to cluster
criminal transactions (see Figure 7 in [12]).

###  2.3 Blockchain Graph Analysis

Early efforts in Blockchain graph analysis used heuristics based on
transaction behavior to de-anonymize and associate Bitcoin addresses [22]. Two
heuristics by Meiklejohn et al. [22] are widely used in Bitcoin address
clustering: Co-spending and Transition. Co-spending posits that "if two
addresses are inputs to the same transaction, they are controlled by the same
user". Transition posits that "if we observe one transaction with addresses A
and B as inputs, and another with addresses B and C as inputs, then we
conclude that A, B, and C all belong to the same user". We did not implement
the third heuristic named Change used in some settings, because the results of
the change heuristics are not definitive; addresses discovered using this
heuristic are not necessarily controlled by the same user.

As the number of e-crime transactions increased on cryptocurrencies,
algorithmic models beyond simple heuristics have been developed to link
bitcoin addresses by using node features on the network [12].

Mark et al. [32] introduced the Elliptic dataset, which is a temporal graph
dataset consisting of over 200,000 Bitcoin node transactions, 234,000 payment
edges, and 49 transaction graphs with unique time steps. Each transaction was
categorized as "licit," "illicit," or "unknown." They employed machine
learning techniques including Logistic Regression (LR), Random Forest (RF),
Graph Convolutional Networks (GCNs) [14], and EvolveGCN [28] for transaction
labeling. The evaluation showed that RF achieved a recall score of 0.67 in the
illicit category, while GCNs achieved a score of 0.51. Elliptic dataset only
contains anonymized transactions and omits address nodes, which we need to
extract Orbits. As a result, we could not apply our Orbit approach to the
dataset. Furthermore, in our experiments, we show that using a neighborhood
approach yields better results as measured by AUC.

Vassallo et al. [31] detected illicit cryptocurrency activities such as scams,
terrorism financing, and Ponzi schemes, and used the Elliptic dataset to
evaluate its effectiveness at both the address and transaction classification.

Feature extraction on the Bitcoin transaction network has many examples.
Chaehyeon et al. [16] employed supervised machine learning algorithms to
classify e-crime addresses in the Bitcoin network. The authors extracted
classification features based on the characteristics of Bitcoin transactions,
such as transaction fees and size. Monamo et al. [23] used nearest-neighbor
methods to detect fraud in the Bitcoin network, using graph centrality
measures and currency features, such as the total amount sent. Our method
avoids domain-specific feature engineering, and creates structural topological
node embeddings that can be applied to any UTXO-based blockchain (See the
appendix for applicability to other blockchains).

Graph neural networks achieve better results and do not require feature
engineering. AlArab et al. [5] proposed a model that combines GCNs and linear
layers to classify illicit nodes in the Elliptic dataset [32]. The results
showed that the model had an overall classification accuracy of 97.40% and a
recall of 0.67 for detecting illicit transactions. In [25], the authors used
an autoencoder with graph embedding to detect mixing and demixing services for
Bitcoin cryptocurrency, and the proposed model effectively performed the
detection of these anomalies. In [29], the authors used unsupervised learning
to identify suspicious nodes in the Bitcoin transaction graph. Unlike these
approaches, our method can operate on larger networks and provide
interpretable and explainable results.

##  3 Chainlets and Orbits

In this section, we first outline UTXO chainlets and give mathematical
background, then define our topological address embeddings, i.e., orbits. We
start with giving our intuition and motivation.

Intuition behind Orbits: UTXO transaction networks are composed of transaction
and address nodes, as shown in Figure 2. The position of address nodes within
a subgraph (i.e., chainlet) determines their structural role, referred to as
an "orbit," which can be leveraged in supervised learning tasks. Our primary
objective is to establish these roles and utilize them in graph machine
learning applications.

Chainlet. Chainlets have been initially introduced in the setting of temporal
predictions to define position invariant versions of occurrence and amount
information of Bitcoin temporal transaction patterns (see overviews by [3,
1]). The key idea of the chainlet approach is to offer a lossless summary of
all transactions by assigning two indices $i,o$ in the $[0,n]$ range to each
transaction with respect to its input and output addresses. A Bitcoin subgraph
$\mathcal{G}^{\prime}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime},\mathcal{B})$
is a subgraph of $\mathcal{G}$, if $\mathcal{V}^{\prime}\subseteq\mathcal{V}$
and $\mathcal{E}^{\prime}\subseteq\mathcal{E}$. If
$\mathcal{G^{\prime}}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime},\mathcal{B})$
is a subgraph of $\mathcal{G}$ and $\mathcal{E}^{\prime}$ contains all edges
$e_{uv}\in\mathcal{E}$ such that $(u,v)\in V^{\prime}$, then
$\mathcal{G}^{\prime}$ is called an induced subgraph of $\mathcal{G}$. Two
graphs
$\mathcal{G^{\prime}}=(\mathcal{V}^{\prime},\mathcal{E}^{\prime},\mathcal{B})$
and
$\mathcal{G^{{}^{\prime\prime}}}=(\mathcal{V}^{{}^{\prime\prime}},\mathcal{E}^{{}^{\prime\prime}},\mathcal{B})$
are called isomorphic if there exists a bijection
$h:\mathcal{V}^{\prime}\to\mathcal{V}^{{}^{\prime\prime}}$ such that all node
pairs $u,v\in\mathcal{V}^{\prime}$ are adjacent in $\mathcal{G}^{\prime}$ if
and only if $h(u)$ and $h(v)$ are adjacent in
$\mathcal{G}^{{}^{\prime\prime}}$.

######  Definition 1 ($k$-chainlet).

Let $k$-chainlet $\mathcal{C}_{k}$ in $\mathcal{G}$ be the subgraph of
$\mathcal{G}$ containing $k$ subsequent transaction nodes, all output nodes of
these $k$ transaction nodes, and all input nodes of the first transaction (See
Figure 2).

A 1-chainlet lists a transaction node with its input and output addresses.
Figure 2 shows a 2-chainlet; transactions $t_{1}$ and $t_{2}$ share the common
address $a_{1}$. A $k$-chainlet where $k>1$ has a natural ordering of
transactions because transactions are ordered in blocks which, in turn, are
ordered in a blockchain. This natural order allows us to assign roles to an
address depending on where the address appears in a 2-chainlet. In the next
section, we will count address roles and list them as address orbits.

Figure 2: A 2-chainlet of five addresses. Each of the two transactions has one
input address, but the $t_{2}$ has three output addresses.

### Orbits

Before we define chainlet orbits, we need to recall a few basic notions from
graph theory. A mapping $\varphi:\mathcal{G}\to\mathcal{H}$ is called a graph
isomorphism if $\varphi:\mathcal{V}_{\mathcal{G}}\to\mathcal{V}_{\mathcal{H}}$
is a bijection such that $u,v\in\mathcal{V}_{\mathcal{G}}$ are adjacent in
$\mathcal{G}$ if and only if $\varphi(u)$ and $\varphi(v)$ are adjacent in
$\mathcal{H}$. If there exists an isomorphism from $\mathcal{C}_{k}$ to
another chainlet $\mathcal{C}_{k}^{\prime}$ of $\mathcal{G}$, we call
$\mathcal{C}_{k}^{\prime}$ an occurrence of $\mathcal{C}_{k}$ in
$\mathcal{G}$. We define the chainlet class of $\mathcal{C}_{k}$ in
$\mathcal{G}$ as the set of all occurrences of $\mathcal{C}_{k}$ as follows:

######  Definition 2 (Chainlet Classes).

Let $\mathcal{A}_{k}(\mathcal{G})$ be the set of all $k$-chainlets in
$\mathcal{G}$. For a given $k$-chainlet
$\mathcal{C}_{k}\in\mathcal{A}_{k}(\mathcal{G})$, the equivalence class of
$\mathcal{C}_{k}$, $\mathcal{E}(\mathcal{C}_{k})$, is the set of all
isomorphic copies (occurrences) of $\mathcal{C}_{k}$ in $\mathcal{A}_{k}$.

While in the previous definition, we define equivalence classes of chainlets
in $\mathcal{G}$, in the next definition we define an equivalence class of
address nodes for a fixed chainlet class.

######  Definition 3 (Orbit of an Address Node).

Let $\mathcal{C}_{k}$ be a $k$-chainlet in $\mathcal{G}$, and $u$ be an
address node in $\mathcal{C}_{k}$. We define the orbit of $u$ with respect to
$\mathcal{C}_{k}$ as the collection of the images $\varphi(u)$ where $\varphi$
is an isomorphism from $\mathcal{C}_{k}$ to another $k$-chainlet
$\mathcal{C}^{\prime}_{k}$ in $\mathcal{G}$, i.e. $O_{u}=\\{\varphi(u)\\}$. We
call $u\in\mathcal{C}_{k}$ and $v=\varphi(u)\in\mathcal{C}^{\prime}_{k}$
similar nodes wrt $\mathcal{C}_{k}$, i.e., $u\sim v\ (\mbox{mod }C_{k})$.

Notice that the similarity of two nodes depends on the fixed chainlet class
$\mathcal{E}(\mathcal{C}_{k})$. In other words, a node can belong to different
chainlets $\mathcal{C}_{k}$ and $\mathcal{C}_{k}^{\prime}$ because of their
relative locations to the transaction nodes. Hence, the orbit of $u$ with
respect to $\mathcal{C}_{k}$ versus with respect to $\mathcal{C}_{k}^{\prime}$
can be completely different. In the remaining text, if the fixed chainlet type
$\mathcal{C}_{k}$ is understood, we will refer to it as simply orbit of u,
$O_{u}$. Note also that if $u\sim v$, then the corresponding chainlets
containing $u$ and $v$ must be in the same equivalence class of chainlets. In
other words, any orbit $O_{u}$ belongs to only one chainlet class
$\mathcal{E}(\mathcal{C}_{k})$. Furthermore, by definition, if two nodes are
similar, then they have the same orbit and vice versa, i.e. $u\sim v\iff
O_{u}=O_{v}$.

The following theorem gives the relationship between the size of the orbit
$O_{u}$ of a node $u\in\mathcal{C}_{k}$, and the size of the equivalence class
of the chainlet $\mathcal{E}(\mathcal{C}_{k})$. Let
$\mathcal{I}(\mathcal{C}_{k})$ be the set of all isomorphisms between any two
$k$-chainlets in $\mathcal{E}(\mathcal{C}_{k})$. $\Lambda(\mathcal{C}_{k})$ be
the group of all automorphisms (self-isomorphisms) of $\mathcal{C}_{k}$. Then,
for any node $u$ in $\mathcal{C}_{k}$, we define the stabilizer subgroup of
$u$ in $\Lambda(\mathcal{C}_{k})$ as
$\mathcal{S}(u)=\\{\varphi\in\Lambda(\mathcal{C}_{k})|\varphi(u)=u\\}$. Let
$|K|$ represent the cardinality of the set $K$. Now, we are ready to state our
theorem.

######  Theorem 1.

Let $\mathcal{C}_{k}$ be a $k$-chainlet in $\mathcal{G}$, and $u$ be a node in
$\mathcal{C}_{k}$. Let $I(\mathcal{C}_{k}),\Lambda(\mathcal{C}_{k}),S(u)$ be
as defined above. Then, we have

$|\mathcal{I}(\mathcal{C}_{k})|=|\mathcal{E}(\mathcal{C}_{k})|^{2}\cdot|\Lambda(\mathcal{C}_{k})|\quad\mbox{and}\quad|O_{u}|=|\mathcal{E}(C_{k})|\cdot\frac{|\Lambda(\mathcal{C}_{k})|}{|\mathcal{S}(u)|}$

###### Proof.

We start with proving the first equation
$|\mathcal{I}(\mathcal{C}_{k})|=|\mathcal{E}(\mathcal{C}_{k})|^{2}\cdot|\Lambda(\mathcal{C}_{k})|$.
Let $|\mathcal{E}(\mathcal{C}_{k})|=m$ and
$\mathcal{E}(\mathcal{C}_{k})=\\{\mathcal{C}_{k}^{1},\mathcal{C}_{k}^{2},\dots,\mathcal{C}^{m}_{k}\\}$
where $\mathcal{C}_{k}=\mathcal{C}_{k}^{1}$. For each $\mathcal{C}_{k}^{i}$,
fix $\varphi_{i}\in\mathcal{I}(\mathcal{C}_{k})$ such that
$\varphi_{i}:\mathcal{C}_{k}^{1}\to\mathcal{C}_{k}^{i}$. Let
$\psi\in\mathcal{I}(\mathcal{C}_{k})$ be any isomorphism, i.e.,
$\psi:\mathcal{C}_{k}^{j_{1}}\to\mathcal{C}_{k}^{j_{2}}$. Then,
$\widehat{\psi}=\varphi^{-1}_{j_{2}}\circ\psi\circ\varphi_{j_{1}}$ is an
automorphism of $\mathcal{C}_{k}$, i.e.
$\widehat{\psi}:\mathcal{C}_{k}\to\mathcal{C}_{k}$. This implies
$\widehat{\psi}\in\Lambda(\mathcal{C}_{k})$. Hence, by using fixed
isomorphisms $\\{\varphi_{1},\dots,\varphi_{m}\\}$, we find a unique
automorphism $\widehat{\psi}\in\Lambda(\mathcal{C}_{k})$ for any isomorphism
$\psi\in\mathcal{I}(\mathcal{C}_{k})$. Notice that this process is reversible,
i.e., for any $1\leq i\leq j\leq m$, and for any
$\psi\in\Lambda(\mathcal{C}_{k})$, we obtain a unique isomorphism
$\widetilde{\psi}_{ij}=\varphi_{j}\circ\psi\circ\varphi_{i}^{-1}:\mathcal{C}^{i}_{k}\to\mathcal{C}_{k}^{j}$.
This implies
$|\mathcal{I}(\mathcal{C}_{k})|=|\mathcal{E}(\mathcal{C}_{k})|^{2}\cdot|\Lambda(\mathcal{C}_{k})|$.
The first equation follows.

To prove the second equation
$|O_{u}|=|\mathcal{E}(C_{k})|\cdot\frac{|\Lambda(\mathcal{C}_{k})|}{|\mathcal{S}(u)|}$,
we first show that $\frac{|\Lambda(\mathcal{C}_{k})|}{|\mathcal{S}(u)|}$ is
equal to the number of similar nodes of $u$ in $\mathcal{C}_{k}$, i.e.
$\frac{|\Lambda(\mathcal{C}_{k})|}{|\mathcal{S}(u)|}=|O_{u}^{1}|=|O_{u}\cap\mathcal{V}_{\mathcal{C}^{1}_{k}}|$
where $O^{i}_{u}$ represent the similar nodes to $u$ belonging to
$\mathcal{C}^{i}_{k}$. Notice that $\Lambda(\mathcal{C}_{k})$ is a group
acting on the set $\mathcal{C}_{k}$. Hence, the size of the orbit of $u$ in
$\mathcal{C}^{1}_{k}$ is equal to the cardinality of the quotient of
$\Lambda(\mathcal{C}_{k})$ by the stabilizer subgroup $\mathcal{S}(u)$ by [15,
Proposition 5.1]. In other words, $O_{u}^{1}$ has one to one correspondence
with the quotient group $\Lambda(\mathcal{C}_{k})/\mathcal{S}(u)$ (# of
cosets). Hence, if $O_{u}^{1}=\\{u_{1},u_{2},\dots,u_{r}\\}$, then
$\Lambda(\mathcal{C}_{k})/\mathcal{S}(u)=\\{\mathcal{S}(u)u_{1},\dots,\mathcal{S}(u)u_{r}\\}$,
i.e., $|O_{u}^{1}|=\frac{|\Lambda(\mathcal{C}_{k})|}{|\mathcal{S}(u)|}$

Now, take any node $v\in O_{u}$. Assume $v\in\mathcal{C}_{k}^{i}$ where
$\mathcal{C}_{k}^{i}\in\mathcal{E}(\mathcal{C}_{k})$. By definition of the
whole orbit $O_{u}$, there exists an isomorphism
$\psi:\mathcal{C}_{k}^{1}\to\mathcal{C}_{k}^{i}$ with $\psi(u)=v$. Since
$\varphi_{i}^{-1}\circ\psi\in\Lambda(\mathcal{C}_{k})$,
$\varphi_{i}^{-1}(v)\in O_{u}^{1}$. This proves there is a one-to-one
correspondence between the sets
$O_{u}^{1}\times\\{\varphi_{1},\dots,\varphi_{m}\\}$ and $O_{u}$. In other
words, $O_{u}=\bigcup_{i=1}^{m}O_{u}^{i}$ where $O_{u}^{i}$ is the orbit of
$u$ in $\mathcal{C}_{k}^{i}$. The proof follows. ∎

Choice on $k$-chainlets. In this article, we focus on $2$-chainlet classes,
and orbits of the addresses defined by these $2$-chainlets because this choice
provides rich enough and effective structural topological embeddings that
capture important transaction behavior. We chose to not utilize one-chainlets
as they only model the receiving behavior of coins, whereas we wanted to model
both the sending and receiving behavior. $k$=2 is a natural choice that suits
all application settings. Note that $k$=2 subsumes $k$=1 (as we show in only-
active and only-passive experiments in Table 6). $k>3$ is not useful because
once bitcoins are received (modeled with $k$=1) and spent (modeled with k=2),
the owner of the coins has no control over what transactions the new buyer of
the coins creates. In this sense, $k$=2 is necessary and sufficient in
understanding spending behavior. Beyond 2, we can only model the global
behavior in the graph. See appendix Figures 6, 7, 8, 8, 9 for all 48 orbits.

When counting the total number of distinct orbits, we exclude the input
addresses of transaction $t_{1}$ (i.e., the first transaction involved in a
$2$-chainlet) as their inclusion leads to a count of more than 120 orbits.
Figure 3 shows an example of the resulting reduced chainlet form. By
disregarding the input information of the first transaction, we reduce the
orbit count to 48 (see Appendix for definitions and shapes). As an address
typically has only 1.5 orbits on average, utilizing this reduced set of 48
orbits mitigates the sparsity issues in the extracted orbit data. Our
experiments, presented in Section 4, demonstrate that our orbit definition
provides strong classification power.

Active and Passive Orbits: We define the orbit of an address as active or
passive based on the role of the address within a $2$-chainlet.

######  Definition 4 (Active Orbit).

Let $\mathcal{C}$ be a $2$-chainlet. An active orbit with respect to
$\mathcal{C}$ is an orbit where its addresses appear as the output of the
first transaction in the corresponding occurrence of $\mathcal{C}$. When
$\mathcal{C}$ is understood, we denote the set of active orbits as $O_{act}$.

Active orbits indicate behavior created by the address owner, as the address
in this orbit initiates a new transaction and sends coins to other addresses.
We define passive orbits as follows:

######  Definition 5 (Passive Orbit).

Let $\mathcal{C}$ be a $2$-chainlet. A passive orbit with respect to
$\mathcal{C}$ is an orbit where it appears as the output of the second
transaction in the corresponding occurrence of $\mathcal{C}$. Again, when
$\mathcal{C}$ is understood, we denote the set of passive orbits as
$O_{pass}$.

Passive orbits for a chainlet class $\mathcal{E}(\mathcal{C})$, on the other
hand, receive coins without shaping any occurrences of $\mathcal{C}$, as the
address in this orbit only receives coins from another address, without
initiating a new transaction. It is worth noting that an address can belong to
an active orbit for one chainlet class while belonging to a passive orbit for
a different chainlet class. This means that an address may exhibit both
initiating and receiving behavior in different $2$-chainlet types. Figure 3
shows the example of an active orbit 5 and a passive orbit 6 in a 2-chainlet.

Figure 3: Examples of an active orbit 5 (in red) and a passive orbit 6 (in
gray) in a 2-chainlet. Both outputs of $t_{2}$ are orbit 6 addresses. Address
with orbit 5 initiates the second transaction, $t_{2}$, in the 2-chainlet,
thus its spending behavior shapes the structure of the chainlet and determines
the orbit type of the associated address.

The distinction between active and passive orbits is significant as it allows
us to focus on specific types of behavior within the transaction graph. Active
orbits, for example, can be used to train a classifier for identifying
spending patterns associated with coins gathered from darknet marketplaces and
ransomware payments. This is because active orbits indicate behavior initiated
by the address owner. On the other hand, passive orbits, which are shaped by
payers, can be used to study the behavior of victims, as these orbits simply
receive coins without shaping any chainlet. By understanding the behavior of
both active and passive orbits, we can gain a more comprehensive understanding
of the transaction graph and the actors within it.

Computational Complexity of Orbit Discovery: Extracting a 2-chainlet requires
three searches on the adjacency matrix. First, we must find all transaction
nodes which has a cost of $\mathcal{O}(\mathcal{V})$. Next, for every
transaction pair $t_{1}$ and $t_{2}$, we must find if $\exists
a\in\\{O(t_{1})\cap I(t_{2})\\}$, where $O(t_{1})$ represents the set of
outputs of transaction $t_{1}$ and $I(t_{2})$ represents the set of inputs of
transaction $t_{2}$. The second step thus has a complexity of
$\mathcal{O}(\mathcal{V}^{3})$. However, 57.40% of Bitcoin transactions have
less than two input and two output addresses. Hence the complexity is reduced
to $k\times\mathcal{O}(\mathcal{V}^{2})$ where $k\approx 2$. As a result,
finding all address orbits of a daily Bitcoin graph (approximately $400K$
transactions) takes around 15 minutes on an off-the-shelf computer. Please
refer to Figure 4 for the visualization of the orbit extraction time in the
daily heterogeneous Bitcoin transaction graph.

##  4 Experiments

Datasets We extract data from the Bitcoin transaction network and use labels
for ransomware and darknet market addresses that have been obtained from
external sources.

Bitcoin Transaction Network. We have downloaded and analyzed the complete
Bitcoin transaction graph from its inception in 2009 until 2023 by utilizing
the Bitcoin Core Wallet and the Bitcoin-ETL library
https://github.com/blockchain-etl/bitcoin-etl. By implementing a 24-hour time
window centered on the Central Standard Time zone, we extract daily
transactions on the network and construct the Bitcoin graph.

Ransomware Addresses. Our ransomware dataset is a union of datasets from three
widely adopted studies: Montreal [27], Princeton [12] and Padua [9]. The
combined dataset contains addresses from 27 ransomware families. 19930 unique
ransomware addresses appear a total of 62641 times between 2009-2023.

Darknet Addresses. We have downloaded the Grams dataset from Darknet Market
Archives [7]. Grams was a platform that primarily focused on facilitating
searches for market listings. The platform utilized a variety of methods to
acquire the listings, including utilizing API exports provided by various
markets. The dataset spans from June 9, 2014, to July 12, 2015. We identify
7557 unique addresses associated with darknet marketplaces a total of 1288100
times.

###  4.1 Experimental Setup

Hardware: We ran experiments on a single machine with a Dell PowerEdge R630,
featuring an Intel Xeon E5-2650 v3 Processor (10-cores, 2.30 GHz, 20MB Cache),
and 192GB of RAM (DDR4-2133MHz).

Preprocessing: We do not apply any normalization or preprocessing to the
address orbits. Because extracting orbits is not demanding in terms of
computational resources, we were able to use the entire transaction network
without removing any edges or nodes. The address orbits, which are 27 GB in
size, are publicly available at ***.org/btc/orbits.

Orbit discovery is an efficient process on the large Bitcoin transaction
network. As Figure 4 shows, on the majority of days, orbit extraction is
completed within 1000 seconds (equivalent to 15 minutes).

Figure 4: Time required to extract orbits from the daily Bitcoin transaction
graph, which consists of two node types (addresses and transactions) and their
edges. Each data point represents the time costs associated with extracting
orbits for a particular day.

Code and Parameters: Our code and details on hyperparameter selection and
models are given at https://github.com/chainletRepo/chainlet.

###  4.2 Results

This section begins by presenting the outcomes of widely employed heuristics
for clustering Bitcoin addresses. Then, we outline the tactics utilized in
e-crime before proceeding to the primary findings of this article, including
the BitcoinHeist benchmark and the full orbit dataset.

Heuristics. By employing the co-spending and transition heuristics (Section 2)
with all history of ransomware addresses, we discover only 40 unique addresses
from CryptoLocker (Padua), CryptoWall (Padua), CryptoTorLocker2015 (Montreal),
and CryptoTorLocker2015 (Padua) families. These low numbers are also because
researchers of the datasets had already used the heuristics to expand the
datasets. As a result, we do not find many more undisclosed addresses.

####  4.2.1 Adversarial Patterns of E-Crime

Table 1: Top-3 most frequent orbit counts of darknet market, ransomware (RS) and white addresses. Orbits that do not appear in the top-3 have been excluded. Percentages are computed within each address class.  | Rank | $\mathbb{O}1$ | $\mathbb{O}9$ | $\mathbb{O}12$ | $\mathbb{O}13$ | $\mathbb{O}14$ | Perc.
---|---|---|---|---|---|---|---
| 1 | 0 | 0 | 0 | 1 | 1 | 2.63
Darknet | 2 | 1 | 0 | 0 | 0 | 0 | 1.13
| 3 | 1 | 0 | 0 | 0 | 1 | 0.97
| 1 | 0 | 1 | 0 | 1 | 1 | 6.69
RS | 2 | 0 | 0 | 0 | 1 | 1 | 6.61
| 3 | 0 | 0 | 1 | 1 | 1 | 3.23
| 1 | 0 | 0 | 1 | 0 | 1 | 5.93
White | 2 | 0 | 0 | 1 | 1 | 1 | 4.89
| 3 | 1 | 0 | 0 | 0 | 0 | 4.40

Our analysis has revealed a total of 22682 unique orbit patterns associated
with addresses linked to ransomware and 546838 unique orbit patterns
associated with addresses associated with darknet marketplaces. Typically, an
address is only connected to a single occurrence within an orbit for any given
day, resulting in an orbit value of 1. Table 1 lists the most frequent orbit
patterns for classes.

Additionally, 2.63% of all darknet market addresses appear in two 2-chainlets
daily, within orbits $\mathbb{O}13$ and $\mathbb{O}14$. However, it should be
noted that these orbits are prevalent across all types of addresses, including
those associated with ransomware. As Table 1 illustrates, the top three
patterns associated with ransomware addresses make up 16.53% of all ransomware
addresses, indicating a lack of diversity in ransomware address behavior. This
implies that ransomware operators tend to use a limited number of patterns,
which makes tracking ransomware payments easier.

Table 2: Top-5 orbit patterns where addresses have the highest likelihood of being associated with ransomware. A checkmark indicates non-zero orbit count. The RS% column in the table represents the percentage of addresses that are identified as ransomware addresses. W (white), DM (darknet market) and RS (ransomware) are counts of addresses with the related pattern. For example, for the first orbit pattern, 95.9% (i.e., 94) of the addresses with that pattern are identified as ransomware addresses, while the remaining addresses are either white or associated with darknet  $\mathbb{O}1$ |  $\mathbb{O}6$ |  $\mathbb{O}8$ |  $\mathbb{O}9$ |  $\mathbb{O}13$ |  $\mathbb{O}14$ |  $\mathbb{O}17$ |  $\mathbb{O}25$ |  $\mathbb{O}28$ |  $\mathbb{O}31$ |  $\mathbb{O}32$ |  $\mathbb{O}34$ |  $\mathbb{O}37$ |  $\mathbb{O}40$ |  $\mathbb{O}42$ |  $\mathbb{O}43$ |  $\mathbb{O}45$ |  $\mathbb{O}47$ | W | DM | RS | RS%
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  ✓ |  |  |  |  |  ✓ |  ✓ |  ✓ |  |  |  ✓ |  |  ✓ |  |  ✓ |  | 1 | 3 | 94 | 95.9
|  |  ✓ |  |  |  |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ | 64 | 3 | 216 | 76.3
✓ |  ✓ |  |  ✓ |  ✓ |  ✓ |  |  |  |  |  |  |  |  |  |  |  |  | 21 | 91 | 81 | 41.9
✓ |  |  |  ✓ |  ✓ |  ✓ |  |  |  |  |  |  |  |  |  |  |  |  | 230 | 186 | 131 | 23.9
✓ |  |  |  ✓ |  ✓ |  ✓ |  |  |  |  |  |  |  |  |  |  |  |  | 106 | 150 | 108 | 29.7

Table 2 further delves into the RS patterns. In the table, we binarize orbit
counts and use a checkmark to indicate that an address has a non-zero count
for the associated orbit. Certain patterns exhibit a strong association with
addresses linked to ransomware. A significant proportion of addresses (95.9%)
with the pattern in the first row of the table are identified as ransomware
addresses. It is highly probable that the remaining white address with this
pattern is also associated with e-crime activities. Table 3 shows the most and
the least likely patterns associated with white addresses.

Table 3: The top-3 and bottom-3 orbit patterns with the lowest and highest likelihood of being white addresses, respectively. A checkmark indicates non-zero orbit count. The Non-White% column in the table represents the percentage of addresses that are NOT identified as white addresses. W (white), DM (darknet market) and RS (ransomware) are counts of addresses with the related pattern. For example, for the first orbit pattern, 99.9% (i.e., 9800) of the addresses with that pattern are identified as ransomware, while the remaining addresses are white. $\mathbb{O}10$ |  $\mathbb{O}12$ |  $\mathbb{O}13$ |  $\mathbb{O}14$ |  $\mathbb{O}25$ |  $\mathbb{O}28$ |  $\mathbb{O}30$ |  $\mathbb{O}31$ |  $\mathbb{O}32$ |  $\mathbb{O}34$ |  $\mathbb{O}37$ |  $\mathbb{O}40$ |  $\mathbb{O}41$ |  $\mathbb{O}43$ |  $\mathbb{O}44$ |  $\mathbb{O}45$ |  $\mathbb{O}46$ |  $\mathbb{O}47$ | W | DM | RS | Non-White%
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  $\checkmark$ | 4 | 0 | 9800 | 99.9
|  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  |  |  |  |  |  |  |  | 61 | 14 | 7316 | 99.0
$\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 116 | 10 | 10666 | 98.8
|  |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ | 12915 | 0 | 1249 | 8.8
$\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 6029 | 1 | 442 | 6.8
|  |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  | 9309 | 0 | 533 | 5.4

Our analysis has provided us with the following insights into e-crime
behavior.

  * •

On average, white addresses have three distinct non-zero orbits, whereas
darknet addresses have six non-zero orbits. Similarly, RS addresses have six
non-zero orbits. These values are higher than expected because the Bitcoin
community discourages address reuse in transactions. As a result, we would
expect most addresses to participate in fewer chainlets. However, the high
values are likely a result of multiple payments being made to the same
address, which indicates a lack of common privacy considerations in these
types of e-crime transactions.

  * •

Orbit 1 in darknet addresses reveals that coins transferred in a transaction
are not re-transferred on the same day, indicating that darknet payments are
retained in the receiving address. This aligns with the previous escrow
practice of darknet markets where payments to sellers were withheld until
buyers confirmed receipt of goods. Orbit 1 does not appear in the top 10 most
frequent orbits of either ransomware or white addresses.

  * •

Orbits 9 and 12 in ransomware addresses indicate immediate forwarding of
payments to new addresses (see Figure 8).

  * •

Orbit 9 indicates merges coins from multiple payments. However, none of these
orbits connect to coin-mixing transactions (see the appendix for coin-mixing
transactions).

  * •

Orbits 30, 31, 32, 39, 40, 41, 46, and 47 are observed in chainlets that may
be involved in coin-mixing transactions. Interestingly, these orbits are not
present in the top 10 most common patterns for any address type. E-crime coins
are first combined in an address and then coin-mixed.

####  4.2.2 BitcoinHeist Benchmark

We use orbits of Bitcoin addresses in a binary (white vs. ransomware) address
classification task on the BitcoinHeist dataset and compare our results to
those reported in the article by Akcora et al. [2]. We obtained 2916697
ransomware and white addresses from the authors of the dataset. The
BitcoinHeist dataset comprises 1048576 transactions from January 2009 to
December 2018. Transactions with amounts less than
$\leavevmode\vtop{\offinterlineskip \vbox{\hbox
to7.08336pt{\hfil\hskip-0.29999pt\vrule height=1.29167pt,width=0.6458pt\hskip
0.80002pt\vrule height=1.29167pt,width=0.6458pt\hfil}\hbox{B}}\hbox
to7.08336pt{\hfil\hskip-0.29999pt\vrule height=1.29167pt,width=0.6458pt\hskip
0.80002pt\vrule height=1.29167pt,width=0.6458pt\hfil}}0.3$ were removed as
ransomware payments are usually significantly larger. Since the dataset only
includes ransomware and white addresses, we present the performance of our
methods in terms of binary classification by using a Random Forest [8] with
300 trees.

In 24 ransomware families, at least one address appears in more than one
24-hour time window. CryptoLocker has 13 addresses that appear more than 100
times each. The CryptoLocker address 1LXrSb67EaH1LGc6d6kWHq8rgv4ZBQAcpU
appears for a maximum of 420 times. Four addresses have conflicting ransomware
labels between Montreal and Padua datasets. APT (Montreal) and Jigsaw (Padua)
ransomware families have two and one multisig addresses, respectively. All
other addresses are ordinary addresses that start with ’1’.

Table 4: AUC scores for address classification using features from the BitcoinHeist dataset and Orbits. White addresses were under-sampled without replacement, while ransomware addresses were not under-sampled. The results indicate that using Orbits improves address classification significantly. AUC@Sample | BitcoinHeist | Orbit | BitcoinHeist $+$ Orbit
---|---|---|---
AUC@1M | 0.791(0.005) | 0.839(0.003) | 0.895(0.006)
AUC@500K | 0.798(0.006) | 0.849(0.002) | 0.903(0.003)
AUC@100K | 0.808(0.003) | 0.856(0.005) | 0.908(0.003)
AUC@50K | 0.807(0.006) | 0.857(0.004) | 0.908(0.003)

Table 4 shows the AUC scores of the ROC curve for the binary address
classification task. In BitcoinHeist+Orbit, we combine feature sets of
addresses. The AUC of the BitcoinHeist and BitcoinHeist+Orbit models improves
as the white sample rate increases from 50K to 100K, but larger sample rates
decrease performance. However, the performance of the Orbit model remains
relatively stable with larger white sampling rates, decreasing only by 0.018
from 50K to 1M. This suggests that orbits are relatively robust against the
class imbalance issues that often occur in blockchain data analytics, as
labeled data is often scarce in these tasks.

The superiority of the BitcoinHeist+Orbit model over the Orbit model can be
attributed to the integration of the income feature from the BitcoinHeist
dataset, which calculates the sum of incoming coin amounts for each address.
Please note that orbits do not consider the coin amounts that are transferred.
As the ransom amounts are typically consistent (e.g.,  B 0.5), the income
provides a good feature to predict the ransomware label. The Mean Decrease in
Impurity [8] identifies income, $\mathbb{O}9$, and $\mathbb{O}12$ as the three
most important features of the classifier. Incorporating the income feature
into the orbits data is computationally efficient (i.e., $O(\mathcal{V})$) as
it only requires summing the incoming amounts for each address.

It’s worth mentioning that the BitcoinHeist study employs graph flow-based
features for each address, which are computationally intensive. To minimize
computational expenses, the authors applied edge filtering with a threshold of
0.3 Bitcoins. On the other hand, orbits are not as computationally demanding
(Section 3), thus, we were not constrained by the computational complexity
issues. This permitted us to extract orbits for all Bitcoin addresses within
the examined period.

Table 5: AUC (and standard deviation in parentheses) values for address classification using BitcoinHeist features and Orbits. Both white (W) and ransomware (RS) addresses are under-sampled without replacement.  Sample |  | GCN | GIN | DGCNN
---|---|---|---|---
20K |  | 0.76(0.002) | 0.90(0.02) | 0.79(0.09)
10K |  | 0.70(0.004) | 0.88(0.01) | 0.75(0.09)
5K |  | 0.70(0.002) | 0.86(0.02) | 0.78(0.09)

Graph Neural Networks on the BitcoinHeist Benchmark. We evaluate the orbits-
based method against four graph classification techniques on the BitcoinHeist
data utilizing graph neural networks: a GCN [14], DGCNN [35], GIN [34] and
GraphSage [11]. Our objective is to evaluate the performance of orbit-based
classifiers by comparing them to Graph Neural Network (GNN) classifiers. While
there are numerous methods, we are not conducting a comprehensive comparison
of all of them. Instead, we aim to provide a good understanding of what GNNs
can achieve. We report the average accuracy of five runs, along with their
standard deviation. In these models, we adopted the grid search parameters
from the benchmark article by Errica et al. [10].

We could not apply existing node (i.e., address) classification approaches
[33] to the Bitcoin transaction network because the Bitcoin transaction
network is too large and contains as few as four e-crime addresses per day
(over ten thousand days). Instead, we re-formulated the address classification
task as the classification of the subgraph of the two-hop neighborhood of an
address. Formally, the neighborhood graph of an address is the combination of
the induced subgraphs of i) $2$-chainlets where the second transaction outputs
coins to the address and ii) $2$-chainlets where the first transaction outputs
coins to the address. An example of a neighborhood graph is shown in Figure 1.

There are several advantages of using the neighborhood approach instead of a
traditional node classification approach. Firstly, the neighborhood approach
enables an inductive learner where an address from any day can be classified
by using its neighborhood graph from that day, while transductive node
classification is restricted to the specific graph on which it was trained and
cannot be applied to new, unseen graphs. Secondly, the neighborhood approach
provides us with the ability to control the number of addresses in the model,
as GNNs can have enormous computational costs when applied to very large
datasets.

Despite utilizing the neighborhood graph, neural networks still face the
challenge of data imbalance. In the case of more than 600K addresses daily,
only a small number of e-crime addresses exist. To address this problem, we
take two steps. Firstly, we create a balanced dataset to evaluate the graph
classification methods where close to half of the addresses are white (i.e.,
57%). Secondly, we restrict the number of addresses fed into the model.

The resulting AUC scores are shown in table 5. In addition to the models
mentioned, we attempted to train a robust GraphSage [34] model, but it did not
perform well as the model AUC did not improve throughout the training. We
believe that this could be due to the sparsity of the Bitcoin transaction
network, where many addresses have only one or two neighbors, and the usage of
sampling-based approaches leads to disconnected graphs.

(a) Sample=100K.

(b) Sample=500K.

(c) Sample=1M.

Figure 5: AUC scores for Orbit-based e-crime address classification in the
period of 2009-2023. White and darknet classes are undersampled. Orbit
performance is quite robust against the increasing sample sizes.

As shown in Table 5, GIN is a powerful model and consistently improves its AUC
score with more training data. However, the complexity of the model limits the
amount of training data that can be used. However, these results are achieved
over balanced data assumptions which do not hold over all data. As we show in
the next section, our orbit method is capable of achieving improved AUC scores
for the challenging task of three-way classification when trained on large
e-crime data.

Orbits vs. GIN. The results of GIN in Table 5 (0.9 AUC) are similar to those
of the Orbit results in Table 4 (0.857 AUC). However, GIN is limited to
smaller datasets of under 20K addresses, while Orbit-based models can utilize
a full day’s worth of addresses (over 500K) to make predictions for the next
day. This allows us to incorporate temporal patterns into our models and give
more importance to recent Orbit patterns.

####  4.2.3 Orbits on E-crime Graph Classification

The goal of this experiment is to generalize the orbit e-crime address
classification performance with a larger dataset. We take a sample of 1% of
the daily white addresses in the Bitcoin history, resulting in 3.16 million
addresses, and also include all 1.28 million darknet market and 64,000
ransomware addresses.

Table 6: AUC scores for active ($O_{act}$) and passive ($O_{pass}$) orbits on 1% of Bitcoin addresses (4.3M addresses). Class |  $O_{act}$ AUC |  $O_{pass}$ AUC
---|---|---
White | 0.731(0.006) | 0.870(0.002)
RS | 0.725(0.004) | 0.736(0.001)
DM | 0.727(0.004) | 0.876(0.001)

Figure 5 illustrates the three-way address classification performance for
white, ransomware, and darknet addresses. The figure tracks the AUC score for
each class with a one-versus-rest Random Forest classifier, as more white data
is added in three steps to the classifier input. Training/test split is 80/20.
The consistently high AUC score, above 0.8, demonstrates the strength of the
classifier.

Furthermore, we investigate the role of different types of orbits in detecting
e-crime payments. We consider two types of orbits as outlined in Section 3:
active orbits, which refer to the behavior of e-crime operators such as
darknet market sellers and ransomware hackers, and passive orbits, which refer
to the behavior of darknet market buyers or ransomware victims. In Table 6, we
evaluate the performance of different feature sets by limiting the data
features to either active or passive orbits and calculating the macro-averaged
AUC performance.

The results show that reducing the feature set to active orbits lead to a
decrease in AUC scores compared to using both feature sets, which has AUC
scores of 0.84 or higher (Figure 5(c)).

However, the results also indicate that the passive orbits yield better AUC
results, meaning that the behavior of victims or buyers is more informative in
detecting e-crime payments. Furthermore, active orbit AUC values per class are
similar, which suggests that the performance of the classifier is consistent
across different classes.

In conclusion, these results highlight the importance of considering both
active and passive orbits in detecting e-crime payments. While passive orbits
provide valuable information, the best classification performance is achieved
by utilizing both active and passive orbits together.

##  5 Conclusions

We propose a topological node embedding model, called "orbit," that captures a
node’s structural role in a UTXO graph. This approach results in compact and
effective embeddings that can be easily integrated into node and graph
classification models which outperforms current state-of-the-art methods in
blockchain graph node classification tasks. Additionally, the orbit patterns
can be visualized and used with interpretable and explainable models.

Our approach is not limited to Bitcoin graphs only and can be applied to UTXO
graphs of other blockchains, such as ZCash and Monero, where e-crime is a
prevalent issue. The orbit discovery process is computationally efficient and
can be applied to very large graphs easily.

Future work could focus on multi-modal graphs where meta attributes of
transactions can be incorporated as node features. Another promising area of
research is developing self-supervised ways to learn node and edge functions
from data to define new orbits.

##  6 Acknowledgements

This work is supported in part by National Science Foundation under Grant No.
ECCS 2039701, ECCS 1824716, DMS 1925346, CNS 1837627, OAC 1828467, IIS
1939728, CNS 2029661 and Canadian NSERC Discovery Grant RGPIN-2020-05665.

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##  7 Appendix

In this part, we give further details of our method and experiments. In
Section 8, chainlet orbits are visualized and described. In Section 9, we
describe how orbits can be applied to other UTXO blockchains. In Section 10,
we explain the coin-mixing behavior and give our related orbits.

##  8 UTXO network and Chainlets

In the context of the Bitcoin blockchain, a transaction can be formally
defined in terms of its input and output as follows: Input: A set of one or
more digital signatures, or "unspent transaction outputs" (UTXOs), that
authorize the transfer of value from the sender’s Bitcoin address to the
recipient’s address. UTXOs are previously-created outputs of transactions that
were not spent yet and they are used as an input in a new transaction. Output:
A set of one or more new UTXOs that represent the transfer of value from the
sender’s address to the recipient’s address. In other words, a Bitcoin
transaction takes one or more UTXOs as input and creates one or more new UTXOs
as output. The total value of the inputs must be greater than or equal to the
total value of the outputs, otherwise the transaction will be considered
invalid.

Figures 6, 7, 8, 9 list all orbits and define address roles in them. We use
the X-M-N notations to describe chainlets where the first chainlet has X
inputs, M outputs whereas the second chainlet has N outputs. In orbits, we are
particularly interested in M and N.

##  9 Applicability to Monero and Zcash

The concept of orbits can be applied to heterogeneous graphs like the UTXO
transaction networks that consist of transaction and address nodes. In the
case of Monero and ZCash, which both utilize the UTXO data model [4], orbits
can be directly applied to the unshielded transactions pool of ZCash without
any modifications.

In Monero, the association between addresses and transactions is concealed in
all transactions, but the input and output counts are not hidden. With orbits
defined for both input and output addresses of transactions, these counts can
be tracked. However, instead of representing addresses as nodes, output hash
IDs would be represented as nodes in this context.

(a) Orbit 0: if $\left|\Gamma_{1}^{o}\right|=1$ and $a\in\Gamma_{1}^{o}$ then
$a\in\mathbb{O}_{0}.$

(b) Orbit 1: if $\left|\Gamma_{1}^{o}\right|=2$ and $a\in\Gamma_{1}^{o}$ then
$a\in\mathbb{O}_{1}.$

(c) Orbit 2: if $\left|\Gamma_{1}^{o}\right|=3$ and $a\in\Gamma_{1}^{o}$ then
$a\in\mathbb{O}_{2}.$

Figure 6: Orbits 0 to 2 involve addresses who do not create a 2-chainlet for
the daily network snapshot. These addresses keep the received coins dormant
for the given day. All addresses have passive orbits.

(d) Orbit 3: if $\left|\Gamma_{1}^{o}\right|=1$,
$\left|\Gamma_{2}^{i}\right|=1$, $\left|\Gamma_{2}^{o}\right|=1$ and
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}_{3}$. Orbit 4:
if $\left|\Gamma_{1}^{o}\right|=1$, $\left|\Gamma_{2}^{i}\right|=1$,
$\left|\Gamma_{2}^{o}\right|=1$ and $a\in\Gamma_{2}^{o}$, then
$a\in\mathbb{O}_{4}$.

(e) Orbit 5: if $\left|\Gamma_{1}^{o}\right|=1$,
$\left|\Gamma_{2}^{i}\right|=1$, $\left|\Gamma_{2}^{o}\right|=2$, and
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}_{5}$. Orbit 6:
if $\left|\Gamma_{1}^{o}\right|=1$, $\left|\Gamma_{2}^{i}\right|=1$,
$\left|\Gamma_{2}^{o}\right|=2$, and $a\in\Gamma_{2}^{o}$ then
$a\in\mathbb{O}_{6}.$

(f) Orbit 7: if $\left|\Gamma_{1}^{o}\right|=1$,
$\left|\Gamma_{2}^{i}\right|=1$, $\left|\Gamma_{2}^{o}\right|=3$, and
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}7$. Orbit 8: if
$\left|\Gamma_{1}^{o}\right|=1$, $\left|\Gamma_{2}^{i}\right|=1$,
$\left|\Gamma_{2}^{o}\right|=3$, and $a\in\Gamma_{2}^{o}$ then
$a\in\mathbb{O}8$.

Figure 7: X-1-N addresses create orbits 3 to 8 where the first transaction has
a single output address and the second transaction receives coins from the
address. Red addresses have active orbits, whereas gray addresses have passive
orbits.

(g) Orbit 9: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$, $\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}9$. Orbit 10:
if $\left|\Gamma_{1}^{o}\right|=2$, $\left|\Gamma_{2}^{i}\right|=1$,
$\left|\Gamma_{2}^{o}\right|=1$, $a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$
then $a\in\mathbb{O}10$. Orbit 11: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$, $\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}11$.

(h) Orbit 12: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$, $\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}12$. Orbit 13:
if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}13$. Orbit
14: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}14$.

(i) Orbit 15: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}15$. Orbit 16:
if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}16$. Orbit
17: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}17$.

(j) Orbit 18: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}18$. Orbit 19:
if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{2}^{o}$, then $a\in\mathbb{O}19$.

(k) Orbit 20: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}20$. Orbit 21:
if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{2}^{o}$, then $a\in\mathbb{O}21$.

(l) Orbit 22: if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}22$. Orbit 23:
if $\left|\Gamma_{1}^{o}\right|=2$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{2}^{o}$, then $a\in\mathbb{O}23$.

Figure 8: X-2-N create orbits 9 to 23 where the first transaction has two
output addresses and the second transaction receives coins from one or two of
the output addresses of the first transaction. Red addresses have active
orbits, whereas gray addresses have passive orbits.

(m) Orbit 24: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}24$. Orbit 25:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}25$. Orbit
26: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{2}^{o}$, then $a\in\mathbb{O}26$.

(n) Orbit 27: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}27$. Orbit 28:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}28$. Orbit
29: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}29$.

(o) Orbit 30: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}30$. Orbit 31:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}31$. Orbit
32: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=1$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}32$.

(p) Orbit 33: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}33$. Orbit 34:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}34$. Orbit
35: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}35$.

(r) Orbit 36: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}36$. Orbit 37:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}37$. Orbit
38: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}38$.

(s) Orbit 39: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}39$. Orbit 40:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\notin\Gamma_{2}^{i}$ then $a\in\mathbb{O}40$. Orbit
41: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=2$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}41$.

(t) Orbit 42: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=3$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}42$. Orbit 43:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=3$,$\left|\Gamma_{2}^{o}\right|=1$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}43$.

(u) Orbit 44: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=3$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}44$. Orbit 45:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=3$,$\left|\Gamma_{2}^{o}\right|=2$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}45$.

(v) Orbit 46: if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=3$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{1}^{o}$, $a\in\Gamma_{2}^{i}$ then $a\in\mathbb{O}46$. Orbit 47:
if $\left|\Gamma_{1}^{o}\right|=3$,
$\left|\Gamma_{2}^{i}\right|=3$,$\left|\Gamma_{2}^{o}\right|=3$,
$a\in\Gamma_{2}^{o}$ then $a\in\mathbb{O}47$.

Figure 9: X-3-N create orbits 24 to 47 where the first transaction has three
output addresses and the second transaction receives coins from one, two or
three of the output addresses of the first transaction. Red addresses have
active orbits, whereas gray addresses have passive orbits.

##  10 Orbits and Coin-Mixing Transactions

A coin-mixing transaction, also known as a coin-join transaction [21], is a
technique used to enhance the privacy and anonymity of cryptocurrency
transactions. In a typical coin-mixing transaction, multiple participants
combine their coins or funds into a single transaction, making it difficult to
trace the original source of the funds.

A coin-mixing transaction works in the following way:

Participants: Several individuals who wish to improve the privacy of their
transactions agree to participate in a coin-mixing process.

Inputs: Each participant selects a specific amount of their cryptocurrency
holdings (coins) to contribute to the coin-mixing transaction. These
individual inputs are usually of equal value to obfuscate the original
ownership.

Mixing Pool: All the selected inputs from the participants are combined into a
mixing pool or a single transaction. This process effectively mixes the coins
together, making it challenging to identify which coins belong to each
participant.

Outputs: The mixing pool generates new outputs or addresses that receive the
mixed coins. The number of outputs is usually the same as the number of inputs
to maintain balance.

Distribution: The newly mixed coins are then distributed back to the
participants, but each participant receives their coins in a randomized
manner. This step further obscures the link between the original inputs and
the final outputs.

The mixing process is repeated multiple times to enhance privacy. It’s worth
noting that while coin-mixing transactions enhance privacy, they can also be
associated with illicit activities if used for money laundering or other
illegal purposes.

In graph terms, the coin mixing transactions create 2-chainlets with many
inputs and outputs. We identify several orbits that may encode coin mixing
transactions. These are orbits 30, 31, 32, 39, 40, 41, 46, and 47 in Figures 8
and 9.


# Multistrand Eigenvalue conjecture and Racah symmetries

Andrey Morozova,b
a NRC Kurchatov Institute
b Institute for Information Transmission Problems

ITEP/TH-26/22
IITP/TH-23/22 e-mail<EMAIL_ADDRESS>
###### Abstract

Racah matrices of quantum algebras are of great interest at present time.
These matrices have a relation with $\mathcal{R}$-matrices, which are much
simpler than the Racah matrices themselves. This relation is known as the
eigenvalue conjecture. In this paper we study symmetries of Racah matrices
which follow from the eigenvalue conjecture for multistrand braids.

##  1 Introduction

Racah matrices and $6j$-symbols have quite a long history in science. They
first appeared in physics through the studies of angular momenta [1], where
they describe transformation between different couplings of three particles.
Racah matrices appearing in this problem are actually quite simple and well
studied. However, in mathematical terms this angular momenta problem actually
corresponds to the multiplication of different representations of the $SU(2)$
algebra. In this language Racah coefficients appear when one takes a product
of three representations and expands it as a sum of irreducible
representations:

$T_{1}\otimes T_{2}\otimes T_{3}=\sum M_{12}^{1,2}Q_{12}\otimes T_{3}=\sum
N^{12,3}M_{12}^{1,2}Q.$ (1)

There is also another way to represent the same product:

$T_{1}\otimes T_{2}\otimes T_{3}=\sum T_{1}\otimes M_{23}^{2,3}Q_{23}=\sum
N^{1,23}M_{23}^{2,3}Q.$ (2)

Map betwenn these two bases is called Racah matrix, which elements are
$6j$-symbols:

$\begin{array}[]{c}U:\ N^{12,3}M_{12}^{1,2}\rightarrow
N^{1,23}M_{23}^{2,3},\\\ \\\
U^{12}_{23}=\left\\{\begin{array}[]{ccc}T_{1}&T_{2}&Q_{12}\\\
T_{3}&Q&Q_{23}\end{array}\right\\}\end{array}$ (3)

What is interesting about such a definition is that it can be easily
generalized to the representations of other algebras, which makes them
applicable to other physical theories such as chromodynamics, where $SU(3)$
group should be used. In this case representations should be enumerated not by
natural numbers as in $SU(2)$ case, but by Young diagrams $T=[T_{1},\ldots
T_{n}]$, $T_{1}\geq T_{2}\geq T_{3}\ldots$. However, one can generalize this
definition to not only other Lie algebras, but also to their deformations such
as quantum groups [2]. Quantum groups are essentially a deformation of
universal enveloping algebras. They are connected among other things
integrable systems [3], lattice gauge theory [4], 3-d quantum gravity [5],
quantum $sl_{N}$ invariants of knots [6], Turaev-Viro invariants of
3-manifolds and topological field theory [7, 8], WZW conformal field theory
and 3d Chern-Simons theory [9, 10]. In the latter case quantum groups describe
the hidden symmetries of the observables, which are also equal to the knot
polynomials from the knot theory.

In recent years we actively studied applications of quantum groups and Racah
matrices to the calculations of knot polynomials and observables in Chern-
Simons theory. The main problem of these approaches is that one has to know
many different Racah matrices to calculate corresponding knot polynomials.
However, at the moment there are only few cases where general answers for
Racah matrices are known. Among these are Racah matrices for $U_{q}(sl_{2})$
group [11] and multiplicity free exclusive Racah matrices [12, 13]. In the
context of the present paper we are interested in Racah matrices which appear
in the calculations using braid representation of a knot, see Fig.1.

Figure 1: Braid representation of a knot $4_{1}$, braid is on the left while
on the right is it closure which is a knot.

In [14] quite a powerful tool to find missing Racah matrices was suggested,
namely the eigenvalue conjecture which relates known eigenvalues of quantum
$\mathcal{R}$-matrices with the Racah matrices for the 3-strand braids. This
conjecture was further studied in [15, 16, 17, 18]. Since eigenvalue
conjecture relates different Racah matrices, it imposes certain symmetries on
Racah matrices, which were studied in [17, 18]. However, symmetries studied in
these papers are the symmetries connected with the three-strand eigenvalue
conjecture. There is also eigenvalue conjecture for higher number of strands
[19]. In the present letter we are interested in studying symmetries implied
by this multi-strand eigenvalue conjecture.

The paper is organized as follows. In Section 2 we briefly remind the
structure of Reshetikhin-Turaev approach to the knot calculus, involving
$\mathcal{R}$ and Racah matrices. In Section 3 we explain why Racah matrices
for multistrand braids possess certain block structure. In Section 4 we remind
the formulation of eigenvalue conjecture. In Section 5 the main result of the
paper, (18) and (19), is introduced and explained.

##  2 Reshetikhin-Turaev approach

One of the most productive approaches to the calculations of knot polynomials
(as well as Wilson-loop averages of the Chern-Simons theory, which are equal
to them) is modern Reshetikhin-Turaev approach [20]-[25]. It is most clearly
defined for the braid representation of a knot (see Fig.1). Then in such a
braid all crossings correspond to $\mathcal{R}$-matrices. These
$\mathcal{R}$-matrices are very simple in their diagonal form. Eigenvalues of
the $\mathcal{R}$-matrices correspond to irreducible representations in a
tensor product of the representations corresponding to two crossing strands.
For the knot case and representation $Q$ from the product of two
representations $T$ the eigenvalue is equal to [26, 27, 28, 2]

$\lambda_{Q}=q^{\varkappa_{Q}-4\varkappa_{T}-N|T|},$ (4)

where $\varkappa_{Q}$ is calculated directly from the Young diagram $Q$:

$\varkappa_{Q}=\sum\limits_{\\{i,j\\}\in Q}(i-j),$ (5)

where $\\{i,j\\}$ enumerates the boxes in the Young diagram.

For two-strand braids, which are parameterised by a number of crossings $n$,
the answer for the knot polynomial is then given by a character expansion:

$H^{T[2,n]}_{T}=\sum\limits_{Q\vdash T\otimes
T}S^{*}_{Q}(A,q)Tr\mathcal{R}^{n}_{Q}=\sum\limits_{Q\vdash T\otimes
T}S^{*}_{Q}(A,q)\lambda^{n}_{Q},$ (6)

where $S^{*}_{Q}(A,q)$ are Shur polynomials (characters of the representation
$Q$ for the quantum group) in a special point [25], which can also be easily
calculated from the Young diagram.

For multi-strand braids situation is more complicated. There are
$\mathcal{R}$-matrices, corresponding to the crossings of different pairs of
strands. They still can be made diagonal and thus provided by the (4), but
then rotation matrices which change basis are needed. These rotation matrices
are Racah matrices. Then answer for polynomials of $m$ strand braids are
provided by the modified version of character expansion:

$H^{\mathcal{K}}_{T}=\sum\limits_{Q\vdash T^{\otimes
m}}S^{*}_{Q}(A,q)Tr\mathcal{B}_{Q},$ (7)

where $\mathcal{B}_{Q}$ is a product of diagonal $\mathcal{R}$-matrices and
Racah matrices, which order can be written down from the braid. This means
that the most difficult quantities to find in this whole approach are Racah
matrices.

##  3 Block structure

Racah matrices and, correspondingly, non-diagonal $\mathcal{R}$-matrices for
the number of strands larger than 3 possess a certain block structure [16,
19].

For 3-strand braid if one wants to contruct non-diagonal $\mathcal{R}$-matrix
for the construction of $\mathcal{B}_{Q}$, the following Racah matrix is
required:

$\mathcal{R}_{2}=U\mathcal{R}U^{\dagger},\ \ \
U=\left[\begin{array}[]{ccc}T&T&Q_{12}\\\ T&Q&Q_{23}\end{array}\right].$ (8)

This Racah matrix can also be described pictorially, see Fig. 2.

$T$$T$$T$$Q_{12}$$Q$$=\sum_{Q_{23}}u^{Q_{12}}_{Q_{23}}$$T$$T$$T$$Q_{23}$$Q$
Figure 2: Pictorial description of 3-strand Racah matrix.

When we move to the 4-strands, two distinct types of Racah matrices are
needed. First, to find $\mathcal{R}_{2}$, according to Fig. 3, similar matrix
$U_{4}$ is needed. One should write irreducible representations expansion for
the product of four representations $T$ again in two ways:

$\begin{array}[]{l}\left((T\otimes T)\otimes T\right)\otimes
T=\\!\\!\\!\sum\limits_{Q_{12}\in T\otimes T}\\!\\!\\!\left(Q_{12}\otimes
T\right)\otimes T=\\\ \\\ =\sum\limits_{Q_{12}\in T\otimes
T}\sum\limits_{Q_{123}\in Q_{12}\otimes T}Q_{123}\otimes T=\sum\limits_{Q\in
T^{\otimes 4}}Q,\end{array}$ (9)

and

$\begin{array}[]{l}\left(T\otimes(T\otimes T)\right)\otimes
T=\\!\\!\\!\sum\limits_{Q_{23}\in T\otimes T}\\!\\!\\!\left(T\otimes
Q_{23}\right)\otimes T=\\\ \\\ =\sum\limits_{Q_{23}\in T\otimes
T}\sum\limits_{Q_{123}\in T\otimes Q_{23}}Q_{123}\otimes T=\sum\limits_{Q\in
T^{\otimes 4}}Q.\end{array}$ (10)

Racah matrix $U_{4}$ which describes rotation between these two bases consists
of blocks, since it does not mix representations, which come from different
$Q_{123}$. This means that in fact these blocks are in fact Racah matrices $U$
from (8), with $Q$ substituted by $Q_{123}$. Then
$\mathcal{R}_{2}=U_{4}\mathcal{R}U^{\dagger}_{4}$.

$T$$T$$T$$Q_{12}$$Q_{123}$$T$Q$=\sum_{Q_{23}}u^{Q_{12}}_{Q_{23}}$$T$$T$$T$$Q_{23}$$Q_{123}$$T$Q
Figure 3: Pictorial description of 4-strand Racah matrix $U$.

To find $\mathcal{R}_{3}$ additional Racah matrix $V$ is required. The natural
way is to look at the matrix corresponding to the Fig.4. In the language of
irreducible representation expansion this matrix connects bases (10) and

$\begin{array}[]{l}T\otimes\left((T\otimes T)\otimes
T\right)=\\!\\!\\!\sum\limits_{Q_{23}\in T\otimes
T}\\!\\!\\!T\otimes\left(Q_{23}\otimes T\right)=\\\ \\\
=\sum\limits_{Q_{23}\in T\otimes T}\sum\limits_{Q_{234}\in T\otimes
Q_{23}}T\otimes Q_{234}=\sum\limits_{Q\in T^{\otimes 4}}Q.\end{array}$ (11)

This matrix again consists of blocks. These blocks do not mix different
representations $Q_{23}$ and correspond to following blocks of $6j$-symbols

$V_{i}=\left[\begin{array}[]{ccc}T&Q_{23}&Q_{123}\\\
T&Q&Q_{234}\end{array}\right].$ (12)

The resulting $\mathcal{R}$-matrix is then equal to

$\mathcal{R}_{4}=U_{4}VU_{4}\mathcal{R}\left(U_{4}VU_{4}\right)^{\dagger}.$
(13)

Another interesting thing about matrix $V$ is that since it does not mix
different representations $Q_{23}$, and eigenvalues of $\mathcal{R}$-matrix
depend on these representations, they in fact commute
$\mathcal{R}V=V\mathcal{R}$. This structure in fact continues further to
larger number of strands with each new strand adding new Racah matrix.

$T$$T$$T$$Q_{23}$$Q_{123}$$T$Q$=\sum_{Q_{234}}v^{Q_{123}}_{Q_{234}}$$T$$T$$T$$Q_{23}$$Q_{234}$$T$Q
Figure 4: Pictorial description of 4-strand Racah matrix $V$.

##  4 Eigenvalue conjecture

The origins if the eigenvalue conjecture comes from studying Yang-Baxter
equation on $\mathcal{R}$-matrices:

$\mathcal{R}_{1}\mathcal{R}_{2}\mathcal{R}_{1}=\mathcal{R}_{2}\mathcal{R}_{1}\mathcal{R}_{2}.$
(14)

If one rewrites it as an equation on diagonal $\mathcal{R}$-matrix and Racah
matrices, then it looks like this:

$\mathcal{R}U\mathcal{R}U^{\dagger}\mathcal{R}=U\mathcal{R}U^{\dagger}\mathcal{R}U\mathcal{R}U^{\dagger}.$
(15)

This equation can be interpreted as an equation on Racah matrices $U$. This
leads to the eigenvalue conjecture [14]:

If normalied eigenvalues of two $\mathcal{R}$-matrices coincide then the
corresponding Racah matrices also coincide.

We are interested only in normalized eigenvalues, since if all
$\mathcal{R}$-matrices in (15) are multiplied by some constant the equation
still holds.

However (15) includes only three-strand Racah matrices. There are also other
Racah matrices which appear for larger number of strands. As it appears [19],
other Yang-Baxter equations such as
$\mathcal{R}_{3}\mathcal{R}_{2}\mathcal{R}_{3}=\mathcal{R}_{2}\mathcal{R}_{3}\mathcal{R}_{2}$
are actually trivially satisfied if the first Yang-Baxter equation is
satisfied. The important equations in the multi-strand case are commutation
relations for $\mathcal{R}$-matrices, such as
$\mathcal{R}_{1}\mathcal{R}_{3}=\mathcal{R}_{3}\mathcal{R}_{1}$, or in the
terms of diagonal $\mathcal{R}$-matrices

$\mathcal{R}UV_{1}U\mathcal{R}(UV_{1}U)^{\dagger}=UV_{1}U\mathcal{R}(UV_{1}U)^{\dagger}\mathcal{R}.$
(16)

Also it is important to note that these Racah matrices possess certain block
structure, as was described in the previous section. This leads to the
following generalization of the eigenvalue conjecture [19]:

If normalized eigenvalues of two $\mathcal{R}$-matrices coincide and the block
structure of the corresponding multistrand Racah matrices coincide then the
Racah matrices themselves also coincide.

##  5 Eigenvalue conjecture and symmetries of Racah matrices

For 3-strand braids in symmetric representations it was first observed [14]
and later proven [17, 18], that from the Eigenvalue conjecture it follows that
following Racah matrices coincide:

$\left[\begin{array}[]{ccc}\ [r]&[r]&[a,b]\\\ \
[r]&[l,m,n]&[x,y]\end{array}\right]=$ (17) $\left[\begin{array}[]{ccc}\
[r+c]&[r+c]&[a+c,b+c]\\\ \ [r+c]&[l+c,m+c,n+c]&[x+c,y+c]\end{array}\right].$

In fact no other Racah matrices appear in this case due to the structure of
representations product, namely, in the tensor product $[r]\otimes[r]$ only
one- and two-row Young diagrams appear. Correspondingly in the tensor cube of
the representation one-, two- and three-row Young diagrams appear. Equation
(17) in fact allows one to find all the 3-strand Racah matrices from the known
$U_{q}(sl_{2})$ answer [11] by setting $c=-n$.

In the context of this letter we discuss braids with larger number of strands.
We will still only talk about symmetric representations, since for non-
symmetric ones some of the $\mathcal{R}$-matrix eigenvalues start to coincide
which makes eigenvalue conjecture unsolvable in the sense of (15).
Nevertheless known answers for Racah matrices show that eigenvalue conjecture
still holds even in this case, but this case is out of the scope of this
paper.

The claim of this paper is that from the 4-strand eigenvalue conjecture the
following Racah matrices should coincide:

$\left[\begin{array}[]{ccc}\ [r]&[p,q]&[a,b,d]\\\ \
[r]&[k,l,m,n]&[x,y,z]\end{array}\right]=$ (18) $\left[\begin{array}[]{ccc}\
[r+c]&[p+c,q+c]&[a+c,b+c,d+c]\\\ \
[r+c]&[k+c,l+c,m+c,n+c]&[x+c,y+c,z+c]\end{array}\right],$

and similarly for larger number of strands:

$\left[\begin{array}[]{ccc}\ [r]&[p_{1},\ldots p_{i}]&[a_{1},\ldots
a_{i+1}]\\\ \ [r]&[k_{1},\ldots k_{i+2}]&[x_{1},\ldots
x_{i+1}]\end{array}\right]=$ (19) $\left[\begin{array}[]{ccc}\
[r+c]&[p_{1}+c,\ldots p_{i}+c]&[a_{1}+c,\ldots a_{i+1}+c]\\\ \
[r+c]&[k_{1}+c,\ldots k_{i+2}+c]&[x_{1}+c,\ldots
x_{i+1}+c]\end{array}\right].$

To explain these relations we need to check two things. First we will explain
why the structure of tensor product of representations and, thusly, block
structure of Racah matrices is the same. Then we will show that normalized
eigenvalues are still the same.

To understand why tensor product of representations remains the same, we prove
the following property of tensor product of representations:

$\begin{array}[]{c}\
[a_{1},\ldots,a_{n+1}]\in[p_{1},\ldots,p_{n}]\otimes[r]\\\ \Updownarrow\\\ \
[a_{1}+1,\ldots,a_{n+1}+1]\in[p_{1}+1,\ldots,p_{n}+1]\otimes[r+1].\end{array}$
(20)

In this formula some of the elements of Young diagrams can be equal to zero.
To prove this statement we have to explain which representations appear in the
tensor product of representations. Basically [2],
$[a_{1},\ldots,a_{n+1}]\in[p_{1},\ldots,p_{n}]\otimes[r]$ if and only if

$\left\\{\begin{array}[]{l}a_{i}\leq p_{i}+r\\\ p_{i-1}\geq a_{i}\geq p_{i}\\\
a_{1}\geq r\\\ \sum a_{i}=r+\sum p_{i}\end{array}\right.$ (21)

In fact the first line is superseded by the last one, since none of the
elements $a_{i}$ can become smaller than $p_{i}$, while the whole size is
changed by $r$. If we add or substract one from each element of Young diagrams
then all of these properties but the first one hold identically, which shows
that (20) indeed holds.

From (20) it follows that if we add one to each element of all the Young
diagrams, then the whole tree of representations, like on Figs. 2, 3, 4,
remains the same, and thus also the block structure of the Racah matrices
remains the same.

To explain the main claim of this paper, (18) and (19), we also need
normalized eigenvalues of the $\mathcal{R}$-matrices to coincide. If one takes
a product of two representations $[r]$ and representation $[2r-a,a]$ from this
product then the corresponding eigenvalue is equal to

$\lambda_{[2r-a,a]}=q^{\varkappa_{[2r-a,a]}-4\varkappa_{[r]}-Nr}=q^{a^{2}-2ra}q^{r-Nr}.$
(22)

If we add one to each element of Young diagram, then it becomes

$\begin{array}[]{r}\lambda_{[2r-a+1,a+1]}=q^{\varkappa_{[2r-a+1,a+1]}-4\varkappa_{[r+1]}-Nr}=\\\
=q^{a^{2}-2ra}q^{-r+1-Nr-N}.\end{array}$ (23)

Since we are interested in normalized eigenvalues, part independent of $a$
does not matter, while dependence on $a$ is the same. Thus the set of
normalized eigenvalues is also the same.

Therefore we showed that the block structure of Racah matrices remain the same
and the set of $\mathcal{R}$-matrix eigenvalues also remains the same if we
add one to each element of all the Young diagrams. Thus the claims (18) and
(19) also hold.

##  6 Conclusion

In this paper we discussed symmetries of Racah matrices and $6j$-symbols
coming from the multi-strand eigenvalue conjecture. This allowed us to write
down new symmetries of quantum $6j$-symbols, which are much harder to study
using simpler 3-strand eigenvalue conjecture. We provide new symmetries
involving non-symmetric representations. This is an important step on the way
to understand the structure of quantum Racah matrices and, therefore, to
understand representation theory of quantum groups. These Racah matrices then
can also be applied to many other problems in theoretical in mathematical
physics, especially in knot theory.

In this paper we discussed only braids with symmetric representations.
However, the cases with non-symmetric representations are much more
interesting. In this case there appear multiplicities in
$\mathcal{R}$-matrices, which means that some eigenvalues start coinciding.
This, of course, makes it much harder to understand the structure of Racah
matrices, due to additional freedom in equation (15). These matrices, however,
seem to also possess block-structure, as was described in [29]. This makes it
look similar to the Racah matrices appearing in this paper. Such connection
remain to be studied.

## Acknowledgements

We are grateful for very useful discussions to N. Tselousov and A. Sleptsov.

This work was supported by Russian Science Foundation grant No 20-12-00195.

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