Complex & Intelligent Systems (2025) 11:271
https://doi.org/10.1007/s40747-025-01913-w
ORIGINAL ARTICLE
Hide and seek in transaction networks: a multi-agent framework for
simulating and detecting money laundering activities
Qianyu Wang1
· Wei-Tek Tsai3,4 · Tianyu Shi5 · Wang Tang6 · Bowen Du1,2
Received: 2 April 2025 / Accepted: 9 April 2025 / Published online: 6 May 2025
© The Author(s) 2025
Abstract
Detecting money laundering within financial networks presents a complex challenge due to the elusive behavior patterns of
laundering agents, often resulting in data gaps. In this research, we propose a ‘Multiverse Simulation’ framework using a
multi-agent system to generate synthetic datasets for anti-money laundering (AML) training and detection. This framework
creates diverse virtual worlds, each with unique parameters to represent varying levels of illicit activity, thus mimicking
the dynamics of money laundering and legitimate transactions. Our framework comprises two main types of agents: (1) the
Detector, trained to identify laundering signs, and (2) Transaction agents, divided into those involved in laundering and those
in legal transactions. These agents interact in a synthetic environment governed by rules that simulate real-world financial
behaviors, enabling the generation of complex, realistic data. In the hide-and-seek multiverse simulation, the Detector learns
to distinguish between licit and illicit transactions, a process refined by the evolving strategies of transaction agents to
avoid detection. This adversarial setup fosters the co-evolution of laundering techniques and detection methods, enhancing
system robustness. We demonstrate the efficacy of this approach by pre-training on synthetic cross-bank data, then evaluating
with real-world data from the Elliptic dataset. Our results show that transfer learning significantly improves AML system
performance, effectively bridging the gap between synthetic and authentic transaction patterns. The ‘Multiverse Simulation’
offers a scalable, dynamic approach to better understand and mitigate the gap between simulation and reality, contributing to
more resilient and intelligent AML solutions.
Keywords Anti-money laundering · Reinforcement learning · Multi-agent system · Transfer learning
Introduction
B Qianyu Wang
qianyu.w@outlook.com
Wei-Tek Tsai
tsai7@yahoo.com
Tianyu Shi
tianyu.s@outlook.com
Wang Tang
tangwang9527@gmail.com
Bowen Du
dubowen@buaa.edu.cn
1
2
3
State Key Laboratory of Software Development Environment,
Beihang University, No. 37 Xueyuan Rd, Beijing 100083,
China
School of Transportation Science and Engineering, Beihang
University, No. 37 Xueyuan Rd, Beijing 100083, China
School of Insurance and Economics, University of
International Business and Economics, No. 10 East Huixin St,
Beijing 100029, China
Money laundering, the process of detecting the origins of
illegal transactions, often involves a complex network of
transactions designed to obfuscate the trail of financial crime
[1]. The detection of money laundering is not only crucial for
upholding the integrity of global financial systems but is also
imperative in the fight against organized crime, terrorism,
and corruption. The urgency to detect and prevent money
laundering is underscored by its impact on economic stability and governance, where the lack of effective detection
4
School of Computer Science, Arizona State University, 336E
Orange St, Tempe, AZ 85281, USA
5
School of Computer Science, University of Toronto, 27
King’s College Circle, Toronto, ON M5S 1A1, Canada
6
School of Computer Science, University of Leeds,
Woodhouse Lane, Leeds, West Yorkshire LS2 9JT, UK
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mechanisms can erode public trust and facilitate the growth
of criminal enterprises.
Current frameworks for detecting money laundering are
primarily rule-based systems [2] or utilize graph learning
techniques [3], which have inherent limitations. One main
limitation is that the deep learning system often involves
complex interactions which makes it have to interpret [4, 5].
Secondly, these systems often suffer from a high volume of
false positives, lack the adaptability to evolving laundering
tactics, and are limited by the static nature of the rules upon
which they are built [6, 7]. Additionally, they struggle with
the sheer complexity and volume of modern financial data,
which necessitates a solution capable of dynamic learning
and adaptation.
Dataset construction is important for machine learningbased detection systems. The quality and realism of data are
pivotal in training models that can accurately differentiate
between normal and suspicious transactions. The challenge
with existing datasets [8] is twofold: (1) they are either too
sanitized, failing to capture the complexities of real-world
financial systems, or (2) they are too sparse, with insufficient
examples of laundering behavior (many unknown labels).
Moreover, privacy concerns and the sensitive nature of financial data often restrict the availability of such datasets for
academic and commercial research, leading to a scarcity of
robust models capable of detecting sophisticated laundering
schemes.
Considering the successful application of deep learning
technology in various fields [9, 10], and inspired by OpenAI’s
hide-and-seek framework [11] as illustrated in Fig. 1, our proposed framework addresses these challenges by introducing
a ’Multiverse Simulation’ model. This multi-agent approach
simulates various laundering scenarios within synthetic environments. This model generates rich datasets by enacting a
game-theoretic hide-and-seek dynamic among three types of
agents-launderers, innocents, and inspectors-each with local
observations and objectives. The launderers aim to conceal
their trails within the network, innocents conduct regular
transactions, and inspectors utilize advanced detection algorithms to uncover illicit activities.
In this mixed cooperative and competitive environment
setup, transaction agents (launderers and innocents) continuously adapt their strategies to either evade or follow detection,
respectively, while the detector agent evolves its detection
capabilities. The result is a co-evolutionary environment that
not only generates transaction data with realistic and varied
laundering patterns but also enhances the detector’s ability
to learn and identify complex laundering signatures.
By employing this simulation, we create datasets that
address the current lack of realistic and varied examples in
AML training sets, enabling the transfer-ability of the detector’s performance to real-world datasets, such as the Elliptic
dataset [8]. The “Multiverse Simulation” framework thus
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Complex & Intelligent Systems (2025) 11:271
presents an innovative solution to the current limitations in
money laundering detection, offering a scalable, adaptable,
and effective approach for the financial system simulation.
To summarize, in this paper, we introduce the strategic interaction and synthetic transaction simulation (SISTS)
framework, an innovative approach to the anti-money laundering (AML) problem. Our main contributions can be
summarized as below:
• We have developed SISTS, a dynamic and evolving synthetic environment that mimics the complex nature of
financial networks. This realistic virtual setting enables
the modeling of a wide range of transaction behaviors,
effectively capturing the subtleties and nuances of money
laundering activities.
• We have designed a multi-agent simulation framework in
which each agent-acting as a launderer, an innocent party,
or a detector-operates with distinct goals and strategies.
This system mimics an adversarial landscape that encourages the evolution of both money laundering strategies
and detection methodologies.
• By structuring the interactions of agents within a gametheoretic framework, we propose an adversarial training
strategy that facilitates the co-evolution of transaction
agents and detector agents. This approach is crucial for
developing sophisticated detection algorithms that are
resilient to the constantly evolving strategies of financial
criminals.
• Our proposed SISTIS framework is not only effective
within its synthetic dataset but also demonstrates capabilities for transfer learning in the real world. Models trained
in the SISTS environment can be fine-tuned with realworld data, such as the Elliptic dataset, thereby enhancing
the practical applicability of our detection mechanisms.
• We also enhance model interpretability through the
strategic interactions within SISTS. Each transaction and
its underlying behavior are generated with an inherent
rationale, contributing to the improved interpretability of
the detection models. This interpretability is critical for
understanding model predictions and providing insights
into the reasoning behind identified suspicious activities.
Related work
In this section, we provide a comprehensive review of
existing anti-money laundering (AML) systems that reveals
a reliance on rule-based and statistical methodologies,
alongside more contemporary machine learning approaches.
These methods, integrated within financial institutions, have
demonstrated varied levels of efficacy in detecting complex
money laundering activities.
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271
Fig. 1 Overview of our
proposed multiverse
multi-actors interaction
simulation framework
Anti-money laundering (AML) system overview
Numerous methodologies have been employed in the of
anti-money laundering, commencing with rule-based systems and evolving towards the utilization of data mining [12] and machine learning algorithms. Additionally,
certain researchers have adopted deep learning [13] models
to identify potential fraudulent activities in financial transactions, which has significantly enhanced the precision of
detection. With the development of blockchain system, some
researchers have also designed frameworks from the perspectives of software testing [14, 15] and model regulation [16,
17] to enhance the security, reliability, and integrity of financial data sources [18].
Despite advances in AML technologies, significant challenges persist [19–21]. High false positive rates, coupled
with a lack of adaptability to innovative laundering tactics,
impede the effectiveness of previous AML systems [22, 23].
The dynamic and evolving strategies employed by launderers necessitate more agile and intelligent detection methods
[24, 25]. The employment of synthetic data(e.g., [26, 27]) for
AML model training has been explored as a means to mitigate
data privacy concerns and the scarcity of real-world laundering examples [8]. Synthetic datasets have proven beneficial,
though their realism and applicability to real-world scenarios remain a subject of ongoing research [2, 28]. However,
how to better the multi-agent behavior is still a remaining
question.
Transfer learning in financial datasets
In the realm of anti-money laundering (AML) research, the
availability and quality of data emerge as significant challenges in the development of effective detection systems.
While traditional methods have leaned on real-world data to
refine and test AML models, instances such as the work by
Starnini et al. [29], offering noteworthy insights. This study
utilized data from a prominent Italian bank, encompassing
180 million transactions over 6 months, to pinpoint specific
laundering behaviors, termed smurfs or motifs. Despite the
extensive dataset, the research was hampered by limitations,
including a narrow focus on just a couple of patterns and the
data’s restricted access to the team of researchers, limiting
wider investigational opportunities.
In another related work, Harris et al. [30] analyzed transaction history summaries from 4469 Scotia Bank customers
in Canada, aiming to evaluate their laundering risk profile.
Though this study operates on a more modest scale, it further
highlights the issues of limited data access and the targeted
examination of laundering patterns [31].
To navigate around the constraints of data scarcity, the
trend has shifted towards generating synthetic data within
various fields. Synthetic data promises to mitigate the challenges of real data’s limited availability and privacy concerns [2]. Nonetheless, existing AML endeavors employing
synthetic data face their own set of issues, such as reliance
on real data for generating synthetic analogs, the absence of
authoritative labels for laundering activities, and the omission of essential attributes present in real transactional data,
underscoring the challenges in accurately modeling AML
scenarios.
The synthetic AML data landscape has embraced agentbased simulations, presenting a varied approach to modeling
laundering activities. AMLworld, in particular, distinguishes
itself with superior capabilities and a detailed modeling
of laundering patterns compared to earlier initiatives like
MLDP and AMLsim [2]. AMLworld’s methodology, which
highlights an increased transaction volume and realistic laundering rates, emphasizes the significance of scale and fidelity
in generating synthetic data for AML purposes.
Furthermore, the integration of graph neural networks
(GNNs) has brought a new perspective to AML efforts
through their adept handling of relational data. Li et al.
[32] introduced FlowScope, a novel approach that models
bank transactions as a multipartite graph to accurately detect
money laundering by tracing the complete flow of illicit funds
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Complex & Intelligent Systems (2025) 11:271
through account chains, significantly outperforming existing
methods in both synthetic and real-world data scenarios.
To summarize, it becomes evident that the AML field
needs innovative methods that incorporate synthetic data
and cutting-edge machine learning technologies like GNNs.
These strategies not only tackle the limitations presented by
real-world data but also provide scalable and flexible solutions for the detection and prevention of money laundering.
As the domain adaptation method continues to progress,
the fusion of synthetic data generation and GNNs promises
to significantly enhance the effectiveness and efficiency of
AML endeavors.
Multi-agent reinforcement learning (MARL)
Multi-agent reinforcement learning (MARL) [33, 34] extends
the single-agent reinforcement learning paradigm to environments where multiple agents interact simultaneously. In these
settings, each agent aims to learn a policy π i : O i → Ai that
maximizes its cumulative reward over time, considering both
the environment’s dynamics and the actions of other agents.
Formally, a MARL problem can be modeled as a partially
observable Markov decision process (POMDP), represented
by a tuple (S, A, P, R, O, N ), where S denotes the state
space, A the set of actions, P the state transition probability, R the reward function, O the observation space, and N
the number of agents. The objective for each agent i is to
maximize its expected cumulative reward, defined as:
max E
πi
γ
t
R (st , ati , at−i ) | π i
i
,
(1)
t=0
where γ ∈ [0, 1) is the discount factor, st ∈ S is the state at
time t, ati is the action taken by agent i, and at−i represents
the actions of all other agents. The challenge in MARL lies in
the agents’ need to learn cooperative or competitive strategies
based on their objectives and the nature of their interactions.
Graph attention networks (GAT)
Graph attention networks (GAT) introduce an attention
mechanism into the propagation step of graph neural networks, allowing nodes to dynamically assign importance to
their neighbors’ information. Given a graph G = (V , E)
with nodes vi ∈ V and edges (vi , v j ) ∈ E, a GAT layer
updates the representation of a node vi by aggregating features from its neighbors, weighted by attention coefficients
αi j . These coefficients are computed using a learnable func-
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αi j = exp(LeakyReLU(a T [Wh i Wh j ]))
,
T
k∈N (i) exp(LeakyReLU(a [Wh i Wh k ]))
(2)
where denotes concatenation, a is a learnable weight vector, W is a weight matrix applied to each node’s features
h, and N (i) is the set of neighbors of node i. GAT’s ability to model relational data makes it particularly suitable for
MARL scenarios, where understanding the complex relationships between agents is crucial for effective strategy
development.
Hierarchical relationship modeling
Preliminaries
∞
tion f of the features of the nodes involved:
In mixed cooperative-competitive environments, the relationships among agents can be complex and multi-faceted,
requiring a nuanced approach to model effectively. Hierarchical relationship modeling aims to capture these dynamics
by constructing a multi-layer graph structure that represents
different levels of interaction among agents. This approach
allows for a more granulated understanding of the agents’
roles and relationships, facilitating the development of strategies that consider both immediate local interactions and
broader global objectives. By integrating hierarchical relationship models with multi-agent reinforcement learning
(MARL) and graph attention networks (GAT), our framework seeks to leverage the strengths of both to address the
unique challenges presented by mixed environments. A hierarchical graph H = (V , E, L) can be defined, where L
represents the levels of hierarchy in the relationships among
agents.
The hierarchical graph is defined as follows:
H = (V , E, L)
(3)
where V represents the set of nodes (agents), E ⊆ V × V is
the set of edges (interactions), and L: V → N is a function
assigning levels to nodes.
The interaction between two agents vi , v j ∈ V is given by
an edge ei j ∈ E. This interaction can be weighted to represent the strength or capacity of interaction. The hierarchical
level of interaction li j between agents vi and v j is given by
the sum of their individual levels:
li j = L(vi ) + L(v j ).
(4)
The strength or influence of the relationship between two
agents can be modeled as:
si j = f (L(vi ), L(v j )), where f : N × N → R≥0 .
(5)
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The global objective is formulated as:
Oglobal =
max(L)
Ol ,
(6)
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networks (GAT) to model the nuanced relationships among
agents hierarchically, facilitating a more balanced and adaptable approach to learning in mixed cooperative-competitive
settings.
l=1
where Ol represents the objective contribution from level l.
The local objective for an agent at level l is:
l
(vi ) =
Olocal
g(si j , ei j ),
(7)
v j :L(v j )=l
where g: R≥0 × E → R being a function mapping the interaction strength and edge to a local objective score.
Agents develop strategies based on their hierarchical level:
Model framework
Our strategic interaction and synthetic transaction simulation
(SISTS) framework is designed to consider the essence of
mixed dynamics through a sophisticated hierarchical graph
structure. This structure not only captures local interactions
among agents but also integrates global information, enhancing strategic decision-making. The framework comprises the
following components:
Problem formulation
• Hierarchical relationship graph (HRG): As the transaction graph usually involves thousands of agents. A multilayered representation delineates the diverse interaction
levels among agents, distinguishing between cooperative
and competitive dynamics effectively.
• Graph attention networks (GAT): We dynamically prioritize the influence of neighboring agents’ actions based
on the current environmental context, thus sharpening
the focus on the most pertinent information for strategy
refinement.
• MARL with HRG and GAT integration: A fusion of
MARL principles with our hierarchical graph construct,
promoting the decentralized enactment of strategies that
are enlightened by both immediate local interactions and
overarching global insights.
In mixed cooperative-competitive environments, agent interactions exhibit complex dynamics, oscillating between cooperation with allies and competition against adversaries.
Traditional multi-agent reinforcement learning (MARL)
approaches often face challenges in balancing these intricate
interactions effectively, leading to strategies that may not
fully adapt to the dynamic nature of such environments [35].
These challenges are further compounded by issues such as:
The model framework is shown in Fig. 3, the SISTS framework aims to advance a more refined comprehension of agent
interplay within mixed environments, enabling the evolution
of flexible and effective strategies can model the intricacies of
cooperation and competition. This approach aims not only to
address the challenges identified but also to set a foundation
for further explorations into the potentials of graph-based
behavior learning in complex multi-agent systems.
• The difficulty of large-scale simulations in achieving converging behaviors due to the complexity and variability
of agent interactions.
• The challenge of simulating complex patterns that accurately reflect real-world scenarios, making it hard for
agents to learn realistic strategies.
• Problems related to gradient updating and relationship
modeling within a multi-agent context, which can hinder
the optimization of agent behaviors and decision-making
processes.
Agent design
l
× Oglobal → A,
π l : Olocal
(8)
where A is the set of possible actions.
The influence of an agent vi on another agent v j is propagated according to:
Ii j = α L(vi )−L(v j ) si j ,
(9)
where α ∈ (0, 1) is a decay factor representing the diminution of influence with distance in the hierarchy.
Proposed method
To overcome these obstacles as shown in Fig. 2, we introduce
a novel framework (SISTS) that employs graph attention
In the SISTS framework, agents are designed to operate
within a mixed cooperative-competitive environment, necessitating distinct roles and capabilities for interaction. Each
agent is characterized by its type (e.g., collaborator or
competitor), which informs its objectives and strategy formulation. Collaborators seek to maximize a shared objective,
while competitors aim to optimize their individual goals,
often at the expense of others.
Agents utilize a graph attention network (GAT) to dynamically understand and adapt to the environment and other
agents’ actions. This approach allows each agent to selec-
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Fig. 2 Overview of the problem
formulation. The main problem
is learning the behavior in
synthetic data framework and
trying to transfer the knowledge
into real data
Fig. 3 The overall architecture of our proposed framework named
SISTS; as the first step, the source task is learned using a skill learning
method through multi-agent graph reinforcement learning, and source
data is gathered for initial behavior learning. Besides, in the second step,
a domain adaptation approach is adapted to find a latent space where the
source and target data can be compared. The inspector is used to detect
money laundering behavior in this simulator. Both two types of agents
evolve together. In the final step, the agent uses this function to learn
raw target skills offline or to find inter-task mapping to leverage the
source learned skills. Then, the target task is learned using pre-learned
skills for money laundering activity detection
tively focus on the most relevant information from its surroundings, facilitating strategic decision-making that accounts
for both local interactions and the broader context of the environment.
representing agents and edges ei j ∈ E representing transactions, the state of an agent vi is defined as a feature vector
si . This vector encapsulates the detailed attributes of transactions such as amounts, frequencies, and historical patterns,
and is formulated as follows:
State, reward, and action definition
si = φ(ai , fi , hi ),
(10)
State representation with graph neural networks
The state in our simulation framework is a function of transaction attributes. For a graph G = (V , E), with nodes vi ∈ V
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where ai represents the amounts of transactions involving
agent vi , fi denotes the frequencies of these transactions,
and hi encapsulates the historical patterns. The function φ
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aggregates these attributes into a state representation that a
graph neural network can process.
For the graph neural network that captures the interactions
among agents, the feature representation of the transactional
interaction between agents vi and v j is defined by an edge
feature vector ei j , which is constructed as follows:
ei j = ψ(ti j , ri j ),
(11)
where ψ is a function that combines the transaction attributes
ti j , such as transaction types or categories, and the relationship strength or capacity ri j between agents vi and v j .
The overall state of the system is then represented by the
set of all node states and edge features:
N
, E = {ei j }i,N j=1 ,
S = {si }i=1
(12)
Reward
The reward function is designed to provide specific incentives
for agents based on their actions and the outcomes of these
actions. For a regulatory agent i, the reward at time t can be
formulated as:
⎧
⎪
+rdetect ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨−r
miss ,
i
Rr eg (st , at ) =
⎪
⎪
⎪
⎪
⎪
⎪−r f alse ,
⎪
⎪
⎪
⎩
if suspicious activity is
correctly identified,
if suspicious activity is
missed,
if normal activity is
is the reward for successfully executing a legitimate transaction, and rdetected is the penalty for a laundering transaction
that is detected.
The reward structure is thus carefully designed to reflect
the complex dynamics of the simulated financial ecosystem,
incentivizing agents to adopt strategies that align with their
roles-either to detect money laundering activities efficiently
or to conduct transactions in a manner that meets their objectives while evading detection.
Action
Our agent’s actions in the simulation framework are represented as vectors where each component corresponds to a
decision or a strategy an agent can take. For an agent vi , the
action vector ai is given by:
ai = [αi1 , αi2 , . . . , αik ],
where N is the number of agents in the system.
(13)
(15)
where each component αi j can represent different decisions, including initiating a transaction, choosing transaction
parameters, or investigating suspicious activities.
For instance, for a transactional agent, the action vector is
defined as :
ai = [initiate, amount, recipient, strategy],
(16)
where “initiate” is a binary decision to start a transaction or
not, “amount” quantifies the funds being transacted, “recipient” identifies the receiving party, and “strategy” represents
any tactic employed to conduct the transaction securely.
Similarly, for a regulatory agent, the action vector is
defined as :
wrongly flagged.
ai = [select_tx, analysis_depth, update_strategy],
Here, rdetect is the reward for correctly identifying a suspicious transaction, rmiss is the penalty for missing a suspicious
transaction, and r f alse is the penalty for incorrectly flagging
a normal transaction as suspicious.
For an agent j representing entities engaging in transactions (either legitimate or for the purpose of laundering
money), the reward can be defined as:
j
Rtrans (st , at ) =
⎧
+rsuccess ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨+r
,
legit
271
if a transaction is
successfully executed without detection,
if a legitimate transaction is
⎪
successfully executed,
⎪
⎪
⎪
⎪
⎪
⎪−rdetected , if a laundering transaction is
⎪
⎩
detected.
(17)
where “select_tx” indicates the transactions selected for
investigation, “analysis_depth” specifies the level of scrutiny
applied, and “update_strategy” involves adjustments to the
agent’s approach to detection.
The action space A for the entire system is the Cartesian
product of all individual action vectors:
A = A1 × A 2 × . . . × A N ,
(18)
where Ai is the set of all possible actions for agent vi , and
N is the number of agents.
Multi-agent graph learning with domain adaptation
(14)
In this equation, rsuccess is the reward for successfully executing a laundering transaction without being detected, rlegit
The main training framework is designed based on MADDPG [35]. To further model complex interactions and relationships among agents, and to facilitate knowledge transfer
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across domains in a graph-structured environment, we extend
our multi-agent graph learning framework by integrating a
domain adaptation mechanism. This mechanism, inspired
by disentanglement-based graph domain adaptation models (DGDA), utilizes graph attention networks (GAT) for
dynamic representation and incorporates a domain adaptation component to bridge domain discrepancies.
Graph attention networks (GAT)
We construct a dynamic graph where nodes represent agents,
and edges encode relationships and interactions. The GAT
mechanism allows agents to dynamically assign importance
to their neighbors based on the current context, optimizing
their actions through:
⎛
hi = σ ⎝
⎞
αi j Wh j ⎠ ,
(19)
j∈N (i)
where hi is the feature vector of agent i, N (i) denotes the
neighbors of agent i, W is a weight matrix, αi j represents
the attention coefficient between agents i and j, and σ is a
non-linear activation function.
Domain adaptation component
The domain adaptation component employs variational
graph auto-encoders to reconstruct and disentangle latent
variables into semantic information, domain characteristics,
and noise. The semantic latent variables (Zs ) aim to capture domain-invariant features, while domain latent variables
(Zd ) and noise (Zn ) represent domain-specific attributes
and data generation uncertainties, respectively. This disentanglement facilitates knowledge transfer by focusing on
domain-invariant information.
The domain adaptation loss, critical for minimizing
domain discrepancies between real and synthetic dataset, is
formulated as:
L D A = EZs ∼q(Zs |X,A) [log p(Yt |Zs )]
−β D K L (q(Zd |X, A)|| p(Zd )),
(20)
where X and A denote the feature matrix and adjacency
matrix of the graph, respectively, Yt is the target domain
label, q represents the encoded distributions of latent variables, p is the prior distribution of domain latent variables,
D K L denotes the Kullback–Leibler divergence, and β is a
hyperparameter balancing the two terms.
This domain adaptation loss guides the optimization process towards minimizing the difference between source and
target domain distributions, ensuring the transferability of
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the learned agent behaviors across different graph-structured
environments.
Adversarial training integration
Adversarial examples are generated by applying perturbations to the original graph data that maximize the prediction
error. These perturbations are constrained to ensure they do
not alter the semantic meaning of the input data significantly.
This adversarial loss is also inspired to mimic the real-world
dataset’s missing values. The adversarial loss for training the
graph attention networks is defined as follows:
Ladv = max E(X+δ,A) [log p(Yt |Zs (X + δ, A))],
δ∈
(21)
where δ represents small perturbations within the permissible
set , and X, A, and Yt are the feature matrix, adjacency matrix, and target labels, respectively. The function
Zs denotes the semantic latent variables affected by the perturbed inputs.
The combined loss function, incorporating domain adaptation and adversarial training, is then:
L = L D A + λLadv ,
(22)
where λ is a tuning parameter that balances the domain adaptation loss and the adversarial loss. The pseudo-code for the
algorithm is illustrated in Algorithm 1.
Algorithm 1 Multi-agent graph reinforcement learning with
GAT and adversarial training
Require: The number of agents N , maximum episodes M, learning
rate α, discount factor γ .
Ensure: Optimized policies for all agents.
1: Initialize graph attention network parameters for all agents θ1:N .
2: Initialize environment and get initial state s0 .
3: for episode = 1, M do
4:
Reset environment and get initial state s.
5:
Generate adversarial examples by perturbing s within limits .
6:
while not done do
7:
for each agent i = 1, N do
8:
Observe local state si from both s and adversarial s.
9:
Determine action ai using policy derived from GAT with
parameters θi .
10:
end for
11:
Execute actions a1:N in the environment.
12:
Observe next state s and reward r1:N .
13:
Update the graph based on s and a1:N .
14:
for each agent i = 1, N do
15:
Calculate expected reward Q(si , ai |θi ).
16:
Calculate target yi = ri + γ maxa Q(si , a |θi ).
17:
Update θi by minimizing combined loss L(θi ) = (yi −
Q(si , ai |θi ))2 + L D A + λLadv .
18:
end for
19:
s ← s.
20:
end while
21: end for
Complex & Intelligent Systems (2025) 11:271
Experiment setting
Our primary objective is to demonstrate the high quality of
our synthetically generated dataset for anti-money laundering (AML) purposes and its effectiveness in enhancing the
training outcomes of various detection models.
Implementation details
For our multi-agent graph learning framework enhanced with
domain adaptation capabilities, Our computational experiments were executed on an NVIDIA Tesla V100 GPU. Model
optimization was carried out using the Adam optimizer, with
a learning rate meticulously set at 1×10−3 . Each experiment
consistently utilized a batch size of 128. We also incorporated a dropout strategy within the graph attention networks
(GAT) layers, applying a dropout rate of 0.5. The architecture’s latent variables, representing semantic information,
domain characteristics, and inherent noise, were dimensionally configured to 256, 64, and 128, respectively. Moreover,
the balancing parameter β for the domain adaptation loss was
empirically determined to be 0.01. In the following section,
we will also evaluate the sensitivity of the selected hyperparameters. We summarized the parameters in the following
Table 1.
Dataset description
AML sim
Altman et al. [2] proposed the multi-agent simulator to
generate a synthetic dataset to model the transaction framework. Our agent is trained on the simulator built upon
AML sim. Using this simulator provides a robust foundation
for initializing our simulation framework because it generates synthetic datasets that realistically mimic real-world
financial transactions. This allows for a controlled testing
environment, essential for developing and evaluating antimoney laundering systems.
Elliptic dataset and Elliptic++ dataset
Weber et al. [36] proposed the Elliptic dataset which acquired
the data based on real-world transactions. Elmougy et al. [8]
further extend the Elliptic transaction dataset as Elliptic++
dataset, including more than 822 k Bitcoin wallet addresses
(nodes), each with 56 features, and 1.27 m temporal interactions. We apply the trained model to test on both Elliptic and
Elliptic++ ’s test datasets.
A brief comparison of those two datasets is illustrated
in Tables 2 and 3. We summarized the statistics of the two
datasets’ different features. We can find that their features
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have discrepancies, which makes sim-to-real transfer[2] difficult.
Baseline models
Graph domain adaptation method
To demonstrate the effectiveness of our proposed framework,
we picked up the following methods as baselines.
• Domain adversarial neural network (DANN) [37]: this
is an adversarial representation framework to distinguish
the domain labels and coupled with gradient reverse layers in an adversarial way.
• Graph domain adaptation A generative view (DGDA)
[38]: this framework proposes a disentanglement-based
unsupervised domain adaptation method for the graphstructured data, which applies variational graph autoencoders to recover these latent variables and disentangles them via three supervised learning modules
• Graph domain adaptation via theory-grounded spectral
regularization (GDA-SpecReg) [39]: this is a theorygrounded algorithm for graph domain adaptation by
deriving a model-based bound related to two spectral
properties of graph neural networks-spectral smoothness
and maximum frequency response, they demonstrated
that transferability can be improved across different
graph domains.
• Graph-relational domain adaptation (GRDA) [40]: it is
a graph-relational domain adaptation method that uses a
domain graph to flexibly align domains based on their
adjacency, enhancing the traditional adversarial learning
framework with graph-structured discriminators to better
handle varying domain relationships and demonstrating
improved performance on both synthetic and real-world
datasets.
Multi-agent methods
For multiagent comparison, we considered ISDAPGN [41],
which considers adam-type updates and importance sampling technique. In this method, they also established rigorous analysis of the convergence of the method. We considered
this as a baseline to compare the multiagent baseline for our
method.
Ablation of SISTS
In our study, we investigate the significance of various components by conducting ablation experiments. Each listed
ablation, denoted by an abbreviation, systematically removes
a specific component to evaluate its impact on our proposed
model:
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Complex & Intelligent Systems (2025) 11:271
Table 1 Parameters of multi-agent graph reinforcement learning
Parameter
Description
Value
MAX_EPISODES
Maximum number of episodes for training
1000
LEARNING_RATE
Learning rate for the Adam optimizer
0.001
DISCOUNT_FACTOR
Discount factor for future rewards
0.99
BATCH_SIZE
Number of samples per gradient update
128
DROPOUT_RATE
Dropout rate for regularization
0.5
LATENT_DIM
Dimensionality of the latent variables
256 (semantic), 64 (domain), 128 (noise)
BETA_DA
Balancing parameter for domain adaptation loss
0.01
ENTROPY_COEF
Coefficient for entropy bonus in policy learning
0.01
GRADIENT_CLIP
Maximum norm for gradient clipping
1.0
Table 2 Statistics of synthetic (AML Sim) & real Elliptic++ dataset
t is given as:
Dataset
AML sim
Elliptic++
# of nodes
19,773
859,254
# of edges
96,531
995,508
# of graphs
1000
2000
⎧
⎪
⎨+runi f or m , for any executed action,
i
(st , at ) = −runi f or m , if any action leads to
Rsimple
⎪
⎩
an undesired outcome.
Avg degree
4.88
1.16
Here, runi f or m is a flat reward for taking any action, which
does not distinguish between the types of transactions
conducted, whether they are legitimate or suspicious.
Table 3 Example feature summary of synthetic (AML Sim) & real
(Elliptic++) dataset
AML Sim
Elliptic++
CUSTOMER_ID
num_txs_as_sender
INIT_BALANCE
num_txs_as_receiver
COUNTRY
first_block_appeared_in
SENDER_ACCOUNT_ID
last_block_appeared_in
Experimental setup
Experiments are structured to assess two main aspects: the
fidelity of the synthetic dataset compared to real transaction
data and the performance improvement of detection models
trained on the synthetic dataset.
Dataset fidelity
• No multi-agent interaction (NMAI): Evaluating the
framework without the agents’ interactive dynamics to
understand the significance of agent interaction.
• No hierarchical relationship graph (NHRG): Observing
the system’s performance when the HRG, which captures
complex agent relationships, is removed.
• No graph attention networks (NGAT): Assessing the
framework by excluding the GAT to measure the impact
of attention-based feature weighting on agent decisionmaking.
• No domain adaptation (NDA): This is trying to understand the role of the domain adaptation module by
removing the domain adaptation loss component.
• Static reward function (SRF): Testing with a non-adaptive
reward signal to quantify how dynamic rewards contribute to the agents’ strategic evolution. The reward is
formulated as below. For any agent i, the reward at time
123
• Statistical analysis: we conduct a comprehensive statistical comparison between the synthetic and real datasets,
focusing on key features relevant to AML detection.
• Feature distribution matching: we use distance metrics
and distribution similarity scores to quantify the likeness
between the synthetic and real datasets’ feature distributions.
Model performance
• Training and validation: Train baseline models on both
the synthetic dataset and a real transaction dataset separately. Evaluate the models’ performance using standard
metrics such as accuracy, precision, recall, and F1 score.
• Transfer learning evaluation: Investigate the transferability of models pre-trained on synthetic data and fine-tuned
on real data, comparing the results with models trained
solely on real data.
Complex & Intelligent Systems (2025) 11:271
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271
Evaluation metrics
In our experiment, we used the AUC-ROC curve to assess the
model’s ability to distinguish between classes under various
threshold settings. We selected Jensen–Shannon divergence
to measure the similarity between the feature distributions of
synthetic and real datasets. To ensure fairness and monitor
the experiment’s progress smoothly, we employed standard
metrics such as accuracy, precision, recall, and F1-score to
evaluate the performance of all anti-money laundering and
illegal transaction detection methods. The calculations for
these standard metrics are presented in the equation.
2·TP
2 · T P + FN + FP
TP +TN
Accuracy =
T P + T N + FP + FN
TP
Pr ecision =
T P + FP
TP
Recall =
T P + FN
F1 =
(23)
(24)
(25)
Fig. 4 Comparison of different agents’ convergence. While fluctuations
are present due to the dynamic interactions among agents, the overall
performance metrics show a trend toward stabilization. These fluctuations represent adaptive strategy shifts rather than strict convergence,
highlighting the agents’ ability to adjust in response to adversarial conditions. By extending the training to more epochs, we observed that the
fluctuations decreased, indicating that the agents’ learning processes
are approaching equilibrium
(26)
where T P, T N , F P and F N indicate true positive, true negative, false positive, false negative, respectively.
Result analysis
Our main research questions for the experiments are:
Research questions
Given the emphasis on simulating realistic transaction data
for anti-money laundering (AML) purposes and the need to
highlight the capability of our model, we formulate the following research questions:
• RQ1: How does the model ensure the simulation of realistic transaction feature distributions in the generated
dataset?
• RQ2: What is the transferability of the detector trained
on the simulated dataset to real-world datasets?
• RQ3: In what ways can improvements in feature simulation contribute to enhancing the detection capabilities of
AML systems?
• RQ4: How does the model adapt to varying degrees of
complexity and subtlety in laundering behavior within
transaction datasets?
Performance comparison and analysis
Learning performance evaluation
Firstly, in the training phase, we compare different agents’
convergence plots, as shown in Fig. 4, and we can find that the
agents’ learning is converged. The plot shows the reward trajectories for the detector, innocent, and launder agents across
training epochs. While fluctuations are observed, particularly
in the early stages, these fluctuations gradually diminish over
time, indicating that each agent is adapting and stabilizing its
strategy within the competitive environment.
As shown in Fig. 5, it reflects the illicit/licit agent ratio and
the detection rate. Initially, the illicit/licit agent ratio starts
high and gradually decreases, suggesting an improvement
in the model’s ability to distinguish between illicit and licit
behaviors as training progresses. Conversely, the detection
rate shows a consistent upward trend and converged until
the end, indicating enhanced detector performance and accuracy in identifying illicit activities. These trends confirm that
the training effectively enhances the model’s discrimination
capabilities, leading toward more accurate and reliable detection in simulated environments. This improvement is critical
for deploying such models in real-world applications.
Performance comparison with baselines
Then, we compare the performance of our proposed method
against various baseline models using established evaluation metrics. As shown in Table 4, our method achieves
optimal results across all metrics-including accuracy, preci-
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sion, recall, F1 score, and AUC-ROC-compared to recent and
traditional baseline methods such as DANN, DGDA, GDASpecRe, GRDA, and ISDAPGN. This demonstrates the
effectiveness of our approach in domain adaptation for graphstructured data. Additionally, the lowest Jensen–Shannon
(JS) divergence among the methods indicates greater diversity in our simulations. Furthermore, we introduce the
Elliptic++ dataset to test our method. Once again, our method
outperforms the baseline models on all metrics. The results
are shown in Table 5.
Ablation study and sensitive analysis
We furthermore compare our full model against several
ablated versions to demonstrate the impact of individual components on overall performance. The Table 6, summarizes the
results.
We find that the full model outperforms each baseline condition, which demonstrate the importance of each component
in achieving high performance across all metrics.
As shown in Table 7, we created various scales of the
dataset by adjusting the number of nodes and edges to
assess model performance under different data complexities. The sensitivity analysis demonstrates the robustness of
our proposed method compared to several baseline models, including DGDA, DANN, GDA-SpecRe, GRDA, and
ISDAPGN. Our method consistently achieved the highest
F1 scores across different scales, showcasing stability and
resilience in handling datasets with varying sizes. In comparison, the second-best baseline, DGDA, demonstrated close
but slightly lower performance, while other baselines such
as DANN and GRDA showed more significant drops in F1
scores as the data complexity increased. This trend suggests
Fig. 5 Comparison of illicit/licit
agent ratios and detection rate
across training epochs. The
model is validated every 10
epochs using saved checkpoints,
showing the progression and
stabilization of detection
performance over time. While
the detection rate plateaus rather
than reaching 100%, this reflects
the competitive environment
where laundering agents adapt
their strategies to evade
detection, creating a realistic
balance that mirrors practical
anti-money laundering scenarios
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Complex & Intelligent Systems (2025) 11:271
that our model not only adapts well to increased data volume but also maintains robust accuracy, precision, and recall
across various conditions.
As shown in Fig. 6, The histograms compare the distribution of features between the real and synthetic datasets,
highlighting the fidelity of the synthetic data generation process. This figure shows that the feature generated aligns well
between the synthetic dataset and the real dataset.
With the random seed now fixed for both the proposed
model and the baseline, we’ve re-generated the plot showing how each model’s F1 score varies with increasing data
complexity. This control ensures that each run produces consistent results, allowing for a fair comparison.
As depicted in Fig. 7, the proposed model demonstrates
superior performance compared to the baseline across all levels of complexity. This consistent outperformance, especially
under higher noise levels, underlines the proposed model’s
robustness and its capability to handle complex scenarios
more effectively than the baseline model. This visualization
solidly supports the advantage of the proposed method in
challenging conditions.
Conclusion and discussion
In this research, we presented a novel approach to simulating
and detecting money laundering activities within financial
networks through the “Multiverse Simulation” framework.
Utilizing a multi-agent system, this framework generates
synthetic datasets that closely resemble real-world transaction patterns, addressing the crucial gap between synthetic
simulations and their real-world applicability in anti-money
laundering (AML) systems.
Complex & Intelligent Systems (2025) 11:271
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Table 4 Performance comparison of our method against baseline models on elliptic dataset
Metric
Our method
DANN [37]
DGDA [38]
GDA-SpecRe [39]
GRDA [40]
ISDAPGN [41]
Accuracy (%)
95.2
92.0
95.1
94.0
93.2
94.5
Precision (%)
94.5
91.0
92.7
93.8
92.9
93.2
Recall (%)
93.8
90.5
92.1
93.5
92.0
92.8
F1 score
94.1
90.7
92.4
93.6
92.3
93.0
AUC-ROC
0.981
0.956
0.969
0.975
0.960
0.970
Jensen–Shannon divergence
0.06
0.12
0.09
0.07
0.11
0.08
Bold values indicate the best scores for each performance metric
Table 5 Performance comparison of our method against baseline models on Elliptic++ Dataset
Metric
Our method
DANN [37]
DGDA [38]
GDA-SpecRe [39]
GRDA [40]
ISDAPGN [41]
Accuracy (%)
94.0
91.0
93.5
92.0
91.8
93.0
Precision (%)
93.2
90.2
91.3
92.5
91.5
92.0
Recall (%)
92.8
89.7
90.8
92.2
90.8
91.5
F1 score
93.0
89.9
91.0
92.3
91.0
91.8
AUC-ROC
0.970
0.950
0.960
0.965
0.955
0.965
Jensen–Shannon divergence
0.08
0.14
0.11
0.09
0.13
0.10
Bold values indicate the best scores for each performance metric
Table 6 Ablation experiment of the proposed full models
Metric
Full model
NMAI
NHRG
NGAT
NDA
SRF
Accuracy (%)
95.2
90.1
91.3
92.6
93.8
89.5
Precision (%)
94.5
89.2
90.4
91.7
93.0
88.7
Recall (%)
93.8
88.9
90.1
91.4
92.7
88.3
F1 score
94.1
89.0
90.2
91.5
92.8
88.5
AUC-ROC
0.981
0.948
0.957
0.965
0.976
0.942
Jensen–Shannon divergence
0.06
0.12
0.11
0.09
0.07
0.13
GRDA
ISDAPGN
Bold values indicate the best scores for each performance metric
Table 7 Sensitivity analysis on different scales relative to full scale for our method and baseline models
Number of nodes/edges
10,000/25,000
100,000/250,000
500,000/1,250,000
Metric
Our method
DGDA
DANN
GDA-SpecRe
Accuracy (%)
91.5
90.8
89.7
90.2
89.5
90.0
Precision (%)
90.9
90.2
89.2
90.0
89.0
89.8
Recall (%)
90.3
89.7
88.5
89.8
88.9
89.3
F1 score
90.6
90.0
88.9
89.9
88.8
89.6
Accuracy (%)
93.0
92.5
91.5
92.0
91.3
92.2
Precision (%)
92.7
92.1
91.0
91.8
90.9
91.5
Recall (%)
92.4
91.9
90.8
91.5
90.7
91.2
F1 score
92.5
92.0
90.9
91.6
90.8
91.3
Accuracy (%)
94.0
93.8
92.8
93.5
92.9
93.3
Precision (%)
93.7
93.6
92.5
93.3
92.6
93.0
Recall (%)
93.5
93.4
92.3
93.1
92.5
92.9
F1 score
93.6
93.5
92.4
93.2
92.6
93.1
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Fig. 6 Comparison of feature distributions between the synthetic and
real datasets. The x-axis represents the normalized range for each feature, transformed to a consistent scale for direct comparison, while
the y-axis shows the frequency of occurrences within each range.
Complex & Intelligent Systems (2025) 11:271
Selected features reflect a combination of high importance to the
model’s decision-making and strong alignment between datasets, indicating that our simulated agents’ behavior is close to the real dataset
Fig. 7 Comparison of different
noise levels for different
methods
Our investigation into the framework’s effectiveness
yielded several key findings: (1) Our proposed framework
can generate synthetic datasets that realistically mimic the
dynamics of real-world financial transactions, including
complex laundering patterns. (2) We also demonstrate the
effectiveness of AML systems trained on these synthetic
datasets, particularly when these models are further applied
to real-world data, highlighting the promising impact of
our domain adaptation techniques. (3) The integration of
adversarial and domain adaptation losses, facilitated by the
interactions between transaction and detection agents, leads
to the co-evolution of more sophisticated laundering and
123
detection strategies. This enhancement enables our framework to outperform other baselines.
However, in our simulation, we assume that agents act
rationally and according to defined strategies. In contrast,
real-world actors might not always behave rationally, which
could limit the model’s effectiveness. Additionally, the
complexity of the multi-agent system and the interactions
between agents can make it challenging to fully understand
and explain the model’s decisions. We aim to further investigate the explainability of our method. Furthermore, while we
currently apply the proposed method only in a money laundering setting, exploring more diverse datasets, such as those
for credit card fraud detection, could also prove beneficial.
Complex & Intelligent Systems (2025) 11:271
In the future, we will extend the framework’s application
across diverse financial systems and regulatory environments, testing its generalizability and adaptability. Additionally, we plan to explore scalability and complexity
management techniques to ensure sustained performance
across larger and more intricate financial networks. And we
will further explore the integration of blockchain technology
and other emerging technologies to enhance the framework’s
detection capabilities and the overall security of financial
transactions.
In conclusion, the ‘Multiverse Simulation’ framework
marks a significant advancement in the domain of AML
research. It not only enhances the accuracy and robustness
of current detection systems but also lays the groundwork
for future innovations in financial security and integrity. As
the financial landscape continues to evolve, the strategies
employed by launderers will necessitate ongoing research
and development in AML technologies to safeguard global
financial systems.
Acknowledgements This work is supported by Beijing Municipal Science and Technology Plan Project (Z241100001324025).
Author Contributions Qianyu Wang: data curation, methodology, conceptualization, software, visualization, writing—original draft, writing—
review and editing. Wei-Tek Tsai: supervision, funding acquisition.
Tianyu Shi: validation, writing - original draft, writing—review and
editing. Wang Tang: software, writing—review and editing. Bowen Du:
resources, supervision.
Data availability The authors confirm that all relevant data and datasets
used in the analysis during this study are available from the corresponding author upon reasonable request.
Declarations
Conflict of interest There is no conflict of interest in this study. All
authors agree to publish.
Open Access This article is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License,
which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted
material derived from this article or parts of it. The images or other
third party material in this article are included in the article’s Creative
Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons
licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission
directly from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by-nc-nd/4.0/.
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