Quantum-Classical Link Prediction on Dark Web Hyperlink Graphs

Vojtěch Novák; Radovan Rečka; Ondřej Knebl; Aktham Youssef; Ivan Zelinka

cs.NIOct 23, 2026

Quantum-Classical Link Prediction on Dark Web Hyperlink Graphs

Authors: Vojtěch Novák, Radovan Rečka, Ondřej Knebl, Aktham Youssef, Ivan Zelinka

Organizations: Department of Computer Science, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, Ostrava, Czech Republic. · IT4Innovations National Supercomputing Center, VSB-Technical University of Ostrava, Ostrava, 708 00, Czech Republic. · Department of Informatics and Statistics, Marine Research Institute, Klaipeda University, Klaipeda, Lithuania.

Abstract

We analyze HTML content and hyperlinks from an offline Tor crawl containing 383 onion hosts and 2,067 directed cross-host links. The study combines semantic analysis with degree-controlled quantum--classical link prediction. Two directory groups emit 94.8% of the observed host links. Directed preferential attachment reaches ROC-AUC 0.976 with random non-edges, but falls to $0.550\pm0.014$ under source-conditioned target-degree matching. With disjoint training and test pairs, multimodal logistic regression reaches $0.672\pm0.021$ , GAE $0.899\pm0.016$ , and VGAE $0.906\pm0.016$ . A four-qubit variational quantum classifier (VQC) comparison tests two encodings across 20 disjoint-pair graph splits under exact and 1,024-shot simulation, with classical controls using the same inputs and training examples. Its best exact mean ROC-AUC is $0.549\pm0.023$ , compared with $0.551\pm0.016$ for RBF SVC and $0.573\pm0.029$ for a 16-parameter MLP on the same inputs and training examples. These results support predictive signal in combined graph/content features and show why degree control, feature compression, and optimization conditions matter in quantum--classical comparisons.

Figures & tables

Quantity	Value
HTTP-200 pages	1,000
Largest HTTP-200 weak component	998 pages
Directed page edges in core	13,515
Onion hosts in core	383
Distinct directed cross-host edges	2,067
Host-graph density	0.0141

Table 1: Summary of the analyzed dataset. Cross-host page edges count page-level hyperlinks, whereas host edges count distinct directed host pairs.

Text view	NMI	AMI	ARI
Raw visible	0.126	0.096	-0.008
Boilerplate clean	0.175	0.145	$\approx 0.000$
Boilerplate + porn filtered	0.175	0.145	$\approx 0.000$

Table 2: Agreement between seven content clusters and seven Louvain host communities.

Descriptive group	Hosts	Internal outgoing weight
French-language local/alternative media	19	—
Russian-language darknet portals/marketplaces	14	26.7%
Drug-related storefronts	13	—
OnionMail / encrypted mail	22	—
Tor navigation directories	8	0.0%
Tor search/link directories	47	6.4%

Table 3: Selected interpretable groups from the exploratory semantic partition. Labels summarize representative pages and top terms and are not ground-truth activity classes. A dash denotes no observed outgoing links, for which the internal fraction is undefined.

Model	Random	Topology-hard	Semantic-hard
Directed preferential attachment	$0.978\pm 0.005$	$0.985\pm 0.005$	$0.989\pm 0.006$
Graph logistic regression	$0.969\pm 0.006$	$0.982\pm 0.004$	$0.930\pm 0.035$
Multimodal logistic regression	$0.967\pm 0.007$	$0.980\pm 0.003$	$0.914\pm 0.032$
Raw-content logistic regression	$0.961\pm 0.004$	$0.981\pm 0.007$	$0.824\pm 0.032$
Boilerplate-clean content	$0.961\pm 0.006$	$0.980\pm 0.007$	$0.874\pm 0.046$
Boilerplate + porn filtered	$0.961\pm 0.005$	$0.980\pm 0.007$	$0.877\pm 0.046$

Table 4: Initial diagnostic benchmark. Values are mean ROC-AUC

\pm

standard deviation across five repeated splits. These regimes are reported as a bias diagnostic; the primary benchmark uses degree-matched negatives.

Negative sampler	PA ROC-AUC	$W_{1}$ source out	$W_{1}$ target in
Random	$0.9761\pm 0.0064$	3.146	0.951
Directed degree-corrected	$0.6648\pm 0.0218$	0.411	0.241
Source-conditioned degree-matched	$0.5498\pm 0.0136$	0.000	0.125

Table 5: Degree-bias audit across 20 held-out splits.

W_{1}

denotes first Wasserstein distance between positive and negative log-degree distributions.

Figure 1: Directed preferential-attachment ROC-AUC under increasingly degree-controlled negative sampling. Error bars show standard deviation over 20 splits.

Model	ROC-AUC	Average precision
Directed multimodal VGAE	$0.9058\pm 0.0159$	$0.8778\pm 0.0230$
Directed multimodal GAE	$0.8986\pm 0.0158$	$0.8673\pm 0.0280$
Multimodal logistic regression	$0.6717\pm 0.0212$	$0.6458\pm 0.0181$
8-D RBF SVC	$0.6429\pm 0.0176$	$0.6205\pm 0.0174$
Content logistic regression	$0.6327\pm 0.0137$	$0.6298\pm 0.0129$
Graph logistic regression	$0.6253\pm 0.0271$	$0.6130\pm 0.0223$

Table 6: Primary source-conditioned target-degree-matched benchmark with disjoint training and test pairs. Values are mean

\pm

standard deviation across 20 graph splits.

Figure 2: Selected flat-feature models under the primary source-conditioned degree-matched protocol. Error bars show standard deviation over 20 graph splits.

PCA dim.	Explained var.	Logistic regression	MLP-16	RBF SVC
2	0.121	$0.515\pm 0.007$	$0.512\pm 0.007$	$0.563\pm 0.019$
4	0.177	$0.511\pm 0.006$	$0.512\pm 0.010$	$0.620\pm 0.018$
6	0.223	$0.515\pm 0.008$	$0.521\pm 0.018$	$0.636\pm 0.016$
8	0.262	$0.514\pm 0.008$	$0.514\pm 0.008$	$0.643\pm 0.018$
12	0.315	$0.511\pm 0.009$	$0.524\pm 0.024$	$0.667\pm 0.022$
16	0.359	$0.511\pm 0.010$	$0.524\pm 0.011$	$0.691\pm 0.021$

Table 7: Sensitivity of classical models to PCA dimension under the primary protocol. Explained variance and ROC-AUC are means over 20 graph splits; AUC values are reported as mean

\pm

standard deviation.

Figure 3: Classical ROC-AUC as the PCA dimension is varied. Error bars show standard deviation over 20 graph splits.

Model	MRR	H@1	H@3	H@10	Median rank
Preferential attachment	$0.1645\pm 0.0148$	0.070	0.143	0.367	16.3
Multimodal logistic regression	$0.0811\pm 0.0135$	0.012	0.058	0.218	30.0
GAE	$0.2013\pm 0.0382$	0.072	0.216	0.504	10.5
VGAE	$0.2137\pm 0.0394$	0.085	0.222	0.528	9.7

Table 8: Query-weighted filtered all-target ranking, averaged over 20 graph splits; GAE/VGAE are additional refits on the same splits.

Figure 4: Filtered ranking metrics for classical baselines and GAE/VGAE refits. Error bars show standard deviation over 20 graph splits.

Model / optimizer	Parameters	Exact ROC-AUC	1,024-shot ROC-AUC
VQC ( $R_{y}$ ) + COBYLA	16	$0.528\pm 0.010$	$0.524\pm 0.019$
VQC ( $R_{y}$ ) + SPSA	16	$0.520\pm 0.027$	$0.529\pm 0.016$
VQC (ZZ) + COBYLA	16	$0.549\pm 0.023$	$0.531\pm 0.014$
VQC (ZZ) + SPSA	16	$0.542\pm 0.019$	$0.538\pm 0.023$
MLP + COBYLA	16	$0.523\pm 0.013$	—
MLP + SPSA	16	$0.513\pm 0.010$	—

Table 9: Four-component comparison with disjoint training and test pairs. All models use the same balanced 256-pair training draws and full test pools. Values are mean

\pm

standard deviation across 20 graph splits after averaging two starts per split. Classical scores are reported once. RBF SVC is parameterized by its support vectors.

Figure 5: Clean four-qubit comparison. Error bars show standard deviation across 20 graph splits, with two starts averaged within each split. The classical reference lines use the same inputs and training draws.

Link Prediction

Link prediction estimates missing or future relationships in graphs, from social ties and knowledge-graph facts to biological interactions.

See all 46 papers

A2QTGN: Adaptive Amplitude Quantum-Integrated Temporal Graph Network for Dynamic Link Prediction
May 21, 2026Nouhaila Innan, M. Murali Karthick, Simeon Kandan Sonar +2Temporal GNNs Quantum Machine Learning
1eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE · 3Indian Institute of Technology Palakkad (IITPKD), Palakkad 678623, India · Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE
How Embeddings Shape Graph Neural Networks: Classical vs Quantum-Oriented Node Representations
Apr 16, 2026Nouhaila Innan, Antonello Rosato, Alberto Marchisio +1Quantum Machine Learning Quantum-Inspired ML
eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE · Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, Rome, Italy · Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE
Quantum Evolution Kernel for Node-Attributed Graphs
Oct 27, 2026Manuel Padovani, Alessandra Di Pierro, Claudia DaffaraQuantum Kernel Methods Graph Representation Learning
Department of Computer Science, University of Verona, Verona, Italy

May 21, 2026J/K move · Enter open · S save

A2QTGN: Adaptive Amplitude Quantum-Integrated Temporal Graph Network for Dynamic Link Prediction

Nouhaila Innan, M. Murali Karthick, Simeon Kandan Sonar, Vivek Chaturvedi+1

1eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE · 3Indian Institute of Technology Palakkad (IITPKD), Palakkad 678623, India · Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE

Temporal GNNs Quantum Machine Learning Hybrid Quantum-Classical ML Temporal Link Prediction

Open paper AskPDF arXiv

Dynamic link prediction is important for modeling evolving interactions in complex systems, including social, communication, financial, and transportation networks. Classical temporal graph models capture sequential dependencies, but they may struggle to represent concurrent and rapidly changing node-edge interactions in large dynamic graphs. We propose A2QTGN (Adaptive Amplitude Quantum-Integrated Temporal Graph Network), a hybrid quantum-classical framework that combines adaptive amplitude encoding with a Temporal Graph Network backbone. The proposed mechanism represents node interaction features as quantum states and selectively refreshes amplitude embeddings based on temporal activity, preserving stable node states while emphasizing meaningful structural changes. This design reduces unnecessary quantum re-encoding and improves temporal representation for link prediction. Experiments on five Temporal Graph Benchmark datasets show that A2QTGN achieves strong predictive and ranking performance across diverse dynamic graphs. Ablation studies confirm the importance of both the quantum embedding module and the adaptive update strategy, while hardware-aware inference using a noisy backend and limited real-device execution supports the feasibility of near-term quantum-assisted temporal graph learning.

Abstract

Link Prediction

Explore similar work

A2QTGN: Adaptive Amplitude Quantum-Integrated Temporal Graph Network for Dynamic Link Prediction

How Embeddings Shape Graph Neural Networks: Classical vs Quantum-Oriented Node Representations

Quantum Evolution Kernel for Node-Attributed Graphs