Efficient Computation for Diagonal of Forest Matrix via Variance-Reduced Forest Sampling

Source: arXiv:2606.26599 · Published 2026-06-25 · By Haoxin Sun, Zhongzhi Zhang

TL;DR

This paper addresses the problem of efficiently computing the diagonal elements of the forest matrix Ω = (I + L)^{-1} for large graphs, including directed graphs, where existing state-of-the-art approaches rely on fast Laplacian solvers that do not extend to digraphs. The authors propose three novel sampling-based algorithms—SCF, SCFV, and SCFV+—that exploit a probability interpretation of the diagonal of the forest matrix and leverage uniform spanning converging forest sampling via an extension of Wilson's algorithm. SCF uses basic uniform sampling; SCFV introduces a variance reduction technique inspired by opinion dynamics modeled by a matrix-vector iteration; finally, SCFV+ proposes a new iteration equation that removes complex cross terms in variance, yielding a tighter theoretical bound and better practical performance. SCFV+ achieves a relative error guarantee with high probability and linear time complexity O(n) in number of nodes, improving over prior methods' theoretical and practical efficiency. Experiments on multiple real-world networks, including massive graphs with over 30 million nodes and over 300 million edges, demonstrate that these approaches outperform previous state-of-the-art baselines in both accuracy and efficiency for both directed and undirected graphs. The methods thus enable scalable diagonal forest matrix estimation where fast Laplacian solvers cannot be applied.

Key findings

• SCF algorithm estimates the diagonal with time complexity O(ln), where l is the sampling number (Lemma 4.2).
• SCFV reduces variance over SCF by incorporating a matrix-vector iteration inspired by the Friedkin-Johnsen model (Lemma 5.2).
• SCFV+ further reduces variance by removing complex cross-product terms, enabling a relative error guarantee with probability ≥ 1 - δ and fixed sample complexity independent of graph size or structure (Theorem 6.3).
• SCFV+ achieves variance ratio Var{ω_ii}/ω_ii² ≤ 1/8, key to the constant sample complexity result (Lemma 6.2).
• State-of-art baselines JLT and UST rely on fast Laplacian solvers and cannot handle directed graphs; proposed methods apply to both directed and undirected graphs.
• Experiments on 9 undirected datasets and 8 directed datasets, including a graph with 33 million nodes and 301 million edges, confirm superior estimation accuracy and time efficiency of SCF, SCFV, and SCFV+ compared to JLT and UST.
• All three proposed algorithms scale linearly with the number of sampled forests l and the number of nodes n, making them practical for massive networks.
• The probabilistic interpretation of the diagonal of forest matrix as root probability in a uniform spanning converging forest underpins the sampling approach.

Threat model

n/a — The paper is research on algorithmic estimation methods for graph matrices without adversarial or attacker considerations. The primary challenges are computational efficiency and accuracy under randomness from sampling, not security threats.

Methodology — deep read

1. Threat Model & Assumptions: The paper considers the computational problem of estimating diagonal entries ω_ii of the forest matrix Ω=(I + L)^(-1) for large digraphs. No adversarial threat is modeled; focus is on efficiency and accuracy under randomness from sampling.

2. Data: The evaluations use multiple publicly available real-world datasets from KONECT and SNAP comprising undirected and directed graphs with node counts ranging from ~16K to over 33M and edges from ~25K to over 300M. Ground truth diagonal values are computed via direct inversion or conjugate gradient solver depending on scale.

3. Architecture / Algorithm: Three algorithms are proposed.

• SCF: Uses a probability interpretation ω_ii = P(node i is a root in a uniform spanning converging forest). An extension of Wilson's algorithm samples l spanning converging forests uniformly. The average indicator of i being a root estimates ω_ii. It runs in O(ln).
• SCFV: Improves on SCF by exploiting the Friedkin-Johnsen opinion dynamics model, encoding opinion updates via z(t+1) = s_i + sum_{j in neighbors} z_j(t) / (1 + d_i). Initial opinion s set from l sampled forests to reduce variance. Running a single iteration yields a new unbiased estimator eω_ii with reduced variance.
• SCFV+: Further generalizes the iteration by inverting opinion flow direction, leading to a new iteration scheme for the i-th row of Ω and estimator ω_ii with variance lacking complex cross-product terms. This permits proving a relative error guarantee with high probability and bounded variance ratio.

4. Training Regime: Not applicable; sampling algorithms run with configurable sampling number l. Hyperparameters l and error bounds ε, δ analyzed theoretically and chosen accordingly.

5. Evaluation Protocol: Benchmarked against JLT and UST on datasets with measured relative estimation error and runtime. Experiments include ablations comparing SCF, SCFV, and SCFV+ methods. Theoretical results use Chernoff bounds and variance analysis for sample complexity and error guarantees.

6. Reproducibility: Algorithms are described in pseudocode (Algorithms 1-3). Datasets are public. Implementation details mention Julia programming language on CPU. Code release status not stated. Evaluation clearly defined with fixed seeds and single-threaded runs.

Example End-to-End: For estimating ω_ii, SCF samples l spanning converging forests using extended Wilson’s algorithm, records indicator whether node i is root, averages over samples forming unbiased estimator. SCFV then initializes opinion vector estimates from multiple SCF samples and applies one iteration of opinion dynamics iteration to compute a refined ω_ii estimator with provably smaller variance. SCFV+ adjusts adjacency direction in iteration, yielding even cleaner variance bounds and a practical algorithm that requires fewer samples for a fixed relative error.

Technical innovations

• Introduces three novel sampling-based algorithms (SCF, SCFV, SCFV+) for estimating diagonal elements of the forest matrix applicable to directed graphs, overcoming limitations of Laplacian solver-based methods.
• Develops a variance reduction technique inspired from opinion dynamics (Friedkin-Johnsen model) applied to spanning forest sampling to derive SCFV, reducing estimator variance over naive sampling.
• Proposes a new iteration equation reversing opinion flow direction to eliminate complex cross-product variance terms, enabling SCFV+ with provable relative error guarantees and constant sample complexity independent of graph size.
• Extends Wilson's algorithm to sample uniform spanning converging forests in directed graphs, enabling probabilistic interpretation and unbiased estimators for forest matrix diagonal entries.

Datasets

• web-webbase-2001 — 16,062 nodes, 25,593 edges — KONECT
• soc-gemsec-RO — 41,773 nodes, 125,826 edges — KONECT
• tech-p2p-gnutella — 62,561 nodes, 147,878 edges — KONECT
• tech-RL-caida — 190,914 nodes, 607,610 edges — KONECT
• soc-twitter-follows — 404,719 nodes, 713,319 edges — KONECT
• soc-delicious — 536,108 nodes, 1,375,961 edges — KONECT
• dblp — 5,624,219 nodes, 12,282,055 edges — SNAP
• livejournal — 7,489,073 nodes, 112,307,315 edges — SNAP
• delicious — 33,777,767 nodes, 301,183,342 edges — SNAP
• wikipedialinks — 17,649 nodes, 296,918 edges — SNAP
• p2p-gnutella31 — 62,586 nodes, 147,892 edges — KONECT
• email-euall — 265,009 nodes, 418,956 edges — SNAP
• web-Stanford — 281,903 nodes, 2,312,500 edges — SNAP
• web-Google — 875,713 nodes, 5,105,039 edges — SNAP
• northwestUSA — 1,207,945 nodes, 2,820,774 edges — SNAP
• wikitalk — 2,394,385 nodes, 5,021,410 edges — SNAP
• greatlakes — 2,758,119 nodes, 6,794,808 edges — SNAP
• fullUSA — 23,947,347 nodes, 57,708,624 edges — SNAP

Baselines vs proposed

• JLT: relative error bound with time complexity O(m ε^{-2} log^{2.5} n log(1/ε) polyloglog(n)) on undirected graphs; proposed SCFV+ achieves relative error with sample number independent of graph size.
• UST: absolute error guarantee with time O(m ε^{-2} log^{3/2} n); SCFV+ outperforms in time and applies to directed graphs.
• On undirected graphs, SCFV+ achieves lower relative estimation errors than JLT and UST across 6 medium-size datasets.
• On directed graphs, SCF, SCFV, and SCFV+ provide accurate diagonal estimates where JLT and UST cannot operate due to Laplacian solver limits.
• Runtime experiments show SCFV+ is fastest among proposed algorithms, scaling linearly with graph size and number of samples, enabling graphs up to over 30 million nodes.

Limitations

• Theoretical guarantees focus on nodes with diagonal elements ω_ii above a threshold σ, but behavior for very low ω_ii nodes is less analyzed.
• Variance reduction in SCFV depends on complex cross-product terms hard to bound, addressed only in SCFV+ with a new iteration equation; how this generalizes to deeper iterations is unclear.
• Experiments run single-threaded; potential parallelization and distributed scalability are not discussed.
• Code release and reproduction instructions are not explicitly mentioned, limiting immediate adoption.
• The approach relies on uniform sampling of spanning converging forests via Wilson's algorithm extension; sampling efficiency on graphs with extreme degree distributions or pathological structures is not extensively studied.
• Analysis assumes unweighted simple graphs; extensions to weighted or dynamic graphs remain open.

Open questions / follow-ons

• Can the new SCFV+ estimator and iteration approach be extended beyond a single iteration to achieve even lower variance or faster convergence?
• How does the method perform on weighted, dynamic, or temporal graphs where edge weights or structures change over time?
• Can the sampling and variance reduction techniques be parallelized or distributed to further scale computing on massive graphs?
• What are the robustness properties of these estimators against graph irregularities such as heavy-tailed degree distributions or disconnected components?

Why it matters for bot defense

For engineers working on bot defense and CAPTCHA systems, the paper introduces efficient and theoretically grounded methods to compute diagonal elements of the forest matrix for large and directed graphs. Since the forest matrix diagonal relates to node centrality and influence metrics, improved diagonal estimation can enable better graph-based user interaction modeling, anomaly detection, or behavioral tracking incorporating directionality of user interactions or referral edges. The sampling-based algorithms, especially SCFV+ with variance reduction and linear complexity, allow practical calculations on massive graphs that are common in large-scale bot detection networks. The applicability to directed graphs removes a common limitation of prior Laplacian-based solvers, broadening real-world usability. However, integration would require adapting the sampling process to evolving or weighted graphs typical in user-session networks. Overall, the results offer a promising avenue to improve feature extraction or graph embeddings for bot detection models using forest-matrix derived metrics.

Cite

@article{arxiv2606_26599,
  title={ Efficient Computation for Diagonal of Forest Matrix via Variance-Reduced Forest Sampling },
  author={ Haoxin Sun and Zhongzhi Zhang },
  journal={arXiv preprint arXiv:2606.26599},
  year={ 2026 },
  url={https://arxiv.org/abs/2606.26599}
}

Read the full paper

• arXiv:2606.26599 (abstract)
• arXiv:2606.26599 (PDF)