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arxiv: 2606.26599 · v1 · pith:CT5RMMFDnew · submitted 2026-06-25 · 💻 cs.SI

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

Haoxin Sun , Zhongzhi Zhang This is my paper

Pith reviewed 2026-06-26 02:29 UTC · model grok-4.3

classification 💻 cs.SI
keywords forest matrixdiagonal computationvariance reductionspanning forestsWilson's algorithmsampling algorithmsdirected graphsnetwork analysis
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The pith

Variance-reduced sampling of spanning forests computes the forest matrix diagonal with relative error guarantees in linear time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The diagonal of the forest matrix carries useful information for network science and machine learning, yet existing fast Laplacian solvers cannot handle directed graphs. The paper first gives a probability interpretation that lets an extension of Wilson's algorithm sample spanning converging forests, yielding the basic SCF estimator. It then introduces two variance-reduction steps: matrix-vector iterations drawn from opinion dynamics produce SCFV, and a new iteration equation produces SCFV+ that removes the cross-product term from the variance expression. The resulting SCFV+ estimator is proved to meet a relative-error bound with high probability while running in time linear in the number of nodes.

Core claim

The paper establishes that SCFV+, obtained by applying a new variance-reduced iteration to the sampling of spanning converging forests, delivers a relative error guarantee with high probability for every diagonal entry of the forest matrix while preserving linear time complexity in the number of nodes, a stronger theoretical guarantee than that of prior Laplacian-solver methods.

What carries the argument

Variance-reduced sampling of spanning converging forests via an extension of Wilson's algorithm together with a new matrix-vector iteration that removes the cross-product variance term.

If this is right

  • The same linear-time procedure works on both directed and undirected graphs.
  • The method scales to graphs containing more than twenty million nodes.
  • Empirical accuracy exceeds that of previous algorithms on real-world networks.
  • No fast Laplacian solver is required, removing the directed-graph limitation.
  • Time complexity remains linear in the number of nodes even after the variance reductions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unbiased sampling primitive could be reused to estimate other entries of the forest matrix or related Green-function quantities.
  • If the iteration can be maintained under edge insertions or deletions, the approach might extend to dynamic or streaming graphs.
  • Because the variance reductions are matrix-vector operations, they may admit straightforward parallel or distributed implementations on very large graphs.
  • The same sampling-plus-iteration pattern might apply to other spectral quantities that admit forest or tree interpretations.

Load-bearing premise

The probability interpretation yields unbiased samples of spanning converging forests, and the two variance-reduction steps preserve that unbiasedness without adding new bias.

What would settle it

Execute SCFV+ on a moderate-sized directed graph, collect many independent runs, and check whether the observed fraction of estimates lying outside the claimed relative-error interval exceeds the failure probability stated in the high-probability bound.

Figures

Figures reproduced from arXiv: 2606.26599 by Haoxin Sun, Zhongzhi Zhang.

Figure 1
Figure 1. Figure 1: A toy digraph and one of its spanning converging [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scatter plot of maximum and average relative er [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scatter plot of maximum and average relative [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

The forest matrix of a graph, particularly its diagonal elements, has far-reaching implications in network science and machine learning. The state-of-the-art algorithms for the diagonal of forest matrix computation are based on the fast Laplacian solver. However, these algorithms encounter limitations when applied to digraphs due to the incapacity of the Laplacian solver. To overcome the issue, in this paper, we propose three novel sampling-based algorithms: SCF, SCFV, and SCFV+. Our first algorithm SCF leverages a probability interpretation of the diagonal of the forest matrix and utilizes an extension of Wilson's algorithm to sample spanning converging forests. To reduce the variance in forest sampling, we develop two novel variance-reduction techniques. The first technique, leading to the SCFV algorithm, is inspired by opinion dynamics in graphs and applies matrix-vector iteration to spanning forest sampling. While SCFV achieves reduced variance compared to SCF, the cross-product term in its variance expression can be complex and potentially large in certain graphs. Therefore, we develop another technique, leading to a new iteration equation and the SCFV+ algorithm. SCFV+ achieves further reduced variance without the cross-product term in the variance of SCFV. We prove that SCFV+ can achieve a relative error guarantee with high probability and maintain linear time complexity relative to the number of nodes in the graph, presenting a superior theoretical result compared to state-of-the-art algorithms. Finally, we conduct extensive experiments on various real-world networks, showing that our algorithms achieve better estimation accuracy and are more time-efficient than the state-of-the-art algorithms. Particularly, our algorithms are scalable to massive graphs with more than twenty million nodes in both undirected and directed graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes three sampling-based algorithms (SCF, SCFV, SCFV+) for estimating the diagonal of the forest matrix on directed and undirected graphs. SCF samples spanning converging forests via an extension of Wilson's algorithm using the probabilistic interpretation of the forest matrix. SCFV reduces variance via a matrix-vector iteration inspired by opinion dynamics. SCFV+ introduces a new recurrence to eliminate the cross-product variance term. The authors claim that SCFV+ achieves relative-error guarantees with high probability while retaining linear time complexity in the number of nodes, outperforming Laplacian-solver baselines, with supporting experiments on real-world networks up to >20M nodes.

Significance. If the unbiasedness of the SCFV+ estimator and the subsequent concentration argument are rigorously established, the work supplies the first linear-time, relative-error method for forest-matrix diagonals on digraphs (where Laplacian solvers do not apply). The explicit variance-reduction analysis and large-scale empirical validation would constitute a practical advance for network-science and ML tasks that rely on forest-matrix quantities.

major comments (2)
  1. [SCFV+ iteration equation] Section introducing the SCFV+ recurrence (the 'new iteration equation'): the relative-error guarantee with high probability rests on the estimator remaining unbiased (E[estimator] = forest-matrix diagonal). The manuscript must explicitly recompute the expectation under the altered recurrence and confirm that the sampling measure is unchanged; any deviation would invalidate both the variance bound and the concentration step used for the high-probability claim.
  2. [Theorem on relative-error guarantee] Proof of Theorem on relative-error guarantee for SCFV+: the argument must separately verify that the two variance-reduction iterations preserve unbiasedness while strictly lowering variance without introducing new bias terms. If the cross-product term removal alters the estimator functional, the claimed superiority over SCFV and prior methods cannot be maintained.
minor comments (2)
  1. Notation for the forest-matrix diagonal and the sampling probabilities should be introduced once with a single consistent symbol set; repeated redefinitions across sections reduce readability.
  2. [Experiments] Experimental section: the description of baseline implementations (fast Laplacian solver on undirected graphs) should state the exact solver library and tolerance used so that timing comparisons are reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comments below and will incorporate clarifications to strengthen the presentation of the unbiasedness arguments.

read point-by-point responses

  1. Referee: [SCFV+ iteration equation] Section introducing the SCFV+ recurrence (the 'new iteration equation'): the relative-error guarantee with high probability rests on the estimator remaining unbiased (E[estimator] = forest-matrix diagonal). The manuscript must explicitly recompute the expectation under the altered recurrence and confirm that the sampling measure is unchanged; any deviation would invalidate both the variance bound and the concentration step used for the high-probability claim.

    Authors: We agree that an explicit recomputation of the expectation is necessary for clarity. The SCFV+ recurrence is constructed so that the sampling measure over spanning converging forests remains identical to that of SCF; only the variance-reduction step is modified. In the revised manuscript we will add a dedicated lemma that recomputes E[SCFV+ estimator] directly from the new iteration equation, confirming it equals the forest-matrix diagonal with no change to the underlying sampling distribution. This will also explicitly link the preserved unbiasedness to the subsequent variance bound and concentration inequality. revision: yes

  2. Referee: [Theorem on relative-error guarantee] Proof of Theorem on relative-error guarantee for SCFV+: the argument must separately verify that the two variance-reduction iterations preserve unbiasedness while strictly lowering variance without introducing new bias terms. If the cross-product term removal alters the estimator functional, the claimed superiority over SCFV and prior methods cannot be maintained.

    Authors: The proof already separates the two iterations: the SCFV matrix-vector step is shown to be an unbiased linear transformation of the original forest samples, and the SCFV+ recurrence is derived as an algebraic rearrangement that cancels the cross-product term while leaving the marginal expectation unchanged. We will expand the proof to include an explicit side-by-side comparison of the three estimators (SCF, SCFV, SCFV+) demonstrating that unbiasedness is preserved at each step and that variance is strictly reduced. No alteration to the estimator functional occurs that would introduce bias; the superiority claim therefore remains valid. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on independent probabilistic sampling and proofs

full rationale

The paper derives SCF/SCFV/SCFV+ from a probability interpretation of the forest-matrix diagonal, an extension of Wilson's algorithm for sampling spanning converging forests, and two variance-reduction iterations whose unbiasedness is asserted and used to prove relative-error guarantees. No equation reduces a claimed result to a fitted parameter or self-citation by construction; the central claims (unbiased estimator, variance bounds, high-probability guarantee, linear time) are presented as proven consequences of the sampling measure and iteration definitions rather than tautological renamings or self-referential fits. This is the normal case of an algorithmic paper whose correctness can be checked externally against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on a probabilistic interpretation of the forest-matrix diagonal and an extension of Wilson's algorithm; no free parameters, invented entities, or non-standard axioms are explicitly introduced.

axioms (1)
  • domain assumption The diagonal of the forest matrix admits a probability interpretation that enables unbiased sampling of spanning converging forests via an extension of Wilson's algorithm.
    Invoked as the foundation for the SCF algorithm in the abstract.

pith-pipeline@v0.9.1-grok · 5828 in / 1245 out tokens · 48026 ms · 2026-06-26T02:29:58.044584+00:00 · methodology

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