Fast Estimation for Forest Matrix of Signed Graphs

Haoxin Sun; Zhongzhi Zhang

arxiv: 2606.26608 · v1 · pith:YMK4SAUKnew · submitted 2026-06-25 · 💻 cs.SI

Fast Estimation for Forest Matrix of Signed Graphs

Haoxin Sun , Zhongzhi Zhang This is my paper

Pith reviewed 2026-06-26 02:21 UTC · model grok-4.3

classification 💻 cs.SI

keywords forestgraphsmatrixsignedalgorithmsconvergingestimatingestimation

0 comments

The pith

Signed forest matrix theorem links generalized spanning converging forests to the matrix and enables linear-time algorithms for signed graphs

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

This paper introduces the signed forest matrix theorem, which connects the forest matrix of a signed graph to its generalized spanning converging forests. It proposes the GSCF algorithm that uses a variant of loop-erased random walks to generate these forests in expected linear time in the number of nodes. Building on the theorem, the FMDE and FMDE+ sampling algorithms estimate the diagonal of the forest matrix with time linear in the number of samples. The work targets signed graphs because prior methods for unsigned graphs do not extend readily, yet signed graphs model positive and negative edges central to network science and opinion dynamics. Experiments confirm the methods produce accurate estimates while scaling to graphs exceeding twenty million nodes.

Core claim

The signed forest matrix theorem establishes the relationship between generalized spanning converging forests and the forest matrix. Based on this result, the GSCF algorithm, built on a variant of loop-erased random walks, generates generalized spanning converging forests in expected O(n) time. The FMDE and FMDE+ sampling algorithms then estimate the diagonal of the forest matrix, both with time complexity O(ln) where l is the number of samples. These procedures achieve high estimation accuracy on various signed graphs and scale to instances with over twenty million nodes.

What carries the argument

The signed forest matrix theorem, which establishes the relationship between generalized spanning converging forests and the forest matrix entries

If this is right

Forest matrix estimation becomes feasible for signed graphs with millions of nodes
The methods deliver high accuracy on tested signed graphs from network applications
Computational time improves significantly over prior approaches that do not handle signed edges
The algorithms apply directly to modeling tasks in social opinion dynamics
Generation of generalized spanning converging forests occurs in expected linear time

Where Pith is reading between the lines

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

The sampling approach might extend to estimating off-diagonal entries of the forest matrix without changing the core generation step
Linear-time forest generation could support repeated queries on evolving signed networks that were previously too slow to analyze
If the distributional invariance holds, the same loop-erased walk variant may apply to other matrix functions defined via forests on signed graphs
The scaling results suggest the method could handle even larger instances once memory for the graph representation is addressed
keywords:[

Load-bearing premise

The variant of loop-erased random walks extends directly to signed graphs while preserving the exact distributional properties needed for the theorem to yield unbiased or consistent estimates of the forest matrix entries

What would settle it

Exact computation of the forest matrix diagonal on a small signed graph followed by comparison to repeated FMDE estimates; consistent mismatch beyond sampling error bounds would falsify the unbiasedness claim

Figures

Figures reproduced from arXiv: 2606.26608 by Haoxin Sun, Zhongzhi Zhang.

**Figure 1.** Figure 1: A toy signed graph G0 with its 12 generalized spanning converging forests. Blue nodes are roots. mas that establish the relationship between the determinant of the matrix I + L and its submatrices, obtained by deleting one column and one row, with the weights of specific generalized spanning converging forests. Lemma 4.1. For a directed signed graph G = (V, E, w), the determinant of matrix I + L is equal … view at source ↗

**Figure 2.** Figure 2: Comparison of average relative errors of the diagonals for algorithms FMDE and FMDE+ on six graphs: Bitcoinotc(a), Wikielec(b), SlashdotZoo(c), Adolescent∗ (d), Gnutella08∗ (e), Wikipedia∗ (f) across three different settings of ϵ. The results displayed in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

The forest matrix of a signed graph plays an important role in network science and social opinion dynamics, yet existing algorithms are mainly designed for unsigned graphs and are difficult to extend to signed graphs. In this paper, we study the problem of efficiently estimating the forest matrix of signed graphs with n nodes and introduce the signed forest matrix theorem, which establishes the relationship between generalized spanning converging forests and the forest matrix. Based on this result, we propose a novel algorithm GSCF, built on a variant of loop-erased random walks, to generate generalized spanning converging forests in expected O(n) time. We further develop two sampling algorithms, FMDE and FMDE+, for estimating the diagonal of the forest matrix, both with time complexity O(ln), where l is the number of samples. Extensive experiments on various signed graphs show that our methods achieve high estimation accuracy, significantly improve computational efficiency, and scale to graphs with over twenty million nodes. Our source code is publicly available on https://github.com/HaoxinSun98/SignedForestDiagonal.

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.

Desk Editor's Note private letter to a colleague

The paper gives a practical O(ln) sampler for the diagonal of the signed forest matrix that scales to 20M-node graphs, but the load-bearing claim is that the signed LERW variant exactly matches the forest distribution required by the new theorem.

read the letter

The main takeaway is that this work supplies a concrete way to estimate the diagonal of the forest matrix on signed graphs at large scale. It rests on a new theorem relating the matrix to generalized spanning converging forests and then uses a loop-erased walk variant to sample those forests quickly.

What is actually new is the signed forest matrix theorem itself and the GSCF procedure that generates the forests in expected linear time. The two follow-on estimators, FMDE and FMDE+, turn those samples into diagonal estimates in O(ln) time. The experiments run on multiple signed graphs, report high accuracy, and reach instances larger than twenty million nodes, with code released publicly. That combination of scaling result and open implementation is the part that could be useful to people who actually need to compute this matrix.

The soft spot is the distributional claim for the signed sampler. The theorem equates forest-matrix entries to probabilities over forests whose weights are products of signed edge weights. For the estimators to be unbiased or consistent, GSCF must produce forests under exactly that measure. Negative weights change transition probabilities and could alter which loops get erased or how the walk terminates. The abstract presents the adaptation as preserving the required properties, but any deviation in the erasure or conditioning rules would break the link between samples and the theorem. That is the step that needs the closest check in the proof. The sample count l is a tunable parameter, which is standard for Monte Carlo methods but means accuracy depends on user choice rather than being automatic.

This paper is for researchers who work with signed networks or opinion dynamics and need the forest matrix on graphs too large for exact methods. It is not a sweeping theoretical result, but it targets a real computational gap. The thinking is clear enough and the literature engagement looks honest, so it deserves a serious referee. Referees should be asked to verify the sampling distribution in detail and to examine whether the experiments control for bias introduced by the signed adaptation.

I would send it out for review.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a signed forest matrix theorem relating entries of the forest matrix to (weighted) probabilities over generalized spanning converging forests in signed graphs. It proposes the GSCF algorithm, which uses a variant of loop-erased random walks to sample such forests in expected O(n) time, and develops the FMDE and FMDE+ sampling estimators for the diagonal entries with O(ln) time complexity (l samples). Experiments on signed graphs up to 20M nodes report high accuracy and improved efficiency over prior methods; source code is released.

Significance. If the sampling procedure exactly matches the measure required by the theorem, the work would enable scalable estimation of forest matrices for large signed graphs arising in opinion dynamics and network science. The public code release is a positive factor for reproducibility.

major comments (1)

[Abstract (GSCF description)] Abstract (GSCF and FMDE descriptions): the claim that GSCF produces forests whose distribution exactly matches the signed forest matrix theorem (i.e., probability proportional to the product of signed edge weights) is load-bearing for unbiasedness of the O(ln) estimators. The manuscript must demonstrate that the signed-edge variant of loop-erased random walks preserves the required transition probabilities and erasure conditioning; negative weights can alter both, and no argument or proof is visible in the provided material.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for recognizing the potential impact of our work on scalable signed graph analysis. We address the single major comment below.

read point-by-point responses

Referee: [Abstract (GSCF description)] Abstract (GSCF and FMDE descriptions): the claim that GSCF produces forests whose distribution exactly matches the signed forest matrix theorem (i.e., probability proportional to the product of signed edge weights) is load-bearing for unbiasedness of the O(ln) estimators. The manuscript must demonstrate that the signed-edge variant of loop-erased random walks preserves the required transition probabilities and erasure conditioning; negative weights can alter both, and no argument or proof is visible in the provided material.

Authors: We agree that an explicit demonstration is required to confirm that the GSCF sampling distribution exactly matches the measure in the signed forest matrix theorem, which underpins the unbiasedness of the FMDE/FMDE+ estimators. The theorem itself appears in Section 3, and Section 4 describes GSCF as a signed variant of loop-erased random walks. However, the manuscript does not contain a dedicated argument or proof addressing how negative edge weights affect transition probabilities and the erasure conditioning. In the revised version we will insert a new subsection (immediately following the algorithm description) that supplies a rigorous proof of correctness for the signed case, showing that the required probabilities are preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain is self-contained

full rationale

The paper introduces the signed forest matrix theorem as establishing a relationship between generalized spanning converging forests and the forest matrix entries, then builds GSCF on a variant of loop-erased random walks claimed to generate samples from the induced distribution in expected O(n) time, followed by FMDE/FMDE+ estimators for the diagonal in O(ln) time. No equation or step reduces by construction to a fitted parameter renamed as a prediction, no self-citation chain is load-bearing for the central theorem or sampling correctness, and no ansatz or uniqueness result is smuggled in via prior author work. The claims rest on the theorem's derivation and the sampling algorithm preserving the exact weighted measure, which are presented as independent content rather than tautological to the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full paper may contain additional parameters or assumptions not visible here. The number of samples l is a user-chosen parameter trading accuracy for speed.

free parameters (1)

l (number of samples)
Controls the accuracy-time tradeoff in FMDE/FMDE+; chosen by user rather than derived.

axioms (1)

domain assumption Loop-erased random walk properties extend to signed graphs while preserving the exact connection to the forest matrix
Central to the signed forest matrix theorem and GSCF algorithm.

pith-pipeline@v0.9.1-grok · 5699 in / 1288 out tokens · 40777 ms · 2026-06-26T02:21:40.376574+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references

[1]

and Gaudilli`ere, A

Avena, L. and Gaudilli`ere, A. Two applications of random spanning forests.Journal of Theoretical Probability, 31 (4):1975–2004,

1975
[2]

Chebotarev, P. Y . and Shamis, E. Matrix-forest theorems. ArXiv, abs/math/0602575,

arXiv
[3]

Opinion maximization in social networks

Gionis, A., Terzi, E., and Tsaparas, P. Opinion maximization in social networks. InProceedings of the 2013 SIAM International Conference on Data Mining, pp. 387–395. SIAM,

2013
[4]

Efficient person- alized pagerank computation: A spanning forests sam- pling based approach

9 Fast Estimation for Forest Matrix of Signed Graphs Liao, M., Li, R.-H., Dai, Q., and Wang, G. Efficient person- alized pagerank computation: A spanning forests sam- pling based approach. InProceedings of the 2022 In- ternational Conference on Management of Data, SIG- MOD/PODS ’22, pp. 2048–2061, New York, NY , USA,

2022
[5]

Sublinear-time opinion estimation in the friedkin–johnsen model

Neumann, S., Dong, Y ., and Peng, P. Sublinear-time opinion estimation in the friedkin–johnsen model. InProceedings of the ACM on Web Conference 2024, pp. 2563–2571,

2024
[6]

Y ., Amblard, P.-O., Barthelme, S., and Tremblay, N

Pilavci, Y . Y ., Amblard, P.-O., Barthelme, S., and Tremblay, N. Variance reduction for inverse trace estimation via ran- dom spanning forests.arXiv preprint arXiv:2206.07421,

arXiv
[7]

and Adhikari, B

Singh, R. and Adhikari, B. Measuring the balance of signed networks and its application to sign prediction.Journal of Statistical Mechanics: Theory and Experiment, 2017 (6):063302,

2017
[8]

and Zhang, Z

Sun, H. and Zhang, Z. Efficient computation for diagonal of forest matrix via variance-reduced forest sampling. InProceedings of the ACM Web Conference 2024, pp. 792–802,

2024
[9]

New approximation algorithms for forest closeness centrality–for individual vertices and vertex groups

van der Grinten, A., Angriman, E., Predari, M., and Mey- erhenke, H. New approximation algorithms for forest closeness centrality–for individual vertices and vertex groups. InProceedings of the 2021 SIAM International Conference on Data Mining, pp. 136–144. SIAM,

2021
[10]

Searching for polarization in signed graphs: a local spectral approach

Xiao, H., Ordozgoiti, B., and Gionis, A. Searching for polarization in signed graphs: a local spectral approach. InProceedings of The Web Conference 2020, pp. 362– 372,

2020
[11]

Fast evaluation for relevant quantities of opinion dynamics

Xu, W., Bao, Q., and Zhang, Z. Fast evaluation for relevant quantities of opinion dynamics. InProceedings of the Web Conference 2021, pp. 2037–2045. ACM,

2021
[12]

LetbL be the matrix D−A + +A −, which is the Laplacian matrix of an unsigned directed graphbG= (V, E,bw) , where bwij =|w ij|= 1

(18) After summing the expected number of visits for all nodes, we obtain:Pn i=1 E(hi)≤trace((I+D−A ++A−)−1(I+D)). LetbL be the matrix D−A + +A −, which is the Laplacian matrix of an unsigned directed graphbG= (V, E,bw) , where bwij =|w ij|= 1 . The entry at row i and column j ofbL is denoted by lij. We have lii =d i and lij = 0 or −1. Furthermore, the su...

2023
[13]

Our experiments are conducted on a diverse range of networks, with node counts ranging from 2,539 to over 23 million and edge counts from 12,969 to 112 million

and SNAP (Leskovec & Sosiˇc, 2016). Our experiments are conducted on a diverse range of networks, with node counts ranging from 2,539 to over 23 million and edge counts from 12,969 to 112 million. Details of these datasets are presented in Table 3, which includes six small graphs along with six medium and large-sized graphs. We utilize both original signe...

2016

[1] [1]

and Gaudilli`ere, A

Avena, L. and Gaudilli`ere, A. Two applications of random spanning forests.Journal of Theoretical Probability, 31 (4):1975–2004,

1975

[2] [2]

Chebotarev, P. Y . and Shamis, E. Matrix-forest theorems. ArXiv, abs/math/0602575,

arXiv

[3] [3]

Opinion maximization in social networks

Gionis, A., Terzi, E., and Tsaparas, P. Opinion maximization in social networks. InProceedings of the 2013 SIAM International Conference on Data Mining, pp. 387–395. SIAM,

2013

[4] [4]

Efficient person- alized pagerank computation: A spanning forests sam- pling based approach

9 Fast Estimation for Forest Matrix of Signed Graphs Liao, M., Li, R.-H., Dai, Q., and Wang, G. Efficient person- alized pagerank computation: A spanning forests sam- pling based approach. InProceedings of the 2022 In- ternational Conference on Management of Data, SIG- MOD/PODS ’22, pp. 2048–2061, New York, NY , USA,

2022

[5] [5]

Sublinear-time opinion estimation in the friedkin–johnsen model

Neumann, S., Dong, Y ., and Peng, P. Sublinear-time opinion estimation in the friedkin–johnsen model. InProceedings of the ACM on Web Conference 2024, pp. 2563–2571,

2024

[6] [6]

Y ., Amblard, P.-O., Barthelme, S., and Tremblay, N

Pilavci, Y . Y ., Amblard, P.-O., Barthelme, S., and Tremblay, N. Variance reduction for inverse trace estimation via ran- dom spanning forests.arXiv preprint arXiv:2206.07421,

arXiv

[7] [7]

and Adhikari, B

Singh, R. and Adhikari, B. Measuring the balance of signed networks and its application to sign prediction.Journal of Statistical Mechanics: Theory and Experiment, 2017 (6):063302,

2017

[8] [8]

and Zhang, Z

Sun, H. and Zhang, Z. Efficient computation for diagonal of forest matrix via variance-reduced forest sampling. InProceedings of the ACM Web Conference 2024, pp. 792–802,

2024

[9] [9]

New approximation algorithms for forest closeness centrality–for individual vertices and vertex groups

van der Grinten, A., Angriman, E., Predari, M., and Mey- erhenke, H. New approximation algorithms for forest closeness centrality–for individual vertices and vertex groups. InProceedings of the 2021 SIAM International Conference on Data Mining, pp. 136–144. SIAM,

2021

[10] [10]

Searching for polarization in signed graphs: a local spectral approach

Xiao, H., Ordozgoiti, B., and Gionis, A. Searching for polarization in signed graphs: a local spectral approach. InProceedings of The Web Conference 2020, pp. 362– 372,

2020

[11] [11]

Fast evaluation for relevant quantities of opinion dynamics

Xu, W., Bao, Q., and Zhang, Z. Fast evaluation for relevant quantities of opinion dynamics. InProceedings of the Web Conference 2021, pp. 2037–2045. ACM,

2021

[12] [12]

LetbL be the matrix D−A + +A −, which is the Laplacian matrix of an unsigned directed graphbG= (V, E,bw) , where bwij =|w ij|= 1

(18) After summing the expected number of visits for all nodes, we obtain:Pn i=1 E(hi)≤trace((I+D−A ++A−)−1(I+D)). LetbL be the matrix D−A + +A −, which is the Laplacian matrix of an unsigned directed graphbG= (V, E,bw) , where bwij =|w ij|= 1 . The entry at row i and column j ofbL is denoted by lij. We have lii =d i and lij = 0 or −1. Furthermore, the su...

2023

[13] [13]

Our experiments are conducted on a diverse range of networks, with node counts ranging from 2,539 to over 23 million and edge counts from 12,969 to 112 million

and SNAP (Leskovec & Sosiˇc, 2016). Our experiments are conducted on a diverse range of networks, with node counts ranging from 2,539 to over 23 million and edge counts from 12,969 to 112 million. Details of these datasets are presented in Table 3, which includes six small graphs along with six medium and large-sized graphs. We utilize both original signe...

2016