Spanning Forests of a Digraph and Their Applications
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Proposition 11 of [2] states that rank(L) = n − d, where d is the in-forest dimension of G, the minimum possible number of converging trees in a spanning converging forest of G.
“Proposition 11 of [2] states that rank( L ) = n - d , where d is the so-called in-forest dimension of G , i.e., the minimum possible number of converging trees in a spanning converging forest of G .”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
Proposition 11 (Section 8.1, p. 12) concerns forest-accessibility measures, not the rank of L. Section 5 (p. 8) gives the dimension of the range of Lᵀ as n − v, where v is the out-forest dimension; Section 2 (p. 3) says the in-forest dimension v′ can differ from v. Thus the cited proposition does not state the claimed equation, and the paper’s rank relation uses a different forest dimension. (LLM, conf=0.94)
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredBy Proposition 11 of [2], rank(J) = d, where d is the in-forest dimension of G.
“by Proposition 11 of [2] rank ) ( J = d , where d is the in-forest dimension of G.”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
Definition 3 (p. 3) defines the paper’s J as a matrix of maximum out forests, while Section 2 (pp. 2–3) distinguishes out-forest dimension from in-forest dimension and says they need not be equal. The proof of Proposition 11 (Conclusion, p. 20) concerns accessibility measures, not the rank of J. Thus Proposition 11 does not establish the claimed equality. (LLM, conf=0.78)
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredThe elements of J were characterized in Theorems 2 and 3 of [2].
“The elements of J were characterized in Theorems ' 2 and 3 of [2].”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
In “Notation and Some Earlier Results” (p. 3), Theorems 2 and 3 establish properties of the matrix J: it is idempotent and satisfies LJ = JL = 0. They do not directly characterize its individual elements; the entry-wise statements appear in Theorem 1 on the same page. (LLM, conf=0.83)
Experimental — accuracy not yet measured(i) of Lemma 3 in [1] coincides with Proposition 12 of [2].
“in particular, (i) of Lemma 3 in [1] coincides with Proposition 12 of [2].”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
Proposition 12 appears in §8.2 (p. 13) and states two properties of out forests. The cited paper therefore supports the reference to Proposition 12, but its text does not establish that the proposition coincides with part (i) of Lemma 3 in [1]. (LLM, conf=0.78)
Experimental — accuracy not yet measuredAccording to Proposition 8, the forest dimension of a digraph is no less than its number of weak components and does not exceed the number of its strong components and the number of its unilateral components.
“In addition, according to Proposition 8, 'the forest dimension of a digraph is no less than its number of weak components 2 and does not exceed the number of its strong components and the number of its unilateral components.'”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
The definition in §2 (p. 2) makes the lower bound follow: an out forest spans the digraph, so it needs at least one tree per weak component. But Proposition 8 (§6, pp. 9–10) concerns the group inverse of the Kirchhoff matrix, not component-count bounds, and the cited passages do not state the claimed upper bounds for strong or unilateral components. (LLM, conf=0.84)
Experimental — accuracy not yet measured
The in-forest dimension of G is equal to the number of its sink SCCs.
“It was also shown (Proposition 6) that the in-forest dimension of G is equal to the number of its sink SCC's”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
Section 2 (pp. 2–3) defines the in-forest dimension and says reversing all arcs exchanges in- and out-forest dimensions. The paper’s discussion of undominated knots (Conclusion, p. 18) supports the corresponding count of sink strongly connected components after reversal. However, its Proposition 6 (Section 4, p. 6) concerns stochastic forest matrices, not this equality, so the proposition number in the citing sentence is unsupported. (LLM, conf=0.82)
Experimental — accuracy not yet measuredAn algebraic method for calculating J was presented in [7].
“An algebraic method for calculating J was presented in [7].”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [7]
The statement of Lemma 2 that all nontrivial eigenvalues of L have positive real parts holds true in the general case, and not only for strongly connected digraphs or digraphs that contain directed spanning trees.
“Fourth, the statement of Lemma 2 that 'all nontrivial eigenvalues of L have positive real parts' holds true in the general case, and not only for strongly connected digraphs or digraphs that contain directed spanning trees.”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [7]
For a strongly connected digraph, rank(L) = n − 1, which coincides with the first statement of Lemma 2.
“Consequently, for a strongly connected digraph, rank( L ) = n -1, which coincides with the first statement of Lemma 2.”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
The forest dimension of a strongly connected digraph is one.
“the forest dimension of a strongly connected digraph is one (Proposition 7).”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
The statement that Lemma 2 holds under a weaker condition of existence of a directed spanning tree for G follows from Proposition 11 of [2].
“so this statement follows from Proposition 11 of [2].”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
Proposition 11 (Forest accessibility, p. 12) concerns properties of forest-based accessibility measures, including a disconnection condition; it does not state that an unspecified Lemma 2 holds whenever G has a directed spanning tree. The copy discusses spanning trees elsewhere (Notation and Some Earlier Results, p. 2), but without the citing paper’s Lemma 2, these passages do not establish the claimed implication. (LLM, conf=0.74)
Experimental — accuracy not yet measuredThe last statement of Lemma 2 becomes valid if one replaces strongly connected components with weakly connected components and additionally requires that these WCCs are strong.
“The statement under consideration becomes valid if one replaces strongly connected components with weakly connected components (WCC) and additionally requires that these WCC's are strong.”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
The paper’s Proof of Lemma 2 (CONCLUSION, p. 20) concerns matrix identities, not strongly or weakly connected components. The discussion of weak components (CONCLUSION, pp. 17–18) does not state that a lemma becomes valid when weak components are required to be strong. I searched for that proposed correction and found no passage addressing it. (LLM, conf=0.75)
Experimental — accuracy not yet measuredA more general correct statement results by substituting sink SCCs for strongly connected components in the last statement of Lemma 2.
“A more general correct statement results by substituting, in the same place, sink SCC's, where a sink strongly connected component is an SCC having no edges directed outwards.”
From Comments on "Consensus and cooperation in networked multi-agent systems" (2010) · cites this paper as [2]
In our copy, Lemma 2 (Section 6, p. 10) concerns commuting matrices, not strongly connected components. Section 2 (pp. 2–3) discusses arc reversal and undominated knots, but does not identify the cited Lemma 2 statement or say that replacing its strongly connected components with sink SCCs corrects it. I searched for that replacement and for sink SCCs. (LLM, conf=0.78)
Experimental — accuracy not yet measured
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Cites
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Research works printed but not linked: 22
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Fouss, F., Pirotte, A., Renders, J.-M., & Saerens, M. (2007) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2007.46 · Cited by 1,265 · see citing works
“However, the nice interpretation in terms of the pseudoinverse of the Laplacian matrix does not apply in this case (still, see [1], [2] for potential generalizations to directed graphs).”
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“For instance, an extension of the Laplacian matrix to directed graphs is proposed in [14] while an extension of the regularized Laplacian kernel to directed graphs appears in [1, 2].”
BackgroundA Graph-Theoretical Approach for Tracing Filamentary Structures in Neuronal and Retinal Images
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“By connecting to the well-established algebraic digraph theory [9], as well as the transductive inference in machine learning [11], local and global contextual information can be considered explicitly.”
BackgroundMethodInfluentialCoordination in multiagent systems and Laplacian spectra of digraphs
Chebotarev, P. Y., & Agaev, R. P. (2009) · Automation and Remote Control · doi:10.1134/s0005117909030126 · Cited by 90 · see citing works
“To compute the matrix J̄ , one can use the algorithm proposed in [19, 20], which reduces to min{n − d − 1, 0} multiplications of matrices of order n.”
MethodSelf-citationForest matrices around the Laplacian matrix
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“For instance, an extension of the Laplacian matrix to directed graphs is proposed in [22] while an extension of the regularized Laplacian kernel to directed graphs appears in [1].”
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Mantrach, A., Yen, L., Callut, J., Françoisse, K., Shimbo, M., & Saerens, M. (2010) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2009.78 · Cited by 54
“For instance, an extension of the Laplacian matrix to directed graphs is proposed in [23] while an extension of the regularized Laplacian kernel to directed graphs appears in [67], [68].”
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Chebotarev, P. (2006) · Discrete Applied Mathematics · doi:10.1016/j.dam.2007.08.030 · Cited by 44
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“A spanning converging forest is in fact an in-forest in (Agaev and Chebotarev 2001; Chebotarev and Agaev 2002).”
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De, J., Zhang, X., Lin, F., & Cheng, L. (2018) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2017.2730871 · Cited by 32 · see citing works
“Spanning Forest of Digraphs [30]: The celebrated MatrixTree theorem has been extended to general digraphs [31], where the quantity Q := ( I + τL )−1 with L := D−W is shown to be the normalized counts of spanning out-forests.”
Efficient Personalized PageRank Computation: A Spanning Forests Sampling Based Approach
Liao, M.-H., Li, R.-H., Dai, Q.-Q., & Wang, G.-R. (2022) · SIGMOD Conference · doi:10.1145/3514221.3526140 · Cited by 24 · see citing works
“Such a theorem has been generalized to spanning forest in early years [1, 11, 12, 38].”
BackgroundEffects of Stubbornness on Opinion Dynamics
Xu, W.-Y., Zhu, L.-W., Guan, J., Zhang, Z.-B., & Zhang, Z.-Z. (2022) · International Conference on Information and Knowledge Management · doi:10.1145/3511808.3557304 · Cited by 23 · see citing works
“A spanning diverging forest is also called a spanning out-forest [3, 10].”
BackgroundTracing Retinal Blood Vessels by Matrix-Forest Theorem of Directed Graphs
Cheng, L., De, J., Zhang, X., Lin, F., & Li, H.-Q. (2014) · International Conference on Medical Image Computing and Computer-Assisted Intervention · doi:10.1007/978-3-319-10404-1_78 · Cited by 23
“which is also termed the matrix of dense forest in [4].”
BackgroundMethodInfluentialChebotarev, P., & Пашаевич, А. Р. (2014) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2014.2304369 · Cited by 27 · see citing works
Self-citationSemi-supervised learning with regularized Laplacian
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Self-citationComments on "Consensus and cooperation in networked multi-agent systems"
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“A spanning converging forest is also called an in-forest [41], [42].”
BackgroundThe projection method for reaching consensus and the regularized power limit of a stochastic matrix
Agaev, R., & Chebotarev, P. (2011) · arXiv.org · doi:10.1134/s0005117911120034 · Cited by 15 · see citing works
Self-citationDistributed Learning of Distributions via Social Sampling
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“…that: • show another facet of the elegant link between random processes on graphs and Laplacian-based numerical linear algebra, such as in [7, 8, 9] • scale linearly in |E| and thus useful on very large graphs • can be implemented in a fully distributed fashion: as long as each node is able…”
BackgroundWhich digraphs with ring structure are essentially cyclic?
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Reply to "Comments on "Consensus and Cooperation in Networked Multi-Agent Systems""
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“have also been made towards KRR by redefining a positive semidefinite Laplacian matrix [3], [4], [8], [9].”
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“We note the works of Agaev and Chebotarev [10, 15] and especially [16], which shows a fundamental theorem.”
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“To this end, they interpreted the overall opinion from the perspective of spanning converging forests [4].”
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“This structure coincides with the notion of an in-forest as introduced by [50] and further studied by [51].”
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“Let us define the Laplacian matrix of an unweighted digraph G as follows: Lemma 1 ([24], [25]) .”
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“Even so, there are still some difficulties in target hunting: the consensus algorithm is the basis for coordination between agents[10].”
BackgroundMatrices of Forests and the Analysis of Digraphs
Chebotarev, P., & Agaev, R. (2005) · arXiv.org · doi:10.48550/arxiv.math/0508171 · Cited by 3
“These measures and the dual measures have been studied in [3] (see also [10, p.”
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The Projection Method for Reaching Consensus in Discrete-Time Multiagent Systems
Chebotarev, P., & Agaev, R. (2013) · IFAC Conference on Manufacturing Modelling, Management, and Control · doi:10.3182/20130619-3-ru-3018.00422 · Cited by 2 · see citing works
“One of the first discrete models of reaching consensus was proposed by DeGroot (1974) and further studied in Berger (1981); Gilardoni and Clayton (1993); Bradley (2007); Agaev and Chebotarev (2012); Cao et al.”
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“It is shown by Agaev and Chebotarev [20] and Fax and Murray [21] that an incoming spanning tree in a communication graph G is the necessary condition.”
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Agaev, R., & Khomutov, D. (2023) · 2023 International Russian Automation Conference (RusAutoCon) · doi:10.1109/rusautocon58002.2023.10272796 · Cited by 1 · see citing works
“𝑛𝑛 × 𝑛𝑛 The following equalities is true [21]: 𝐿𝐿 = lim 𝑡𝑡→∞ 𝑒𝑒 .”
Self-citationMatrices of Forests, Analysis of Networks, and Ranking Problems
Chebotarev, P., & Agaev, R. (2013) · International Conference on Information Technology and Quantitative Management · doi:10.1016/j.procs.2013.05.145 · Cited by 1
“Another proof of Theorem 3 can be found in [17]. A review on forest representations of Markov chain probabilities is given in [18]. For an interpretation of J(τ) in terms of Markov chains we refer to [19]. Theorem 3 will be used in Section 5. 4. Forest based accessibility measures Formally, by an accessibility measure…”
Self-citationKammerdiner, A. R., & Pasiliao, E. (2011) · Optimization Letters · doi:10.1007/s11590-011-0396-x · Cited by 1
“Definition 2 [1] The number of out trees (i.”
BackgroundMethodInfluentialThe forest consensus theorem and asymptotic properties of coordination protocols
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Sun, Y.-B., Sun, H.-X., Zehmakan, A. N., & Zhang, Z.-Z. (2026) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2026.3730450
“For a DWU graph G = ( V, E, w ) with w : E → R + being the edge weight function where w e represents the weight of edge e , we use F to denote the set of all rooted spanning forests [34], [35].”
Agaev, R., & Khomutov, D. K. (2024) · Automation and remote control · doi:10.1134/s000511792406002x
“Proposition 1 [31].”
Self-citationFurtat, I., Gushchin, P., & Nguyen, H. B. (2022) · 2022 6th Scientific School Dynamics of Complex Networks and their Applications (DCNA) · doi:10.1109/dcna56428.2022.9923212
“Assume, a digraph corresponded to grid has a directed spanning tree [16], [17].”
BackgroundMost, And Least, Compact Spanning Trees of a Graph
Ranjan, G., Saurabh, N., & Ashutosh, A. (2022) · arXiv.org · doi:10.48550/arxiv.2206.07104
“Indeed, there exists an analogous matrix of relative forest accessibilities for the directed case [6, 7, 13].”
BackgroundClustering in Directed Graph Using the Laplacian Matrices and Their Eigenprojections
Agaev, R. (2020) · 2020 13th International Conference "Management of large-scale system development" (MLSD) · doi:10.1109/mlsd49919.2020.9247701
“One of these methods is the LeVerrier-Faddeev method ( (cid:1843) (cid:2868) = (cid:1835) ); In [5] it was proved that all matrices , (cid:1863) = 0, … , (cid:1866) − − 1, are Laplacian matrices, and = 0.”
BackgroundMethodInfluentialSelf-citationDe, J. (2016) · doi:10.32657/10356/65993
“It has been shown in [88, 89] that each (i, j)-th entry of Q = [qij] concerns the normalized counts of spanning out-forests, as qij = (Fji) (F) , with (·) denoting the count of specified forests.”
BackgroundFurther results on digraphs with completely real Laplacian spectra
Li, H.-H., & Su, L. (2015) · Applied Mathematics and Computation · doi:10.1016/j.amc.2015.05.102
Transduction on Directed Graphs via Absorbing Random Walks
De, J., Zhang, X., & Cheng, L. (2014) · arXiv.org · doi:10.48550/arxiv.1402.4566
“ote the forest matrix Q:= I+ ˝L 1 . Suppose Fis the set of all spanning forests of digraph Gand Fij the set of spanning forests containing edge i j. It has been shown in (Chebotarev and Shamis, 1997; Agaev and Chebotarev, 2001) that each (i;j)-th entry of Q= [q ij] concerns the normalized counts of spanning out-forest…”
BackgroundEfficient Algorithm for Opinion Optimization via Leader Selection in Directed Social Networks
Zhou, X.-N., & Zhang, Z.-Z. (2026) · IEEE Transactions on Networking · doi:10.1109/ton.2026.3732512
“Lemma 3 ([25], [26]): Let G = ( V , E ) be a connected graph and S ⊆ V be a subset representing the root set.”
Control of Electrical Generators Based on Low-pass Filter and Artificial Time-delay
Furtat, I., Gushchin, P., & Nguyen, H. B. (2022) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2022.07.337
- Authors
- Agaev, R. P., & Chebotarev, P. Y.
- Affiliations
- Russian Academy of Sciences (RU)
- ORCID
- Chebotarev, P. Yu.
- Venue
- Automation and Remote Control
- Published
- 2001-03
- Volume
- 62
- Issue
- 3
- Pages
- 443-466
- ISSN
- 0005-1179 (ISSN-L)
- 1608-3032
- OpenAlex
- W1479963817
- Cited by
- 57
- References
- 21
- Citations by year
- 2026: 2 · 2025: 1 · 2024: 7 · 2023: 2 · 2022: 4 · 2021: 2 · 2020: 3 · 2019: 2 · 2018: 1 · 2017: 3 · 2016: 2 · 2015: 2 · 2014: 3 · 2013: 3 · 2012: 3
- FWCI
- 1.10
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Automation and Remote Control (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Graph theory and applications
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Graph Theory and Algorithms
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Keywords
- spanning forests, directed graphs, graph Laplacian, Moore-Penrose inverse, group inverse, Markov chains, observation model, annihilating polynomials, accessibility measures
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