On the Strong Structural Controllability of Matrix-Weighted Networks

Lanhao Zhao

REVIEW 2 major objections 6 minor 13 references

This paper claims a layer-specific squeeze theorem that bounds the strong structural controllable subspace of matrix-weighted networks tightly from both sides.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 00:15 UTC pith:3SIRUHUV

load-bearing objection Nice layer-decomposition idea for matrix-weighted SSC, but the central squeeze theorem's lower bound is false as stated — a 2-agent d=2 counterexample gives dimension 3 against a claimed bound of 4 — and the upper-bound proof has a configuration-transfer gap. the 2 major comments →

arxiv 2607.27852 v1 pith:3SIRUHUV submitted 2026-07-30 eess.SY cs.SYmath.OC

On the Strong Structural Controllability of Matrix-Weighted Networks

Lanhao Zhao This is my paper

classification eess.SY cs.SYmath.OC MSC 93B0593B0793C0505C50

keywords strong structural controllabilitymatrix-weighted networksmulti-agent systemsequitable partitionslayer-specific distance partitioncontrollable subspace boundsWeisfeiler-Lehman refinementgeneric rank

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

For multi-agent networks whose couplings carry matrix weights (so each agent has a multidimensional state), this paper asks how much of the state space can be controlled no matter what the nonzero edge weights are. It answers with a squeeze theorem: the dimension of the strong structural controllable subspace is at least the sum of layer-specific distance-partition sizes and at most the size of a minimal equitable partition times the state dimension. To get there, it decomposes each matrix weight into scalar layers, so singular or asymmetric couplings no longer block controllability analysis. It also gives an algebraic way to choose the projection basis that makes the two bounds meet most tightly, proves that such a basis exists for almost all parameter values, and finds the required partitions in polynomial time using color refinement.

Core claim

The central discovery is Theorem 3, the Layer-specific Squeeze Theorem. For a matrix-weighted network whose d mutually orthogonal basis layers drive the dynamics, the dimension of the strong structural controllable subspace W' satisfies Σ_{m∈D} |π_LDP^(m)| ≤ dim(W') ≤ |π_jm| × d. The upper bound comes from a generalized equitable partition: nodes that are indistinguishable under weighted in-degree sums collapse into the same state-space limitation, so the controllable subspace sits inside the image of that partition's characteristic matrix. The lower bound uses a new layer-specific distance partition: within each scalar layer, signals propagate only along edges present in that layer, so the

What carries the argument

The central object is the layer-specific distance partition (LDP), defined per scalar layer m as the partition of vertices by shortest-path distance from the leader set in the layer's subgraph; its cell count |π_LDP^(m)| = d_max^(m)+1 lower-bounds the controllable dimension contributed by that layer. The paper couples this with a matrix-space basis decomposition, writing each matrix weight as a combination of d^2 basis matrices, which converts the matrix-weighted network into multiplex scalar layers. The upper bound uses a generalized equitable partition, whose characteristic matrix spans an L-invariant subspace containing the controllable subspace. Finally, the optimal basis is characterize

Load-bearing premise

The lower-bound half of the squeeze theorem assumes that the controllable subspaces contributed by the mutually orthogonal basis layers are linearly independent, so their dimensions add; if those layer subspaces overlap, the bound can overestimate the true dimension.

What would settle it

Take the 4-node, d=2 network from Example 3 and compute the exact controllability-matrix rank for generic nonzero weights matching the structural pattern (with the shortcut edge zero on layer 1). If the rank is 6 or less, the claimed lower bound of 7 fails; if it is 7 or more, the LDP bound is validated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If Theorem 3 holds, the dimension of the strong structural controllable subspace of any matrix-weighted network can be certified by computing two purely topological quantities, without enumerating edge-weight realizations.
The upper bound |π_jm| × d gives a quantitative symmetry index: when it is less than nd, an invariant uncontrollable subspace of positive dimension provably exists for every admissible weight choice.
The lower bound improves on the classical distance-partition bound in exactly the cases where a shortcut edge exists macroscopically but is absent in the layer that carries the control signal, so network designers can see where controllability is genuinely degraded.
The null-space selection criterion plus generic-rank guarantee implies that a single random projection basis consistent with the structural pattern will almost surely achieve the tightest bounds, matching the parameter-independence requirement of strong structural controllability.
By duality, the same bounds carry over to strong structural observability, giving an upper bound on the observable subspace from an equitable partition of the transposed network.

Where Pith is reading between the lines

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

In this reading, the lower bound of Theorem 3 rests on a direct-sum assumption for the layer subspaces; if that independence fails, the additive bound could overcount. One testable fix would be to subtract an overlap term based on the dimension of intersections between pairwise layer controllable subspaces.
The polynomial-time WL discovery suggests the same pipeline could be applied online for time-varying topologies, a direction the paper only names as future work; one could re-run color refinement after each topology change and recompute the bounds.
The generic-rank result has a measure-theoretic corollary the paper does not state explicitly: for almost every assignment of edge weights consistent with the pattern, the optimal basis exists, so the bounds are not fragile artifacts of particular weight choices.
The LDP lower bound could be sharpened further by considering inter-layer switches, where a signal may move from one layer to another along a shared dimension; this is a natural extension beyond the paper's strictly per-layer analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Nice layer-decomposition idea for matrix-weighted SSC, but the central squeeze theorem's lower bound is false as stated — a 2-agent d=2 counterexample gives dimension 3 against a claimed bound of 4 — and the upper-bound proof has a configuration-transfer gap.

read the letter

Let me get to the point: this paper has a useful layer-decomposition intuition, but the central result is not established, and the lower bound is false as stated. The idea is to take the known scalar-network bounds for strong structural controllability — equitable partitions for upper, distance partitions for lower — and extend them to matrix-weighted networks where agent states have dimension d>1. The layer-specific projection (decompose each weight matrix on a basis B_m, study each scalar layer, define a layer-specific distance partition) is the right kind of move. The null-space formulation for picking a projection basis and the WL-based discovery algorithm are reasonable directions, and the numerical examples are at least consistent with the authors' intent.

The problem is Theorem 3. The proof asserts that orthogonality of the basis matrices B_m gives a direct sum of the lifted controllable subspaces in R^{nd}. It doesn't. Trace-orthogonality says the matrices are orthogonal as vectors in R^{d×d}; it says nothing about whether the subspaces they generate in the full state space are disjoint. Concrete failure: d=2, B1=diag(1,0), B2=[[0,1],[0,0]], single leader, A21=B1+B2. Both scalar layer graphs are single edges, so each LDP lower bound is 2 and the claimed lower bound is 4. But the actual strong structural controllable subspace has dimension 3 for every nonzero choice of the two free parameters. So the claimed squeeze is not merely missing a proof step; it is false for the singular/asymmetric case the paper says it covers.

There is also a gap in Theorem 1. It uses Lemma 1, which requires the containment W_j ⊆ im(P_πjm) for every admissible weight configuration, but the proof only establishes containment for the configuration used to define π_jm. Equitable partitions are weight-dependent here, so the invariance does not transfer. The upper bound is therefore incomplete as written, even if it is likely repairable. The WL claim about returning the maximal equitable partition overstates what color refinement gives, and Theorem 5's “almost everywhere” basis is doing less work than it appears, since a basis that works for individual parameter values need not be a valid structural basis.

No code or machine-checked proof accompanies the paper, and the numerical examples don't include the failure mode above. I'd still send this to a serious referee rather than desk-reject: the layer-specific direction is worth vetting, and the counterexample is a clean falsification the author should see. But my recommendation would be reject as is, with an invitation to repair — add a genuine disjointness/direct-sum condition for the lower bound and fix the configuration-transfer step in Theorem 1. Right now I would not cite it in my own work. Reading-group value: maybe, mostly as a case study in why matrix orthogonality doesn't imply subspace orthogonality.

Referee Report

2 major / 6 minor

Summary. The paper studies strong structural controllability of matrix-weighted multi-agent networks with d-dimensional states. It introduces a generalized equitable partition upper bound (Theorem 1), a layered decomposition of matrix weights into scalar layers (Theorem 2), and a 'layer-specific distance partition' (LDP) lower bound summarized as a Squeeze Theorem (Theorem 3). It then proposes an optimal-basis algorithm via null-space intersection with a generic-rank existence result (Theorems 4 and 5), a WL-refinement algorithm for automatic partition/shortcut discovery (Theorem 6), and extensions to observability and invariant subspaces (Theorems 7-8). The main claimed contribution is closed-form, layer-aware upper and lower bounds on the dimension of the strong structural controllable subspace.

Significance. The layer-decomposition idea is potentially useful: it attempts to handle singular and asymmetric matrix weights by projecting onto scalar layers, and the automated WL-based partition discovery addresses a real combinatorial bottleneck. If the bounds were correct, they would provide practical estimates for strong structural controllability in matrix-weighted networks, an area with few quantitative tools. However, the central lower bound (Theorem 3) is false in the singular case the paper explicitly targets, and the proof of the upper bound (Theorem 1) does not establish containment across all admissible weight configurations. The paper provides no machine-checked proofs or reproducible code; its numerical examples mostly illustrate the formulas rather than test the bounds.

major comments (2)

[Section III.C, Eq. (19)] The lower bound is false. The proof asserts 'Due to orthogonality, the overall controllable subspace structurally admits a direct sum' and concludes dim(W') >= sum_m dim(W'_m). Trace/Frobenius orthogonality of the B_m does not imply that the lifted layer subspaces are disjoint in R^{nd}; trace-orthogonal matrices can share the same column space. Counterexample: n=d=2, internal dynamics A=0, leader v1, B1=[[1,0],[0,0]], B2=[[0,1],[0,0]] (trace-orthogonal, both with im = span{e1}), and A21 = alpha B1 + beta B2 with alpha,beta arbitrary nonzero. The controllability matrix of (L,M) has rank 3 for all nonzero alpha,beta, so dim(W')=3. Each scalar layer is a single edge, hence |pi_LDP^(1)|=|pi_LDP^(2)|=2, giving lower bound 4. Thus Eq. (19) fails in exactly the singular/asymmetric setting the paper claims to cover.
[Section III.A, proof of Theorem 1] The deduction W' ⊆ im(P_{π_jm}) uses Lemma 1, but Lemma 1 requires containment W_j ⊆ im(P_{π_jm}) for every admissible weight configuration j. The proof establishes this containment only for the particular Laplacian L used to define π_jm; for another weight configuration the partition π_jm need not be equitable, so im(P_{π_jm}) need not be L-invariant. Therefore the upper bound dim(W') ≤ |π_jm| d is not proved for the strong structural (all-configuration) subspace.

minor comments (6)

[Section III.C, indexing] The layer indexing is inconsistent. Eq. (14) uses d^2 basis layers, while Theorem 3 sums over D={1,...,d}. Please state how the d active layers are selected and whether the result is meant to hold for an arbitrary subset of d orthogonal basis matrices.
[Section III.D, Definition 4/Theorem 5] Since g-rank is defined as the maximum rank over all λ, the condition g-rank(Hbar)<d^2 already implies rank deficiency at every λ. The conclusion 'the optimal basis exists almost everywhere' is therefore a restatement of the definition; the algebraic-geometry discussion is redundant for that implication.
[Section IV.B, Example 2] The example only checks that layer 1 satisfies the generalized equitable partition condition; it does not compute the claimed upper/lower dimension or compare it with an actual strong structural controllable subspace. It is therefore not a numerical verification of the bound.
[Section III.F, Theorem 7] The argument that the orthogonal complement of W' is non-empty does not establish a fixed invariant uncontrollable subspace. Since W' is defined as a minimum dimension over configurations, it is not a fixed subspace; one must exhibit a nonzero vector that is outside W_i for all i and lies in a common L_i-invariant subspace.
[Section III.E, Algorithm 1] The claim of convergence to the 'unique maximal structural equitable partition' is not proved for directed labeled graphs; WL stabilization gives a color refinement partition, not necessarily the coarsest equitable partition. The shortcut detection also uses macroscopic BFS levels, which is at odds with the layer-specific LDP construction.
[General presentation] References [2] and [6] are duplicates; [14] is cited as a prior result but appears to be an unreviewed arXiv preprint. Some equations, e.g., Eq. (10)-(12), have notation that is hard to parse.

Circularity Check

1 steps flagged

No significant circularity in the central bounds; Theorem 5 is a minor definitional restatement of generic rank.

specific steps

self definitional [Definition 4 and Theorem 5, Section III.D]
"Definition 4 (Generic Rank): The generic rank (g-rank) of the structural pattern matrix ¯H(λ) is defined as the maximum rank it can achieve over the entire parameter space R^q: g-rank( ¯H) = max_{λ∈R^q} rank( ¯H(λ)). ... Theorem 5: If the structural pattern matrix satisfies g-rank( ¯H)<d^2, then the optimal projection basis p_m exists almost everywhere ... Given the structural premise g-rank( ¯H)<d^2, the maximum possible rank over the entire space R^q is strictly less than d^2."

The theorem's hypothesis is exactly Definition 4 restated: if the maximum rank over all λ is < d^2, then rank( ¯H(λ))<d^2 for every λ, so a non-trivial left null-space exists for all admissible parameter realizations. The 'almost everywhere' conclusion therefore follows immediately from the definition of g-rank as a maximum; no independent structural or parameter-independent argument is supplied beyond that definition. This is a definitional tautology in a supporting existence claim, not a fitted prediction and not the source of the main Squeeze Theorem bounds.

full rationale

The central derivation chain is not circular: Theorem 1 derives an upper bound from L-invariance of im(P_π) under equitable partitions; Theorem 3 aggregates layer-wise distance partitions into a lower bound; Theorem 4 obtains optimal-basis existence from rank-nullity on an augmented matrix. None of these steps fits parameters to data, renames a known result, or rests on a self-citation for its load-bearing content; [12] and [14] are used contextually for observability/invariant extensions. The one genuinely circular-looking item is Theorem 5, where the generic-rank condition is defined as the maximum rank over the parameter space, making the asserted parameter-independent null-space existence a direct unpacking of the definition. I treat that as a minor definitional tautology and score it accordingly. The disputed 'orthogonality implies direct sum' step in Theorem 3 is a mathematical-support/correctness concern about the lower bound, not a reduction of an output to an input of the kind this circularity pass targets, so it is not scored as circularity here.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 3 invented entities

The central bounds rest on standard EP and distance-partition machinery plus two ad hoc premises: the direct-sum independence of layers in the lower bound, and the claim that WL color refinement outputs the maximal equitable partition. Both are unproven and questionable. There are no fitted physical constants; the basis and the target partition are user-chosen inputs that the algorithms are meant to discover.

free parameters (3)

Basis matrices B_m = chosen by hand in Examples 2-4 (e.g., B1 = I, B2 = [[0,1],[-1,0]])
The LDP lower bound and bound gap depend on the chosen basis. The examples select bases that make a shortcut weight vanish in one layer and expose symmetry; the optimal-basis algorithm is supposed to find such bases automatically.
Target equitable partition π and shortcut set E_short = Example 4 fixes π = {{v1},{v2,v3},{v4}} and E_short = {(v1,v4)}
These are inputs to the null-space optimization in Theorem 4. They are chosen manually in the examples, and the tightness of the resulting bound depends on their choice.
d mutually orthogonal basis layers D = Example 3 uses d = 2 layers B1, B2
Theorem 3 assumes the dynamics are driven by d orthogonal basis layers rather than d^2 layers from the full matrix-space decomposition; this choice is not derived or reconciled with Eq. (14).

axioms (5)

domain assumption Strong structural controllability requires a property to hold for every admissible nonzero weight configuration.
Used throughout; the paper's definition of W′ as min_i dim(W_i) conflates the dimension-minimizing configuration with the strong structural controllable subspace.
standard math Equitable partitions induce L-invariant subspaces (Lemma 2).
Invoked for the upper bound; the proof sketch assumes no self-loops and constant block row sums, but is otherwise a standard quotient-subspace argument.
domain assumption Scalar distance partitions give lower bounds on controllable subspace dimension (from [5]-[9]).
Borrowed without proof and applied independently to each scalar layer in Theorem 3; the paper does not address cancellation in Krylov products or the difference between generic and all-nonzero weight requirements.
ad hoc to paper Matrix orthogonality implies direct-sum independence of the lifted layer subspaces.
This is the load-bearing premise of the lower bound in Theorem 3; it is not proven and is not generally true for trace-orthogonal matrix bases.
ad hoc to paper WL color refinement returns the unique maximal structural equitable partition.
Claimed in Theorem 6; standard color refinement produces a bisimulation partition, which is generally finer than the coarsest equitable partition when edge weights differ but cellwise sums coincide.

invented entities (3)

Layer-specific distance partition (LDP) no independent evidence
purpose: Defines per-layer distance sets from leaders in the scalar subgraph of each basis layer, used for the lower bound.
A new mathematical construct with no external falsifiable prediction beyond the paper's own bounds.
Optimal projection basis p_m no independent evidence
purpose: A vector whose associated basis matrix simultaneously satisfies symmetry and shortcut-cutting constraints to tighten the bound gap.
Constructed by null-space intersection; its almost-everywhere existence is claimed via generic rank.
Multi-layer WL color refinement partition no independent evidence
purpose: Automated discovery of the target equitable partition and shortcut set.
An algorithmic variant whose claimed equivalence to the maximal equitable partition is not established.

pith-pipeline@v1.3.0-daily-deepseek · 13814 in / 17390 out tokens · 159263 ms · 2026-08-01T00:15:19.064091+00:00 · methodology

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read the original abstract

This paper investigates the strong structural controllability of multi-agent networks. Based on the definition of equitable partitions, an upper bound for the strong structural controllable subspace (SSCS) is established. To reflect the physical significance of matrix weights where the state dimension is greater than one, the multi-agent system is modeled using higher-order dynamics. Furthermore, to address matrix singularity and asymmetric couplings, a matrix space basis decomposition method is proposed to transform the matrix-weighted network into layered scalar networks. Additionally, by extending this basis decomposition to the lower bound estimation, a layer-specific distance partition (LDP) is introduced. This formulation establishes a tighter Squeeze Theorem, narrowing the mathematical boundaries for the controllable subspace by capturing layer-specific structural delays. To systematically identify the optimal basis that minimizes the bounds gap, an algebraic algorithm based on null-space projection is formulated. Furthermore, by introducing pattern matrices and generic rank, the almost-everywhere existence of this optimal basis in the parameter space is rigorously proved, perfectly aligning with the definition of strong structural controllability. To break the NP-hard combinatorial bottleneck of manually pre-defining the targets, a polynomial-time automated discovery algorithm based on the multi-layer Weisfeiler-Lehman (WL) color refinement is proposed. Finally, the strong structural observability and invariant attributes of the network are evaluated. Numerical examples with asymmetric matrix weights and directed multi-layer topologies are provided to verify the derived theorems.

Figures

Figures reproduced from arXiv: 2607.27852 by Lanhao Zhao.

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 1 linked inside Pith

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Determination of the dimensions of strong structural controllable subspaces,

J. C. Jarczyk, F. Svaricek, and B. Alt, “Determination of the dimensions of strong structural controllable subspaces,”IFAC Proceedings Volumes, vol. 43, no. 21, pp. 131-137, 2010

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On strong structural controllability of networked systems: A constrained matching approach,

A. Chapman and M. Mesbahi, “On strong structural controllability of networked systems: A constrained matching approach,” in2013 American Control Conference. IEEE, 2013, pp. 6126-6131

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On the computation of the distance-based lower bound on strong structural controllability in networks,

M. Shabbir, W. Abbas, and Y . Yazıcıo ˘glu, “On the computation of the distance-based lower bound on strong structural controllability in networks,” in2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, 2019, pp. 5468-5473

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Strong structural controllability of diffusively coupled networks: Comparison of bounds based on distances and zero forcing,

Y . Yazıcıo ˘glu, M. Shabbir, W. Abbas, and X. Koutsoukos, “Strong structural controllability of diffusively coupled networks: Comparison of bounds based on distances and zero forcing,” in2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 566-571

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S. Zhang, M. Cao, and M. K. Camlibel, “Upper and lower bounds for controllable subspaces of networks of diffusively coupled agents,”IEEE Transactions on Automatic control, vol. 59, no. 3, pp. 745-750, 2013

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Controllability of multiagent networks with antagonistic interactions,

C. Sun, G. Hu, and L. Xie, “Controllability of multiagent networks with antagonistic interactions,”IEEE transactions on automatic control, vol. 62, no. 10, pp. 5457-5462, 2017

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On the controllability of matrix-weighted networks,

L. Pan, H. Shao, M. Mesbahi, Y . Xi, and D. Li, “On the controllability of matrix-weighted networks,”IEEE Control Systems Letters, vol. 4, no. 3, pp. 572-577, 2020

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Controllability of general linear dis- crete multiagent systems with directed and weighted signed networks,

L. Zhao, Z. Ji, Y . Liu, and C. Lin, “Controllability of general linear dis- crete multiagent systems with directed and weighted signed networks,” Journal of Systems Science and Complexity, 2022

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Equitable partitions in the controllability of undirected signed graphs,

H. Gao, Z. Ji, and T. Hou, “Equitable partitions in the controllability of undirected signed graphs,” in2018 IEEE 14th International Conference on Control and Automation (ICCA), IEEE, 2018, pp. 532-537

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[14]

Controllability and observability of linear multi-agent systems over matrix-weighted signed networks,

L. Zhao, Z. Ji, Y . Liu, and C. Lin, “Controllability and observability of linear multi-agent systems over matrix-weighted signed networks,” arXiv. 2204.00995, 2022

Pith/arXiv arXiv 2022

[1] [1]

Strong structural controllability,

H. Mayeda and T. Yamada, “Strong structural controllability,”SIAM Journal on Control and Optimization, vol. 17, no. 1, pp. 123-138, 1979

1979

[2] [3]

On the structural and strong structural controllability of undirected networks,

S. S. Mousavi, M. Haeri, and M. Mesbahi, “On the structural and strong structural controllability of undirected networks,”IEEE Transactions on Automatic Control, vol. 63, no. 7, pp. 2234-2241, 2017

2017

[3] [4]

A uni- fying framework for strong structural controllability,

J. Jia, H. J. van Waarde, H. L. Trentelman, and M. K. Camlibel, “A uni- fying framework for strong structural controllability,”IEEE Transactions on Automatic Control, vol. 66, no. 1, pp. 391-398, 2020

2020

[4] [5]

Determination of the dimensions of strong structural controllable subspaces,

J. C. Jarczyk, F. Svaricek, and B. Alt, “Determination of the dimensions of strong structural controllable subspaces,”IFAC Proceedings Volumes, vol. 43, no. 21, pp. 131-137, 2010

2010

[5] [6]

On strong structural controllability of networked systems: A constrained matching approach,

A. Chapman and M. Mesbahi, “On strong structural controllability of networked systems: A constrained matching approach,” in2013 American Control Conference. IEEE, 2013, pp. 6126-6131

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On the computation of the distance-based lower bound on strong structural controllability in networks,

M. Shabbir, W. Abbas, and Y . Yazıcıo ˘glu, “On the computation of the distance-based lower bound on strong structural controllability in networks,” in2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, 2019, pp. 5468-5473

2019

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Y . Yazıcıo ˘glu, M. Shabbir, W. Abbas, and X. Koutsoukos, “Strong structural controllability of diffusively coupled networks: Comparison of bounds based on distances and zero forcing,” in2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 566-571

2020

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Upper and lower bounds for controllable subspaces of networks of diffusively coupled agents,

S. Zhang, M. Cao, and M. K. Camlibel, “Upper and lower bounds for controllable subspaces of networks of diffusively coupled agents,”IEEE Transactions on Automatic control, vol. 59, no. 3, pp. 745-750, 2013

2013

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Controllability of multiagent networks with antagonistic interactions,

C. Sun, G. Hu, and L. Xie, “Controllability of multiagent networks with antagonistic interactions,”IEEE transactions on automatic control, vol. 62, no. 10, pp. 5457-5462, 2017

2017

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On the controllability of matrix-weighted networks,

L. Pan, H. Shao, M. Mesbahi, Y . Xi, and D. Li, “On the controllability of matrix-weighted networks,”IEEE Control Systems Letters, vol. 4, no. 3, pp. 572-577, 2020

2020

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Controllability of general linear dis- crete multiagent systems with directed and weighted signed networks,

L. Zhao, Z. Ji, Y . Liu, and C. Lin, “Controllability of general linear dis- crete multiagent systems with directed and weighted signed networks,” Journal of Systems Science and Complexity, 2022

2022

[12] [13]

Equitable partitions in the controllability of undirected signed graphs,

H. Gao, Z. Ji, and T. Hou, “Equitable partitions in the controllability of undirected signed graphs,” in2018 IEEE 14th International Conference on Control and Automation (ICCA), IEEE, 2018, pp. 532-537

2018

[13] [14]

Controllability and observability of linear multi-agent systems over matrix-weighted signed networks,

L. Zhao, Z. Ji, Y . Liu, and C. Lin, “Controllability and observability of linear multi-agent systems over matrix-weighted signed networks,” arXiv. 2204.00995, 2022

Pith/arXiv arXiv 2022