pith. machine review for the scientific record. sign in

arxiv: 2604.14205 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

Recognition: no theorem link

Consensus and Synchronization of Multi-agent Systems over Finite Fields -- Graph Topologies

Kristian Hengster-Movri\'c , \v{S}imon Lehk\'y , Farnaz Adib Yaghmaie
Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:05 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords multi-agent systemsconsensusfinite fieldsgraph topologiessynchronizationNP-hard problemcommunication topologyalgorithms
0
0 comments X

The pith

Two algorithms generate the communication topologies needed for consensus and synchronization in multi-agent systems over finite fields.

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

Multi-agent systems with agents restricted to finite state spaces require specific communication topologies to achieve consensus or synchronization. These topologies are hard to find because the underlying construction problem is NP-hard. The paper shows that by systematically exploring subsets of admissible matrices one can generate valid topologies efficiently. Two new algorithms implement this subset search for both scalar single-integrator consensus and general linear time-invariant synchronization. Simulations confirm that the generated topologies satisfy the required finite-field conditions.

Core claim

By efficiently exploring subsets of admissible matrices, two new algorithms can construct the communication topologies required for consensus in single-integrator agents and synchronization in general LTI agents over finite fields.

What carries the argument

Subset-exploration algorithms that search admissible matrices to produce valid finite-field graph topologies for multi-agent coordination.

If this is right

  • Enables design of consensus protocols for agents that store and transmit only a finite alphabet.
  • Yields explicit topologies for both scalar and higher-order LTI agent dynamics.
  • Produces networks that remain functional under communication noise.
  • Reduces the practical barrier to deploying finite-state multi-agent coordination.

Where Pith is reading between the lines

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

  • The same subset-search idea could be applied to other finite-alphabet coordination tasks such as formation control.
  • Scalability hinges on how quickly the subset enumeration prunes infeasible matrices as network size grows.
  • Algebraic properties of finite fields may allow further analytic shortcuts beyond the current numerical search.

Load-bearing premise

That searching subsets of admissible matrices will produce practical algorithms for the NP-hard problem of building the required communication topologies.

What would settle it

The algorithms return no valid topology after exhaustive search on a small system instance already known to possess admissible topologies.

Figures

Figures reproduced from arXiv: 2604.14205 by Farnaz Adib Yaghmaie, Kristian Hengster-Movri\'c, \v{S}imon Lehk\'y.

Figure 2
Figure 2. Figure 2: The cardinality |URS N,p| of the upper triangular matrices based on the matrix dimension N [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The cardinality |URS N,p| of the upper triangular matrices based on the field size p. VI. SIMULATION RESULTS This section provides a numerical validation of the proposed framework and algorithms using a small-scale system. Con￾sider a network of N = 2 agents operating over F3. A. Matrix Generation and Density The space of all row-stochastic matrices MRS 2,3 has a dimen￾sion of N(N −1) = 2, resulting in |MR… view at source ↗
read the original abstract

This paper brings cooperative protocols for multi-agent systems with agents having a finite state-space. Both scalar single-integrator consensus and general LTI systems synchronization are considered. Systems having a finite state-space describe agents with minimal memory capacity processing only a finite alphabet. Such systems are remarkably resilient to communication noise. The crucial problem, however, is to construct the admissible communication topology, which is NP-hard. We address this by efficiently exploring the subsets of admissible matrices and propose two new algorithms to generate the topologies. Simulations validate the proposed approach.

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 manuscript considers consensus and synchronization in multi-agent systems where agents operate over finite fields with finite state spaces. It notes that admissible communication topologies are NP-hard to construct and proposes two new algorithms that efficiently explore subsets of admissible matrices to generate such topologies, with validation provided by simulations.

Significance. If the algorithms deliver practical performance on instances where NP-hardness is relevant, the work could advance noise-resilient designs for memory-limited multi-agent systems. The emphasis on finite-alphabet resilience is a clear strength, but the absence of complexity bounds or scaling results in the available text limits the assessed contribution.

major comments (2)
  1. [Abstract] Abstract: the claim that the two algorithms 'efficiently' solve the NP-hard topology construction problem is unsupported; no runtime bounds, pseudocode, worst-case analysis, or simulation metrics (e.g., runtime vs. number of agents or field size) are supplied to demonstrate sub-exponential scaling.
  2. [Abstract] The central claim that the algorithms generate admissible topologies for both scalar consensus and general LTI synchronization rests on unspecified simulations; no performance metrics, error analysis, or comparison to existing methods appear in the provided text, preventing verification that the approach is load-bearing.
minor comments (2)
  1. Clarify the precise definition of 'admissible matrices' and the finite-field operations used in the consensus/synchronization protocols.
  2. The abstract would benefit from stating the range of agent counts and field sizes tested in the simulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below and will revise the manuscript to provide the requested details on algorithm performance and validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the two algorithms 'efficiently' solve the NP-hard topology construction problem is unsupported; no runtime bounds, pseudocode, worst-case analysis, or simulation metrics (e.g., runtime vs. number of agents or field size) are supplied to demonstrate sub-exponential scaling.

    Authors: The full manuscript contains pseudocode for both algorithms and simulation results on finite-field instances. We acknowledge the abstract's use of 'efficiently' lacks supporting quantitative details. In revision we will add a dedicated subsection with empirical runtime metrics (average and worst-case runtimes versus number of agents and field size), clarify that efficiency is demonstrated practically via subset exploration rather than worst-case sub-exponential guarantees, and include any derivable complexity observations while respecting the underlying NP-hardness. revision: yes

  2. Referee: [Abstract] The central claim that the algorithms generate admissible topologies for both scalar consensus and general LTI synchronization rests on unspecified simulations; no performance metrics, error analysis, or comparison to existing methods appear in the provided text, preventing verification that the approach is load-bearing.

    Authors: The simulations in the manuscript demonstrate successful generation of admissible topologies for both scalar consensus and LTI synchronization cases. We agree more detail is required. The revision will expand the results section with explicit performance metrics (success rates, convergence error norms, noise resilience), error analysis, and direct comparisons against baselines such as random admissible matrices and exhaustive search on small instances. These additions will be summarized in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain; algorithmic proposal lacks equations or self-referential reductions

full rationale

The paper proposes two algorithms for constructing admissible communication topologies in finite-field multi-agent systems, noting that the problem is NP-hard and claiming efficiency via subset exploration of admissible matrices, with validation by simulations. No equations, derivations, fitted parameters, or first-principles results appear in the provided text. There are no self-definitional steps where a claimed output is defined in terms of itself, no fitted inputs relabeled as predictions, and no load-bearing self-citations or uniqueness theorems imported from prior author work. The central claim rests on the existence and performance of the proposed algorithms rather than any tautological reduction to inputs, making the derivation self-contained against external benchmarks such as simulation results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract.

pith-pipeline@v0.9.0 · 5397 in / 881 out tokens · 27217 ms · 2026-05-13T19:05:12.551046+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    A programmable galois field processor for the internet of things,

    Y . Chen, S. Lu, C. Fu, D. Blaauw, R. Dreslinski, T. Mudge, and H.- S. Kim, “A programmable galois field processor for the internet of things,”ACM SIGARCH Computer Architecture News, vol. 45, pp. 55–68, 06 2017

    work page 2017

  2. [2]

    Consensus networks over finite fields,

    F. Pasqualetti, D. Borra, and F. Bullo, “Consensus networks over finite fields,”Automatica, vol. 50, no. 2, pp. 349–358, 2014. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0005109813005281

    work page 2014

  3. [3]

    Consensus problems on networks with antagonistic in- teractions,

    C. Altafini, “Consensus problems on networks with antagonistic in- teractions,”Automatic Control, IEEE Transactions on, vol. 58, pp. 935–946, 04 2013

    work page 2013

  4. [4]

    Bipartite consensus of linear multi- agent systems over signed digraphs: An output feedback control approach,

    H. Zhang and J. Chen, “Bipartite consensus of linear multi- agent systems over signed digraphs: An output feedback control approach,”IFAC Proceedings Volumes, vol. 47, no. 3, pp. 4681– 4686, 2014, 19th IFAC World Congress. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S1474667016423372

    work page 2014

  5. [5]

    Distributed surrounding design of target region with complex adjacency matrices,

    Y . Lou, “Distributed surrounding design of target region with complex adjacency matrices,”IEEE Transactions on Automatic Control, vol. 60, pp. 283–288, 01 2015

    work page 2015

  6. [6]

    Complex laplacians and applications in multi- agent systems,

    J.-G. Dong and L. Qiu, “Complex laplacians and applications in multi- agent systems,”arXiv preprint arXiv:1406.1862, 2014

    work page arXiv 2014

  7. [7]

    Multiparty consensus of linear heterogeneous multiagent systems,

    F. A. Yaghmaie, R. Su, F. L. Lewis, and L. Xie, “Multiparty consensus of linear heterogeneous multiagent systems,”IEEE Transactions on Automatic Control, vol. 62, no. 11, pp. 5578–5589, 2017

    work page 2017

  8. [8]

    Bipartite and cooperative output synchronizations of linear heterogeneous agents,

    F. Adib Yaghmaie, R. Su, F. L. Lewis, and S. Olaru, “Bipartite and cooperative output synchronizations of linear heterogeneous agents,”Automatica, vol. 80, no. C, pp. 172–176, 06 2017. [Online]. Available: https://doi.org/10.1016/j.automatica.2017.02.033

    work page doi:10.1016/j.automatica.2017.02.033 2017

  9. [9]

    Booth,Sequential Machines and Automata Theory

    T. Booth,Sequential Machines and Automata Theory. Wiley, 1967. [Online]. Available: https://books.google.cz/books? id=WOxQAAAAMAAJ

    work page 1967

  10. [10]

    The theory of autonomous linear sequential networks,

    B. Elspas, “The theory of autonomous linear sequential networks,” IRE Transactions on Circuit Theory, vol. 6, no. 1, pp. 45–60, 1959

    work page 1959

  11. [11]

    Linear modular sequential circuits,

    B. Friedland, “Linear modular sequential circuits,”IRE Transactions on Circuit Theory, vol. 6, no. 1, pp. 61–68, 1959

    work page 1959

  12. [12]

    Linear multivalued sequential coding networks,

    J. Hartmanis, “Linear multivalued sequential coding networks,”IEEE Transactions on Circuits and Systems I-regular Papers, vol. 6, pp. 69–74, 1959. [Online]. Available: https://api.semanticscholar.org/ CorpusID:122426782

    work page 1959

  13. [13]

    Leader-following consensus of multi-agent systems over finite fields,

    X. Xu and Y . Hong, “Leader-following consensus of multi-agent systems over finite fields,”Proceedings of the IEEE Conference on Decision and Control, vol. 2015, 04 2014

    work page 2015

  14. [14]

    Synchronization of networks over finite fields,

    M. Meng, X. Li, and G. Xiao, “Synchronization of networks over finite fields,”Automatica, vol. 115, p. 108877, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0005109820300753

    work page 2020

  15. [15]

    Leader–follower consensus of multiagent systems with time delays over finite fields,

    Y . Li, H. Li, X. Ding, and G. Zhao, “Leader–follower consensus of multiagent systems with time delays over finite fields,”IEEE Transactions on Cybernetics, vol. 49, no. 8, pp. 3203–3208, 2019

    work page 2019

  16. [16]

    Notes on algebra,

    D. Arapura, “Notes on algebra,” 2010, available at https://www.math. purdue.edu/∼arapura/algebra/algebra.pdf

    work page 2010