From Consensus to Robust Clustering: Multi-Agent Systems with Nonlinear Interactions

Submitted to IEEE Transactions on Automatic Control

Authors

Anthony Couthures

Gustave Bainier

Vineeth Satheeskumar Varma

Samson Lasaulce

Irinel-Constantin Morărescu

Published

November 2025

Abstract
This paper establishes a theoretical framework to describe the transition from consensus to stable clustering in multi-agent systems with nonlinear, cooperative interactions. We first establish a sharp threshold for consensus. For a broad class of non-decreasing, Lipschitz-continuous interactions, an explicit inequality linking the interaction’s Lipschitz constant to the second-largest eigenvalue of the normalized adjacency matrix of the interaction graph confines all system equilibria to the synchronization manifold. This condition is shown to be a sharp threshold, as its violation permits the emergence of non-synchronized equilibria. We also demonstrate that such clustered states can only arise if the interaction law itself possesses specific structural properties, such as unstable fixed points. For the clustered states that emerge, we introduce a formal framework using Input-to-State Stability (ISS) theory to quantify their robustness. This approach allows us to prove that the internal cohesion of a cluster is robust to perturbations from the rest of the network. The analysis reveals a fundamental principle: cluster coherence is limited not by the magnitude of external influence, but by its heterogeneity across internal nodes. This unified framework, explaining both the sharp breakdown of consensus and the quantifiable robustness of the resulting modular structures, is validated on Zachary’s Karate Club network, used as a classic benchmark for community structure.

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BibTeX citation:
@misc{couthures2025,
  author = {Couthures, Anthony and Bainier, Gustave and Satheeskumar
    Varma, Vineeth and Lasaulce, Samson and Morărescu,
    Irinel-Constantin},
  title = {From {Consensus} to {Robust} {Clustering:} {Multi-Agent}
    {Systems} with {Nonlinear} {Interactions}},
  date = {2025-11},
  url = {https://anthonycouthures.github.io/publications/2025_preprint_consensus-robust-clustering.html},
  doi = {10.48550/arXiv.2511.21228},
  langid = {en},
  abstract = {This paper establishes a theoretical framework to describe
    the transition from consensus to stable clustering in multi-agent
    systems with nonlinear, cooperative interactions. We first establish
    a sharp threshold for consensus. For a broad class of
    non-decreasing, Lipschitz-continuous interactions, an explicit
    inequality linking the interaction’s Lipschitz constant to the
    second-largest eigenvalue of the normalized adjacency matrix of the
    interaction graph confines all system equilibria to the
    synchronization manifold. This condition is shown to be a sharp
    threshold, as its violation permits the emergence of
    non-synchronized equilibria. We also demonstrate that such clustered
    states can only arise if the interaction law itself possesses
    specific structural properties, such as unstable fixed points. For
    the clustered states that emerge, we introduce a formal framework
    using Input-to-State Stability (ISS) theory to quantify their
    robustness. This approach allows us to prove that the internal
    cohesion of a cluster is robust to perturbations from the rest of
    the network. The analysis reveals a fundamental principle: cluster
    coherence is limited not by the magnitude of external influence, but
    by its heterogeneity across internal nodes. This unified framework,
    explaining both the sharp breakdown of consensus and the
    quantifiable robustness of the resulting modular structures, is
    validated on Zachary’s Karate Club network, used as a classic
    benchmark for community structure.}
}
For attribution, please cite this work as:
Anthony Couthures, Gustave Bainier, Vineeth Satheeskumar Varma, Samson Lasaulce, and Irinel-Constantin Morărescu. 2025. “From Consensus to Robust Clustering: Multi-Agent Systems with Nonlinear Interactions.” arXiv preprint. https://doi.org/10.48550/arXiv.2511.21228.