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In an open-loop system, such as a basic thermostat, the poles of the transfer function influence the system's response but do not determine its stability. However, when feedback is introduced to form a closed-loop system, such as an advanced thermostat that adjusts heating based on room temperature, stability is governed by the new poles of the closed-loop transfer function.
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Controllability Robustness of Simplicial Complexes.

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    Network controllability robustness in simplicial complexes is significantly impacted by higher-order structures. The quantity and arrangement of 2-simplices critically regulate network resilience against attacks.

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    Area of Science:

    • Network Science
    • Complex Systems
    • Dynamical Systems Theory

    Background:

    • Understanding network robustness is crucial for designing resilient systems.
    • Simplicial complexes offer a framework to model complex interactions beyond pairwise connections.
    • Controllability analysis assesses a network's ability to reach desired states.

    Purpose of the Study:

    • To investigate the controllability robustness of simplicial complexes under node and edge attacks.
    • To develop a universal nodal dynamical model for arbitrary dimensional simplicial complexes.
    • To identify key structural features influencing network resilience.

    Main Methods:

    • Formulated a universal nodal dynamical model for simplicial complexes.
    • Conducted quantitative analysis considering network topology and higher-order interactions.
    • Examined the impact of node-based and edge-based attacks on controllability.

    Main Results:

    • The quantity and spatial distribution of 2-simplices significantly regulate network controllability robustness.
    • Second-order interaction structures are critical for overall network resilience.
    • Higher-order topological connectivity and dynamical synergy influence robustness.

    Conclusions:

    • Network controllability robustness is intrinsically linked to higher-order interaction structures in simplicial complexes.
    • The findings provide insights into designing more robust complex networks.
    • The proposed model and analysis can be extended to higher dimensional simplices ($q > 2$).