Openness-Driven Mode Interactions in Complex Systems

• Openness-Driven Mode Interactions are defined by the ability of subsystems to exchange energy, information, or control through coupling, affecting system stability and evolution.
• Mathematical and categorical frameworks, including nonlinear coupling and graph-based models, formalize how openness mediates interactions in diverse systems.
• Applications span microresonator physics, opinion dynamics, economic strategy, and AI governance, where minor changes in openness can trigger major phase transitions and innovation shifts.

Openness‐driven mode interactions refer to the diverse ways in which the property of “openness”—whether conceptual, technical, or social—drives, mediates, or shapes the interactions between modes or components within dynamical, social, technical, or economic systems. The literature demonstrates that openness is a multidimensional property influencing system evolution, stability, and inter-component coupling, with technical instances ranging from microresonator physics to structural opinion dynamics and strategic behavior in economics and AI governance.

1. Mathematical and Physical Foundations of Openness‐Driven Mode Interactions

In physical systems, openness is often instantiated as the possibility for subsystems or modes to exchange energy, information, or control through explicitly defined coupling mechanisms.

A prototypical example is the interaction between parametrically and directly excited vibration modes in a clamped–clamped microresonator (Westra et al., 2011). The dynamical equation for a single mode subjected to both direct and parametric drive is:

$$m \ddot{u} + \left(\frac{m \omega_R}{Q}\right) \dot{u} + \left[m \omega_R^2 + k_p \sin(2\Omega t + \varphi)\right] u + \alpha u^3 = F \cos(\Omega t)$$

where $k_p$ is the strength of parametric modulation, $\varphi$ is the phase difference, and $\alpha$ encodes cubic (Duffing) nonlinearity.

Nonlinear coupling arises as the displacement-induced tension from one resonant mode alters the effective restoring force of another, yielding a quadratic frequency shift:

$f_{R,i} \propto |A_j|^2$

with $A_j$ being the amplitude of the parametrically excited mode. The interplay of openness parameters (e.g., amplitude control via the parametric pump) thus drives linear relationships between observable modal properties, exemplifying openness-driven mode interaction as a mechanism for transducing energy and information across internal degrees of freedom.

2. Openness Parameters in Social and Cognitive Dynamical Systems

In socio-cognitive models, openness takes the form of control parameters that modulate the receptivity and dynamics of agent interactions. The bounded confidence opinion model (Huet et al., 2014) introduces two thresholds:

• $u_m$ (attraction threshold) controlling agent openness on the primary opinion dimension
• $u_s$ (rejection threshold) gating rejection on the secondary dimension

The opinion update rules are:

• Convergence if $|x_m - y_m| \leq u_m$: both main and secondary opinions are attracted
• Rejection if $|x_m - y_m| > u_m$ and $|x_s - y_s| \leq u_s$: secondary opinions are repelled

The openness parameter $u_m$ directly controls cluster formation and stability; low $u_m$ induces fragmentation and volatility on secondary modes, while higher $u_m$ yields large stable clusters. Mode interactions, therefore, are openness-driven, as the system's stability or volatility is controlled by the level of agent openness, which determines both primary aggregation and the fluctuation spectrum on orthogonal opinion modes.

3. Formal and Categorical Models of Openness‐Driven Interactions

Category-theoretic and graph-based frameworks formalize openness as a structural property of dynamical systems. In the theory of open graphic dynamics (Dugowson, 2015) and open sub-functorial dynamics (Dugowson, 2016):

• Dynamics are modeled as morphisms on graphs or categories, equipped with clock structures and parameter spaces.
• Interactions are formalized as multirelations (e.g., $R \in \mathbb{BM}_{(\mathcal{S}, \mathcal{L})}$) that couple system realizations and parameters across multiple dynamics.
• The process of “engendering” new dynamics from interacting open dynamics involves operations such as parameter reduction, synchronization of clocks (either rigid or flexible), and the quotienting of parameter bundles.

A representative formula for synchronization is:

$$\tau_T \circ e^{\alpha_\mu} \subseteq e^h \circ \tau_S$$

and state space construction for the engendered dynamic is:

$$S^\beta = \left\{ (a_i)_{i \in I} \mid a_0 \in S^{\alpha_0}, \forall i \neq 0, a_i \in (\Delta_i S)^{\alpha_i}, \tau_i(a_i) = \delta_i(\tau_0(a_0)) \right\}$$

This categorical perspective provides a structural lens for how openness (parameter freedom, synchronization, relational coupling) determines what combinations and coordination of internal modes are possible in complex multi-agent and multi-modal systems.

4. Openness as a Driver of Network Formation and Economic Strategic Interaction

In economic models of innovation and AI governance, openness is elevated from a technical or organizational property to a variable in a game-theoretic landscape, with direct impacts on market equilibrium and welfare.

The network innovation model (Dasaratha, 2019) defines openness via an agent-level parameter $q_i$ overseen as a probability of interaction (link probability $\iota(q_i, q_j) = q_i q_j$), encoding the firm's strategic balance between learning (innovation) and secrecy (protection).

Key findings include:

• Equilibrium interaction rates are pinned at a critical threshold, with openness-driven transitions marking the emergence of large connected components in the network (percolation threshold).
• The resulting openness-driven mode interactions produce discontinuous (phase-transition-like) responses in innovation rates and network structure: marginal changes in $q$ can shift the system from a sparse (low innovation, high secrecy) to a dense (high innovation, low secrecy) regime.
• Policy interventions (public innovators/altering openness incentives) are only effective when they manage to cross this criticality.

A formal model of AI openness regulation (Qiu et al., 14 Jul 2025) defines the openness level $\omega \in [0,1]$ as a strategic choice by the model provider, coupled to regulatory threshold $\theta$ and penalty $p$. The cost functions for the generalist and specialist are:

$$\varphi_G(\omega | \theta) = \alpha_0 \omega + c_\omega \alpha_1 (1 - \omega) + p \cdot \mathbb{1}[\omega < \theta]$$

$$\varphi_D(\omega) = \frac{(\alpha_1 - \alpha_0)^2}{\omega} + c_\omega \alpha_1 \omega$$

Market equilibria emerge from mode interactions set by openness choices, strategic bargaining ($\delta$ for revenue sharing), and regulatory design, revealing how upstream openness shapes downstream specialization and welfare.

5. Taxonomies and Frameworks for Mode Interactions in AI and Digital Systems

In contemporary AI, openness is a layered, multidimensional property affecting the interactions between system components, actors, and ethical objectives. Recent work articulates comprehensive taxonomies and frameworks:

• Dimensions of AI openness: spanning technical layers (datasets, code, weights, infrastructure, UX), documentation, safeguards, and licensing (Basdevant et al., 17 May 2024).
• The MusGO framework for music-generative AI (Batlle-Roca et al., 4 Jul 2025)—13 interacting categories (8 essential, 5 desirable)—captures the composite nature of model openness as a function of source code availability, training data transparency, evaluation procedures, licensing, and user-oriented documentation. The openness score is defined as:

$$O = \frac{2(E_1 + E_2 + E_3) + E_4 + E_5 + E_6 + E_7 + E_8}{\text{Max Score}} \times 100$$

• The taxonomy in (Paris et al., 9 May 2025) organizes openness by themes (interactivity, freedom, inclusiveness) and approaches (intrinsic properties, afforded actions, desired effects), mapping how system properties (modularity, permeability) interact with cherished social outcomes (democracy, accountability).

Diagrammatic representations (e.g., Fig. 5 and 12 in (Basdevant et al., 17 May 2024)) make explicit that openness at each layer of the stack (e.g., weights, code, data; interface, moderation) enables or constrains translations, integrations, and risk mitigations between modes. Frameworks such as the Model Openness Framework (MOF) provide standardization for comparing and reinforcing mode interactions.

6. Dynamical Consequences and Applications

Openness-driven mode interactions have empirically and theoretically significant effects:

• In microresonators, direct tuning of one mode via parametric openness in another enables reconfigurable signal processing and sensitive transduction (Westra et al., 2011).
• In opinion dynamics, openness levels determine whether social groups are cohesive or fragmented, and which dimensions become volatile (Huet et al., 2014).
• In boundary-driven particle systems, changing openness (boundary conditions) can induce qualitative changes in dynamical relaxation modes, producing transitions with no static counterpart (Botto et al., 2020).
• In technical AI systems, openness in model weights or supporting infrastructures enables or restricts downstream innovation, auditability, and ethical oversight (Basdevant et al., 17 May 2024, Paris et al., 9 May 2025), while inappropriate or unmanaged openness can enable risk propagation or system-level vulnerabilities.
• In strategic regulatory settings, openness thresholds and penalties determine equilibrium strategies, specialization incentives, and the distribution of welfare and innovation benefits (Qiu et al., 14 Jul 2025).

7. Challenges, Constraints, and Future Directions

While openness-driven interactions offer pathways to enhanced adaptability, transparency, and collective innovation, they also introduce challenges:

• Risk–reward tradeoffs: Increased openness may enhance learning opportunities but heighten vulnerabilities (intellectual property, security, safety).
• Structural tension: Complementarity (synergy from multiple openness modes) is not guaranteed; some openness attributes (e.g., non-isolation, unrestricted reuse) may undermine inclusiveness or democratization if not accompanied by adequate safeguards or resource redistribution (Paris et al., 9 May 2025).
• Regulatory ambiguity: The lack of precise, standardized definitions of “openness” for AI models and systems complicates compliance and the design of effective interventions (Qiu et al., 14 Jul 2025).
• Ethical entanglements: Openness can propagate bias, inequity, or misuse unless managed via robust oversight and inclusive practices.

Ongoing research addresses the formalization of openness at multiple system levels, advances in categorical and multi-agent modeling, refinement of openness leaderboards (e.g., MusGO (Batlle-Roca et al., 4 Jul 2025)), and integrated regulatory strategies that optimize for innovation while mitigating risk.

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In summary, openness-driven mode interactions constitute a unifying conceptual and mathematical theme across physics, social science, economics, and digital system design, providing a paradigm for understanding and engineering complex systems whose behavior emerges from the coordinated—or conflictual—interplay of heterogeneous components, mediated by diverse forms of openness.