System Relaxation Algorithm (SRA)

• System Relaxation Algorithm (SRA) is an adaptive network control paradigm that uses physical relaxation principles to balance load through historical stress measurements.
• It integrates normalized betweenness centrality with a leaky integrator mechanism, optimizing throughput in heterogeneous networks and enhancing resilience in homogeneous ones.
• Rigorous non-smooth dynamics analysis and Lyapunov methods prove SRA's global convergence and stability, offering transparent, tunable control for diverse network scenarios.

The System Relaxation Algorithm (SRA) is a computational paradigm for network control that replaces traditional, static shortest-path methods with an adaptive, topology-aware process modeled on physical relaxation principles. The algorithm operates as a discrete-time, non-smooth dynamical system, driving networks toward load-balanced dynamical equilibrium via feedback mechanisms based on structural stress, quantified by betweenness centrality and aggregated through leaky integrators. SRA's design emphasizes interpretability ("white-box" control), predictable performance guarantees, and topology-dependent objectives: it optimizes throughput in heterogeneous networks and resilience in homogeneous ones. The approach includes rigorous proofs of global convergence and practical stability via non-smooth dynamics theory, offering a mathematically grounded alternative to reactive or black-box routing strategies (Ren et al., 21 Sep 2025).

1. Motivations and Objective

Prevailing network control approaches, based on static shortest-path routing, concentrate traffic load ("stress concentration") on high-centrality nodes, causing bottlenecks that impair capacity and resilience. This concentration is quantifiable by elevated betweenness centrality (BC), indicating excessive reliance on a small subset of nodes. SRA counteracts this tendency by continuously redistributing load in response to accumulated stress, seeking a smooth global equilibrium of load distribution. The objective is to minimize peak centrality and, hence, bottleneck probability, without relying solely on instantaneous congestion cues (e.g., queue length or latency metrics).

SRA achieves this by integrating historical stress (accumulated BC) into node cost calculations, thereby guiding dynamical path selection toward a load-balanced steady state. The algorithm is not merely congestion-reactive; it embodies a control-theoretic feedback mechanism that leverages the network's topological structure for proactive governance.

2. Algorithmic Structure and Dynamics

SRA is formalized as a discrete-time feedback control system operating on a graph with $N$ nodes. Core components are:

• Sensing and Normalization: At each iteration $k$, node betweenness centralities $\kappa_i^{(k)}$ are computed and normalized:

$$\mathrm{norm}(\kappa(S)) = \begin{cases} \kappa(S)/\max_j \kappa_j(S), & \text{if } \max_j \kappa_j(S) > 0 \ 0, & \text{otherwise} \end{cases}$$

• Accumulated Pressure ("S" vector): Updated via a leaky integrator with learning rate $0 < \alpha < 1$:

$$S^{(k+1)}_i = (1 - \alpha) S^{(k)}_i + \alpha \cdot \mathrm{norm}(\kappa_i^{(k)})$$

This process ensures that transient fluctuations are damped and persistent stress patterns are reflected in the pressure variable.

• Cost Transformation: Routing cost for node $i$ is set as:

$C_i^{(k+1)} = 1 + \beta_I S^{(k+1)}_i$

where $\beta_I$ is the pressure-to-cost conversion factor. Path selection is thereby modulated not just by physical length, but by accumulated stress.

• Anti-Chattering via Dwell Time: A minimum dwell time $T_d$ is imposed between path-switch events to prevent persistent oscillations ("chattering"), ensuring practical stability and allowing the network to settle between adjustments.
• Feedback Loop: The algorithm senses, computes pressure and cost, enforces minimum dwell time, and updates the path configuration in a closed-loop policy.

3. Performance Metrics and Outcomes

SRA's efficacy is measured via key metrics:

• Peak Centrality Reduction: In heterogeneous network topologies (Barabási–Albert graphs), SRA reduces the peak node centrality by over 80%. This manifests as a substantial decrease in bottleneck vulnerability.
• High-Load Throughput Enhancement: By routing traffic away from overstressed nodes (even at the cost of longer paths), SRA achieves a throughput increase of more than 45% in empirical studies under high-load conditions.
• Resilience Metrics: In homogeneous topologies (Erdős–Rényi, Watts–Strogatz), SRA does not dramatically change throughput but decreases packet loss (e.g., from ~7% to ~2%), trading minor increases in latency for improved reliability.
• Capacity Bounds: The relationship between network capacity and stress is formalized via a lower-bound function:

$L(\kappa) = \mu_{\min} / \kappa_{\max}$

where $\mu_{\min}$ is the minimum service rate and $\kappa_{\max}$ is the maximum centrality. SRA increases $L$ by reducing $\kappa_{\max}$.

A theoretical implication is that SRA can shift the fundamental limits of network capacity by reorganizing stress distribution, enhancing robustness to dynamic congestion and targeted attacks.

4. Topological Dependence and Emergent Behavior

SRA's behavior is topology-dependent:

• Heterogeneous Networks: Such graphs have a few high-degree hubs. SRA efficiently "relaxes" these hubs, shaving peak stress and channeling traffic more equitably. Resultant performance gains include both throughput and robustness to localized failures.
• Homogeneous Networks: The lack of dominant hubs means SRA's impact focuses on enhancing resilience and reducing packet loss, with minimal change to global throughput. The algorithm adapts objectives in response to network uniformity.
• Adaptive Trade-Offs: SRA intelligently negotiates a trade-off between average latency, throughput, and reliability. Slightly suboptimal path choices (with higher cost) are offset by gains in load balancing and reduced risk of catastrophic node overload.

This suggests that SRA's "emergent objective" fundamentally shifts according to structural properties, operating as a throughput optimizer in skewed topologies and as a resilience enhancer in uniform settings.

5. Mathematical Foundations: Non-Smooth Dynamics and Stability

Rigorous convergence and stability analysis is provided using non-smooth dynamical systems and Lyapunov theory:

• Global Attractor: SRA is modeled by the set-valued discrete inclusion

$$S^{(k+1)} \in F(S^{(k)}) = (1-\alpha) S^{(k)} + \alpha \mathcal{N}(S^{(k)})$$

with $\mathcal{N}(S)$ a Krasovskii regularization of normalized centrality. The state space $[0,1]^N$ is positively invariant, and trajectories converge to a compact global attractor $\Omega$.

• Piecewise Lyapunov Function: Defined as $V(S) = \min_{c\in \mathcal{N}(S)} \|S-c\|_2^2$. The following "drop-jump" inequality governs convergence:

$$V(S^{(k+1)}) \leq (1-\alpha)V(S^{(k)}) + \frac{1}{\alpha}\tilde{V}^2$$

where $\tilde{V}$ is an upper bound for path-switch induced jumps. This ensures that despite possible discrete jumps, energy decays and the state ultimately remains bounded near the attractor.

• Practical Stability via Dwell Time: By imposing a minimum dwell time $T_d$, SRA bounds the number of switch events and guarantees rate of convergence to an $\varepsilon$-tube surrounding $\Omega$, preventing cycling and oscillatory instability.

Together, these results establish that SRA's non-smooth update dynamics possess mathematically predictable robustness and eventual stability under broad initial conditions.

6. Applications in Network Control

SRA's architecture and guarantees enable deployment in demanding network control scenarios:

• Software-Defined Networks (SDN): SRA aligns naturally with centralized SDN controllers, which possess global views and can enforce dynamic cost policies. Its white-box logic provides interpretable and tunable control over node cost adaptation.
• Traffic Engineering: By leveraging historical load data, SRA enables proactive re-routing before congestion occurs, especially valuable in backbone and data center networks where peak capacity determines overall performance.
• Resilient Design: SRA's topology-aware behavior ensures that critical infrastructure is protected against sudden demand surges and targeted failures—by maintaining balanced stress, it reduces the probability of catastrophic loss.
• Operator Interactivity: Explicit parameters ($\alpha$, $\beta_I$, $T_d$) grant operators fine-grained control over the trade-off between performance optimization and resilience goal-setting, adapting to network conditions and policy requirements.

A plausible implication is that as network complexity grows, SRA's model-driven paradigm may supplant heuristic or black-box control logic, offering provable performance with transparent adaptation.

7. Summary and Significance

The System Relaxation Algorithm (SRA) constitutes a rigorous, interpretable framework for network control that:

• Proactively balances load by integrating historical stress data into node cost calculations.
• Achieves substantial improvements in throughput and resilience, sharply dependent on network topology.
• Possesses rigorous global convergence and stability guarantees via non-smooth dynamical analysis and Lyapunov techniques.
• Provides direct tunability and transparency for network operators, with unified white-box control logic.
• Opens new directions for adaptive and robust regimes in both centralized and distributed network settings.

SRA signals a fundamental shift from reactive, statically optimal path selection to adaptive, equilibrated network governance that accommodates both performance and reliability objectives in large-scale, heterogeneous infrastructure (Ren et al., 21 Sep 2025).