Spectral Graph Analysis for Predicting QoE Fairness Sensitivity in Wireless Communication Networks

TL;DR

This work addresses how QoE fairness is shaped by both topology and SLA settings, proposing a predictive, interpretable framework based on spectral graph theory. It establishes a topology-dependent exponential upper bound on the QoE imbalance index $I$, linking it to the spectral gap $\lambda_2$ and a service-stringency parameter $a$ with a threshold activation at $r_*$. The main theoretical contribution is an exponential bound $I \le \frac{C\,(1+aH)}{\ln M}\exp(-\min\{a, c\lambda_2\}H)$, where $H=h_0-r_*$, plus a bottleneck interpretation and Fiedler-vector-guided structural interventions to raise $\lambda_2$. The paper also provides practical engineering tools, including a data-driven certificate for rapid upper bounds and a reverse-design framework to achieve target QoE fairness, all validated across synthetic graphs and real-world topologies. Together, these results enable proactive, topology-aware design and optimization for QoE fairness in next-generation networks.

Abstract

The evaluation of Quality of Experience (QoE) fairness depends not only on its current state but, more critically, on its sensitivity to changes in Service Level Agreement (SLA) parameters. However, the academic community has long lacked a predictive method connecting underlying topology to high-level service fairness. To bridge this gap, this paper analyzes a QoE imbalance index ($I$) through the lens of spectral graph theory.Our core contribution is the proof of a novel exponential spectral upper bound. This bound reveals that the improvement of QoE fairness exhibits an exponential decay behavior only above a performance threshold determined jointly by network size and connectivity. Its core decay rate is dominated by the weaker of two factors: the SLA stringency ($a$) and the network's spectral gap ($cλ_2$). The upper bound unifies the service protocol and the topological bottleneck within a single performance bound formula for the first time.This theoretical relationship also reveals a clear bottleneck effect, where the system's fairness ceiling is determined by the weaker link between service parameters and network structure. This finding provides a bottleneck-driven principle for resource optimization in network design and enables goal-driven reverse engineering. Extensive numerical experiments on various random graph models and real-world network topologies robustly validate the correctness and universality of our analytical framework.

Figures (9)

• Figure 1: Network traffic distribution graph. Subgraph (a) shows the traffic distribution under loose SLA parameters, while subgraph (b) shows the distribution under strict SLA parameters.
• Figure 2: Closed-loop optimization flowchart
• Figure 3: Spectral upper bound envelope verification plot. The plot includes simulation results from all network models. Each data point represents a measurement from an independent combination (network, a, $h_0$). The red dashed line indicates the theoretical upper bound with a slope of -1. All data points are strictly located below this boundary and exhibit a significant correlation.
• Figure 4: Design phase diagram. The x-axis and y-axis represent the service evaluation stringency $a$ and network connectivity $\lambda_2$ (scaled by $c$), respectively, both on a logarithmic scale. The color represents the magnitude of the exponential decay rate $\gamma$.
• Figure 5: Validation of the data-driven certificate's effectiveness. This bar chart compares the actual $I$ values with the data-driven upper bound calculated according to Proposition \ref{['prop:data_driven_cert']} for four network models.

Theorems & Definitions (27)

• Theorem 4.1: Spectral Upper Bound
• Proposition 4.2: Bottleneck Effect
• Corollary 4.3: Convergence of Expander Graph Families
• Corollary 4.4: Reverse Design: Threshold Lower Bound
• Corollary 4.5: Reverse Design: Spectral Gap Lower Bound
• Theorem 5.1: Fiedler Vector-Guided Structural Intervention
• Proposition 5.2: Data-Driven Performance Certificate
• Lemma B.1: Spectral Tail Bound
• proof
• Lemma C.1: Tomographic Representation