Decoupling Structural Heterogeneity from Functional Fairness in Complex Networks: A Theoretical Framework based on the Imbalance Metric

Abstract: Performance evaluation of complex networks has traditionally focused on structural integrity or average transmission efficiency, perspectives that often overlook the dimension of functional fairness. This raises a central question: Under certain conditions, structurally heterogeneous networks can exhibit high functional fairness. To systematically address this issue, we introduce a new metric, Network Imbalance (I), designed to quantitatively assess end-to-end accessibility fairness from a perceived QoS perspective. By combining a tunable sigmoid function with a global Shannon entropy framework, the I metric quantifies the uniformity of connection experiences between all node pairs. We analyze the mathematical properties of this metric and validate its explanatory power on various classical network models. Our findings reveal that low imbalance (i.e., high functional fairness) can be achieved through two distinct mechanisms: one via topological symmetry (e.g., in a complete graph) and the other via extreme connection efficiency driven by structural inequality (e.g., in a scale-free network). This decoupling of structure and function provides a new theoretical perspective for network performance evaluation and offers an effective quantitative tool for balancing efficiency and fairness in network design.