D2R: A distance metric for exploring network structural robustness enhancement potential

Ensuring the continued operation of a networked system under various structural disruptions relies heavily on the effective robustness of the system, desirably with optimization. To make this process more efficient, assessing the robustness enhancement potential (REP) in advance helps conserve resources and reduce design and operational costs. This paper proposes a distance to regularity (D2R) structural metric that measures the distance between a network’s degree sequence and its closest regular variant so as to capture REP. Two variants, based on Euclidean distance and Kullback-Leibler divergence, are implemented; both exhibit strong correlations with robustness enhancement under connectedness and controllability measures. Experimental results verify that D2R can effectively capture structural heterogeneity relevant to REP and enhance the performances of machine learning models in predicting REP. Compared to deep neural network approaches, D2R-based models achieve lower prediction errors while offering improved computational efficiency. Feature selection analysis further confirms the consistent improvements of D2R over other benchmarks. These findings establish D2R as a reliable lightweight descriptor for robustness-aware network analysis. © 2025 The Authors.