Depth Fragility and Skeletal Universality: Decoupling Topology and Function in Deep Neural Networks

Deep neural networks (DNNs) are traditionally analyzed as black-box function approximators, yet their internal structure exhibits phase transitions characteristic of complex physical systems. In this study, we investigate topological–functional decoupling —the phenomenon whereby a network retains full graph connectivity while losing computational function—in trained neural networks through the lens of percolation theory. By subjecting three distinct architectures (Shallow, Deep, and Wide MLPs) to a unified edge-pruning analysis on Fashion-MNIST, we uncover a fundamental divergence between structural integrity and computational capacity in this experimental setting. We report three key phenomena observed in these experiments: (1) the zombie network state under stochastic pruning, where the system retains global connectivity ( P ∞ ≈ 1.0 ) yet suffers a catastrophic functional collapse (accuracy falls below 50% of baseline at prunning ratio p f ≈ 0.35 – 0.68 depending on depth), proves that graph reachability does not imply computational capability; (2) depth fragility , where increased network depth triggers multiplicative signal decay (the avalanche effect), rendering deep architectures exponentially more vulnerable to random edge removal than shallow ones ( p f deep ≈ 0.35 vs. p f shallow ≈ 0.68 ); and (3) scale-free universality , observed under magnitude-based pruning, where a robust functional skeleton maintains accuracy near the baseline (∼89%) up to extreme sparsity ( p f ≈ 0.85 – 0.95 ) across all three architectures. Robustness stems not from holographic redundancy in the overall connection count but from the emergent heavy-tailed rich-club organization of weight magnitudes—a sparse set of high-magnitude synapses that form the functional backbone of the network, decoupled from the redundant topological mass. These findings offer new physical constraints for the design of resilient neuromorphic hardware.