The Effective Depth Paradox: Topology and Trainability in Deep CNNs

This paper presents a controlled comparative study of convolutional neural network (CNN) topology and image classification performance across the architectural families VGG, ResNet, and GoogLeNet, evaluated on CIFAR-10 under a unified training protocol. We formalize the distinction between nominal depth (Dnom), the physical count of weight-bearing layers, and effective depth (Deff), an operational metric quantifying the expected length of forward information paths, extending the path-ensemble interpretation of residual networks introduced by Veit et al. (2016) into closed-form, pre-training proxies spanning sequential, residual, and multi-branch topologies. We validate this proxy against a gradient-weighted variant computed from observed backpropagation signal. Across eight representative models (VGG-11/13/16/19, ResNet-18/34/50, GoogLeNet), plain VGG-style stacks show early accuracy saturation as Deff increases, whereas ResNet and GoogLeNet continue to benefit from added depth by keeping Deff low relative to Dnom - a pattern we term the "Effective Depth Paradox". A pooled correlation analysis shows both Dnom and Deff are strongly, significantly associated with accuracy (r = 0.94 and r = 0.93; both p < 0.01); given the small family-clustered sample, this alone cannot cleanly separate the two metrics, so we treat gradient-norm evidence as complementary mechanistic support rather than decisive statistical proof. We conclude that architectural topology, not layer count alone, governs trainability and scaling efficiency in deep CNNs. All claims are scoped to CIFAR-10-scale training of the three families studied; we do not claim validation at ImageNet scale or generalization to modern architectures such as EfficientNet, ConvNeXt, or Vision Transformers, which we identify as necessary future work.