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Efficient model selection for identifying a suitable pre-trained neural network to a downstream task is a fundamental yet challenging task in deep learning. Current practice requires expensive computational costs in model training for performance prediction. In this paper, we propose a novel framework for neural network selection by analyzing the governing dynamics over synaptic connections (edges) during training. Our framework is built on the fact that back-propagation during neural network training is equivalent to the dynamical evolution of synaptic connections. Therefore, a converged neural network is associated with an equilibrium state of a networked system composed of those edges. To this end, we construct a network mapping $ϕ$, converting a neural network $G_A$ to a directed line graph $G_B$ that is defined on those edges in $G_A$. Next, we derive a neural capacitance metric $β_{\rm eff}$ as a predictive measure universally capturing the generalization capability of $G_A$ on the downstream task using only a handful of early training results. We carried out extensive experiments using 17 popular pre-trained ImageNet models and five benchmark datasets, including CIFAR10, CIFAR100, SVHN, Fashion MNIST and Birds, to evaluate the fine-tuning performance of our framework. Our neural capacitance metric is shown to be a powerful indicator for model selection based only on early training results and is more efficient than state-of-the-art methods.