PaI methods manage to find trainable subnetworks that outperform random pruning, their performance in terms of both accuracy and computational reduction is far from satisfactory compared to post-training pruning and the understanding of PaI is missing.

PaI methods manage to find trainable subnetworks that outperform random pruning, their performance in terms of both accuracy and computational reduction is far from satisfactory compared to post-training pruning and the understanding of PaI is missing.

DSN primarily seeks to transfer knowledge to the new coming task from the learned tasks by selecting the affiliated weights of a small set of neurons to be activated, including the reused neurons from prior tasks via neuron-wise masks. And it also transfers possibly valuable knowledge to the earlier tasks via data-free replay.

Pruning at initialization (PaI) aims to remove weights of neural networks before training in pursuit of training efficiency besides the inference. While off-the-shelf PaI methods manage to find trainable subnetworks that outperform random pruning, their performance in terms of both accuracy and computational reduction is far from satisfactory compared to post-training pruning and the understanding of PaI is missing. For instance, recent studies show that existing PaI methods only able to find good layerwise sparsities not weights, as the discovered subnetworks are surprisingly resilient against layerwise random mask shuffling and weight re-initialization.In this paper, we study PaI from a brand-new perspective -- the topology of subnetworks. In particular, we propose a principled framework for analyzing the performance of Pruning and Initialization (PaI) methods with two quantities, namely, the number of effective paths and effective nodes. These quantities allow for a more comprehensive understanding of PaI methods, giving us an accurate assessment of different subnetworks at initialization. We systematically analyze the behavior of various PaI methods through our framework and observe a guiding principle for constructing effective subnetworks: *at a specific sparsity, the top-performing subnetwork always presents a good balance between the number of effective nodes and the number of effective paths.*Inspired