We introduce the popularity-driven random walk graph convolutional network (PDRW-GCN), a topology-aware graph neural network that incorporates degree-biased diffusion to exploit the heavy-tailed connectivity of scale-free networks. By incorporating a structural bias parameter, $\beta$, the model dynamically reweights neighborhood propagation based on node degree. We identify a topology-dependent diffusion transition in heterophilic graph learning: Sparse heterophilic networks benefit from hub-amplified transport ($\beta >0$), whereas dense heterophilic networks require hub-suppressed diffusion ($\beta <0$) to prevent noisy neighborhood overmixing. Experiments across homophilic and heterophilic benchmark networks demonstrate consistent improvements in node classification performance measured by F1 score and receiver operating characteristic—area under the curve. Moreover, the model exhibits diminishing performance gains on non-scale-free topologies, indicating that $\beta$ acts primarily as a topology-dependent routing mechanism rather than a generic regularizer.

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