Message-passing graph neural networks (MPGNNs) dominate modern graph learning. Typical efforts enhance MPGNN's expressive power by enriching the adjacency-based aggregation. In contrast, we introduce a graph-level aggregation over walk incidence-based matrices that are constructed to deliberately trade off some expressivity for stronger and more structured inductive bias. This approach allows for gradual scaling between classical message-passing and simpler methods based on walks. Drawing on weighted automata, we ensure our gradual scaling is polynomial-time and avoids the combinatorial explosion of colored walks. Our second result characterizes the expressive power at each scale using homomorphism counts over a hierarchy of generalized caterpillar graphs. Based on these foundations, we propose Caterpillar GNNs that substantially reduce the number of nodes in the hidden layers of the computational graphs on real-world datasets. This gradual reduction does not obstruct learning, enabling a systematic study of lower-order expressivity. We further describe benchmark settings where task-aligned expressivity supports learning.

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