Oversquashing Meets Resistance: Theoretical Link
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Oversquashing is tightly bounded by effective resistance between nodes.
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The Jacobian norm quantifies how much one node's features influence another.
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Lower resistance = stronger influence, less squashing.
\(\left\| \frac{\partial h_u^{(r)}}{\partial x_v} \right\| \leq (2\alpha\beta)^r \frac{d_{\max}}{2} \left( \frac{2}{d_{\min}} \left( r+1+\frac{\mu^{r+1}}{1-\mu} \right) - R_{u,v} \right)\)
where h_u^{(r)} is the feature vector at node u after r layers,
x_v is the initial feature at node v,
α and β bound the Jacobians of the GNN functions,
d_min and d_max are the minimum and maximum degrees of u and v,
μ is the largest nontrivial eigenvalue of the normalized adjacency,
and R_{u,v} is the effective resistance.