Resistance Curvature: Recognition, Polyhedral Structure, and Graph Products

What the math gives to ML

The paper turns effective-resistance curvature into a constructive polyhedral certificate: a graph is resistance nonnegative exactly when it admits a spanning-tree distribution whose expected degree at every vertex is at most two, equivalently when the relative interior of the spanning-tree polytope intersects twice the matching polytope. This provides a principled way to generate connected, aggressively sparse edge distributions rather than using independent edge dropout or unconstrained attention pruning. The strongest neural-network transfer is a graph-message-passing or graph-attention module whose edge mask is a fractional spanning-tree point, optionally sampled as actual spanning trees for sparse execution. The method is most relevant to graph neural networks with expensive dense neighborhoods, rather than generic non-graph architectures.