Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cost-Sensitive Neighborhood Aggregation for Heterophilous Graphs: When Does Per-Edge Routing Help?

Eyal Weiss

Abstract

Recent work distinguishes two heterophily regimes: adversarial, where cross-class edges dilute class signal and harm classification, and informative, where the heterophilous structure itself carries useful signal. We ask: when does per-edge message routing help, and when is a uniform spectral channel sufficient? To operationalize this question we introduce Cost-Sensitive Neighborhood Aggregation (CSNA), a GNN layer that computes pairwise distance in a learned projection and uses it to soft-route each message through concordant and discordant channels with independent transformations. Under a contextual stochastic block model we show that mean aggregation can reverse the label-aligned signal direction under heterophily, and that cost-sensitive weighting with $w_+/w_- > q/p$ preserves the correct sign. On six benchmarks with uniform tuning, CSNA is competitive with state-of-the-art methods on adversarial-heterophily datasets (Texas, Wisconsin, Cornell, Actor) but underperforms on informative-heterophily datasets (Chameleon, Squirrel) -- precisely the regime where per-edge routing has no useful decomposition to exploit. The pattern is itself the finding: the cost function's ability to separate edge types serves as a diagnostic for the heterophily regime, revealing when fine-grained routing adds value over uniform channels and when it does not. Code is available at https://github.com/eyal-weiss/CSNA-public .

Cost-Sensitive Neighborhood Aggregation for Heterophilous Graphs: When Does Per-Edge Routing Help?

Abstract

Recent work distinguishes two heterophily regimes: adversarial, where cross-class edges dilute class signal and harm classification, and informative, where the heterophilous structure itself carries useful signal. We ask: when does per-edge message routing help, and when is a uniform spectral channel sufficient? To operationalize this question we introduce Cost-Sensitive Neighborhood Aggregation (CSNA), a GNN layer that computes pairwise distance in a learned projection and uses it to soft-route each message through concordant and discordant channels with independent transformations. Under a contextual stochastic block model we show that mean aggregation can reverse the label-aligned signal direction under heterophily, and that cost-sensitive weighting with preserves the correct sign. On six benchmarks with uniform tuning, CSNA is competitive with state-of-the-art methods on adversarial-heterophily datasets (Texas, Wisconsin, Cornell, Actor) but underperforms on informative-heterophily datasets (Chameleon, Squirrel) -- precisely the regime where per-edge routing has no useful decomposition to exploit. The pattern is itself the finding: the cost function's ability to separate edge types serves as a diagnostic for the heterophily regime, revealing when fine-grained routing adds value over uniform channels and when it does not. Code is available at https://github.com/eyal-weiss/CSNA-public .

Paper Structure

This paper contains 46 sections, 2 theorems, 16 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Heterophily-aware GNNs.
  5. Distinction from ACM-GNN.
  6. When per-edge routing helps: a toy example.
  7. Over-smoothing and over-squashing.
  8. Method
  9. Preliminaries
  10. Cost-Sensitive Neighborhood Aggregation
  11. Step 1: Edge cost (observed divergence).
  12. Step 2: Concordance routing.
  13. Step 3: Dual-channel aggregation.
  14. Step 4: Gated combination.
  15. Extended variant: $g + h$.
  16. ...and 31 more sections

Key Result

theorem 1

Let $\mathcal{G} \sim \mathrm{CSBM}(n, 2, p, q, \mu)$ with constant $p, q \in (0,1)$ and equal class sizes. After one round of symmetrically normalized aggregation with self-loops, $\mathbf{H}^{(1)} = \tilde{\mathbf{A}} \mathbf{X}$, where $\tilde{\mathbf{A}} = \mathbf{D}^{-1/2}(\mathbf{A}+\mathbf{I} where $\bar{\mathbf{h}}_c^{(1)}$ is the mean representation of class $c$. The scaling factor $\lamb

Figures (2)

  • Figure 1: Toy example: a heterophilous graph where cross-class edges have mixed utility. Left: ACM-GNN's uniform high-pass channel assigns identical weight to all cross-class edges. Right: CSNA's per-edge cost routing upweights helpful cross-class edges (thick green) and downweights harmful ones (thin dashed red). This distinction is possible only when the cost function can separate edge types---the adversarial-heterophily regime.
  • Figure 2: Learned concordance scores $s_{ij}$ for same-class (green) and different-class (red) edges on three datasets. On adversarial-heterophily datasets (Texas, Cornell), the distributions are well separated; on Actor, overlap is larger. Generated using the extended ($g+h$) variant; similar separation is observed with the lite version.

Theorems & Definitions (7)

  • definition 1: Contextual Stochastic Block Model
  • theorem 1: Signal distortion under mean aggregation
  • proof
  • theorem 2: Cost-sensitive aggregation preserves signal direction
  • proof
  • remark 1: Applicability to CSNA
  • remark 2: Scope and limitations