Title: Graph Logic Flows: Geometry-Driven, Certificate-Carrying Reasoning on Dynamic Graphs
Primary area: learning on graphs and other geometries & topologies
Keywords: Logical Constraints, Graph Learning, Verifiable Machine Learning

Abstract: We introduce Graph Logic Flows (GLF), a framework that replaces stacked message passing with a single implicit update: one Jordan Kinderlehrer Otto (JKO) step of a reflected Wasserstein flow on a learned transport geometry. Task and domain rules such as shortest path consistency, triangle inequalities, conservation, or temporal smoothness are compiled into convex barriers where applicable and smooth surrogate constraints, and a lightweight runtime judge enforces them with different activations at train and test time. Each prediction returns numerical certificates including energy descent, KKT residuals, and logic residuals, yielding certificate carrying outputs. Under standard convexity assumptions, we establish theory for EVI contraction ensuring stability, finite step barrier invariance with strict feasibility in the small step limit for convex barriers, tracking under metric drift with ODE grade rates, and a sensitivity bound against oversquashing governed by the learned geometry rather than network depth. GLF unifies nonlocal reasoning, logic enforcement, and label free test time adaptation within a single convex integration step. Empirical case studies on dynamic graph benchmarks demonstrate the framework in practice, highlighting audit trails and constraint monitoring even under challenging predictive performance.