Convergent Graph Solvers

TL;DR

The paper introduces Convergent Graph Solver (CGS), a deep learning approach that reliably predicts stationary properties of graph systems by learning iterative fixed-point mappings with guaranteed convergence, applicable to various graph problems.

Contribution

CGS is a novel deep learning method that systematically computes fixed points of graph systems with guaranteed convergence, without prior knowledge of solutions or intermediate states.

Findings

CGS accurately predicts stationary properties of linear and nonlinear graph systems.

CGS demonstrates high performance in graph classification tasks with ambiguous fixed points.

CGS is a versatile architecture applicable to diverse graph analysis problems.

Abstract

We propose the convergent graph solver (CGS), a deep learning method that learns iterative mappings to predict the properties of a graph system at its stationary state (fixed point) with guaranteed convergence. CGS systematically computes the fixed points of a target graph system and decodes them to estimate the stationary properties of the system without the prior knowledge of existing solvers or intermediate solutions. The forward propagation of CGS proceeds in three steps: (1) constructing the input dependent linear contracting iterative maps, (2) computing the fixed-points of the linear maps, and (3) decoding the fixed-points to estimate the properties. The contractivity of the constructed linear maps guarantees the existence and uniqueness of the fixed points following the Banach fixed point theorem. To train CGS efficiently, we also derive a tractable analytical expression for its gradient by leveraging the implicit function theorem. We evaluate the performance of CGS by applying it to various network-analytic and graph benchmark problems. The results indicate that CGS has competitive capabilities for predicting the stationary properties of graph systems, irrespective of whether the target systems are linear or non-linear. CGS also shows high performance for graph classification problems where the existence or the meaning of a fixed point is hard to be clearly defined, which highlights the potential of CGS as a general graph neural network architecture.