Reinforcement Learning for Microcanonical Graph Ensemble with Assortativity Constraints

How network structure determines function is a fundamental question, and it can be investigated by graph ensembles with precisely controlled structural properties. Canonical approaches, formulated as exponential random graph models (ERGMs), enforce constraints only in expectation, allowing individual realizations to fluctuate around the target. Conversely, microcanonical ensembles impose hard constraints exactly, but practical sampling methods beyond fixing the degree sequence have remained out of reach. Here we introduce the Deep Microcanonical Graph Generator (DMGG), a reinforcement learning (RL) framework that transforms any given graph through degree-preserving rewirings to exactly reach a prescribed assortativity, which characterizes the degree--degree correlation of adjacent nodes. Instead of relying on the entropically dominated Metropolis--Hastings dynamics of the ERGM, DMGG employs a policy-guided search that maximally alters the joint-degree matrix. This eliminates exhaustive parameter tuning and accelerates generation by at least an order of magnitude while preserving configurational diversity. As DMGG generalizes across various graph sizes, sparsities, and topologies, it provides exact null models that allow for the quantitative isolation of secondary observables, such as the clustering coefficient. These results establish RL as a practical and powerful paradigm for generating hard-constrained graphs, opening avenues to investigate structure-function relationships free from ensemble artifacts.

Neural Graph Simulator for Complex Systems

Numerical simulation is a predominant tool for studying the dynamics in complex systems, but large-scale simulations are often intractable due to computational limitations. Here, we introduce the Neural Graph Simulator (NGS) for simulating time-invariant autonomous systems on graphs. Utilizing a graph neural network, the NGS provides a unified framework to simulate diverse dynamical systems with varying topologies and sizes without constraints on evaluation times through its non-uniform time step and autoregressive approach. The NGS offers significant advantages over numerical solvers by not requiring prior knowledge of governing equations and effectively handling noisy or missing data with a robust training scheme. It demonstrates superior computational efficiency over conventional methods, improving performance by over $10^5$ times in stiff problems. Furthermore, it is applied to real traffic data, forecasting traffic flow with state-of-the-art accuracy. The versatility of the NGS extends beyond the presented cases, offering numerous potential avenues for enhancement.

GNRK: Graph Neural Runge-Kutta method for solving partial differential equations

Neural networks have proven to be efficient surrogate models for tackling partial differential equations (PDEs). However, their applicability is often confined to specific PDEs under certain constraints, in contrast to classical PDE solvers that rely on numerical differentiation. Striking a balance between efficiency and versatility, this study introduces a novel approach called Graph Neural Runge-Kutta (GNRK), which integrates graph neural network modules with a recurrent structure inspired by the classical solvers. The GNRK operates on graph structures, ensuring its resilience to changes in spatial and temporal resolutions during domain discretization. Moreover, it demonstrates the capability to address general PDEs, irrespective of initial conditions or PDE coefficients. To assess its performance, we benchmark the GNRK against existing neural network based PDE solvers using the 2-dimensional Burgers' equation, revealing the GNRK's superiority in terms of model size and accuracy. Additionally, this graph-based methodology offers a straightforward extension for solving coupled differential equations, typically necessitating more intricate models.