We present an approach for analyzing message passing graph neural networks (MPNNs) based on an extension of graphon analysis to a so called graphon-signal analysis. AMPNN is a function that takes a graph and a signal on the graph (a graph-signal) and returns some value. Since the input space of MPNNs is non-Euclidean, i.e., graphs can be of any size and topology, properties such as generalization are less well understood for MPNNs than for Euclidean neural networks. We claim that one important missing ingredient in past work is a meaningful notion of graph-signal similarity measure, that endows the space of inputs to MPNNs with a regular structure. We present such a similarity measure, called the graphon-signal cut distance, which makes the space of all graph-signals a dense subset of a compact metric space - the graphon-signal space.

A recent paper, ``A Graphon-Signal Analysis of Graph Neural Networks'', by Levie, analyzed message passing graph neural networks (MPNNs) by embedding the input space of MPNNs, i.e., attributed graphs (graph-signals), to a space of attributed graphons (graphon-signals). Based on extensions of standard results in graphon analysis to graphon-signals, the paper proved a generalization bound and a sampling lemma for MPNNs. However, there are some missing ingredients in that paper, limiting its applicability in practical settings of graph machine learning. In the current paper, we introduce several refinements and extensions to existing results that address these shortcomings. In detail, 1) we extend the main results in the paper to graphon-signals with multidimensional signals (rather than 1D signals), 2) we extend the Lipschitz continuity to MPNNs with readout with respect to cut distance (rather than MPNNs without readout with respect to cut metric), 3) we improve the generalization bound by utilizing robustness-type generalization bounds, and 4) we extend the analysis to non-symmetric graphons and kernels.

Laboratory of Computational Science and Modeling, Institut des Mat eriaux, Ecole Polytechnique F ed erale de Lausanne, 1015 Lausanne, Switzerland (Dated: August 5, 2025) The equations of classical mechanics can be used to model the time evolution of countless physical systems, from the astrophysical to the atomic scale. Accurate numerical integration requires small time steps, which limits the computational efficiency - especially in cases such as molecular dynamics that span wildly different time scales. Using machine-learning (ML) algorithms to predict trajectories allows one to greatly extend the integration time step, at the cost of introducing artifacts such as lack of energy conservation and loss of equipartition between different degrees of freedom of a system. We propose learning data-driven structure-preserving (symplectic and time-reversible) maps to generate long-time-step classical dynamics, showing that this method is equivalent to learning the mechanical action of the system of interest. We show that an action-derived ML integrator eliminates the pathological behavior of non-structure-preserving ML predictors, and that the method can be applied iteratively, serving as a correction to computationally cheaper direct predictors. Simulating classical mechanical systems with high accuracy and efficiency is a long-standing challenge in computational physics [31, 32]. Traditional numerical methods typically rely on small time steps to propagate the equations of motion for the dynamical system in order to provide accurate integration.

We present an approach for analyzing message passing graph neural networks (MPNNs) based on an extension of graphon analysis to a so called graphon-signal analysis. A MPNN is a function that takes a graph and a signal on the graph (a graph-signal) and returns some value. Since the input space of MPNNs is non-Euclidean, i.e., graphs can be of any size and topology, properties such as generalization are less well understood for MPNNs than for Euclidean neural networks. We claim that one important missing ingredient in past work is a meaningful notion of graph-signal similarity measure, that endows the space of inputs to MPNNs with a regular structure. We present such a similarity measure, called the graphon-signal cut distance, which makes the space of all graph-signals a dense subset of a compact metric space -- the graphon-signal space. Informally, two deterministic graph-signals are close in cut distance if they ``look like'' they were sampled from the same random graph-signal model. Hence, our cut distance is a natural notion of graph-signal similarity, which allows comparing any pair of graph-signals of any size and topology. We prove that MPNNs are Lipschitz continuous functions over the graphon-signal metric space. We then give two applications of this result: 1) a generalization bound for MPNNs, and, 2) the stability of MPNNs to subsampling of graph-signals. Our results apply to any regular enough MPNN on any distribution of graph-signals, making the analysis rather universal.

Continuous surveillance of a spatial region using distributed robots and sensors is a well-studied application in the area of multi-agent systems. This paper investigates a practically-relevant scenario where robotic sensors are introduced asynchronously and inter-robot communication is discrete, event-driven, local and asynchronous. Furthermore, we work with lazy robots; i.e., the robots seek to minimize their area of responsibility by equipartitioning the domain to be covered. We adapt a well-known algorithm which is practicable and known to generally work well for coverage problems. For a specially chosen geometry of the spatial domain, we show that there exists a non-trivial sequence of inter-robot communication events which leads to an instantaneous loss of coverage when the number of robots exceeds a certain threshold. The same sequence of events preserves coverage and, further, leads to an equipartition of the domain when the number of robots is smaller than the threshold. This result demonstrates that coverage guarantees for a given algorithm might be sensitive to the number of robots and, therefore, may not scale in obvious ways. It also suggests that when such algorithms are to be verified and validated prior to field deployment, the number of robots or sensors used in test scenarios should match that deployed on the field.

Given a high-dimensional data set we often wish to find the strongest relationships within it. A common strategy is to evaluate a measure of dependence on every variable pair and retain the highest-scoring pairs for follow-up. This strategy works well if the statistic used is equitable [Reshef et al. 2015a], i.e., if, for some measure of noise, it assigns similar scores to equally noisy relationships regardless of relationship type (e.g., linear, exponential, periodic). In this paper, we introduce and characterize a population measure of dependence called MIC*. We show three ways that MIC* can be viewed: as the population value of MIC, a highly equitable statistic from [Reshef et al. 2011], as a canonical "smoothing" of mutual information, and as the supremum of an infinite sequence defined in terms of optimal one-dimensional partitions of the marginals of the joint distribution. Based on this theory, we introduce an efficient approach for computing MIC* from the density of a pair of random variables, and we define a new consistent estimator MICe for MIC* that is efficiently computable. In contrast, there is no known polynomial-time algorithm for computing the original equitable statistic MIC. We show through simulations that MICe has better bias-variance properties than MIC. We then introduce and prove the consistency of a second statistic, TICe, that is a trivial side-product of the computation of MICe and whose goal is powerful independence testing rather than equitability. We show in simulations that MICe and TICe have good equitability and power against independence respectively. The analyses here complement a more in-depth empirical evaluation of several leading measures of dependence [Reshef et al. 2015b] that shows state-of-the-art performance for MICe and TICe.

To appear in Neural Computation.