In this paper, we propose a novel pooling layer for graph neural networks based on maximizing the mutual information between the pooled graph and the input graph. Since the maximum mutual information is difficult to compute, we employ the Shannon capacity of a graph as an inductive bias to our pooling method. More precisely, we show that the input graph to the pooling layer can be viewed as a representation of a noisy communication channel. For such a channel, sending the symbols belonging to an independent set of the graph yields a reliable and errorfree transmission of information. We show that reaching the maximum mutual information is equivalent to finding a maximum weight independent set of the graph where the weights convey entropy contents. Through this communication theoretic standpoint, we provide a distinct perspective for posing the problem of graph pooling as maximizing the information transmission rate across a noisy communication channel, implemented by a graph neural network. We evaluate our method, referred to as Maximum Entropy Weighted Independent Set Pooling (MEWISPool), on graph classification tasks and the combinatorial optimization problem of the maximum independent set. Empirical results demonstrate that our method achieves the state-of-the-art and competitive results on graph classification tasks and the maximum independent set problem in several benchmark datasets. † Both authors have equally contributed.Preprint. Under review.
Evolving networks are complex data structures that emerge in a wide range of systems in science and engineering. Learning expressive representations for such networks that encode their structural connectivity and temporal evolution is essential for downstream data analytics and machine learning applications. In this study, we introduce a self-supervised method for learning representations of temporal networks and employ these representations in the dynamic link prediction task. While temporal networks are typically characterized as a sequence of interactions over the continuous time domain, our study focuses on their discrete-time versions. This enables us to balance the trade-off between computational complexity and precise modeling of the interactions. We propose a recurrent message-passing neural network architecture for modeling the information flow over time-respecting paths of temporal networks. The key feature of our method is the contrastive training objective of the model, which is a combination of three loss functions: link prediction, graph reconstruction, and contrastive predictive coding losses. The contrastive predictive coding objective is implemented using infoNCE losses at both local and global scales of the input graphs. We empirically show that the additional self-supervised losses enhance the training and improve the model's performance in the dynamic link prediction task. The proposed method is tested on Enron, COLAB, and Facebook datasets and exhibits superior results compared to existing models. 1
We employ neural networks to tackle inverse partial differential equations on discretized Riemann surfaces with boundary. To this end, we introduce the concept of a graph with boundary which models these surfaces in a natural way. Our method uses a message passing technique to keep track of an unknown differential operator while using neural ODE solvers through the method of lines to capture the evolution in time. As training data, we use noisy and incomplete observations of sheaves on graphs at various timestamps. The novelty of this approach is in working with manifolds with nontrivial topology and utilizing the data on the graph boundary through a teacher forcing technique. Despite the increasing interest in learning dynamical systems from finite observations, many current methods are limited in two general ways: first, they work with topologically trivial spaces, and second, they fail to handle the boundary data on the ground space in a systematic way. The present work is an attempt at addressing these limitations. We run experiments with synthetic data of linear and nonlinear diffusion systems on orientable surfaces with positive genus and boundary, and moreover, provide evidences for improvements upon the existing paradigms.Preprint. Under review.
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