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Over the past decade, Graph Neural Networks (GNNs) have transformed graph representation learning. In the widely adopted message-passing GNN framework, nodes refine their representations by aggregating information from neighboring nodes iteratively. While GNNs excel in various domains, recent theoretical studies have raised concerns about their capabilities. GNNs aim to address various graph-related tasks by utilizing such node representations, however, this one-size-fits-all approach proves suboptimal for diverse tasks. Motivated by these observations, we conduct empirical tests to compare the performance of current GNN models with more conventional and direct methods in link prediction tasks. Introducing our model, PROXI, which leverages proximity information of node pairs in both graph and attribute spaces, we find that standard machine learning (ML) models perform competitively, even outperforming cutting-edge GNN models when applied to these proximity metrics derived from node neighborhoods and attributes. This holds true across both homophilic and heterophilic networks, as well as small and large benchmark datasets, including those from the Open Graph Benchmark (OGB). Moreover, we show that augmenting traditional GNNs with PROXI significantly boosts their link prediction performance. Our empirical findings corroborate the previously mentioned theoretical observations and imply that there exists ample room for enhancement in current GNN models to reach their potential. 

Many real-world networks evolve over time, and predicting the evolution of such networks remains a challenging task. Graph Neural Networks (GNNs) have shown empirical success for learning on static graphs, but they lack the ability to effectively learn from nodes and edges with different timestamps. Consequently, the prediction of future properties in temporal graphs remains a relatively under-explored area. In this paper, we aim to bridge this gap by introducing a principled framework, named GraphPulse. The framework combines two important techniques for the analysis of temporal graphs within a Newtonian framework. First, we employ the Mapper method, a key tool in topological data analysis, to extract essential clustering information from graph nodes. Next, we harness the sequential modeling capabilities of Recurrent Neural Networks (RNNs) for temporal reasoning regarding the graph's evolution. Through extensive experimentation, we demonstrate that our model enhances the ROC-AUC metric by 10.2% in comparison to the top-performing state-of-the-art method across various temporal networks. We provide the implementation of GraphPulse at https://github.com/kiarashamsi/GraphPulse. 

Learning time-evolving objects such as multivariate time series and dynamic networks requires the development of novel knowledge representation mechanisms and neural network architectures, which allow for capturing implicit time-dependent information contained in the data. Such information is typically not directly observed but plays a key role in the learning task performance. In turn, lack of time dimension in knowledge encoding mechanisms for time-dependent data leads to frequent model updates, poor learning performance, and, as a result, subpar decision-making. Here we propose a new approach to a time-aware knowledge representation mechanism that notably focuses on implicit time-dependent topological information along multiple geometric dimensions. In particular, we propose a new approach, named Temporal MultiPersistence (TMP), which produces multidimensional topological fingerprints of the data by using the existing single parameter topological summaries. The main idea behind TMP is to merge the two newest directions in topological representation learning, that is, multi-persistence which simultaneously describes data shape evolution along multiple key parameters, and zigzag persistence to enable us to extract the most salient data shape information over time. We derive theoretical guarantees of TMP vectorizations and show its utility, in application to forecasting on benchmark traffic flow, Ethereum blockchain, and electrocardiogram datasets, demonstrating the competitive performance, especially, in scenarios of limited data records. In addition, our TMP method improves the computational efficiency of the state-of-the-art multipersistence summaries up to 59.5 times. 

We study the asymptotic Plateau problem in H2 × R for area-minimizing surfaces, and give a fairly complete solution for finite curves. 

Learning time-evolving objects such as multivariate time series and dynamic networks requires the development of novel knowledge representation mechanisms and neural network architectures, which allow for capturing implicit timedependent information contained in the data. Such information is typically not directly observed but plays a key role in the learning task performance. In turn, lack of time dimension in knowledge encoding mechanisms for time-dependent data leads to frequent model updates, poor learning performance, and, as a result, subpar decision-making. Here we propose a new approach to a time-aware knowledge representation mechanism that notably focuses on implicit timedependent topological information along multiple geometric dimensions. In particular, we propose a new approach, named Temporal MultiPersistence (TMP), which produces multidimensional topological fingerprints of the data by using the existing single parameter topological summaries. The main idea behind TMP is to merge the two newest directions in topological representation learning, that is, multi-persistence which simultaneously describes data shape evolution along multiple key parameters, and zigzag persistence to enable us to extract the most salient data shape information over time.We derive theoretical guarantees of TMP vectorizations and show its utility, in application to forecasting on benchmark traffic flow, Ethereum blockchain, and electrocardiogram datasets, demonstrating the competitive performance, especially, in scenarios of limited data records. In addition, our TMP method improves the computational efficiency of the state-of-the-art multipersistence summaries up to 59.5 times. 

Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an exclusive topological imprint of data by tracing the evolution of latent structures as a scale parameter changes. Present PH tools are confined to analyzing data through a single filter parameter. However, many scenarios necessitate the consideration of multiple relevant parameters to attain finer insights into the data. We address this issue by introducing the Effective Multidimensional Persistence (EMP) framework. This framework empowers the exploration of data by simultaneously varying multiple scale parameters. The framework integrates descriptor functions into the analysis process, yielding a highly expressive data summary. It seamlessly integrates established single PH summaries into multidimensional counterparts like EMP Landscapes, Silhouettes, Images, and Surfaces. These summaries represent data’s multidimensional aspects as matrices and arrays, aligning effectively with diverse ML models. We provide theoretical guarantees and stability proofs for EMP summaries. We demonstrate EMP’s utility in graph classification tasks, showing its effectiveness. Results reveal EMP enhances various single PH descriptors, outperforming cutting-edge methods on multiple benchmark datasets.