Low-dimensional embeddings, from classical spectral embeddings to modern neural-net-inspired methods, are a cornerstone in the modeling and analysis of complex networks. Recent work by Seshadhri et al. (PNAS 2020) suggests that such embeddings cannot capture local structure arising in complex networks. In particular, they show that any network generated from a natural low-dimensional model cannot be both sparse and have high triangle density (high clustering coefficient), two hallmark properties of many real-world networks. In this work we show that the results of Seshadhri et al. are intimately connected to the model they use rather than the low-dimensional structure of complex networks. Specifically, we prove that a minor relaxation of their model can generate sparse graphs with high triangle density. Surprisingly, we show that this same model leads to of many real-world networks. We give a simple algorithm based on logistic principal component analysis (LPCA) that succeeds in finding such exact embeddings. Finally, we perform a large number of experiments that verify the ability of very low-dimensional embeddings to capture local structure in real-world networks.

In recent years, traffic prediction has achieved remarkable success and has become an integral component of intelligent transportation systems. However, traffic data is typically distributed among multiple data owners, and privacy constraints prevent the direct utilization of these isolated datasets for traffic prediction. Most existing federated traffic prediction methods focus on designing communication mechanisms that allow models to leverage information from other clients in order to improve prediction accuracy. Unfortunately, such approaches often incur substantial communication overhead, and the resulting transmission delays significantly slow down the training process. As the volume of traffic data continues to grow, this issue becomes increasingly critical, making the resource consumption of current methods unsustainable. To address this challenge, we propose a novel variable relationship modeling paradigm for federated traffic prediction, termed the Channel-Independent Paradigm(CIP). Unlike traditional approaches, CIP eliminates the need for inter-client communication by enabling each node to perform efficient and accurate predictions using only local information. Based on the CIP, we further develop Fed-CI, an efficient federated learning framework, allowing each client to process its own data independently while effectively mitigating the information loss caused by the lack of direct data sharing among clients. Fed-CI significantly reduces communication overhead, accelerates the training process, and achieves state-of-the-art performance while complying with privacy regulations. Extensive experiments on multiple real-world datasets demonstrate that Fed-CI consistently outperforms existing methods across all datasets and federated settings. It achieves improvements of 8%, 14%, and 16% in RMSE, MAE, and MAPE, respectively, while also substantially reducing communication costs.

Graph layouts and node embeddings are two distinct paradigms for non-parametric graph representation learning. In the former, nodes are embedded into 2D space for visualization purposes. In the latter, nodes are embedded into a high-dimensional vector space for downstream processing. State-of-the-art algorithms for these two paradigms, force-directed layouts and random-walk-based contrastive learning (such as DeepWalk and node2vec), have little in common. In this work, we show that both paradigms can be approached with a single coherent framework based on established neighbor embedding methods. Specifically, we introduce graph t-SNE, a neighbor embedding method for two-dimensional graph layouts, and graph CNE, a contrastive neighbor embedding method that produces high-dimensional node representations by optimizing the InfoNCE objective. We show that both graph t-SNE and graph CNE strongly outperform state-of-the-art algorithms in terms of local structure preservation, while being conceptually simpler.

Low-dimensional embeddings, from classical spectral embeddings to modern neural-net-inspired methods, are a cornerstone in the modeling and analysis of complex networks. Recent work by Seshadhri et al. (PNAS 2020) suggests that such embeddings cannot capture local structure arising in complex networks. In particular, they show that any network generated from a natural low-dimensional model cannot be both sparse and have high triangle density (high clustering coefficient), two hallmark properties of many real-world networks. In this work we show that the results of Seshadhri et al. are intimately connected to the model they use rather than the low-dimensional structure of complex networks. Specifically, we prove that a minor relaxation of their model can generate sparse graphs with high triangle density.

This approach directly targets and GNNs [30] are a category of artificial neural networks specifically rectifies imbalances at the data level. The proposed framework resolves designed to handle data as graphs. GNNs display remarkable adaptability issues such as neglecting graph structures during data generation in handling highly interconnected data of diverse sizes. This and creating synthetic structures usable with GNN-based classifiers versatility makes them suitable for a broad spectrum of domains in downstream tasks. It processes node and edge information and problem scenarios. Graphs can be categorized as either homogeneous concurrently, improving edge balance through node augmentation or heterogeneous based on the variety of nodes and edges and subgraph sampling. Additionally, our framework integrates a they encompass. Both types have been extensively researched for threshold strategy, aiding in determining optimal edge thresholds homogeneous graphs. Examples include the Graph Convolutional during training without time-consuming parameter adjustments.

In the graph node embedding problem, embedding spaces can vary significantly for different data types, leading to the need for different GNN model types. In this paper, we model the embedding update of a node feature as a Hamiltonian orbit over time. Since the Hamiltonian orbits generalize the exponential maps, this approach allows us to learn the underlying manifold of the graph in training, in contrast to most of the existing literature that assumes a fixed graph embedding manifold with a closed exponential map solution. Our proposed node embedding strategy can automatically learn, without extensive tuning, the underlying geometry of any given graph dataset even if it has diverse geometries. We test Hamiltonian functions of different forms and verify the performance of our approach on two graph node embedding downstream tasks: node classification and link prediction. Numerical experiments demonstrate that our approach adapts better to different types of graph datasets than popular state-of-the-art graph node embedding GNNs. The code is available at \url{https://github.com/zknus/Hamiltonian-GNN}.

Instead of using traditional machine learning classification tasks, we can consider using graph neural network (GNN) to perform node classification problems. By providing an explicit link of nodes, this classification problem is no longer classified as an independent problem but leveraging graph structures such as the degree of nodes. The usefulness of graph properties assumes that individual nodes are correlated with other similar nodes. Typically example is a social media network. Imagine how Facebook connects you and somebody else based on what post you like, where you check-in etc. A graph is capable to represent this kind of relationship and we can leverage it to train GNN.

In this work, we present a method for node embedding in temporal graphs. We propose an algorithm that learns the evolution of a temporal graph's nodes and edges over time and incorporates this dynamics in a temporal node embedding framework for different graph prediction tasks. We present a joint loss function that creates a temporal embedding of a node by learning to combine its historical temporal embeddings, such that it optimizes per given task (e.g., link prediction). The algorithm is initialized using static node embeddings, which are then aligned over the representations of a node at different time points, and eventually adapted for the given task in a joint optimization. We evaluate the effectiveness of our approach over a variety of temporal graphs for the two fundamental tasks of temporal link prediction and multi-label node classification, comparing to competitive baselines and algorithmic alternatives. Our algorithm shows performance improvements across many of the datasets and baselines and is found particularly effective for graphs that are less cohesive, with a lower clustering coefficient.