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1. t-HGSP: Hypergraph Signal Processing Using t-Product Tensor Decompositions

   https://doi.org/10.1109/TSIPN.2023.3276687  

   Pena-Pena, Karelia; Lau, Daniel L; Arce, Gonzalo R (January 2023, IEEE Transactions on Signal and Information Processing over Networks)

   Graph signal processing (GSP) techniques are powerful tools that model complex relationships within large datasets, being now used in a myriad of applications in different areas including data science, communication networks, epidemiology, and sociology. Simple graphs can only model pairwise relationships among data which prevents their application in modeling networks with higher-order relationships. For this reason, some efforts have been made to generalize well-known graph signal processing techniques to more complex graphs such as hypergraphs, which allow capturing higher-order relationships among data. In this article, we provide a new hypergraph signal processing framework (t-HGSP) based on a novel tensor-tensor product algebra that has emerged as a powerful tool for preserving the intrinsic structures of tensors. The proposed framework allows the generalization of traditional GSP techniques while keeping the dimensionality characteristic of the complex systems represented by hypergraphs. To this end, the core elements of the t-HGSP framework are introduced, including the shifting operators and the hypergraph signal. The hypergraph Fourier space is also defined, followed by the concept of bandlimited signals and sampling. In our experiments, we demonstrate the benefits of our approach in applications such as clustering and denoising. 

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2. A simplicial epidemic model for COVID-19 spread analysis

   https://doi.org/10.1073/pnas.2313171120  

   Chen, Yuzhou; Gel, Yulia R.; Marathe, Madhav V.; Poor, H. Vincent (January 2024, Proceedings of the National Academy of Sciences)

   Networks allow us to describe a wide range of interaction phenomena that occur in complex systems arising in such diverse fields of knowledge as neuroscience, engineering, ecology, finance, and social sciences. Until very recently, the primary focus of network models and tools has been on describing the pairwise relationships between system entities. However, increasingly more studies indicate that polyadic or higher-order group relationships among multiple network entities may be the key toward better understanding of the intrinsic mechanisms behind the functionality of complex systems. Such group interactions can be, in turn, described in a holistic manner by simplicial complexes of graphs. Inspired by these recently emerging results on the utility of the simplicial geometry of complex networks for contagion propagation and armed with a large-scale synthetic social contact network (also known as a digital twin) of the population in the U.S. state of Virginia, in this paper, we aim to glean insights into the role of higher-order social interactions and the associated varying social group determinants on COVID-19 propagation and mitigation measures. 

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3. Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes

   https://doi.org/10.1038/s42005-022-01062-3  

   Kilic, Bengier Ülgen; Taylor, Dane (December 2022, Communications Physics)

   Abstract Cascades over networks (e.g., neuronal avalanches, social contagions, and system failures) often involve higher-order dependencies, yet theory development has largely focused on pairwise-interaction models. Here, we develop a ‘simplicial threshold model’ (STM) for cascades over simplicial complexes that encode dyadic, triadic and higher-order interactions. Focusing on small-world models containing both short- and long-range k -simplices, we explore spatio-temporal patterns that manifest as a frustration between local and nonlocal propagations. We show that higher-order interactions and nonlinear thresholding coordinate to robustly guide cascades along a k -dimensional generalization of paths that we call ‘geometrical channels’. We also find this coordination to enhance the diversity and efficiency of cascades over a simplicial-complex model for a neuronal network, or ‘neuronal complex’. We support these findings with bifurcation theory and data-driven approaches based on latent geometry. Our findings provide fruitful directions for uncovering the multiscale, multidimensional mechanisms that orchestrate the spatio-temporal patterns of nonlinear cascades. 

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4. BScNets: Block Simplicial Complex Neural Networks

   https://doi.org/10.1609/aaai.v36i6.20583  

   Chen, Yuzhou; Gel, Yulia R.; Poor, H. Vincent (June 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   Simplicial neural networks (SNNs) have recently emerged as a new direction in graph learning which expands the idea of convolutional architectures from node space to simplicial complexes on graphs. Instead of predominantly assessing pairwise relations among nodes as in the current practice, simplicial complexes allow us to describe higher-order interactions and multi-node graph structures. By building upon connection between the convolution operation and the new block Hodge-Laplacian, we propose the first SNN for link prediction. Our new Block Simplicial Complex Neural Networks (BScNets) model generalizes existing graph convolutional network (GCN) frameworks by systematically incorporating salient interactions among multiple higher-order graph structures of different dimensions. We discuss theoretical foundations behind BScNets and illustrate its utility for link prediction on eight real-world and synthetic datasets. Our experiments indicate that BScNets outperforms the state-of-the-art models by a significant margin while maintaining low computation costs. Finally, we show utility of BScNets as a new promising alternative for tracking spread of infectious diseases such as COVID-19 and measuring the effectiveness of the healthcare risk mitigation strategies. 

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5. Learning embeddings for multiplex networks using triplet loss

   https://doi.org/10.1007/s41109-019-0242-0  

   Seyedsaeed Hajiseyedjavadi, Yu-Ru Lin (January 2019, Applied network science)

   null (Ed.)

   Learning low-dimensional representations of graphs has facilitated the use of traditional machine learning techniques to solving classic network analysis tasks such as link prediction, node classification, community detection, etc. However, to date, the vast majority of these learning tasks are focused on traditional single-layer/unimodal networks and largely ignore the case of multiplex networks. A multiplex network is a suitable structure to model multi-dimensional real-world complex systems. It consists of multiple layers where each layer represents a different relationship among the network nodes. In this work, we propose MUNEM, a novel approach for learning a low-dimensional representation of a multiplex network using a triplet loss objective function. In our approach, we preserve the global structure of each layer, while at the same time fusing knowledge among different layers during the learning process. We evaluate the effectiveness of our proposed method by testing and comparing on real-world multiplex networks from different domains, such as collaboration network, protein-protein interaction network, online social network. Finally, in order to deliberately examine the effect of our model’s parameters we conduct extensive experiments on synthetic multiplex networks. 

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