An official website of the United States government
Network data has become widespread, larger, and more complex over the years. Traditional network data is dyadic, capturing the relations among pairs of entities. With the need to model interactions among more than two entities, significant research has focused on higher-order networks and ways to represent, analyze, and learn from them. There are two main directions to studying higher-order networks. One direction has focused on capturing higher-order patterns in traditional (dyadic) graphs by changing the basic unit of study from nodes to small frequently observed subgraphs, called motifs. As most existing network data comes in the form of pairwise dyadic relationships, studying higher-order structures within such graphs may uncover new insights. The second direction aims to directly model higher-order interactions using new and more complex representations such as simplicial complexes or hypergraphs. Some of these models have long been proposed, but improvements in computational power and the advent of new computational techniques have increased their popularity. Our goal in this paper is to provide a succinct yet comprehensive summary of the advanced higher-order network analysis techniques. We provide a systematic review of the foundations and algorithms, along with use cases and applications of higher-order networks in various scientific domains.
more »
« less
Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes
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.
more »
« less
- Award ID(s):
- 2052720
- PAR ID:
- 10397882
- Date Published:
- Journal Name:
- Communications Physics
- Volume:
- 5
- Issue:
- 1
- ISSN:
- 2399-3650
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications.more » « less
-
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.more » « less
-
Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, [Formula: see text]-dimensional “simplices”) and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together.more » « less
-
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.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1038/s42005-022-01062-3
