Paper defines untangling number to measure entanglement complexity in 3-periodic networks.
problem Measuring the complexity of entanglement in 3-periodic networks.
method Defining ground states through knot-theoretic crossing diagrams and measuring untangling number.
result Introduced untangling number as a measure of entanglement complexity.
Paper introduces untangling number to quantify 3-periodic tangle complexity.
problem Quantifying the complexity of 3-periodic tangles in biological, chemical, and physical systems.
method Introduces untangling number, a measure of minimum distance to ground state through diagrammatic operations.
result For infinite open curves, generic ground states are crystallographic rod packings.
In the present paper we construct a Z^{3}-periodic surface in R^{3} whose almost all plane sections of a certain direction consist of exactly one connected component. This question originates from a problem of Novikov on the semi- classical motion of an electron in strong magnetic field. Our main tool is the Rips machi…
A lower bound for the number of 3-periodical billiard trajectories in a manifold embedded in Euclidean space is obtained.
Simple proof shows forecasts can be calibrated in a few periods.
problem Ensuring forecasts are calibrated over multiple periods.
method Uses minimax theorem to prove existence and guarantees calibration error.
result Calibration can be achieved in N3 periods with error at most 1/N. Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has ze…
The paper finds linked periodic orbits in disc homeomorphisms using braids.
problem Finding linked periodic orbits in disc homeomorphisms.
method Interpreting linking of orbits by induced braids and using forcing relations.
result New examples of linked orbits of periods at most 4 for pseudo-Anosov braid types.
In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic …
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …
This paper is devoted to prove the existence of q-periodic alternating projections of prime alternating q-periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping theorem. Let K be an oriented prime alternating knot that is q-periodic with q≥3, i.e. K admits a symmetry that is a rotation of…
Gradient descent dynamics in quadratic regression models are analyzed, revealing five phases: monotonic, catapult, periodic, chaotic, and divergent.
problem Analyzing the dynamics of gradient descent in quadratic regression models.
method Fine-grained bifurcation analysis of gradient descent dynamics using a cubic map parameterized by the step-size.
result Gradient descent dynamics in quadratic regression models exhibit five distinct phases: monotonic, catapult, periodic, chaotic, and divergent.
New diagrams classify triply periodic entanglements.
problem Classifying triply periodic entanglements.
method Introducing diagrams for links in 3-torus and defining additional moves.
result New diagrams allow classification of non-isotopic links with same preimage.
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.
RFN improves GCNs for road networks, outperforming state-of-the-art by 21%-40%.
problem Leveraging the structure of road networks effectively in machine learning tasks.
method Introducing RFN, a novel GCN specifically designed for road networks.
result RFN outperforms state-of-the-art GCNs by 21%-40% on road network tasks.
This survey clarifies dynamic network terminology and reviews GNN models for dynamic networks.
problem Ambiguity in dynamic network terminology and lack of GNN models for dynamic networks.
method Established consistent terminology and notation for dynamic networks, reviewed GNN models.
result Comprehensive survey of dynamic graph neural network models.
From the perspective of network analysis, the ubiquitous networks are comprised of regular and irregular components, which makes uncovering the complexity of network structures to be a fundamental challenge. Exploring the regular information and identifying the roles of microscopic elements in network data can help us …
DCNs mimic neuronal networks for improved neural classification.
problem Lack of topological similarity between DNNs and biological neural networks.
method Developed DCNs with topologies inspired by real-world neuronal networks.
result High classification accuracy achieved by DCNs.
Chemical networks outperform spiking neural networks in classification tasks.
problem Learning tasks with spiking neural networks require hidden layers, which are computationally expensive.
method Used deterministic mass-action kinetics to prove chemical reaction networks without hidden layers can solve tasks previously solved by spiking neural networks.
result A chemical reaction network without hidden layers outperforms a spiking neural network with hidden layers in a handwritten digit classification task.
Deep ReLU networks can be simplified to a three-layer model.
problem Understanding the behavior of deep neural networks.
method Constructive proof and algorithm to transform deep networks into shallow ones.
result Deep ReLU networks can be represented by a simpler three-layer structure.
Tackles network structure inference from time series data using GNN.
problem Inferring network structure from incomplete or no information.
method Gumbel Graph Network (GGN) model for network reconstruction and completion.
result GGN can reconstruct up to 100% network structure and infer missing parts with up to 90% accuracy.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
New approach learns latent motifs in networks for mesoscale structure analysis.
problem Understanding large-scale behavior in complex systems through mesoscale structures.
method Network dictionary learning (NDL) combining network sampling and nonnegative matrix factorization.
result Networks can be approximated using a small set of latent motifs.
A challenging problem in complex networks is the network reconstruction problem from data. This work deals with a class of networks denoted as conserved networks, in which a flow associated with every edge and the flows are conserved at all non-source and non-sink nodes. We propose a novel polynomial time algorithm to …
Social network analysis is an important problem in data mining. A fundamental step for analyzing social networks is to encode network data into low-dimensional representations, i.e., network embeddings, so that the network topology structure and other attribute information can be effectively preserved. Network represen…
Taking inspiration from biological evolution, we explore the idea of "Can deep neural networks evolve naturally over successive generations into highly efficient deep neural networks?" by introducing the notion of synthesizing new highly efficient, yet powerful deep neural networks over successive generations via an ev…
SyNGLER generates synthetic networks efficiently while preserving key structural properties.
problem Efficiently generating realistic synthetic networks with preserved structural properties.
method SyNGLER uses latent space network models to learn and reconstruct node embeddings, then generates synthetic networks.
result SyNGLER produces synthetic networks that better preserve key network characteristics than existing approaches.
Secret neural networks hidden within trained models.
problem Excess capacity in neural networks allows embedding secret models.
method Novel framework for hiding secret neural networks within carrier networks.
result Detection of hidden networks is computationally infeasible.
Natural graph networks are a new class of graph neural networks that are more flexible and scalable.
problem Traditional graph neural networks are limited by equivariance to node permutations.
method Introduced natural graph networks, which are more flexible and scalable than conventional graph neural networks.
result Natural graph networks are as scalable as conventional message passing graph neural networks but more flexible.
The structure of complex networks has been of interest in many scientific and engineering disciplines over the decades. A number of studies in the field have been focused on finding the common properties among different kinds of networks such as heavy-tail degree distribution, small-worldness and modular structure and …
Convolutional networks outperform fully-connected ones in certain tasks.
problem Understanding the computational advantage of convolutional networks over fully-connected networks.
method Demonstrated a computational advantage through a specific problem class.
result Convolutional networks can solve certain problems that fully-connected networks cannot, even with gradient descent.
Proposes a graph neural network for traffic forecasting in WANs.
problem Traffic forecasting challenges in WANs due to dynamic and large data volumes.
method Dynamic diffusion convolutional recurrent neural networks for multistep traffic forecasting.
result Significant improvements in forecasting accuracy compared to classical methods.
Researchers derive exact priors for finite Bayesian neural networks.
problem Understanding non-Gaussian priors in finite Bayesian neural networks.
method Analytical derivation of function space priors for finite fully-connected feedforward networks.
result Exact solutions for priors of finite networks, including Meijer G-function for linear networks and mixtures for ReLU networks.
i-cNRL learns network differences with interpretability.
problem Comparing unique network characteristics.
method Contrastive network representation learning (cNRL) integrating machine learning schemes.
result i-cNRL reveals unique network patterns with interpretability.
Deep and wide networks are shown to be equivalent in terms of their capability.
problem The relationship between the width and depth of neural networks.
method Formulated transforms to map networks, used polynomial representations.
result Deep and wide networks are quasi-equivalent with an arbitrarily small error.
We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thu…
DCGANs generate drainage networks quickly from samples.
problem High computational costs in generating large numbers of drainage networks.
method DCGANs trained with connectivity-informed directional information.
result Connectivity-informed DCGANs outperform other methods in reproducing accurate drainage networks.
A network embedding consists of a vector representation for each node in the network. Its usefulness has been shown in many real-world application domains, such as social networks and web networks. Directed networks with text associated with each node, such as software package dependency networks, are commonplace. Howe…
Quantum neural network and tensor network models outperform classical models in Japanese stock market predictions.
problem Improving stock return predictions using quantum and quantum-inspired machine learning.
method Evaluation of quantum neural network and tensor network models against classical models like linear and neural networks.
result Tensor network model outperforms classical models in Japanese stock market, including linear and neural network models.
Study deep maxout networks and their equivalence to Gaussian processes.
problem Understanding neural networks with infinite width.
method Derive equivalence between deep maxout networks and Gaussian processes, characterize maxout kernel, and provide efficient numerical implementation.
result Bayesian inference based on deep maxout network kernel leads to competitive results compared to finite-width counterparts and deep neural network kernels.
Machine learning improves network classification and model selection.
problem Quantifying suitability of generative models for network structures.
method Interpretable machine learning to classify simulated networks based on features and interactions.
result Specific network features and their interactions are crucial for distinguishing generative models.
Study shows effective resistance distance yields more accurate network barycenter than Hamming distance.
problem Identifying the best metric for computing the Fréchet mean network.
method Compared the effectiveness of Hamming distance and effective resistance distance in capturing network topology.
result Effective resistance distance produces a more accurate Fréchet mean network.
Model-based neural networks generalize better than ReLU networks for sparse recovery.
problem Understanding and quantifying the superior generalization of model-based neural networks.
method Using complexity measures like global and local Rademacher complexities, the paper provides theoretical bounds on generalization and estimation errors.
result Model-based neural networks exhibit higher generalization capabilities for sparse recovery problems compared to ReLU networks.
A new method predicts links better across various networks.
problem Adaptive link prediction for diverse network types.
method MOLI method using local information from neighbors of different distances.
result MOLI outperforms other link prediction algorithms.
Recent works reveal that network embedding techniques enable many machine learning models to handle diverse downstream tasks on graph structured data. However, as previous methods usually focus on learning embeddings for a single network, they can not learn representations transferable on multiple networks. Hence, it i…
This paper proposes network recasting as a general method for network architecture transformation. The primary goal of this method is to accelerate the inference process through the transformation, but there can be many other practical applications. The method is based on block-wise recasting; it recasts each source bl…
Road networks are a type of spatial network, where edges may be associated with qualitative information such as road type and speed limit. Unfortunately, such information is often incomplete; for instance, OpenStreetMap only has speed limits for 13% of all Danish road segments. This is problematic for analysis tasks th…
With the widespread use of information technologies, information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, telecommunication networks, and biological networks. Analyzing these networks sheds light on different aspe…
Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLUk neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.