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Disease risk prediction is a rising challenge in the medical domain. Researchers have widely used machine learning algorithms to solve this challenge. The k -nearest neighbour (KNN) algorithm is the most frequently used among the wide range of machine learning algorithms. This paper presents a study on different KNN variants (Classic one, Adaptive, Locally adaptive, k-means clustering, Fuzzy, Mutual, Ensemble, Hassanat and Generalised mean distance) and their performance comparison for disease prediction. This study analysed these variants in-depth through implementations and experimentations using eight machine learning benchmark datasets obtained from Kaggle, UCI Machine learning repository and OpenML. The datasets were related to different disease contexts. We considered the performance measures of accuracy, precision and recall for comparative analysis. The average accuracy values of these variants ranged from 64.22% to 83.62%. The Hassanaat KNN showed the highest average accuracy (83.62%), followed by the ensemble approach KNN (82.34%). A relative performance index is also proposed based on each performance measure to assess each variant and compare the results. This study identified Hassanat KNN as the best performing variant based on the accuracy-based version of this index, followed by the ensemble approach KNN. This study also provided a relative comparison among KNN variants based on precision and recall measures. Finally, this paper summarises which KNN variant is the most promising candidate to follow under the consideration of three performance measures (accuracy, precision and recall) for disease prediction. Healthcare researchers and stakeholders could use the findings of this study to select the appropriate KNN variant for predictive disease risk analytics.

Community detection is often used to understand the structure of large and complex networks. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. To address this problem, we introduce the Leiden algorithm. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. In addition, we prove that, when the Leiden algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are locally optimally assigned. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees.

Rich data are revealing that complex dependencies between the nodes of a network may not be captured by models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems.Rich data are revealing that complex dependencies between the nodes of a network may not be captured by models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems.

Improving the efficiency of algorithms for fundamental computations can have a widespread impact, as it can affect the overall speed of a large amount of computations. Matrix multiplication is one such primitive task, occurring in many systems—from neural networks to scientific computing routines. The automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. However, automating the algorithm discovery procedure is intricate, as the space of possible algorithms is enormous. Here we report a deep reinforcement learning approach based on AlphaZero 1 for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. Our agent, AlphaTensor, is trained to play a single-player game where the objective is finding tensor decompositions within a finite factor space. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago 2 . We further showcase the flexibility of AlphaTensor through different use-cases: algorithms with state-of-the-art complexity for structured matrix multiplication and improved practical efficiency by optimizing matrix multiplication for runtime on specific hardware. Our results highlight AlphaTensor’s ability to accelerate the process of algorithmic discovery on a range of problems, and to optimize for different criteria. A reinforcement learning approach based on AlphaZero is used to discover efficient and provably correct algorithms for matrix multiplication, finding faster algorithms for a variety of matrix sizes.

Complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units. Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems. Here, we highlight recent evidence of collective behaviours induced by higher-order interactions, and we outline three key challenges for the physics of higher-order systems.Network representations of complex systems are limited to pairwise interactions, but real-world systems often involve higher-order interactions. This Perspective looks at the new physics emerging from attempts to characterize these interactions.

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter 1 – 6 , non-equilibrium systems 7 – 9 , networks of neurons 10 , 11 , social groups with conformist and contrarian members 12 , directional interface growth phenomena 13 – 15 and metamaterials 16 – 20 . Although wave propagation in non-reciprocal media has recently been closely studied 1 , 16 – 20 , less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points 21 . We describe the emergence of these phases using insights from bifurcation theory 22 , 23 and non-Hermitian quantum mechanics 24 , 25 . Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle. A theoretical study of non-reciprocity in collective phenomena reveals the emergence of time-dependent phases heralded by exceptional points in contexts ranging from synchronization and flocking to pattern formation.

Reservoir computing originates in the early 2000s, the core idea being to utilize dynamical systems as reservoirs (nonlinear generalizations of standard bases) to adaptively learn spatiotemporal features and hidden patterns in complex time series. Shown to have the potential of achieving higher-precision prediction in chaotic systems, those pioneering works led to a great amount of interest and follow-ups in the community of nonlinear dynamics and complex systems. To unlock the full capabilities of reservoir computing towards a fast, lightweight, and significantly more interpretable learning framework for temporal dynamical systems, substantially more research is needed. This Perspective intends to elucidate the parallel progress of mathematical theory, algorithm design and experimental realizations of reservoir computing, and identify emerging opportunities as well as existing challenges for large-scale industrial adoption of reservoir computing, together with a few ideas and viewpoints on how some of those challenges might be resolved with joint efforts by academic and industrial researchers across multiple disciplines. Reservoir Computing has shown advantageous performance in signal processing and learning tasks due to compact design and ability for fast training. Here, the authors discuss the parallel progress of mathematical theory, algorithm design and experimental realizations of Reservoir Computers, and identify emerging opportunities as well as existing challenges for their large-scale industrial adoption.

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear. The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al. develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.

Non-Hermitian theory is a theoretical framework used to describe open systems. It offers a powerful tool in the characterization of both the intrinsic degrees of freedom of a system and the interactions with the external environment. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. These structures not only underpin novel approaches for precisely tailoring non-Hermitian systems for applications but also give rise to topologies not found in Hermitian systems. In this Review, we provide an overview of non-Hermitian topology by establishing its relationship with the behaviours of complex eigenvalues and biorthogonal eigenvectors. Special attention is given to exceptional points — branch-point singularities on the complex eigenvalue manifolds that exhibit nontrivial topological properties. We also discuss recent developments in non-Hermitian band topology, such as the non-Hermitian skin effect and non-Hermitian topological classifications.Non-Hermitian theory consists of mathematical structures that are used to describe open systems, which can give rise to non-Hermitian topology not found in Hermitian systems. This Review provides an overview of non-Hermitian band topology and discusses recent developments, such as the non-Hermitian skin effect and non-Hermitian topological classifications.

Current learning machines have successfully solved hard application problems, reaching high accuracy and displaying seemingly intelligent behavior. Here we apply recent techniques for explaining decisions of state-of-the-art learning machines and analyze various tasks from computer vision and arcade games. This showcases a spectrum of problem-solving behaviors ranging from naive and short-sighted, to well-informed and strategic. We observe that standard performance evaluation metrics can be oblivious to distinguishing these diverse problem solving behaviors. Furthermore, we propose our semi-automated Spectral Relevance Analysis that provides a practically effective way of characterizing and validating the behavior of nonlinear learning machines. This helps to assess whether a learned model indeed delivers reliably for the problem that it was conceived for. Furthermore, our work intends to add a voice of caution to the ongoing excitement about machine intelligence and pledges to evaluate and judge some of these recent successes in a more nuanced manner. Nonlinear machine learning methods have good predictive ability but the lack of transparency of the algorithms can limit their use. Here the authors investigate how these methods approach learning in order to assess the dependability of their decision making.