A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called ``detectability limit'', no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erdős--Rényi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.

We provide an overview of high dimensional dynamical systems driven by random matrices, focusing on applications to simple models of learning and generalization in machine learning theory. Using both cavity method arguments and path integrals, we review how the behavior of a coupled infinite dimensional system can be characterized as a stochastic process for each single site of the system. We provide a pedagogical treatment of dynamical mean field theory (DMFT), a framework that can be flexibly applied to these settings. The DMFT single site stochastic process is fully characterized by a set of (two-time) correlation and response functions. For linear time-invariant systems, we illustrate connections between random matrix resolvents and the DMFT response. We demonstrate applications of these ideas to machine learning models such as gradient flow, stochastic gradient descent on random feature models and deep linear networks in the feature learning regime trained on random data. We demonstrate how bias and variance decompositions (analysis of ensembling/bagging etc) can be computed by averaging over subsets of the DMFT noise variables. From our formalism we also investigate how linear systems driven with random non-Hermitian matrices (such as random feature models) can exhibit non-monotonic loss curves with training time, while Hermitian matrices with the matching spectra do not, highlighting a different mechanism for non-monotonicity than small eigenvalues causing instability to label noise. Lastly, we provide asymptotic descriptions of the training and test loss dynamics for randomly initialized deep linear neural networks trained in the feature learning regime with high-dimensional random data. In this case, the time translation invariance structure is lost and the hidden layer weights are characterized as spiked random matrices.

Sophie Turner reveals intense 8-hour daily training for upcoming "Tomb Raider" film as Lara Croft. The "Game of Thrones" actress discusses her physical preparation since February 2025.

Science Archaeology 3D map of Easter Island takes you places visitors aren't allowed One of the world's most isolated islands is open to virtual tourists. Breakthroughs, discoveries, and DIY tips sent every weekday. Nestled in the South Pacific Ocean, some 6,000 people live on the most isolated, inhabited island in the world: Rapa Nui. Known to many as Easter Island, a name Dutch explorer Jacob Roggeveen coined after landing on the island on Easter Sunday 1722, Rapa Nui is roughly double the size of Disney World, or 63.2 square miles. And every year, some 100,000 people visit the remote island to see the famed 13-foot-tall moai statues or Easter Island heads .

Things to Do in L.A. Tap to enable a layout that focuses on the article. Voters are silhouetted near the American flag while casting ballots in the California special election at the Huntington Beach Central Library in Huntington Beach on Nov. 4. This is read by an automated voice. Please report any issues or inconsistencies here . A mysterious news site called the California Courier floods Facebook with conservative-leaning stories attacking Democrats.

Ornate medieval ring discovered in Norway's oldest town Scientists are still investigating if the ring's center stone is a sapphire or colored glass. Breakthroughs, discoveries, and DIY tips sent every weekday. Last summer, Linda Åsheim found a ring so beautiful it looks like it could have been made yesterday. But Åsheim is an archaeologist, and she found the rare artifact while excavating in a Norwegian town believed to be the oldest in the country. The gorgeous golden ring is decorated with a gemstone and filigree décor--and is over 800 years old.

Wing's drone deliveries are coming to 150 more Walmarts The service expansion will reach Walmart customers in Los Angeles, St. Louis, Cincinnati, Miami and other US metro areas. Don't be surprised if you see even more drones delivering groceries across the US since the Alphabet-owned Wing announced another service expansion with Walmart over the next year. The partnership said that drone delivery services will be available at 150 more Walmart locations in Los Angeles, St. Louis, Cincinnati, Miami and more metros that have yet to be announced. According to Wing, its top 25 percent of customers have ordered its delivery drones up to three times a week. To meet growing demand, Wing and Walmart said it will serve up to 40 million US customers and build up a network of 270 delivery locations by 2027.

Even the Empire State Building would eventually crumble. Breakthroughs, discoveries, and DIY tips sent every weekday. New York City is one of the noisiest cities in the world. With a population of eight and a half million people, the city is a nonstop symphony of car honks, yelling, and ambulance sirens. Now, imagine if all that noise and all those people suddenly disappeared overnight. Just how quickly would nature move into abandoned apartments? Well in a new episode of's podcast, we explore just that. So, yes, there's a reason cats love boxes and no, hot workout classes usually aren't better . If you have a question for us, send us a note .

Kurt'CyberGuy' Knutsson joins'Fox & Friends' to discuss grocery stores collecting biometric data, including facial recognition and eye scans, as Wegmans confirms limited use in higher-risk locations. NEW You can now listen to Fox News articles! Protecting your personal information online starts with understanding the language behind your apps, devices and accounts. We'll break down five essential tech terms that directly impact your digital privacy, from app permissions and location tracking to VPNs and cross-app advertising. Learning these concepts will help you limit data exposure and stay in control of who can see what. Stay tuned for more in this series as we dive deeper into privacy-related tech terms and other essential concepts, answering the top questions we get from readers like you!

Zurich-based firm taps into latest robot tech to'fibre-spray' high-end sports shoes worn by the likes of Roger Federer A robot leg whirs around in a complex ballet as an almost invisible spray of "flying fibre" builds a hi-tech £300 sports shoe at its foot. This nearly entirely automated process - like a sci-fi future brought to life - is part of the gameplan from On, the Swiss sports brand that is taking on the sector's mighty champions Nike and Adidas with a mix of technology and chutzpah. The brand is expanding rapidly after teaming up with the former tennis pro Roger Federer to create shoes suitable for the Swiss star's sport and a mix of fashion-led collaborations including with the luxury brand LOEWE, actor Zendaya and singers FKA twigs and Burna Boy. In China, sales have doubled year-on-year. Growth has been strong in the US and mainland Europe and this month On will open its fourth London store, in Kensington.