17,081 research outputs found

    Band width estimates with lower scalar curvature bounds

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    A band is a connected compact manifold X together with a decomposition ∂X = ∂−X t ∂+X where ∂±X are non-empty unions of boundary components. If X is equipped with a Riemannian metric, the pair (X, g) is called a Riemannian band and the width of (X, g) is defined to be the distance between ∂−X and ∂+X with respect to g. Following Gromov’s seminal work on metric inequalities with scalar curvature, the study of Riemannian bands with lower curvature bounds has been an active field of research in recent years, which led to several breakthroughs on longstanding open problems in positive scalar curvature geometry and to a better understanding of the positive mass theorem in general relativity In the first part of this thesis we combine ideas of Gromov and Cecchini-Zeidler and use the variational calculus surrounding so called ”-bubbles to establish a scalar and mean curvature comparison principle for Riemannian bands with the property that no closed embedded hypersurface which separates the two ends of the band admits a metric of positive scalar curvature. The model spaces we use for this comparison are warped product over scalar flat manifolds with log-concave warping functions. We employ ideas from surgery and bordism theory to deduce that, if Y is a closed orientable manifold which does not admit a metric of positive scalar curvature, dim(Y ) 6= 4 and Xn≀7 = Y ×[−1, 1], the width of X with respect to any Riemannian metric with scalar curvature ≄ n(n − 1) is bounded from above by 2π n. This solves, up to dimension 7, a conjecture due to Gromov in the orientable case. Furthermore, we adapt and extend our methods to show that, if Y is as before and Mn≀7 = Y × R, then M does not admit a metric of positive scalar curvature. This solves, up to dimension 7 a conjecture due to Rosenberg and Stolz in the orientable case. In the second part of this thesis we explore how these results transfer to the setting where the lower scalar curvature bound is replaced by a lower bound on the macroscopic scalar curvature of a Riemannian band. This curvature condition amounts to an upper bound on the volumes of all unit balls in the universal cover of the band. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If Y is filling enlargeable, Xn = Y × [−1, 1] and g is a Riemannian metric on X with the property that the volumes of all unit balls in the universal cover of (X, g) are bounded from above by a small dimensional constant Δn, then width(X, g) ≀ 1. Finally, we establish that whether or not a closed orientable manifold is filling enlargeable or not depends on the image of the fundamental class under the classifying map of the universal cover

    Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region

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    In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincar\'e half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.Comment: To appear in Communications in Nonlinear Science and Numerical Simulatio

    A Scalable Linear-Time Algorithm for Horizontal Visibility Graph Construction Over Long Sequences

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    Annals [...].

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    Pedometrics: innovation in tropics; Legacy data: how turn it useful?; Advances in soil sensing; Pedometric guidelines to systematic soil surveys.Evento online. Coordenado por: Waldir de Carvalho Junior, Helena Saraiva Koenow Pinheiro, Ricardo SimĂŁo Diniz Dalmolin

    Towards a non-equilibrium thermodynamic theory of ecosystem assembly and development

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    Non-equilibrium thermodynamics has had a significant historic influence on the development of theoretical ecology, even informing the very concept of an ecosystem. Much of this influence has manifested as proposed extremal principles. These principles hold that systems will tend to maximise certain thermodynamic quantities, subject to the other constraints they operate under. A particularly notable extremal principle is the maximum entropy production principle (MaxEPP); that systems maximise their rate of entropy production. However, these principles are not robustly based in physical theory, and suffer from treating complex ecosystems in an extremely coarse manner. To address this gap, this thesis derives a limited but physically justified extremal principle, as well as carrying out a detailed investigation of the impact of non-equilibrium thermodynamic constraints on the assembly of microbial communities. The extremal principle we obtain pertains to the switching between states in simple bistable systems, with switching paths that generate more entropy being favoured. Our detailed investigation into microbial communities involved developing a novel thermodynamic microbial community model, using which we found the rate of ecosystem development to be set by the availability of free-energy. Further investigation was carried out using this model, demonstrating the way that trade-offs emerging from fundamental thermodynamic constraints impact the dynamics of assembling microbial communities. Taken together our results demonstrate that theory can be developed from non-equilibrium thermodynamics, that is both ecologically relevant and physically well grounded. We find that broad extremal principles are unlikely to be obtained, absent significant advances in the field of stochastic thermodynamics, limiting their applicability to ecology. However, we find that detailed consideration of the non-equilibrium thermodynamic mechanisms that impact microbial communities can broaden our understanding of their assembly and functioning.Open Acces

    Machine learning for managing structured and semi-structured data

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    As the digitalization of private, commercial, and public sectors advances rapidly, an increasing amount of data is becoming available. In order to gain insights or knowledge from these enormous amounts of raw data, a deep analysis is essential. The immense volume requires highly automated processes with minimal manual interaction. In recent years, machine learning methods have taken on a central role in this task. In addition to the individual data points, their interrelationships often play a decisive role, e.g. whether two patients are related to each other or whether they are treated by the same physician. Hence, relational learning is an important branch of research, which studies how to harness this explicitly available structural information between different data points. Recently, graph neural networks have gained importance. These can be considered an extension of convolutional neural networks from regular grids to general (irregular) graphs. Knowledge graphs play an essential role in representing facts about entities in a machine-readable way. While great efforts are made to store as many facts as possible in these graphs, they often remain incomplete, i.e., true facts are missing. Manual verification and expansion of the graphs is becoming increasingly difficult due to the large volume of data and must therefore be assisted or substituted by automated procedures which predict missing facts. The field of knowledge graph completion can be roughly divided into two categories: Link Prediction and Entity Alignment. In Link Prediction, machine learning models are trained to predict unknown facts between entities based on the known facts. Entity Alignment aims at identifying shared entities between graphs in order to link several such knowledge graphs based on some provided seed alignment pairs. In this thesis, we present important advances in the field of knowledge graph completion. For Entity Alignment, we show how to reduce the number of required seed alignments while maintaining performance by novel active learning techniques. We also discuss the power of textual features and show that graph-neural-network-based methods have difficulties with noisy alignment data. For Link Prediction, we demonstrate how to improve the prediction for unknown entities at training time by exploiting additional metadata on individual statements, often available in modern graphs. Supported with results from a large-scale experimental study, we present an analysis of the effect of individual components of machine learning models, e.g., the interaction function or loss criterion, on the task of link prediction. We also introduce a software library that simplifies the implementation and study of such components and makes them accessible to a wide research community, ranging from relational learning researchers to applied fields, such as life sciences. Finally, we propose a novel metric for evaluating ranking results, as used for both completion tasks. It allows for easier interpretation and comparison, especially in cases with different numbers of ranking candidates, as encountered in the de-facto standard evaluation protocols for both tasks.Mit der rasant fortschreitenden Digitalisierung des privaten, kommerziellen und öffentlichen Sektors werden immer grĂ¶ĂŸere Datenmengen verfĂŒgbar. Um aus diesen enormen Mengen an Rohdaten Erkenntnisse oder Wissen zu gewinnen, ist eine tiefgehende Analyse unerlĂ€sslich. Das immense Volumen erfordert hochautomatisierte Prozesse mit minimaler manueller Interaktion. In den letzten Jahren haben Methoden des maschinellen Lernens eine zentrale Rolle bei dieser Aufgabe eingenommen. Neben den einzelnen Datenpunkten spielen oft auch deren ZusammenhĂ€nge eine entscheidende Rolle, z.B. ob zwei Patienten miteinander verwandt sind oder ob sie vom selben Arzt behandelt werden. Daher ist das relationale Lernen ein wichtiger Forschungszweig, der untersucht, wie diese explizit verfĂŒgbaren strukturellen Informationen zwischen verschiedenen Datenpunkten nutzbar gemacht werden können. In letzter Zeit haben Graph Neural Networks an Bedeutung gewonnen. Diese können als eine Erweiterung von CNNs von regelmĂ€ĂŸigen Gittern auf allgemeine (unregelmĂ€ĂŸige) Graphen betrachtet werden. Wissensgraphen spielen eine wesentliche Rolle bei der Darstellung von Fakten ĂŒber EntitĂ€ten in maschinenlesbaren Form. Obwohl große Anstrengungen unternommen werden, so viele Fakten wie möglich in diesen Graphen zu speichern, bleiben sie oft unvollstĂ€ndig, d. h. es fehlen Fakten. Die manuelle ÜberprĂŒfung und Erweiterung der Graphen wird aufgrund der großen Datenmengen immer schwieriger und muss daher durch automatisierte Verfahren unterstĂŒtzt oder ersetzt werden, die fehlende Fakten vorhersagen. Das Gebiet der WissensgraphenvervollstĂ€ndigung lĂ€sst sich grob in zwei Kategorien einteilen: Link Prediction und Entity Alignment. Bei der Link Prediction werden maschinelle Lernmodelle trainiert, um unbekannte Fakten zwischen EntitĂ€ten auf der Grundlage der bekannten Fakten vorherzusagen. Entity Alignment zielt darauf ab, gemeinsame EntitĂ€ten zwischen Graphen zu identifizieren, um mehrere solcher Wissensgraphen auf der Grundlage einiger vorgegebener Paare zu verknĂŒpfen. In dieser Arbeit stellen wir wichtige Fortschritte auf dem Gebiet der VervollstĂ€ndigung von Wissensgraphen vor. FĂŒr das Entity Alignment zeigen wir, wie die Anzahl der benötigten Paare reduziert werden kann, wĂ€hrend die Leistung durch neuartige aktive Lerntechniken erhalten bleibt. Wir erörtern auch die LeistungsfĂ€higkeit von Textmerkmalen und zeigen, dass auf Graph-Neural-Networks basierende Methoden Schwierigkeiten mit verrauschten Paar-Daten haben. FĂŒr die Link Prediction demonstrieren wir, wie die Vorhersage fĂŒr unbekannte EntitĂ€ten zur Trainingszeit verbessert werden kann, indem zusĂ€tzliche Metadaten zu einzelnen Aussagen genutzt werden, die oft in modernen Graphen verfĂŒgbar sind. GestĂŒtzt auf Ergebnisse einer groß angelegten experimentellen Studie prĂ€sentieren wir eine Analyse der Auswirkungen einzelner Komponenten von Modellen des maschinellen Lernens, z. B. der Interaktionsfunktion oder des Verlustkriteriums, auf die Aufgabe der Link Prediction. Außerdem stellen wir eine Softwarebibliothek vor, die die Implementierung und Untersuchung solcher Komponenten vereinfacht und sie einer breiten Forschungsgemeinschaft zugĂ€nglich macht, die von Forschern im Bereich des relationalen Lernens bis hin zu angewandten Bereichen wie den Biowissenschaften reicht. Schließlich schlagen wir eine neuartige Metrik fĂŒr die Bewertung von Ranking-Ergebnissen vor, wie sie fĂŒr beide Aufgaben verwendet wird. Sie ermöglicht eine einfachere Interpretation und einen leichteren Vergleich, insbesondere in FĂ€llen mit einer unterschiedlichen Anzahl von Kandidaten, wie sie in den de-facto Standardbewertungsprotokollen fĂŒr beide Aufgaben vorkommen

    Influence of sensorimotor ” rhythm phase and power on motor cortex excitability and plasticity induction, assessed with EEG-triggered TMS

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    In dieser Arbeit werden zwei Experimente vorgestellt, bei denen EEG-getriggerte transkranielle Magnetstimulation (TMS) an gesunden Probanden eingesetzt wurde, um die Rolle des sensomotorischen 8-14Hz ”-Rhythmus auf die kortikospinale Erregbarkeit (CSE) und die Induktion positiver PlastizitĂ€t zu untersuchen. Unser Ziel war es, fĂŒr PlastizitĂ€tsinduktion gĂŒnstige Zeitpunkte im EEG zu identifizieren, um in Zukunft die EffektivitĂ€t solcher zurzeit oft noch unzuverlĂ€ssigen Anwendungen zu steigern. Unser EEG-TMS System interpretierte Oszillationen im EEG in Echtzeit und löste einen Stimulus aus, wenn bestimmte, vorher festgelegte Eigenschaften zutrafen. Die ‘Gehirnwellen’ im EEG entstehen durch synchronisierte Fluktuationen des Membranpotentials kortikaler Neurone, welche aufgrund ihrer intrakortikalen Kommunikationsfunktion wertvolle Informationen ĂŒber neuronale Erregbarkeit vermitteln. Im Gegensatz zu “open-loop” TMS ermöglicht EEG-TMS nicht nur eine prĂ€zisere Erforschung der Funktion von Gehirnwellen, sondern auch die Umsetzung der gewonnenen Erkenntnisse in effizientere therapeutische Anwendungen. Speziell Oszillationen im Alpha-Frequenzbereich (8-14Hz) spielen eine bedeutsame Rolle, indem sie den Informationsfluss im Gehirn durch Hemmung aktuell irrelevanter Areale steuern, und zwar laut einer fĂŒhrenden Theorie als “asymmetrisch gepulste Inhibition” mit einem Maximum der Hemmung wĂ€hrend der Hochpunkte (“Peaks”) und wĂ€hrend hoher “Power” (∌ Amplitude). Der “”-Rhythmus”, Wellen in alpha-Frequenz ĂŒber dem sensomotorischen Kortex, scheint fĂŒr diese Areale eine analoge Rolle wie das okzipitale Alpha fĂŒr den visuellen Kortex zu spielen. Die CSE lĂ€sst sich durch die Amplitude der ausgelösten kontralateralen Muskelzuckungen (MEPs im EMG) quantifizieren. Im Vorexperiment erforschten wir den Einfluss der Power der ”-Wellen auf die CSE. 16 Teilnehmer wurden in einer Sitzung mit Einzelpuls-TMS des linken M1 stimuliert. Die Pulse wurden durch die momentane Power ausgelöst, 10 Dezile des individuellen ”-Powerspektrums wurden in pseudorandomisierter Reihenfolge angesteuert, verteilt auf 4 Stimulationsblöcke. Nach BerĂŒcksichtigung der “Inter-Trial-Intervalle” (ITIs, bekannter “Confounder”) und Normalisierung pro Block zeigten unsere Daten eine schwache positiv-lineare Korrelation zwischen ” Power und MEP-Amplitude, welche somit im Widerspruch zur angenommenen hemmenden Wirkung von ” steht, aber mittlerweile in mehreren anderen Studien repliziert wurde. Diese Diskrepanz kann z.B. durch eine tatsĂ€chlich fazilitatorische Wirkung erklĂ€rt werden, oder auch durch eine anatomisch dem sensorischen Kortex (S1) zuzuordnende Quelle der angesteuerten ”-Wellen, was ĂŒber hem- 83mende Interneurone von S1 auf M1 zu einer ‘Vorzeichenumkehrung’ der Effektrichtung fĂŒhren könnte. Weiterhin wird eine AbhĂ€ngigkeit der ‘erregbarsten’ Power-Werte von der StimulusstĂ€rke diskutiert. Im Hauptexperiment sollte mit ‘paarig-assoziativer Stimulation’ (PAS) (intervallsensitive Kombination von Elektrostimulation des rechten Nervus medianus mit TMS des linken M1) positive PlastizitĂ€t (die Intervention ĂŒberdauernde StĂ€rkung von Synapsen) induziert werden. Dem ging ein umfangreiches “Screening” zur Identifikation geeigneter Probanden mit ausgeprĂ€gtem ”-Rhythmus (fĂŒr prĂ€zise EEGTriggerung) voraus. Letztlich absolvierten 16 Teilnehmer je 4 Sitzungen (eine pro Trigger-Bedingung). Unsere Hypothese war hierbei, mehr PlastizitĂ€t nach Stimulation wĂ€hrend der Tiefpunkte (“Troughs”) als wĂ€hrend der Peaks zu erzielen, also mehr synaptische ‘Formbarkeit’ wĂ€hrend höherer Erregbarkeit. In Anbetracht der schwachen Ergebnisse des Vorexperiments sowie einer widersprĂŒchlichen Beweislage bezĂŒglich einer fazilitatorischen oder inhibitorischen Funktion wurden hohe und niedrige Power nicht explizit miteinander verglichen. TMS wĂ€hrend PAS wurde durch (1) ”-Peaks, (2) ”-Troughs, (3) mittlere ”-Power und (4) open-loop getriggert. (3) und (4) dienten jeweils als Kontrollbedingung. PAS konnte, unabhĂ€ngig von der EEG-Bedingung, keine signifikante VerĂ€nderung der MEP-Amplituden vom Ausgangswert hervorrufen. Die fehlende Wirkung könnte durch intra- und interindividuelle Schwankungen gewisser Parameter zwischen den Sitzungen erklĂ€rt werden (z.B. MEP-Ausgangswerte, absolute ”-Power wĂ€hrend PAS), die sich jedoch nicht als systematische Confounder zwischen EEG-Bedingungen herausstellten. Die, im Gegensatz zu open-loop-Studien, schwankenden ITIs wĂ€hrend der PAS könnten die Wirkung ebenfalls beeintrĂ€chtigt haben. Weiterhin waren zwei verschiedene Kortexareale (S1 und M1) am Protokoll beteiligt, was die Identifikation einer relevanten EEG-Eigenschaft erschwerte. GegenwĂ€rtig rufen PlastizitĂ€ts-induzierende TMS-Protokolle in der Forschung und in Studien mit Schlaganfallpatienten schwankende und zeitlich begrenzte Wirkungen hervor. Durch EEG-Triggerung und / oder die Kombination mit klassischer Physiotherapie könnte eine verbesserte EffektivitĂ€t und somit eine routinemĂ€ĂŸige Anwendung erreicht werden. Trotz unserer negativen Ergebnisse bleibt EEG-getriggerte TMS ein vielversprechendes Instrument in Forschung und Klinik.This thesis presents two experiments employing real-time EEG-triggered transcranial magnetic stimulation (TMS) on healthy volunteers to investigate the role of sensorimotor 8-14Hz ” rhythm in EEG at rest on corticospinal excitability and induction of positive plasticity. We intended to identify brain states favorable to induction of positive plasticity to inform development of more efficient TMS protocols for clinical application e.g. in stroke patients. Applying TMS triggered by pre-determined EEG brain states in real time (opposed to open-loop TMS with post-hoc trial sorting) offers not only more precise research into the role of certain brain waves, but also translation into more efficient therapies. The membrane potential of superficial cortical neurons fluctuates rhythmically, visible as oscillations in surface EEG. Different brain areas seem to communicate through these synchronized fluctuations. ‘Brain waves’ therefore convey valuable information about the excitability of said areas. Oscillations in the alpha frequency range (8-14Hz) play a crucial role in this, gating information by inhibiting brain areas irrelevant to the current task. According to an influential hypothesis, this function is exerted as an ‘asymmetric pulsed inhibition’, with a maximum of inhibition during the peaks and during high alpha power (∌ amplitude). Sensorimotor alpha frequency waves (” rhythm) play a similar role as the well-researched occipital alpha does for the visual cortex. The primary motor cortex (M1) provides a quantifiable measure of (corticospinal) excitability, the amplitude of TMS-elicited contralateral muscle twitches (appearing as MEPs in the EMG). The first experiment investigated the role of ” power for M1 excitability. 16 participants underwent one session of single-pulse TMS of the left M1, triggered by overall 10 individual power deciles in pseudorandomized order, partitioned into 4 ‘blocks’ of stimulation over time. The data revealed, after stratification for confounding inter-trial-intervals (ITIs) and normalization to block average, a weak positive linear relationship contrary to the proposed inhibitory role of ”, which has however since been replicated several times in other studies. This discrepancy can be explained e.g. by an in fact facilitatory nature of ”, by a postcentral and thus sensory cortical (S1) source of the targeted oscillations, reversing the inhibitory effect in sign to a facilitatory one through S1-to-M1 feedforward inhibition, or by a shift of most excitable power values dependent on stimulus strength. For the main experiment, we applied a paired associative stimulation (PAS) pro- 81tocol intended to induce positive plasticity (strengthening of synaptic connection outlasting the intervention), combining electrical stimulation of the right median nerve at the wrist with a TMS of the left M1 in a temporally sensitive manner. After an extensive screening to pre-select suitable subjects with a sufficiently strong ” rhythm (to ensure accurate performance of the real-time EEG targeting), 16 participants completed 4 sessions (one condition each). We expected to induce more positive plasticity during more excitable brain states, i.e., ” troughs rather than ” peaks. In light of our findings on ” power from the first experiment (weak influence as compared to ITIs and intrinsic variability over time) and overall contradictory evidence as to its (facilitatory versus inhibitory) role, high vs. low power were not explicitly compared. TMS during PAS was applied at (1) ” peaks, (2) ” troughs, (3) at medium ” powers and (4) open-loop. (3) and (4) both served as controls. The intervention failed to evoke a significant change in MEP amplitudes from baseline irrespective of condition. Possible explanations can be found in the intra- and interindividual variability of decisive parameters across sessions (e.g. baseline amplitudes and absolute ” powers during PAS), which however did not significantly depend on the targeted condition and were thus not true confounders. The number of sessions might still have introduced a further measure of variability. Varying PAS ITIs (due to EEG-triggering) could have also impeded plasticity induction, and the involvement of two cortical regions (S1 and M1) might have complicated the identification of one relevant brain state. Currently, plasticity-inducing TMS protocols in research and clinical trials evoke variable and transient effects. Improvements to enable routine application might come from EEG-triggering and/or combining with traditional motor training (physiotherapy). Regardless of our nil results in plasticity induction, EEG-triggered TMS remains a promising instrument in research and therapy

    Closing the Gap Between Directed Hopsets and Shortcut Sets

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    For an n-vertex directed graph G=(V,E)G = (V,E), a ÎČ\beta-\emph{shortcut set} HH is a set of additional edges H⊆V×VH \subseteq V \times V such that GâˆȘHG \cup H has the same transitive closure as GG, and for every pair u,v∈Vu,v \in V, there is a uvuv-path in GâˆȘHG \cup H with at most ÎČ\beta edges. A natural generalization of shortcut sets to distances is a (ÎČ,Ï”)(\beta,\epsilon)-\emph{hopset} H⊆V×VH \subseteq V \times V, where the requirement is that HH and GâˆȘHG \cup H have the same shortest-path distances, and for every u,v∈Vu,v \in V, there is a (1+Ï”)(1+\epsilon)-approximate shortest path in GâˆȘHG \cup H with at most ÎČ\beta edges. There is a large literature on the tradeoff between the size of a shortcut set / hopset and the value of ÎČ\beta. We highlight the most natural point on this tradeoff: what is the minimum value of ÎČ\beta, such that for any graph GG, there exists a ÎČ\beta-shortcut set (or a (ÎČ,Ï”)(\beta,\epsilon)-hopset) with O(n)O(n) edges? Not only is this a natural structural question in its own right, but shortcuts sets / hopsets form the core of many distributed, parallel, and dynamic algorithms for reachability / shortest paths. Until very recently the best known upper bound was a folklore construction showing ÎČ=O(n1/2)\beta = O(n^{1/2}), but in a breakthrough result Kogan and Parter [SODA 2022] improve this to ÎČ=O~(n1/3)\beta = \tilde{O}(n^{1/3}) for shortcut sets and O~(n2/5)\tilde{O}(n^{2/5}) for hopsets. Our result is to close the gap between shortcut sets and hopsets. That is, we show that for any graph GG and any fixed Ï”\epsilon there is a (O~(n1/3),Ï”)(\tilde{O}(n^{1/3}),\epsilon) hopset with O(n)O(n) edges. More generally, we achieve a smooth tradeoff between hopset size and ÎČ\beta which exactly matches the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors). Using a very recent black-box reduction of Kogan and Parter, our new hopset implies improved bounds for approximate distance preservers.Comment: Abstract shortened to meet arXiv requirements, v2: fixed a typ

    Network Geometry

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    Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, self-similarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity
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