17,081 research outputs found
Band width estimates with lower scalar curvature bounds
A band is a connected compact manifold X together with a decomposition âX = ââX t â+X where â±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 ââX and â+X with respect to g.
Following Gromovâs 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 Xnâ€7 = Y Ă[â1, 1], the width of X with respect to any Riemannian metric with scalar curvature â„ n(n â 1) is bounded from above by 2Ï 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 Mnâ€7 = Y Ă 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 Ă [â1, 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
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
Annals [...].
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
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
Recommended from our members
MODELING CHAIN PACKING IN COMPLEX PHASES OF SELF-ASSEMBLED BLOCK COPOLYMERS
Block copolymer (BCP) melts undergo microphase seperation and form ordered soft matter crystals with varying domain shapes and symmetries. We study the con- nection between diblock copolymer molecular designs and thermodynamic selection of ordered crystals by modeling features of variable sub-domain geometry filled with individual blocks within non-canonical sphere-like and network phases that together with layered, cylindrical and canonical spherical phases forms ânatural formsâ of self- assembled amphiphilic soft matter at large. First, we present a model to revise our understanding of optimal Frank-Kasper sphere-like morphologies by advancing the- ory to account for varying domain volumes. We then develop generic approaches to quantify local changes to domain thickness or packing frustration using medial sets and show its application to morphologies with arbitrary domain topologies and sym- metries in both theoretical models and experimental data. We further use medial sets as a proxy for terminal boundaries of blocks within different domains and revise thermodynamic models of BCP assembly in the strong segregation limit. Finally, we use this revised model to study effect of elastic stiffness asymmetry on relaxing packing frustration experienced by BCPs in tubular and matrix domains leading to equilibrium double gyroid network morphology in diblock copolymers
Machine learning for managing structured and semi-structured data
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 gröĂere 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€Ăigen Gittern auf allgemeine (unregelmĂ€Ăige) Graphen betrachtet werden.
Wissensgraphen spielen eine wesentliche Rolle bei der Darstellung von Fakten ĂŒber EntitĂ€ten in maschinenlesbaren Form. Obwohl groĂe 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 ĂberprĂŒfung und Erweiterung der Graphen wird aufgrund der groĂen 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 groĂ 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. AuĂerdem 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. SchlieĂlich 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
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 PlastizitÀt zu untersuchen. Unser
Ziel war es, fĂŒr PlastizitĂ€tsinduktion gĂŒnstige Zeitpunkte im EEG zu identifizieren,
um in Zukunft die EffektivitÀt solcher zurzeit oft noch unzuverlÀssigen 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 âGehirnwellenâ im EEG entstehen durch synchronisierte
Fluktuationen des Membranpotentials kortikaler Neurone, welche aufgrund ihrer
intrakortikalen Kommunikationsfunktion wertvolle Informationen ĂŒber neuronale
Erregbarkeit vermitteln. Im Gegensatz zu âopen-loopâ TMS ermöglicht EEG-TMS
nicht nur eine prÀzisere 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 âasymmetrisch gepulste Inhibitionâ mit einem Maximum der Hemmung
wĂ€hrend der Hochpunkte (âPeaksâ) und wĂ€hrend hoher âPowerâ (⌠Amplitude).
Der â”-Rhythmusâ, 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 lÀsst 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
âInter-Trial-Intervalleâ (ITIs, bekannter âConfounderâ) 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 tatsÀchlich fazilitatorische
Wirkung erklÀrt 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 âVorzeichenumkehrungâ der Effektrichtung fĂŒhren könnte. Weiterhin wird eine AbhĂ€ngigkeit der âerregbarstenâ
Power-Werte von der StimulusstÀrke diskutiert.
Im Hauptexperiment sollte mit âpaarig-assoziativer Stimulationâ (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 âScreeningâ 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 (âTroughsâ) als wĂ€hrend der Peaks zu erzielen,
also mehr synaptische âFormbarkeitâ 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
wÀhrend PAS wurde durch (1) ”-Peaks, (2) ”-Troughs, (3) mittlere ”-Power und
(4) open-loop getriggert. (3) und (4) dienten jeweils als Kontrollbedingung. PAS
konnte, unabhÀngig von der EEG-Bedingung, keine signifikante VerÀnderung der
MEP-Amplituden vom Ausgangswert hervorrufen. Die fehlende Wirkung könnte durch intra- und interindividuelle Schwankungen gewisser Parameter zwischen den Sitzungen erklÀrt werden (z.B. MEP-Ausgangswerte, absolute ”-Power
wÀhrend PAS), die sich jedoch nicht als systematische Confounder zwischen
EEG-Bedingungen herausstellten.
Die, im Gegensatz zu open-loop-Studien, schwankenden ITIs wÀhrend der PAS
könnten die Wirkung ebenfalls beeintrÀchtigt haben. Weiterhin waren zwei verschiedene Kortexareale (S1 und M1) am Protokoll beteiligt, was die Identifikation
einer relevanten EEG-Eigenschaft erschwerte.
GegenwÀrtig rufen PlastizitÀts-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€Ăige
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. âBrain wavesâ 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 âasymmetric pulsed inhibitionâ, 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 âblocksâ 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
For an n-vertex directed graph , a -\emph{shortcut set}
is a set of additional edges such that has
the same transitive closure as , and for every pair , there is a
-path in with at most edges. A natural generalization of
shortcut sets to distances is a -\emph{hopset} , where the requirement is that and have the same
shortest-path distances, and for every , there is a
-approximate shortest path in with at most
edges.
There is a large literature on the tradeoff between the size of a shortcut
set / hopset and the value of . We highlight the most natural point on
this tradeoff: what is the minimum value of , such that for any graph
, there exists a -shortcut set (or a -hopset) with
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 , but in a breakthrough result Kogan and Parter [SODA 2022] improve
this to for shortcut sets and
for hopsets.
Our result is to close the gap between shortcut sets and hopsets. That is, we
show that for any graph and any fixed there is a
hopset with edges. More generally, we
achieve a smooth tradeoff between hopset size and 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
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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