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Complex time-varying networks are prominent models for a wide variety of spatiotemporal phenomena. The functioning of networks depends crucially on their connectivity, yet reliable techniques for determining communities in spacetime networks remain elusive. We adapt successful spectral techniques from continuous-time dynamics on manifolds to the graph setting to fill this gap. We formulate an inflated dynamic Laplacian for graphs and develop a spectral theory to underpin the corresponding algorithmic realisations. We develop spectral clustering approaches for both multiplex and non-multiplex networks, based on the eigenvectors of the inflated dynamic Laplacian and specialised Sparse EigenBasis Approximation (SEBA) post-processing of these eigenvectors. We demonstrate that our approach can outperform the Leiden algorithm applied both in spacetime and layer-by-layer, and we analyse voting data from the US senate (where senators come and go as congresses evolve) to quantify increasing polarisation in time., Comment: 47 pages, 8 figures

Misinformation poses a significant challenge studied extensively by researchers, yet acquiring data to identify primary sharers is costly and challenging. To address this, we propose a low-barrier approach to differentiate social media users who are more likely to share misinformation from those who are less likely. Leveraging insights from previous studies, we demonstrate that easy-access online social network metrics -- average daily tweet count, and account age -- can be leveraged to help identify potential low factuality content spreaders on X (previously known as Twitter). We find that higher tweet frequency is positively associated with low factuality in shared content, while account age is negatively associated with it. We also find that some of the effects, namely the effect of the number of accounts followed and the number of tweets produced, differ depending on the number of followers a user has. Our findings show that relying on these easy-access social network metrics could serve as a low-barrier approach for initial identification of users who are more likely to spread misinformation, and therefore contribute to combating misinformation effectively on social media platforms.

Nodal loop semimetals are topological materials with drumhead surface states characterized by reduced kinetic energy. If the Fermi energy of such a system is near these nondispersive states interaction among charge carriers substantially impacts their electronic structure. The emergence of magnetism in these surface states is one of the possible consequences. Ca3P2 an already synthesized material possesses a remarkably large nodal loop which is situated exactly at the Fermi level of the bulk system. In the present work, we investigate how surface magnetism is impacted by surface reconstruction and lattice termination in finite slabs in this material. We show that a slight deviation from the stoichiometric occupation of Ca sites results in the stabilization of magnetic patterns.

We study the multiplier ideals and the corresponding jumping numbers and multiplicities $\{m(c)\}_{c\in \mathbb{R}}$ in the following context: $(X,o)$ is a complex analytic normal surface singularity, ${\mathfrak a}\subset \mathcal{O}_{X,o}$ is an ${\mathfrak m}_{X,o}$--primary ideal, $\phi:\widetilde{X}\to X$ is a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)$, for some nonzero effective divisor $F$ supported on $\phi^{-1}(0)$. We show that $\{m(c)\}_{c>0}$ is combinatorially computable from $F$ and the resolution graph $\Gamma$ of $\phi$, and we provide several formulae. We also extend Budur's result (valid for $(X,o)=(\mathbb{C}^2,0)$), which makes an identification of $\sum_{c\in[0,1]}m(c)t^c$ with a certain Hodge spectrum. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that $\{m(c)\}_{c\leq 0}$ usually depends on the analytic type of $(X,o)$. However, for some distinguished analytic types we determine it concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated with the central vertex), we recover $\sum_cm(c)t^c$ from the Poincar\'e series of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover $\sum_cm(c)t^c$ from the multivariable topological Poincar\'e (zeta) function of $\Gamma$., Comment: 23 pages, some minor corrections, list of references extended

Coherent sets are time-dependent regions in the physical space of nonautonomous flows that exhibit little mixing with their neighborhoods, robustly under small random perturbations of the flow. They thus characterize the global long-term transport behavior of the system. We propose a framework to extract such time-dependent families of coherent sets for nonautonomous systems with an ergodic driving dynamics and (small) Brownian noise in physical space. Our construction involves the assembly and analysis of an operator on functions over the augmented space of the associated skew product that, for each fixed state of the driving, propagates distributions on the corresponding physical-space fibre according to the dynamics. This time-dependent operator has the structure of a semigroup (it is called the Mather semigroup), and we show that a spectral analysis of its generator allows for a trajectory-free computation of coherent families, simultaneously for all states of the driving. Additionally, for quasi-periodically driven torus flows, we propose a tailored Fourier discretization scheme for this generator and demonstrate our method by means of three examples of two-dimensional flows.

Lagrangian methods continue to stand at the forefront of the analysis of time-dependent dynamical systems. Most Lagrangian methods have criteria that must be fulfilled by trajectories as they are followed throughout a given finite flow duration. This key strength of Lagrangian methods can also be a limitation in more complex evolving environments. It places a high importance on selecting a time window that produces useful results, and these results may vary significantly with changes in the flow duration. We show how to overcome this drawback in the tracking of coherent flow features. Finite-time coherent sets (FTCS) are material objects that strongly resist mixing in complicated nonlinear flows. Like other materially coherent objects, by definition they must retain their coherence properties throughout the specified flow duration. Recent work [Froyland and Koltai, CPAM, 2023] introduced the notion of semi-material FTCS, whereby a balance is struck between the material nature and the coherence properties of FTCS. This balance provides the flexibility for FTCS to come and go, merge and separate, or undergo other changes as the governing unsteady flow experiences dramatic shifts. The purpose of this work is to illustrate the utility of the inflated dynamic Laplacian introduced in [Froyland and Koltai, CPAM, 2023] in a range of dynamical systems that are challenging to analyse by standard Lagrangian means, and to provide an efficient meshfree numerical approach for the discretisation of the inflated dynamic Laplacian., Comment: 25 figures

The layered van der Waals material ZrTe$_5$ is known as a candidate topological insulator (TI), however its topological phase and the relation with other properties such as an apparent Dirac semimetallic state is still a subject of debate. We employ a semiclassical multicarrier transport (MCT) model to analyze the magnetotransport of ZrTe$_5$ nanodevices at hydrostatic pressures up to 2 GPa. The temperature dependence of the MCT results between 10 and 300 K is assessed in the context of thermal activation, and we obtain the positions of conduction and valence band edges in the vicinity of the chemical potential. We find evidence of the closing and re-opening of the band gap with increasing pressure, which is consistent with a phase transition from weak to strong TI. This matches expectations from ab initio band structure calculations, as well as previous observations that CVT-grown ZrTe$_5$ is a weak TI in ambient conditions., Comment: Main Text: 11 pages, 5 figures; Supporting Information: 13 pages, 9 figures

We extend a construction of Stipsicz-Szab\'{o} of infinitely many irreducible exotic smooth structures of some closed four-manifolds with even $b_2^+$ and fundamental group $\mathbb{Z}/2\mathbb{Z}$. We use the double node surgery and rational blow down constructions of Fintushel-Stern on some elliptic fibrations equipped with a free involution. The construction is done in an equivariant manner and the factor manifolds are distinguished by the Seiberg-Witten invariants of their universal covers., Comment: 14 pages, 7 figures, to appear in Michigan Journal of Mathematics

Understanding the topological aspects of the band structure of solids has fundamentally changed our appreciation of their properties. The layered, van der Waals transition-metal pentatelluride ZrTe$_5$ has proven on numerous occasions to be an excellent candidate for the study of controllable topological phase transitions. Here, we investigate the topological phase diagrams of monolayer and bilayer forms of ZrTe$_5$, under mechanical deformations using \textit{ab initio} techniques. We find that mechanical deformation can close the monolayer's topological gap, while the bilayer exhibits richer phase diagram, including both topological insulating, trivial metallic and insulating phases. The bilayer is predicted to be on the topological phase boundary. We also address the preparation of monolayers, using \emph{ab initio} simulations and experimental scanning tunneling microscopy measurements. We confirm that while monolayer ZrTe$_5$ is difficult to exfoliate without compromising its crystalline structure, bilayers offer a more stable alternative, revealing the complexities and limitations of using gold substrates for monolayer exfoliation.

The direct bandgap found in hexagonal germanium and some of its alloys with silicon allows for an optically active material within the group-IV semiconductor family with various potential technological applications. However, there remain some unanswered questions regarding several aspects of the band structiure, including the strength of the electric dipole transitions at the center of the Brillouin zone. Using the $\mathbf{k\cdot p}$ method near the $\Gamma$ point, including 10 bands, and taking spin-orbit coupling into account, we obtain a self-consistent model that produces the correct band curvatures, with previously unknown inverse effective mass parameters, to describe 2H-Ge via fitting to {\it ab initio} data and to calculate effective masses for electrons and holes. To understand the weak dipole coupling between the lowest conduction band and the top valance band, we start from a spinless 12-band model and show that when adding spin-orbit coupling, the lowest conduction band hybridizes with a higher-lying conduction band, which cannot be explained by the spinful 10-band model. With the help of L\"owdin's partitioning, we derive the effective low-energy Hamiltonian for the conduction bands for the possible spin dynamics and nanostructure studies and in a similar manner, we give the best fit parameters for the valance-band-only model that can be used in the transport studies. Finally, using the self-consistent 10-band model, we include the effects of a magnetic field and predict the electron and hole g-factor of the conduction and valance bands., Comment: 11 pages, 4 figures, v2: minor corrections

We analyze connections between two low rank modeling approaches from the last decade for treating dynamical data. The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called Direct Bayesian Model Reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of nonnegative matrix factorization, namely the Frobenius norm and the generalized Kullback--Leibler divergence, and suggests new links between likelihood-based and projection-based estimation of probabilistic models.

We generalise the dynamic Laplacian introduced in (Froyland, 2015) to a dynamic $p$-Laplacian, in analogy to the generalisation of the standard $2$-Laplacian to the standard $p$-Laplacian for $p>1$. Spectral properties of the dynamic Laplacian are connected to the geometric problem of finding "coherent" sets with persistently small boundaries under dynamical evolution, and we show that the dynamic $p$-Laplacian shares similar geometric connections. In particular, we prove that the first eigenvalue of the dynamic $p$-Laplacian with Dirichlet boundary conditions exists and converges to a dynamic version of the Cheeger constant introduced in (Froyland, 2015) as $p\rightarrow 1$. We develop a numerical scheme to estimate the leading eigenfunctions of the (nonlinear) dynamic $p$-Laplacian, and through a series of examples we investigate the behaviour of the level sets of these eigenfunctions. These level sets define the boundaries of sets in the domain of the dynamics that remain coherent under the dynamical evolution.

The least squares linear filter, also called the Wiener filter, is a popular tool to predict the next element(s) of time series by linear combination of time-delayed observations. We consider observation sequences of deterministic dynamics, and ask: Which pairs of observation function and dynamics are predictable? If one allows for nonlinear mappings of time-delayed observations, then Takens' well-known theorem implies that a set of pairs, large in a specific topological sense, exists for which an exact prediction is possible. We show that a similar statement applies for the linear least squares filter in the infinite-delay limit, by considering the forecast problem for invertible measure-preserving maps and the Koopman operator on square-integrable functions.

Individuals socio-demographic and economic characteristics crucially shape the spread of an epidemic by largely determining the exposure level to the virus and the severity of the disease for those who got infected. While the complex interplay between individual characteristics and epidemic dynamics is widely recognized, traditional mathematical models often overlook these factors. In this study, we examine two important aspects of human behavior relevant to epidemics: contact patterns and vaccination uptake. Using data collected during the Covid-19 pandemic in Hungary, we first identify the dimensions along which individuals exhibit the greatest variation in their contact patterns and vaccination attitudes. We find that generally privileged groups of the population have higher number of contact and a higher vaccination uptake with respect to disadvantaged groups. Subsequently, we propose a data-driven epidemiological model that incorporates these behavioral differences. Finally, we apply our model to analyze the fourth wave of Covid-19 in Hungary, providing valuable insights into real-world scenarios. By bridging the gap between individual characteristics and epidemic spread, our research contributes to a more comprehensive understanding of disease dynamics and informs effective public health strategies., Comment: 33 pages, 22 figures

In this paper we establish a connection between the bulk topological structure and the magnetic properties of drumhead surface states of nodal loop semimetals. We identify the magnetic characteristics of the surface states and compute the system's magnon spectrum by treating electron-electron interactions on a mean-field level. We draw attention to a subtle connection between a Lifshitz-like transition of the surface states driven by mechanical distortions and the magnetic characteristics of the system. Our findings may be experimentally verified e.g. by spin polarized electron energy loss spectroscopy of nodal semimetal surfaces.

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this work, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barab\'asi model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

We consider linear cocycles taking values in $\textup{SL}_d(\mathbb{R})$ driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is natural in the sense that the base is extended to an associated suspension flow and that the dimension of the cocycle does not change. Further, we refine our general result for the case of (quasi-)periodic driving. As an example, we present a discrete-time cocycle due to Michael Herman. The Furstenberg--Kesten limits of this cocycle do not exist everywhere and its Oseledets splitting is discontinuous. Our results on the continuous-time extension of discrete-time cocycles allow us to construct a continuous-time cocycle with analogous properties.

The dynamics in a confined turbulent convection flow is dominated by multiple long-lived macroscopic circulation states, which are visited subsequently by the system in a Markov-type hopping process. In the present work, we analyze the short transition paths between these subsequent macroscopic system states by a data-driven learning algorithm that extracts the low-dimensional transition manifold and the related new coordinates, which we term collective variables, in the state space of the complex turbulent flow. We therefore transfer and extend concepts for conformation transitions in stochastic microscopic systems, such as in the dynamics of macromolecules, to a deterministic macroscopic flow. Our analysis is based on long-term direct numerical simulation trajectories of turbulent convection in a closed cubic cell at a Prandtl number $Pr = 0.7$ and Rayleigh numbers $Ra = 10^6$ and $10^7$ for a time lag of $10^5$ convective free-fall time units. The simulations resolve vortices and plumes of all physically relevant scales resulting in a state space spanned by more than 3.5 million degrees of freedom. The transition dynamics between the large-scale circulation states can be captured by the transition manifold analysis with only two collective variables which implies a reduction of the data dimension by a factor of more than a million. Our method demonstrates that cessations and subsequent reversals of the large-scale flow are unlikely in the present setup and thus paves the way to the development of efficient reduced-order models of the macroscopic complex nonlinear dynamical system., Comment: 24 pages, 12 Figures, 1 table

Carbon nanotubes can serve as one-dimensional nanoreactors for the in-tube synthesis of various nanostructures. Experimental observations have shown that chains, inner tubes, or nanoribbons can grow by the thermal decomposition of organic/organometallic molecules encapsulated in carbon nanotubes. The result of the process depends on the temperature, the diameter of the nanotube, and the type and amount of material introduced inside the tube. Nanoribbons are particularly promising materials for nanoelectronics. Motivated by recent experimental results observing the formation of carbon nanoribbons inside carbon nanotubes, molecular dynamics calculations were performed with the open source LAMMPS code to investigate the reactions between carbon atoms confined within a single-walled carbon nanotube. Our results show that the interatomic potentials behave differently in quasi-one-dimensional simulations of nanotube-confined space than in three-dimensional simulations. In particular, the Tersoff potential performs better than the widely used Reactive Force Field potential in describing the formation of carbon nanoribbons inside nanotubes. We also found a temperature window where the nanoribbons were formed with the fewest defects, i.e., with the largest flatness and the most hexagons, which is in agreement with the experimental temperature range., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in (J. Chem. Phys. 158, 224304 (2023)) and may be found at (https://doi.org/10.1063/5.0151276)

We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called ``voter model'' on Erd\H{o}s-R\'enyi random graphs, on the stochastic block model, and on random regular graphs. Moreover, a heterogeneous population of agents is studied.

The layered material ZrTe$_5$ appears to exhibit several exotic behaviors which resulted in significant interest recently, although the exact properties are still highly debated. Among these we find a Dirac/Weyl semimetallic behavior, nontrivial spin textures revealed by low temperature transport, and a potential weak or strong topological phase. The anomalous behavior of resistivity has been recently elucidated as originating from band shifting in the electronic structure. Our work examines magnetotransport behavior in ZrTe$_5$ samples in the context of multicarrier transport. The results, in conjunction with ab-initio band structure calculations, indicate that many of the transport features of ZrTe$_5$ across the majority of the temperature range can be adequately explained by the semiclassical multicarrier transport model originating from a complex Fermi surface., Comment: Main Text: 10 pages, 5 figures; Supporting Information: 10 pages, 7 figures

Near-real time estimations of the effective reproduction number are among the most important tools to track the progression of a pandemic and to inform policy makers and the general public. However, these estimations rely on reported case numbers, commonly recorded with significant biases. The epidemic outcome is strongly influenced by the dynamics of social contacts, which are neglected in conventional surveillance systems as their real-time observation is challenging. Here, we propose a concept using online and offline behavioral data, recording age-stratified contact matrices at a daily rate. Modeling the epidemic using the reconstructed matrices we dynamically estimate the effective reproduction number during the two first waves of the COVID-19 pandemic in Hungary. Our results demonstrate how behavioral data can be used to build alternative monitoring systems complementing the established public health surveillance. They can identify and provide better signals during periods when official estimates appear unreliable due to observational biases.

Topological materials host robust properties, unaffected by microscopic perturbations, owing to the global topological properties of the bulk electron system. Materials in which the topological invariant can be changed by easily tuning external parameters are especially sought after. Zirconium pentatelluride (ZrTe$_5$) is one of a few experimentally available materials that reside close to the boundary of a topological phase transition, allowing the switching of its invariant by mechanical strain. Here, we unambiguously identify a topological insulator - metal transition as a function of strain, by a combination of ab initio calculations and direct measurements of the local charge density. Our model quantitatively describes the response to complex strain patterns found in bubbles of few layer ZrTe$_5$ without fitting parameters, reproducing the mechanical deformation dependent closing of the band gap observed using scanning tunneling microscopy. We calculate the topological phase diagram of ZrTe$_5$ and identify the phase at equilibrium, enabling the design of device architectures which exploit the unique topological switching characteristics of the system., Comment: 18 pages, 4 figures

In crystalline solids the interactions of charge and spin can result in a variety of emergent quantum ground states, especially in partially filled, topological flat bands such as Landau levels or in 'magic-angle' bilayer graphene. Much less explored is rhombohedral graphite (RG), perhaps the simplest and structurally most perfect condensed matter system to host a flat band protected by symmetry. By scanning tunneling microscopy we map the flat band charge density of 8, 10 and 17 layers and identify a domain structure emerging from a competition between a sublattice antiferromagnetic insulator and a gapless correlated paramagnet. Our density-matrix renormalization group calculations explain the observed features and demonstrate that the correlations are fundamentally different from graphene based magnetism identified until now, forming the ground state of a quantum magnet. Our work establishes RG as a new platform to study many-body interactions beyond the mean-field approach, where quantum fluctuations and entanglement dominate., Comment: supplementary information included