Stable Graphs Research Articles

Incidence varieties in the projectivized kth Hodge bundle over curves with rational tails

Over the moduli space of pointed smooth algebraic curves, the projectivized [Formula: see text]th Hodge bundle is the space of [Formula: see text]-canonical divisors. The incidence loci are defined by requiring the [Formula: see text]-canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized [Formula: see text]th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails.

Using Foliation Leaves To Extract Reeb Graphs On Surfaces.

For Reeb graph extraction on surfaces, existing methods always use the isolines of a function defined on the surface to detect the surface components and the neighboring relationships between them. Since such detection is unstable, it is still a challenge for the extracted Reeb graphs to stably and concisely encode the topological information of the surface. In this paper, we address this challenge by using foliation leaves to extract Reeb graphs. In particular, we employ a method for generating measured harmonic foliations by defining loops for foliation initialization and diffusing leaves from loops over the surface. We demonstrate that when the loops are determined, the neighboring relationships between the leaves from different loops are fixed. Thus, we can use loops to represent surface components for robustly detecting the interrelationships between surface components. As a result, we are able to extract stable and concise Reeb graphs. We developed novel measures for loop determination and improved foliation generation, and our method allows the user to manually prescribe loops for generating Reeb graphs with desired structures. Therefore, the potential of Reeb graphs for representing surfaces is enhanced, including conveniently representing the symmetries of the surface and ignoring topological noise. This is verified by our experimental results which indicate that our Reeb graphs are compact and expressive, promoting shape analysis.

The stable graph: The metric space scaling limit of a critical random graph with i.i.d. power-law degrees

We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent α+1, where α∈(1,2). The limiting components are constructed from random R-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive α-stable Lévy process. These spanning R-trees are measure-changed α-stable trees. In each such R-tree, we make a random number of vertex identifications, whose locations are determined by an auxiliary Poisson process. This generalises results, which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős–Rényi random graph) and where the role of the α-stable Lévy process is played by a Brownian motion.

MRI biomarkers of freezing of gait development in Parkinson’s disease

This study investigated longitudinal clinical, structural and functional brain alterations in Parkinson’s disease patients with freezing of gait (PD-FoG) and in those developing (PD-FoG-converters) and not developing FoG (PD-non-converters) over two years. Moreover, this study explored if any clinical and/or MRI metric predicts FoG development. Thirty PD-FoG, 11 PD-FoG-converters and 11 PD-non-converters were followed for two years. Thirty healthy controls were included at baseline. Participants underwent clinical and MRI visits. Cortical thickness, basal ganglia volumes and functional network graph metrics were evaluated at baseline and over time. In PD groups, correlations between baseline MRI and clinical worsening were tested. A ROC curve analysis investigated if baseline clinical and MRI measures, selected using a stepwise model procedure, could differentiate PD-FoG-converters from PD-non-converters. At baseline, PD-FoG patients had widespread cortical/subcortical atrophy, while PD-FoG-converters and non-converters showed atrophy in sensorimotor areas and basal ganglia relative to controls. Over time, PD-non-converters accumulated cortical thinning of left temporal pole and pallidum without significant clinical changes. PD-FoG-converters showed worsening of disease severity, executive functions, and mood together with an accumulation of occipital atrophy, similarly to PD-FoG. At baseline, PD-FoG-converters relative to controls and PD-FoG showed higher global and parietal clustering coefficient and global local efficiency. Over time, PD-FoG-converters showed reduced parietal clustering coefficient and sensorimotor local efficiency, PD-non-converters showed increased sensorimotor path length, while PD-FoG patients showed stable graph metrics. Stepwise prediction model including dyskinesia, postural instability and gait disorders scores and parietal clustering coefficient was the best predictor of FoG conversion. Combining clinical and MRI data, ROC curves provided the highest classification power to predict the conversion (AUC = 0.95, 95%CI: 0.86–1). Structural MRI is a useful tool to monitor PD progression, while functional MRI together with clinical features may be helpful to identify FoG conversion early.

Stable graphs: distributions and line-breaking construction

For $\alpha \in (1,2]$, the $\alpha$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given $\alpha$-dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an $\mathbb R$-tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning $\mathbb R$-tree, which is a biased version of the $\alpha$-stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals; we will observe that these distributions are related to the distributions of some configuration models (2) determine the distribution of the $\alpha$-stable graph as a collection of $\alpha$-stable trees glued onto its kernel and (3) present a line-breaking construction, in the same spirit as Aldous' line-breaking construction of the Brownian continuum random tree.

Topological recursion for Masur–Veech volumes

We study the Masur–Veech volumes M V g , n $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g $g$ with n $n$ punctures. We show that the volumes M V g , n $MV_{g,n}$ are the constant terms of a family of polynomials in n $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. We also obtain an expression of the area Siegel–Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur–Veech volumes, and thus of area Siegel–Veech constants, for low g $g$ and n $n$ , which leads us to propose conjectural formulae for low g $g$ but all n $n$ . We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

Isolated Rupture in Composite Networks

Computer networks are prone to targeted attacks and random failures. Robustness is a measure of an ability of a network to continue functioning when part of the network is either naturally damaged or targeted for attack. The study of network robustness is a critical tool in the characterization and understanding of complex interconnected systems. There are several proposed graph metrics that predicates network resilience against such attacks. Isolated rupture degree is a novel graph-theoretic concept defined as a measure of network vulnerability. Isolated rupture degree is argued as an appropriate measure for modelling the robustness of network topologies in the face of possible node destruction. In this paper, the relationships between isolated rupture degree and some other graph parameters such as connectivity, covering number, minimum vertex degree are established. The isolated rupture degrees of [Formula: see text]-free graphs, middle graphs, corona graphs of a middle graph and a complete graph [Formula: see text] on two vertices are evaluated, then compared and the more stable graph types are reported. A sharp upper bound for the isolated rupture degree of middle graphs is established.

Microbial fuel cell (MFC) performance with pigments bikaverin and congo red as electrochemical mediators for optmization power energy

For decades, non-renewable energy resources have been used indiscriminately, but their slow depletion and extremely harmful impacts on the environment have shifted the focus to sustainable and renewable energy sources. Among the renewable energy sources, biofuel cells are defined as devices that convert chemical energy present in chemical bonds into electrical energy. Biocells are classified into two broad categories of enzymatic fuel cells, which employ enzymes as biocatalysts, and microbial fuel cells, which use microorganisms as biocatalysts. An important requirement in the functioning of a biofuel cell is the transfer of electrons from inside an active site of an enzyme to the outside, as the electrodes being solid cannot penetrate the enzymes. A wide range of molecules can be used as electrochemical mediators, some with high toxicity and many non-toxic fungal substances having an enormous potential to be used as electrochemical mediators. In this work, the fungal pigment bikaverin was compared to the synthetic dye Congo red, in order to obtain the best energy-optimizing molecule in an enzymatic fuel cell. Congo red presented a higher current density of 273 mA.cm-2 compared to bikaverin, 230 mA.cm-2, but because it presents a more stable chronoamperometric graph and does not have high toxicity, the fungal biopigment proved to be the best option for optimization. on the potential of energy generated in an enzymatic fuel cell.

Subgraph-aware graph structure revision for spatial–temporal graph modeling

Spatial–temporal graph modeling has been widely studied in many fields, such as traffic forecasting and energy analysis, where data has time and space properties. Existing methods focus on capturing stable and dynamic spatial correlations by constructing physical and virtual graphs along with graph convolution and temporal modeling. However, existing methods tending to smooth node features may obscure the spatial–temporal patterns among nodes. Worse, the graph structure is not always available in some fields, while the manually constructed stable or dynamic graphs cannot necessarily reflect the true spatial correlations either. This paper proposes a Subgraph-Aware Graph Structure Revision network (SAGSR) to overcome these limitations. Architecturally, a subgraph-aware structure revision graph convolution module (SASR-GCM) is designed, which revises the learned stable graph to obtain a dynamic one to automatically infer the dynamics of spatial correlations. Each of these two graphs is separated into one homophilic subgraph and one heterophilic subgraph by a subgraph-aware graph convolution mechanism, which aggregates similar nodes in the homophilic subgraph with positive weights, while keeping nodes with dissimilar features in the heterophilic subgraph mutually away with negative aggregation weights to avoid pattern obfuscation. By combining a gated multi-scale temporal convolution module (GMS-TCM) for temporal modeling, SAGSR can efficiently capture the spatial–temporal correlations and extract complex spatial–temporal graph features. Extensive experiments, conducted on two specific tasks: traffic flow forecasting and energy consumption forecasting, indicate the effectiveness and superiority of our proposed approach over several competitive baselines.

Robust graph fusion and recognition framework for fingerprint and finger‐vein

The human finger is the essential carrier of biometric features. The finger itself contains multi-modal traits, including fingerprint and finger-vein, which provides convenience and practicality for finger bi-modal fusion recognition. The scale inconsistency and feature space mismatch of finger bi-modal images are important reasons for the fusion effect. The feature extraction method based on graph structure can well solve the problem of feature space mismatch for the finger bi-modalities, and the end-to-end fusion recognition can be realised based on graph convolutional neural networks (GCNs). However, this fusion recognition strategy based on GCNs still has two urgent problems: first, lack of stable and efficient graph fusion method; second, over-smoothing problem of GCNs will lead to the degradation of recognition performance. A novel fusion method is proposed to integrate the graph features of fingerprint (FP) and finger-vein (FV). Furthermore, we analyse the inner relationship between the information transmission process and the over-smoothing problem in GCNs from an optimisation perspective, and point out that the differentiated information between neighbouring nodes decreases as the number of layers increases, which is the direct reason for the over-smoothing problem. A modified deep graph convolution neural network is proposed, aiming to alleviate the over-smoothing problem. The intuition is that the differentiated features of the nodes should be properly preserved to ensure the uniqueness of the nodes themselves. Thus, a constraint term to the objective function of the GCN is added to emphasise the differentiation features of the nodes themselves. The experimental results show that the proposed fusion method can achieve more satisfied performance in finger bi-modal biometric recognition, and the proposed constrained GCN can well alleviate the problem of over-smoothing.

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

Adaptive sign algorithm for graph signal processing

Efficient and robust online processing techniques for irregularly structured data are crucial in the current era of data abundance. In this paper, we propose a graph/network version of the classical adaptive Sign algorithm for online graph signal estimation under impulsive noise. The recently introduced graph adaptive least mean squares algorithm is unstable under non-Gaussian impulsive noise and has high computational complexity. The Graph-Sign algorithm proposed in this work is based on the minimum dispersion criterion and therefore impulsive noise does not hinder its estimation quality. Unlike the recently proposed graph adaptive least mean pth power algorithm, our Graph-Sign algorithm can operate without prior knowledge of the noise distribution. The proposed Graph-Sign algorithm has a faster run time because of its low computational complexity compared to the existing adaptive graph signal processing algorithms. Experimenting on steady-state and time-varying graph signals estimation utilizing spectral properties of bandlimitedness and sampling, the Graph-Sign algorithm demonstrates fast, stable, and robust graph signal estimation performance under impulsive noise modeled by alpha stable, Cauchy, Student’s t, or Laplace distributions.

A novel way to formalize stable graph cores by using matching-graphs

The increasing amount of data available and the rate at which it is collected leads to rapid developments of systems for intelligent information processing and pattern recognition. Often the underlying data is inherently complex, making it difficult to represent it by linear, vectorial data structures. This is where graphs offer a versatile alternative for formal data representation. Actually, quite an amount of graph-based methods for pattern recognition has been proposed. A considerable part of these methods rely on graph matching. In the present paper, we propose a novel encoding of specific graph matching information. The basic idea is to formalize the stable cores of individual classes of graphs – discovered during intra-class matchings – by means of so called matching-graphs. We evaluate the benefit of these matching-graphs by researching two classification approaches that rely on this novel data structure. The first approach is a distance based classifier focusing on the matching-graphs during dissimilarity computation. For the second approach, we propose to use sets of matching-graphs to embed input graphs into a vector space. The basic idea is to produce hundreds of matching-graphs first, and then represent each graph g as a vector that shows the occurrence of, or the distance to, each matching-graph. In a thorough experimental evaluation on seven real world data sets we empirically confirm that our novel approaches are able to improve the classification accuracy of systems that rely on comparable information as well as state-of-the-art methods.

Molecular sexing in Bos taurus using quantitative polymerase chain reaction (qPCR) method

Cattle are animals that are widely found all over the world which is the domestication of bulls. In Indonesia, domesticated cattle are used as livestock, could be meat cattle or dairy cattle. The sex of the cattle is very influential in terms of selling, especially the type of bull. Calf bull is a product of the process of artificial insemination with spermatozoa resulting in sexing. Mammals have X and Y chromosomes to distinguish their sex. Each chromosome has a special gene such as the SRY gene on the Y chromosome and the PLP gene on the X chromosome. Molecular methods are effective methods of determining the chromosomes carried by spermatozoa. Both special genes are found in the home box (GADPH) of mammals. Primer is made from the mRNA SRY (sex-determining region Y) sequence from Bos taurus with a size above 200bp (SRY B) and below 200bp (SRY A) so that optimal results are obtained between the two. The type of sample that has been used in this research was sexing semen from limousin bulls that are treated from 4 pairs of primers each. The results of the qPCR from both primers are viewed from the melt-peak graph and comparison analysis. The differences between SRY A and SRY-B primer indicates that SRY-A produces a more stable melt-peak graph than SRY B. This indicates that the SRY A primer is more stable when used for molecular verification processes using qPCR.

Processing-While-Transmitting: Cost-Minimized Transmission in SDN-Based STINs

Existing Space-Terrestrial Integrated Network (STIN) applications collect all data from multiple satellites and terrestrial nodes to the specific analyze center on the earth for processing, which wastes lots of network resources. To save these resources, we propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">processing-while-transmitting</i> pattern in the SDN-based STIN architecture. Through a logically centralized control plane, it cooperatively processes a complex task on appropriate nodes during data transmission. Here, the key point is to jointly determine the transmission path and place subtasks adaptive to data distributions, heterogeneous link costs, task characteristics, the dynamic topology, and network resources. In this paper, we firstly formulate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Transmission-cost-minimized joint Routing and Tasks placement Problem (TRTP)</i> in time-varying STINs. We prove it is NP-hard and has no Polynomial-Time Approximation Scheme (PTAS). To solve the problem, we propose the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Joint Routing and Task Placement (JRTP)</i> algorithm. It first converts the time-varying STIN to a stable graph to cope with the network dynamics, according to the topology and resources during task processing. Then, it jointly decides the routing and task placement through a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">task-topology graph model</i> , which converts the TRTP problem on the stable graph to the classic shortest path problem. We prove that the performance of JRTP is bounded in cases when transmission resources are sufficient and further improve it through the idea of reinforcement. The experimental results show that our processing pattern can significantly decrease the transmission cost and delay, and our algorithms outperform most related ones.

Infinite stable graphs with large chromatic number

We prove that if G = ( V , E ) G=(V,E) is an ω \omega -stable (respectively, superstable) graph with χ ( G ) &gt; ℵ 0 \chi (G)&gt;\aleph _0 (respectively, 2 ℵ 0 2^{\aleph _0} ) then G G contains all the finite subgraphs of the shift graph S h n ( ω ) \mathrm {Sh}_n(\omega ) for some n n . We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if G G is ω \omega -stable with U ⁡ ( G ) ≤ 2 \operatorname {U}(G)\leq 2 we prove that n ≤ 2 n\leq 2 suffices.

NOTES ON THE STABLE REGULARITY LEMMA

AbstractThis is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.

PEEK-Barium sulfate composite for three-dimensional virtual reconstruction of a printed human in vitro model using CT

PurposeThe purpose of this study is to test the concept of a relatively low cost but biocompatible customized surgical guide printing method using a new composite material for the FDM process to support accurate virtual model reconstruction in CT.Design/methodology/approachCurrent additive manufacturing printed surgical guides have problems of scanning artifacts or low computed tomography (CT) values for virtual model reconstruction in CT-assisted surgical operations. These tools always face difficulties in precise positioning due to the effect of human soft tissues and manually made unstable landmarks. To solve this problem, this paper proposes a modified material, polyetheretherketone powder mixed with barium sulfate powder, for printing customized surgical guides with relatively low cost to support a synchronized scanning strategy, for the accurate reconstruction of human tissues and in vitro models.FindingsA set of benchmarking experiments and clinical simulation cases were conducted. The results showed that the proposed solution can be used to print surgical guides to form stable and clear CT graphs for three-dimensional digital model reconstruction. Human tissues and in vitro models can be accurately reconstructed using clear CT graphs without any scanning artifacts or difficulties in image segmentation for virtual model reconstruction, thus facilitating accurate operation guidance and positioning.Originality/valueThis method has wide application potential for printing modular or customized surgical guides with low cost and reusability, especially for surgical operations using CT-assisted navigation systems in underdeveloped regions where medical device costs are a critical issue.

The equivalence between Feynman transform and Verdier duality

The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and the Feynman transform. Specifically, for each twisted modular operad $$\mathcal {P}$$ (taking values in dg-vector spaces over a field k of characteristic 0), there is a certain sheaf $$\mathcal {F}$$ associated with it on the moduli space of stable metric graphs such that the Verdier dual sheaf $$D\mathcal {F}$$ is associated with the Feynman transform $$F\mathcal {P}$$ of $$\mathcal {P}$$ . In the course of the proof, we also prove a relation between cyclic operads and modular operads originally proposed in the pioneering work of Getzler and Kapranov; however, to the best knowledge of the author, no proof has appeared. This geometric interpretation in operad theory is of fundamental importance. We believe this result will illuminate many aspects of the theory of modular operads and find many applications in the future. We illustrate an application of this result, giving another proof on the homotopy properties of the Feynman transform, which is quite intuitive and simpler than the original proof.

ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.