Topology

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.

Accurate road environment modeling is fundamental to the simulation and validation of automated driving systems. However, constructing road maps in standardized formats such as ASAM OpenDRIVE from real-world sensor data remains a time-consuming and costly process. Mobile mapping LiDAR captures accurate lane-level geometry but is confined to the driven corridor, while OpenStreetMap (OSM) provides broad road network topology but lacks geometric precision at the lane level. To address this, an automated workflow is proposed to fuse LiDAR point clouds with OSM data to generate georeferenced ASAM OpenDRIVE maps of highway environments, requiring minimal manual intervention. The pipeline reconstructs mainline roads from LiDAR-derived measurements and infers ramp geometry and topology from the OSM road graph, enabling complete highway interchange modeling without full sensor coverage. Experiments demonstrate a mean lateral RMSE of 0.740 m, and the generated maps are directly usable in mainstream simulation platforms including IPG CarMaker and Esmini. These results validate the effectiveness of combining measurement-derived geometry with map-derived topology for automated OpenDRIVE digital twin generation. The project code is available at https://github.com/ftgTUGraz/opendrive-digital-twin-generator

Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce topological flow matching, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.

Rare events in time series are critical to model but hard to learn due to data scarcity. Current generative models struggle with extreme values. We observe that rare events leave distinct topological fingerprints - transitions in Betti numbers from point-cloud embeddings - that are more stable and discriminative than statistical moments. We introduce PHINN, a flow-matching framework using dynamic Betti curves as conditioning signals and a persistence landscape loss for homology consistency. It scales to multivariate data, includes a natural-language interface to set Betti targets, supports cross-domain meta-learning and few-shot generation, and provides certified adversarial robustness. On financial, epidemiological, and multi-modal benchmarks, PHINN outperforms statistical and diffusion baselines in topological fidelity (beta-RMSE down 41-63%, transition accuracy up 84%) and matches jump-diffusion models in tail coverage while exceeding them in shape fidelity. All results have 95% confidence intervals.

Can a learned model capture how faults propagate through a large-scale network and use this knowledge to causally attribute customer impact to its underlying root cause? Existing root cause analysis techniques often rely on static rules, correlation heuristics, or topology-local reasoning, which struggle to generalize in dynamic environments where faults propagate across complex physical and logical dependencies. We present NetCause, a self-supervised learning-based framework that models network incidents as graph-temporal processes and uses counterfactual simulation to rank candidate root causes. This approach produces an interpretable ranking of root cause hypotheses and integrates naturally with operator-defined mitigation and remediation actions. We train the model on over 1,500 incidents collected over six months from a leading cloud provider's production network and evaluate it on 31 expert-labeled incidents. NetCause consistently improves root cause ranking quality in the regime most relevant to operational decision-making, achieving a 16.1% accuracy improvement over a rule-based heuristic baseline. While training is computationally intensive, inference is lightweight, requiring only seconds of GPU runtime per incident (well below typical telemetry collection latencies).

Recent research is exploring self-organised traffic management as a solution for scaling to complex real-world networks. In such a system, trains predict their neighbourhood, produce traffic plan hypotheses, and agree via consensus with neighbours on a future traffic plan to be implemented. This paper investigates a structural parameter within this pipeline: the predictive neighbourhood horizon. The horizon is used by trains to identify future potential conflicts with neighbours, and to establish the local interaction topology, that is, the subset of trains to negotiate with. As the primary design variable, the horizon directly determines the size and density of the social interaction graph, whereas its impact on the complexity of local sub-problems and the distributed consensus dynamics represents a trade-off to be explored. Through a closed-loop simulation framework the study evaluates how variations of the horizon impact the overall decentralised coordination process, from initial conflict detection to distributed schedule consensus. The analysis focuses on investigating the potential trade-off introduced by the horizon choice: balancing local tractability and computational responsiveness with the need for global schedule coherence and feasibility in safety-critical environments. Contrary to intuition, our empirical results indicate that the short time horizons suffice, while long values compromise local tractability and computational responsiveness with no gain in global schedule optimality.

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value $p = d - b$ is a fully observed time-to-event, and the persistence survival function $S(t) = \mathbb{P}(p > t)$ is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

Large language models (LLMs) have shown impressive performance on diverse reasoning tasks, yet their capacity for structural reasoning in graphs remains unclear. We investigate whether LLMs can genuinely understand graph isomorphism -a fundamental problem in graph theory. While LLMs achieve near-perfect accuracy on isomorphism detection, we show this performance is illusory. When identical graphs are presented with permuted node labels, LLMs fail to identify their isomorphism. This finding suggests that LLMs exploit patterns rather than reasoning about abstract graph structure. Since permutation invariance is a fundamental requirement for valid structural reasoning, these results indicate that success on graph reasoning benchmarks should not be interpreted as evidence of genuine topological understanding.

Multimodal federated graph learning (MM-FGL) aims to collaboratively learn from decentralized graphs with text and images. However, real-world clients may not share a common modality basis: a visual-search client may contain image--interaction graphs but no seller descriptions, while a catalog client may provide text but no product images. We refer to this practical setting as client-level modality deficiency. Unlike random instance-wise missingness, a deficient client lacks the local semantic basis needed to reconstruct the absent modality. More importantly, in graph learning, incomplete representations initialize message passing, so imputation errors can be filtered, mixed, and amplified by the receiving topology. To address this gap, we propose \textbf{PRISM} (\textbf{P}roactive \textbf{R}etrieval and \textbf{I}mputation via \textbf{S}tructural \textbf{M}eta-prompting), a topology-aware federated cross-modal imputation framework. Rather than reconstructing the missing modality solely from local observations, PRISM recovers missing-modality semantics from the federation and introduces them into local graph propagation under topology-aware control. Experiments on six multimodal graph datasets across graph-centric and modality-centric tasks show that PRISM consistently improves modality-deficient clients, outperforming state-of-the-art baselines by \textbf{4.48}% on average.

Decentralized SGD is a fundamental algorithm in decentralized learning, although the influence of an underlying network topology on its convergence behavior is not yet fully understood. Existing convergence analyses have shown that topologies with a small spectral gap significantly deteriorate the convergence rate of Decentralized SGD in both homogeneous and heterogeneous cases. However, many prior papers have reported that indeed the choice of the topology has a significant experimental impact in the heterogeneous case, but has little experimental impact on training behavior in the homogeneous case. In this paper, we present a tighter convergence analysis of Decentralized SGD, offering a more precise understanding of how topologies affect the convergence rate than the prior analysis. Specifically, unlike existing convergence analyses that used only the spectral gap as a property of the topology, our novel analysis shows that all eigenvalues of the mixing matrix affect the convergence rate. Throughout the experiments, we carefully evaluated the convergence behavior of Decentralized SGD and demonstrated that our novel convergence analysis can more accurately describe the effect of topology on the convergence rate.

Multi-agent LLM systems increasingly rely on communication protocols for coordination, yet their robustness under adversarial and structural constraints remains poorly understood. Building on prior work showing that cheap-talk channels enable cooperation in LLM coordination games, we investigate two vulnerability classes in a 4-player Stag Hunt across six model families and 720 trials. First, when Byzantine agents signal cooperation but defect, non-Byzantine agents detect the betrayal within one round yet fail to adapt collectively: a substantial fraction continue cooperating despite repeated exploitation, unable to recover coordination due to the game's unanimity payoff structure. Second, explicitly restricting communication topology collapses cooperation, while applying identical restrictions silently preserves near-perfect cooperation. This establishes that coordination failure stems from agents' meta-reasoning about hidden information, not information loss itself. We identify two stable behavioral archetypes that replicate across all model cohorts: Defection-Prone models that switch permanently after betrayal, and Cooperation-Persistent models that continue cooperating at significant individual cost. These findings reveal concrete security vulnerabilities: communication channels can be exploited as adversarial injection vectors, and disclosing network topology to agents can degrade coordination even without any adversary present.

Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation. We present a constructive mathematical framework that resolves this mismatch for all manifolds that admit a product covering space. These are manifolds expressible as products of elementary factors (circles, intervals, or lines) or as quotients of such products by a finite symmetry group. The class includes cylinders, tori, Möbius strips, Klein bottles, and real projective spaces. Factorized distributions over the elementary factors yield product topologies with closed-form, decoupled KL divergences, so that each latent factor can be shaped independently while keeping training tractable. We catalogue reparametrizable encoder-prior pairs for periodic, bounded, and unbounded supports, and provide coordinate transformations that allow standard neural networks to output non-Euclidean parameters with smooth gradients. For quotient manifolds, the decoder receives group-invariant features of the covering-space coordinates, so that identified points produce identical outputs. Anchor constraints fix the coordinate system relative to the data or create soft topological holes. Experiments on synthetic manifolds and real-image datasets (rotated and cyclically shifted MNIST) confirm that a topology-matched prior aligns KL regularization with the data manifold. The resulting topology-aware models outperform the Gaussian baseline at all practically relevant regularization strengths. The code is available at https://github.com/JvHulst/VAE-Topology.

As malware illustrates a complex structure and behavior, detection of these has been a significant challenge in the domain of cybersecurity along with related services in daily life. So, it becomes crucial to have a reliable and adaptive solution to address the issue. Among the several detection methods developed over the years, one of the most reliable ones is studying and analyzing the structural and behavioral patterns of malware. These patterns of sophisticated malware can be obtained with the help of Function Call Graphs (FCGs). However, to effectively cover numerous groups of families of malware, it is required to have a sufficiently large dataset for the system to operate on. In order to ensure accuracy and robustness of the system, the dataset should comprise samples of different malwares and a benign application for secure execution of the detection process. This paper introduces AMD-FCG, an enhanced Function Call Graph dataset integrated with topological features of malwares. The framework enhances the detection procedure, streamlining the workflow for cybersecurity professionals and also eliminating the need for dynamic analysis and extensive processing. Therefore, it can be used to develop and deploy more efficient and innovative malware detection systems.

Topological Data Analysis (TDA) offers a principled, intrinsic lens for comparing neural representations. However, existing paired topological divergences (e.g., RTD) are limited by heuristic asymmetry and, more critically, unbounded scores that depend on sample size, hindering reliable cross-scenario benchmarking. To address these challenges, we develop a unified topological toolkit serving two complementary needs: fine-grained structural diagnosis and robust, standardized evaluation. First, we complete the RTD framework by introducing Symmetric Representation Topology Divergence (SRTD) and its efficient variant SRTD-lite. Beyond resolving the theoretical asymmetry of prior variants, SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature. This allows for precise localization of structural discrepancies and serves as an effective optimization objective without the overhead of dual directional computations. Second, to enable reliable benchmarking across heterogeneous settings, we propose Normalized Topological Similarity (NTS). By measuring the rank correlation of hierarchical merge orders, NTS yields a scale-invariant metric bounded between -1 and 1, effectively overcoming the scale and sample-dependence of unnormalized divergences. Experiments across synthetic and real-world deep learning settings demonstrate that our toolkit captures functional shifts in CNNs missed by geometric measures and robustly maps LLM genealogy even under distance saturation, offering a rigorous, topology-aware perspective that complements measures like CKA.

In recent years, list replicability has emerged as a framework for formalizing reproducibility in learning theory. A central question is how the required list size relates to the accuracy parameter and natural complexity measures of the hypothesis class. To achieve sharp bounds on list replicability, we prove a novel topological sphere covering theorem, derived from the Borsuk-Ulam theorem. Specifically, if the $d$-sphere is covered by open sets, each of which lies in an open hemisphere, then $d+1$ of these sets must have a common intersection. Using this result, we obtain a sharp bound on the relationship between list size and accuracy for VC classes. We also show that for large-margin half-spaces, provided the margin is not too large, the optimal list size equals the ambient dimension. However, when the margin is taken to be very large, we devise a replicable algorithm achieving the minimal list size of $\lceil d/2 \rceil + 1$.

We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis. Each DNA sequence is encoded along two complementary axes: a $p$-adic distance on $k$-mer prefixes, which captures hierarchical positional structure, and a compositional $L_1$ distance on $k$-mer frequencies, which captures local sequence content. The two distances jointly parameterise a bi-filtered Vietoris--Rips complex, and per-sequence topological summaries from this bi-filtration serve as features for standard machine learning classifiers. We establish theoretical guarantees for the construction: stability under metric perturbations and invariance to the choice of prime, alongside a result that explains why a single $p$-adic axis is topologically uninformative and why the bi-filtration recovers nontrivial homology. On twelve genomic benchmarks ($28$ to $500$ sequences, $3$ to $7$ classes), pVR outperforms four established alignment-free baselines on three of six low-sample datasets, with gains of up to $21$ percentage points; it underperforms only on a SARS-CoV-2 variant benchmark whose point-mutation divergence violates the hierarchical assumption, and all methods saturate in the large-sample regime. pVR also outperforms zero-shot frozen embeddings from the 500M-parameter Nucleotide Transformer v2 by $6.7$ to $11.4$ percentage points on three low-sample benchmarks. The pVR codebase is publicly available at https://github.com/MAHI-Group/pVR.

Medical image segmentation is often framed as a search for stronger architectures, but this can obscure a more fundamental question: what does the dataset require from the model? In medical imaging, this requirement is shaped by foreground occupancy, morphology, boundary ambiguity, topology sensitivity, annotation quality, acquisition variation, and operating point. This paper introduces the Medical Segmentation Dataset Knowledge Card (MS-DKC), a framework for making these factors explicit. MS-DKC records dataset evidence through image/acquisition, morphology, supervision, context-dependence, and deployment-risk descriptors. These descriptors are mapped to failure modes, design priors, and risk-aligned criteria, making segmentation design more traceable than architecture-first comparison. We evaluate MS-DKC on DRIVE, ISIC2018, and ACDC, representing distinct regimes. DRIVE contains sparse, thin, branching vessels, favoring detail-preserving models, sensitivity-aware optimization, threshold analysis, and topology-aware metrics. DKC-TNet-v2 achieved Dice 0.8044 and IoU 0.6730 with 35103 parameters, while SA-UNetv2-DKC-AmbRef reached Dice 0.8141, IoU 0.6865, sensitivity 0.8265, specificity 0.9804, and AUC 0.9853. ISIC2018 involves compact but appearance-variable lesions; validation-constrained score-function selection on Att-Next-Topo/ATTNext produced MS-DKC-AttNextTopo-VCSF-NoAug with Dice 0.8872, IoU 0.8214, precision 0.9173, Boundary F1 0.4878, and ASSD 4.13, while plausible additions failed to improve the risk-aligned profile. ACDC provides a multi-class cardiac case, where MS-DKC recommends four-class softmax segmentation, class-balanced Dice/CE supervision, and class-wise surface evaluation. Overall, the results support dataset-conditioned design: different datasets require different priors, operating points, and evidence before a model can be judged appropriate.

Reliable crack assessment requires not only accurate pixel-level masks but also connected crack geometry and confidence estimates that remain stable under domain shift. However, existing segmentation models can achieve high overlap scores while fragmenting cracks, missing fine branches, and providing no calibrated uncertainty. To address this gap, this paper proposes CrackGeoFM, a multi-task framework that combines a frozen visual foundation backbone with crack-specific adaptation for mask prediction, skeleton reconstruction, and uncertainty estimation. The framework integrates a Frequency-Guided Crack Enhancement Module (FCEM) to enhance high-frequency crack cues, a Crack-Domain Feature Adaptation Module (CFAM) to adapt frozen backbone features to crack-domain patterns, and a Structure-Aware Multi-Task Decoder (SMTD) to jointly decode masks, skeletons, and uncertainty. Across 20 crack datasets, CrackGeoFM achieves state-of-the-art segmentation, improved topology preservation, calibrated uncertainty, and effective few-shot adaptation with only five labeled images. These results support reliable, generalizable, and engineering-oriented crack analysis for infrastructure assessment.

We present Flash Cubical, a highly efficient computation of cubical persistence on a V-filtration for 2D and 3D images over $\mathbb{F}_2$. The implementation is built around three core ideas. First, cubical complexes satisfy properties that allow for the computation of persistence of the highest dimension via union-find and duality. Second, pruning of certain edges allows for a fast and efficient implementation of union-find. Third, the use of a lookup table, which exploits the regularity of cubical complexes to pre-compute local information. This avoids the need to compute local information at run time. To the best of our knowledge, this is the most efficient implementation of cubical persistence with a V-filtration, both in terms of time and memory costs. Although the paper focuses on persistence for V-filtration cubical complexes, the underlying ideas generalise naturally to T-filtrations on cubical complexes and suggest promising directions for other complexes.

How much should an LLM agent remember, and how should multi-agent systems be connected when trying to reach consensus? We show these two design choices interact in a way that flips the sign of memory's effect on coordination. Across 432 simulation runs of a networked Naming Game on eight fixed 16-agent topologies, we vary memory depth and network structure. Longer memory slows the time to reach steady state in decentralized networks but accelerates it in centralized ones; the same parameter pushes the system in opposite directions depending on topology. Critically, "faster settling" in centralized networks means locking in to a fragmented plateau more quickly, not reaching system-wide consensus, which can be used to generate diverging opinions. We further document a memory-mediated speed-unity trade-off: centralized networks consistently preserve more competing conventions than decentralized networks, but their settling speed depends sharply on memory. At the agent level, within-network analyses show that high-betweenness bridges suffer a brokerage penalty while agents in locally clustered neighborhoods achieve higher coordination success. Finally, in search of analytically tractable generative mechanisms, we find that agents' choices are well captured by Fictitious Play, indicating belief-based rather than reward-based adaptation. The practical implication: memory depth and communication topology should be co-designed, not optimized in isolation.

Molecular dynamics (MD) simulations generate trajectories in a high-dimensional configuration space whose analysis critically depends on molecular descriptors, typically handcrafted observables or learned kinetic embeddings. Designing descriptors that are both expressive and broadly applicable, however, remains challenging. We study persistent homology (PH) as a general-purpose representation for MD and introduce the masked Flood complex, a protein-tailored modification of a recently introduced simplicial complex construction that emphasizes inter-residue structure at low computational cost. Vectorized persistence diagrams then provide information-rich, geometry-aware summaries of protein conformations, which we evaluate on protein class prediction, frame-level observable regression, and Markov state model (MSM) estimation from learned low-dimensional coordinates in a single shared representation space. Results on the mdCATH dataset show that PH-based descriptors are competitive across tasks, with masked Flood PH yielding the most consistent overall performance. Further, when using topologically-informed MSMs as a drop-in replacement within the recent MarS-FM framework for generative modeling of protein conformations, we obtain consistently better ensemble statistics than MSMs based on physical observables. Finally, we explore the transferability of the generative model to qualitatively different, fast folding, proteins.

Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.

This study introduces an automated deep learning framework for predicting brain injury (BI) in preterm infants from T2-weighted MRI (dHCP dataset). We propose GloResNet, a lightweight 3D CNN based on ResNet-10, pretrained on MedicalNet to address data scarcity. A global manifold mapping strategy first resamples each 3D volume to 128x128x128 and then applies subject-wise z-score intensity normalization, thereby preserving global topology while standardizing appearance. Training integrates mixup, class weighting, and test-time augmentation for robustness. In 5-fold cross-validation, GloResNet achieved 75.18% average accuracy (peak 81.82%), with specificity 0.81 and sensitivity 0.76. Results demonstrate that a topology-aware lightweight CNN has the capability to effectively predict neonatal BI, offering a non-invasive screening tool. The source code of this paper can be obtained from the GitHub repository: https://github.com/ICL-SUST/GloResNet-Preterm-Brain

We propose a novel computational toolbox that integrates Topological Data Analysis (TDA), Differential Box Counting (DBC), Multifractal Partition (MFP), and Local Binary Patterns (LBP), applied to time-lapse super-resolution STED microscopy images of sodium caseinate gelation induced by glucono-delta-lactone (GDL) at 30 °C and 40 °C and two GDL concentrations (1.8% and 3.5% w/v). TDA tracked topological loops, closed ring-like structures reflecting protein network interconnectivity, via max-Betti-1 curves, which revealed a lag phase of dispersed aggregates, a sharp decay coinciding with network percolation and the rheologically observed sol-gel transition, and a post-gelation increase corresponding to network rearrangements. These topological transitions were corroborated by DBC and MFP as these methods were able to resolve changes in structural complexity and spatial heterogeneity. The toolbox was validated on simulated fractal images prior to experimental application. Together, these descriptors provided sensitivity to subtle microstructural transitions that bulk rheology captured as averaged bulk mechanical responses. This integrated approach provides a robust quantitative tool for characterizing complex microstructure in food and material science with evolving microstructural dynamics. Code is available at https://github.com/Zahratabatabaei/Delifood_CV_paper.git

Large language model-driven multi-agent systems enhance the reliability of complex reasoning tasks through multi-round deliberation, role specialization, and cross-validation. However, existing multi-agent debate and collaboration frameworks typically adopt fully connected communication, causing the number of messages, token costs, and end-to-end latency to grow approximately quadratically with the number of agents; although fixed sparse topologies reduce overhead, they cannot adapt communication relationships to different task instances or intermediate reasoning states, making them prone either to preserving low-value interactions or to losing critical error-correction information. To address this problem, this paper proposes DySCo (Dynamic Sparse Consensus), a dynamic trust-aware sparse consensus mechanism. In each round of reasoning, DySCo estimates the value of communication edges based on agent reliability, answer divergence, and task relevance, and selects a small number of high-value edges for message exchange under budget constraints; it then aggregates the answers of different agents through dynamic trust weights and terminates the discussion early once consensus stabilizes. This mechanism replaces universal broadcasting with on-demand communication, thereby reducing communication overhead while preserving essential cross-validation information. We further present analyses of communication complexity and consensus stability, and evaluate the performance of DySCo on mathematical reasoning, logical reasoning, and factual question-answering tasks.

Embodied visual navigation, where an agent perceives a complex environment and acts to reach a goal from raw sensory input, underpins a wide range of applications such as household service robotics, assistive robotics, and large-scale autonomous exploration. However, recent attempts to unify vision-and-language navigation (VLN) and object goal navigation (ObjNav) remain at the level of architectural fusion, mixed-task training, and large vision-language pretraining, without examining whether independently trained vision and language encoders may already share a common semantic structure. Moreover, even object-centric topological maps still ground language goals through explicit cross-modal supervision such as CLIP or large vision-language models, leaving open whether such grounding is possible from a purely vision-built map. To address these challenges, we extend the Platonic Representation Hypothesis to embodied navigation and recast vision-only ObjNav, cross-modal ObjNav, and VLN as three different interfaces to the same object-centric semantic manifold. We further introduce PlatonicNav, a training-free framework whose Platonic Topological Map fuses geometric and semantic node distances from a self-supervised visual encoder, and grounds language goals via blind matching without any paired vision-language data. Extensive experiments on simulation benchmarks including HM3D-IIN, OVON, and R2R-CE on MP3D, together with deployment on Unitree Go2, demonstrate that PlatonicNav generalizes across tasks, modalities, and embodiments without explicit cross-modal training. Code: https://github.com/AIGeeksGroup/PlatonicNav. Website: https://aigeeksgroup.github.io/PlatonicNav.

Accurate brain tumor classification from magnetic resonance imaging (MRI) is a key requirement for early diagnosis and clinical decision-making. Vision Transformers (ViTs) have shown strong performance in medical image analysis by learning global contextual representations, but they often fail to capture intrinsic structural and topological patterns present in tumor regions. To address this limitation, we propose a fusion framework that combines Topological Data Analysis (TDA) features with pretrained Vision Transformer representations for four-class brain tumor classification. In the proposed method, TDA is used to extract complementary topological descriptors that capture geometric structure, connectivity, and shape information from MRI images. In parallel, a pretrained ViT model learns high-level semantic representations from the same images. These two feature spaces are then fused to form a unified and more discriminative representation for classification. The model is evaluated on the BRISC2025 dataset, which contains four brain tumor classes: glioma, meningioma, pituitary tumor, and non-tumor cases. Experimental results show that combining topological and transformer-based features significantly improves performance compared to using either approach alone. The proposed TDA-ViT fusion model achieves an accuracy of 99.10%, precision of 99.27%, recall of 99.15%, F1-score of 99.21%, and an AUC of 99.98%. It also outperforms several state-of-the-art models, including ResNet50, ResNet101, EfficientNetB2, and standalone Vision Transformers. These results demonstrate that topological features provide valuable complementary information that enhances deep representation learning, leading to a robust and highly accurate framework for automated brain tumor classification.

Transfer learning presumes that a representation learned on source tasks carries structure that remains usable on related target tasks. Standard evaluations probe this through target accuracy or distributional discrepancy, yet leave unspecified which structural invariant is meant to transfer. We supply that invariant categorically. A source task category $\mathcal A$, a target task category $\mathcal B$, and a task-change functor $J:\mathcal A\to\mathcal B$ determine, for every invariant-valued source representation $F:\mathcal A\to\mathcal V$, the universal transferred invariant $\operatorname{Lan}J F$. Given a target invariant $G:\mathcal B\to\mathcal V$, we define the transfer discrepancy $\operatorname{Comp}J(F,G)=\sup{b\in\operatorname{Ob}(\mathcal B)} d{\mathcal V}\bigl((\operatorname{Lan}_J F)(b),G(b)\bigr)$, evaluating transfer not by an objectwise comparison of source and target, but by comparing the target invariant against the one forced by the prescribed task transformation. We prove finite cokernel formulas for $(\operatorname{Lan}_J F)(b)$ in chain complexes and persistence modules, indexed by the comma category $J\downarrow b$. For persistence-valued finite-type one-parameter invariants, the discrepancy is computed exactly by bottleneck distances between barcodes. Controlled experiments on neural latent point clouds then test whether the score recovers the correct task functor and flags representation collapses that preserve classification accuracy while destroying transfer-relevant topology.

Integrating dynamic 3D reconstructions into physics simulation requires fixed mesh topology for efficient collision detection, but state-of-the-art methods like DG-Mesh produce varying topology optimized for geometric quality. We investigate whether topology conversion can enable physics integration while preserving reconstruction fidelity. We propose a dual-representation framework combining fixed-topology meshes for physics with Gaussian splatting for rendering, achieving 4.65$\times$ speedup over varying-topology baselines through runtime vertex buffer updates. We evaluate two conversion strategies, temporal correspondence tracking and template-based projection, against native fixed-topology methods (MaGS) on the DG-Mesh dataset. Our evaluation reveals that both conversion approaches incur 65-80% geometric degradation, producing results inferior to MaGS despite DG-Mesh's superior initial quality. This demonstrates that high-quality reconstruction and physics-compatible topology represent fundamentally distinct objectives that cannot be reconciled through post-processing. Our findings inform future development of physics-aware reconstruction methods and our framework enables real-time simulation with any fixed-topology approach.

State abstraction in reinforcement learning is usually formulated as a partition of states based on reward and transition similarity. This excludes a common structural pattern in navigation, graph, and hierarchical decision problems: interface states such as doors, hubs, and bottlenecks naturally participate in more than one region. We introduce \emph{tangle-core abstraction}, an overlapping state-abstraction framework based on graph tangles of empirical transition graphs. The method constructs abstract states from consistently oriented low-order separations and represents shared interfaces through a membership kernel rather than a hard partition. We give value-preservation guarantees for the induced overlapping abstract MDP under an explicit action-consistency condition, identify an interior-homogeneity/boundary-leakage error decomposition, and prove a quantitative interface-overlap result showing when hard partitions incur an avoidable boundary error. Empirically, tangle-core abstractions achieve favorable compression--return tradeoffs against reward-aware, learned, topological-map, and graph-partitioning baselines across bottlenecked tabular domains, procedurally generated mazes, and MiniGrid representations. We also identify a clear failure regime in which transition topology is uninformative, where tangles predictably offer little benefit. These results position graph tangles as an effective topology-aware abstraction prior for decision problems with shared interface structure.

Large Language Model (LLM)-based navigation systems commonly construct explicit spatial representations (e.g., topological graphs, semantic raster maps) and translate them into textual descriptions as LLMs' inputs. However, the linguistic structures of such text-based spatial representations and the choices of contextual features (e.g., topology, geometry) they contain are often treated as neutral engineering decisions rather than key factors that shape LLMs' behavior. To fill the gap, we propose a dual-interventional framework that disentangles linguistic structures from different contextual cues to evaluate the linguistic inductive bias of LLMs for navigation planning. In the framework, representation intervention varies the linguistic format and the degree of linguistic compression, clarifying when linguistic representations support or inhibit navigation planning. Context intervention, combined with contextual feature combination and conflict probing, explicitly clarifies the preferences and weaknesses of LLMs when processing different contextual cues. Experiments across diverse spatial reasoning tasks and multiple model scales reveal a consistent pattern: topological information is a sturdy shield and the backbone of robust planning; linguistic format is a double-edged sword whose effect depends on model size, task demands, and the compression level; and semantic information is a fatal Achilles' heel -- incorrect semantic cues can systematically derail the planning process. Overall, our study shows that effective text-based spatial representations in LLM-based navigation should preserve topological integrity, calibrate representational compression to model capacity, and ensure semantic correctness, rather than simply adopting a single representation. Our code is publicly available at https://github.com/jonesdong150/LLM-Navigation-Inductive-Bias.

Ordinal Regression (OR) aims to predict target values with inherent order, underpinning critical applications across diverse domains, from recommender systems to computer vision. Though having evolved from naive regression to discretization-based classification and generation, existing paradigms remain fundamentally constrained by quantization artifacts and the lack of global ordinal topological perception. These methods typically enforce rigid boundary delineations, failing to capture the non-stationary semantic transitions inherent to ordinal data. In this paper, we propose a novel paradigm where OR is formulated as a Continuous Generative Ordinal Regression task. Under the novel paradigm, we introduce DiffOR, a unified framework that leverages diffusion models to recover continuous ordinal values via iterative denoising, thereby enabling the dynamic learning of soft semantic transitions. To explicitly preserve ordinal topology, we devise a Dual-Decoupling Strategy: Spatially, Multi-scale Increment Aggregation decomposes targets into hierarchical continuous increments; Temporally, Dynamic Denoising Perception synchronizes denoising steps with feature frequencies, ensuring robust coarse-to-fine refinement. Theoretically, we show that the proposed method can significantly enhance both representation capability and mechanistic interpretability. Extensive experiments on 12 benchmarks across four domains validate DiffOR's consistent superiority over state-of-the-art methods, establishing a new standard that demonstrates strong potential as a general-purpose solution for universal ordinal regression.

We propose a topological framework for comparing trained Graph Neural Networks (GNNs) by mapping the Stochastic Block Models (SBMs) induced on the graphon-signal space of a Message Passing Neural Network (MPNN) onto the unit $n$-sphere $\sphere^{n-1}\subset\R^n$. The construction rests on three classical pillars: the \emph{compactness} of the cut-distance graphon space $(\Wo,\cutdist)$ \citep{lovasz2006limits,lovasz2012large}, the Frieze--Kannan \emph{weak regularity lemma} together with its graphon-signal extension due to \citet{levie2023graphon}, and the Lipschitz continuity of MPNNs with respect to the cut-distance. We show that, for any prescribed tolerance $\varepsilon>0$, a trained MPNN $Φ$ acting on a sufficiently large graph factors (up to $\varepsilon$) through a step-graphon-signal of bounded complexity, and we construct an explicit measure-preserving map $Ψ_n\colon[0,1]\to\sphere^{n-1}$ that places the SBM regions on disjoint spherical caps. This produces a problem-agnostic, low-dimensional ``fingerprint'' of a trained GNN that is amenable to visual inspection and to nearest-neighbour search across model zoos, enabling \emph{transfer-learning candidate retrieval} without retraining. We discuss the obstruction posed by concentration of measure in high dimension -- a phenomenon directly relevant to LLM-scale embeddings. We close with five concrete future research directions: hyperbolic and Grassmannian alternatives to the spherical model, Gromov--Wasserstein distances on graphon-signals as an isometry-free alternative to the $n$-sphere map, an information-geometric (Fisher) reformulation of the SBM manifold, persistent-homology fingerprints of layer-wise embedding clouds, and a spectral-distance baseline derived from the graphon eigendecomposition.

Generating realistic graph-structured data is challenging due to discrete connectivity, varying graph sizes, and class-specific structural patterns. Recent Generative Adversarial Networks (GAN)-based graph generation methods improve edge modelling by learning connectivity and matching class-specific density distributions. However these models still exhibit noticeable deviations such as in degree and spectral distribution when compared to real graphs, indicating that important structural properties are not fully preserved. This work aims to reduce these deviations by refining the graphs produced by an existing GAN-based graph generator framework with a Genetic Algorithm (GA). In the GAN framework, the generator produces both node features and connectivity patterns, while a GNN-based critic evaluates graph realism and class consistency to ensure global structural and class alignment. Building on this foundation, we apply a GA to refine the edges of generated graphs. The refinement process guides synthetic graphs toward closer agreement with real data, while preserving diversity and novelty. Experimental results show that the GA refinement consistently lowers combined Maximum Mean Discrepancy (MMD) compared to the base model, leading to graphs that more closely match real structural patterns. This demonstrates that evolutionary refinement is an effective and flexible way to correct residual structural deviations in GAN-based graph generators, improving their suitability for realistic graph synthesis and data augmentation.

Mobile robot path planning methods are often constrained by vast search spaces, resulting in latency in samplingbased algorithms. Learning-based approaches frequently suffer from local region fragmentation and global topological inconsistency. To tackle the problem, we present the Connectivity- Preserving Region Proposal Network (CP-RPN), a segmentationguided model designed to predict compact and topologically connected candidate regions, significantly compressing the search space. Specifically, we design a segmentation model that leverages a Deformable Attention Transformer (DAT) to capture long-range dependencies for global connectivity, with a Deconvolutional decoder to preserve fine-grained spatial details. To guarantee the connectivity of the predicted mask, we design a composite loss function that combines Cross-Entropy loss for pixelwise supervision, a Connectivity-Aware loss to enhance local coherence, and a Topological Continuity loss based on persistent homology to enforce global connectivity. Building on these highconnectivity corridor-like regions, the Voronoi diagram is used to plan the path, backed by a local A* fallback mechanism to ensure robustness. Experimental results demonstrate that CPRPN reduces the candidate region size by over 60.13% compared to the MPT baseline and achieves deterministic low-latency planning (avg. 0.11s) with a 99.60% success rate, outperforming traditional sampling-based algorithms in stability.

Flowcharts are widely used in industrial requirements, but usually remain embedded as static images. Vision Language Models (VLMs) show promise in the conversion of these flowcharts into machine-readable models for RE activities, yet, when directly applied to flowchart conversion, they often fail on topology-critical visual details. To address this, we propose EdgeFlow that augments a VLM's original input with a deterministically extracted Canny edge map-acting as a structural prior-to improve flowchart-to-Mermaid conversion, without requiring annotated training data or domain-specific model fine-tuning. We evaluate EdgeFlow on IndusReqFlow, a dataset sourced from real-world requirements. Compared with off-the-shelf VLMs, EdgeFlow improves node-level F1 by 17.39 percentage points and edge-level F1 by 16.94 percentage points. At the path level, EdgeFlow improves path F1 by 11.06 percentage points, enabling better support for model-based testing. These results demonstrate that EdgeFlow provides a practical, training-free means to improve topology-preserving flowchart-to-Mermaid conversion for industrial RE. Cross-dataset evaluation results on a public synthetic benchmark show no significant improvement; this highlights the need for diverse benchmarks incorporating industrial data for the comprehensive evaluation of future VLM-based RE tools.

Advancements in 3D curve skeletonization are accelerating progress across a wide range of applications. However, developing robust skeletonization algorithms that capture intricate object details remains challenging. Skeletonization via Local Separators (LS) offers an efficient graph-based approach but suffers from representation inaccuracies due to its discrete nature. To address this, we introduce CSCD, a novel framework for Curve Skeletonization in the Continuous Domain, generalizing LS to manifolds. Specifically, we present two realizations: CSCD-M for meshes and CSCD-PC for point clouds. CSCD-M leverages the intrinsic triangulation of a mesh for resilience to noise and improved topological preservation, while CSCD-PC employs tufted Laplacians for enhanced robustness. To our knowledge, CSCD-M is the first intrinsic method for curve skeletonization. Our results show CSCD-M matches LS performance across diverse meshes and outperforms LS (TOG'21) on benchmarks like Thingi10k dataset. CSCD-PC qualitatively outperforms CoverageAxis++ (Eurographics'24) and EPCS (CAG'23). Finally, we demonstrate the efficacy of CSCD in a few downstream tasks: object classification, shape segmentation, identifying handles, tunnels, and constrictions in objects. Project Website: https://cscd-skel.pages.dev

Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different models, layers, or training conditions produce similar representations for the same inputs, with important implications for model interpretation, selection, and robustness analysis. Existing approaches to measure alignment primarily rely on geometric properties, such as neighborhood and cluster similarity, offering limited insight into the global organization of representations. In this work, we present TopoAlign, a topology-aware framework for visually comparing model representations from a structural perspective. Leveraging mapper graphs from topological data analysis, TopoAlign jointly analyzes graphs constructed from representations of shared inputs across different models or layers. The framework supports a top-down comparative workflow: it first performs global structure alignment via joint force-directed optimization to produce coordinated graph layouts; it then identifies local correspondences through automated detection of structurally matching regions, visualized with Bubble Sets; and finally it enables fine-grained pattern inspection through motif-based queries and membrane-inspired visualizations. We demonstrate TopoAlign through case studies on language and multimodal models, complemented by expert feedback. Our results show that TopoAlign provides meaningful insights into representation structure and alignment from a topological perspective.

Vascular circulation follows fundamental biophysical principles that optimize mass transport and metabolic energy expenditure, which can be effectively modeled by Murray's law. However, contemporary deep learning methods for vascular segmentation often neglect these biophysical constraints. This leads to physiologically implausible branching and misclassification vascular trees, rendering. These automated segmentation results are unreliable unreliable for downstream clinical tasks such as blood flow simulation or disease quantification. In this paper, we introduce MARVEL (Universal MurrAy's law-infoRmed Vessel sEgmentation and topoLogy estimation), a backbone-agnostic framework that integrates biophysical priors into vascular tree extraction. MARVEL combines per-pixel supervision with explicit radius predictions to enforce local bifurcation constraints derived from an empirical width-exponent mapping. We implement these constraints as differentiable regularizers during training to guide models toward physiologically consistent reconstructions. We evaluate MARVEL on eight public datasets across multiple vascular modalities and segmentation backbones. Results demonstrate MARVEL's superior performance in segmentation accuracy, topological consistency, and physiological plausibility. By converting segmented masks into graph-based hemodynamic simulations, we demonstrate that MARVEL preserves the subtle pathological narrowing and topological connectivity required to distinguish hypertensive from normotensive eyes. Results show that MARVEL significantly improves the classification of hypertension via arteriovenous pressure differences in the eye (p < 0.001), outperforming baseline models in both topological consistency and clinical predictive value.

Robotic manipulation of deformable linear objects (DLOs) presents significant challenges due to complex dynamics and frequent self-occlusions. Existing robotic knot tying methods typically rely on precise topological state tracking with ordered keypoints and explicit edge connectivity. This reliance makes them prone to failures due to tracking drift and topology mismatch caused by repeated bending and crossings during knot formation.To address these limitations, we introduce RoboHitch, a novel framework that learns to perform hitch knot tying from human demonstrations using only disordered 3D keypoints and RGB images. This eliminates the need for explicit topological order, allowing for more flexible manipulation. Our method employs a dynamic Graph Autoencoder to extract geometric features from untracked keypoints, complemented by a Convolutional Autoencoder that captures essential visual context. A bidirectional cross-attention mechanism then fuses these modalities to jointly predict pick and place affordances, facilitating implicit reasoning about the rope's state and enabling knot tying under occlusion.Real-world experiments demonstrate the effectiveness and generalizability of our approach, successfully completing hitch knots in scenarios with self-occlusions.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

AI-native wireless receivers based on deep learning exhibit remarkable performance under stationary channel conditions, yet their resilience to distributional shifts remains poorly characterized by conventional metrics such as bit error rate (BER). To overcome these limitations, this paper proposes a novel real-time metric, the Topological Resilience Index (TRI), grounded in persistent homology and persistence exponents. TRI quantifies the structural stability of a neural network receiver's parameter space during online adaptation to non-stationary channels. Specifically, TRI captures resilience through three complementary dimensions: (i) validation-loss resilience measuring model-channel mismatch, grounded in the topological persistence of loss-landscape sublevel sets; (ii) channel impulse response (CIR) distribution shift, tracking geometric drift of CIR vectors from the calibration reference distribution; and (iii) channel manifold topology, quantified by the spectral gap of the Gaussian kernel matrix normalized by the Olivier-Ricci curvature norm. We establish theoretical guarantees showing that TRI is bounded, monotonic under performance degradation, and Lipschitz-stable with respect to perturbations in channel distributions measured in Wasserstein distance. Simulation results for an OFDM deep-learning receiver adapting across ten ITU-R inter-environment transitions at three shift rates demonstrate that TRI provides a consistent mean warning lead of more than one OFDM symbol over gradient-norm and validation-loss baselines, whereas the gradient-norm baseline achieves zero lead in every scenario. Furthermore, the proposed TRI-guided burst re-adaptation reduces post-shift BER by 80% relative to no adaptation within 200 OFDM symbols.

Phylogenetic trees are hybrid objects: branch lengths vary continuously, while topologies change discretely through edge contractions and expansions. Billera-Holmes-Vogtmann (BHV) tree space provides a canonical geometry for this structure, representing each resolved topology as a Euclidean orthant and topological changes as motion across shared lower-dimensional boundaries. We introduce PhylaFlow, a hybrid flow-matching model that learns posterior-basin transport in BHV tree space. PhylaFlow is trained on BHV geodesic paths from random starting trees to short-run posterior samples, coupling continuous branch-length motion within orthants with learned boundary events and discrete topology transitions. We evaluate the learned geometry operationally: if the flow reaches posterior-relevant regions, finite-budget Bayesian refinement initialized from, or guided by, its terminal trees should recover posterior-supported topologies more efficiently. Across DS1-DS8 phylogenetic posterior benchmarks, PhylaFlow substantially reduces initial Tree-KL relative to classical initializers. After finite-budget MrBayes refinement, direct PhylaFlow improves early and intermediate topology-recovery trajectories on most datasets, while split-guided PhylaFlow-MCMC obtains the strongest hard-case results. The best PhylaFlow variant outperforms short-warmup on seven of eight datasets and PhyloGFN on five of eight under the same refinement budget. In a joint sequence-conditioned experiment, sequence embeddings steer posterior split recovery, although exact posterior topology recovery remains preliminary. These results show that hybrid flow matching can learn actionable transport in BHV tree space and provide a geometry-aware proposal mechanism for Bayesian phylogenetic inference.

The exponential growth of global academic output has confronted researchers and AI agents with an unprecedented information explosion,'' where fragmented and unstructured knowledge organization impedes deep interdisciplinary integration. Current academic retrieval tools predominantly rely on superficial keyword matching or vector-space semantic retrieval, which lack the topological reasoning capabilities required to navigate complex logical connections. Agentic deep-research-based frameworks are often prone to logical hallucinations and consuming high inference costs. To bridge this gap, in this report, we introduce SciAtlas, a large-scale, multi-disciplinary, heterogeneous academic resource knowledge graph designed as a panoramic scientific evolution network. By integrating over 43M papers from 26 disciplines, and a total of 157M entities and 3B triplets, SciAtlas provides a structured topological cognitive substrate that dismantles disciplinary barriers and furnishes AI agents with a global perspective. Furthermore, we develop a neuro-symbolic retrieval algorithm featuring tri-path collaborative recall and graph reranking, achieving a seamless transition from simple semantic matching to deterministic association discovery. We also present key application directions of SciAtlas, including literature review, automated research trend synthesis, idea positioning, and academic trajectory exploration, to demonstrate that SciAtlas can serve as an effective cognitive map'' to empower the full loop of automated scientific research while significantly reducing reasoning costs. We have released the interfaces for KG retrieval and various downstream tasks in our GitHub repo.

AI agents deployed into SRE workflows currently derive their understanding of environment state from raw observability telemetry at query time, paying a semantic-interpretation tax in tokens, latency, and inferential reliability. We propose Causely, a causal intelligence layer that maintains a structured representation of environment topology, attribute dependencies, and causal relationships that are anchroed to a ontological representation of the managed environment. Causely transforms raw telemetry into a live, queryable model providing the semantic and causal foundation AI agents require to diagnose, evaluate impact, and act safely in production. We evaluate this value proposition through a benchmark study conducted in a controlled setting with injected faults in a 24-microservice OpenTelemetry demo application. Our experiments compare four agent configurations (Claude Code, OpenAI Codex, HolmesGPT with Sonnet and Gemini backends). Experiments are run with and without access to Causely under two scenarios: an active incident and a healthy baseline. On the active-fault scenario, causal grounding reduces mean time-to-diagnosis by 63%, mean token consumption by 60%, and mean tool-call count by 78%, compressing the investigation footprint by 4.8$\times$ and lowering direct API cost per run by 57%; root-cause-diagnosis accuracy rises from 75% to 100%.

We introduce \textbf{Prism}, a framework for structural symmetry diagnosis in complex networks. Given a graph Laplacian $L$ and a duality operator $P$ (a symmetric involution), Prism computes the \emph{duality defect} $δ(L,P) = \|LP - PL\|_F / \|L\|_F$ -- a scalar measuring how far the network deviates from structural self-consistency. When $P$ encodes the network's true symmetry, $δ$ starts near zero and rises monotonically as structure degrades; an arbitrary $P$ gives noise. We prove that the optimal $L'$ satisfying $[L', P] = 0$ is given by a closed-form block-diagonal projection, and provide an unsupervised alternating optimization that learns $P$ from the graph's own Fiedler vector. Experiments on synthetic networks show the true-$P$ defect is $3.38\times$ more sensitive to structural degradation than an index-reversal baseline and more sensitive than modularity. On Zachary's Karate Club with edge noise, Prism achieves $94.5\%$ community detection accuracy at $5\%$ noise versus $76.6\%$ for the raw Laplacian baseline. Applied to live S&P~500 data (2026-05-17), Prism detects rising structural stress (defect $0.43 \to 0.73$ over 90 days) while surface correlations remain low -- a signal invisible to correlation-based methods. In a historical backtest spanning five major stress events (2011--2020), the duality defect exhibits a consistent pattern: it reaches elevated levels \emph{before} the correlation spike that accompanies each crisis, and sustains high readings during periods of structural fragility that conventional metrics classify as calm. The duality defect is a first-principles structural admissibility condition, requiring no training data and computable in milliseconds.

Flow in porous media is difficult to address using standard analytical or numerical methods due to its complexity. However, since synthetic representations of porous media are easy to produce and data from physical experiments are becoming more widely available, the problem is well-suited to studies that include machine learning (ML) techniques. We discuss a number of features that can be extracted from such data, and their utility as input variables into a standard ML algorithm. These features include structural measures describing the geometry of the porous media, topological measures describing the connectivity, and network measures obtained by modeling the porous media as simplified pore networks. These features enable the prediction of the permeability of the considered (synthetic) porous materials using ML techniques that also leverage the separately computed exact permeability (ground truth). Comparing results obtained using different input variables helps develop a better understanding of the utility of various measures for predicting permeability based on the porous media structure. We show, in particular, that topological data analysis (TDA) provides a useful set of features that can be easily combined with ML to yield meaningful results.

Current topology aware diffusion models face an architectural mismatch by using Gaussian noise for corruption while recovering structural features through conditional side channels To fix this we introduce PFlow T a generative model that bases its forward process entirely on persistent homology In PFlow T time measures the destruction of H1 topological features like holes rather than Gaussian noise injection This forward process eliminates features based on their persistence The reverse network then directly inverts this structured corruption to predict the clean state in one step Tests on MNIST digits zero one and eight show PFlow T significantly outperforms a baseline model in generating requested Betti numbers and handling out of distribution tasks PFlow T is the first generative architecture using persistent homology for the forward process although we note it is currently limited to low resolution pixel space proxies

Coordination graphs are a central abstraction in cooperative multi-agent reinforcement learning (MARL), yet existing sparse-graph learners lack a theoretically grounded mechanism to decide which edges should exist and how much information each edge should carry. Current methods rely on heuristic criteria that offer no formal guarantee on the learned topology, and no principled way to allocate different communication capacities to structurally different agent relationships. To address this, we propose Heterogeneous Information-Bottleneck Coordination Graphs (HIBCG), which learns a group-aware sparse graph in which both edge existence and message capacity are theoretically justified. With the graph information bottleneck (GIB) serving as the underlying tool, HIBCG first constructs a group-aligned block-diagonal prior that provides a closed-form criterion for edge retention -- determining which edges should exist and at what density per group block -- and then controls per-agent feature bandwidth on the resulting topology, compressing messages to retain only task-relevant content. We prove that the group-aligned prior strictly tightens the variational bound on topology learning, that the objective decomposes per group block, enabling differential edge control, and that capacity allocation follows a water-filling principle.

Dimensionality reduction algorithms map high-dimensional data into visualizable 2D or 3D spaces, but traditionally rely on a discrete point-cloud paradigm. This discrete abstraction is susceptible to visual occlusion and artificial discontinuities, often failing to represent the continuous density of the underlying manifold. To address these limitations, we introduce Topo-GS, a framework that repurposes 3D Gaussian Splatting (3DGS) to cast multidimensional projection as a meshless volumetric reconstruction process. Instead of standard photometric losses, Topo-GS is driven by local geometric constraints. By solving orthogonal Procrustes targets, the optimization enforces an As-Rigid-As-Possible prior while explicitly aligning the spatial covariance of each Gaussian to the local tangent space. Recognizing that unrolling data of varying intrinsic dimensionalities requires distinct spatial treatments, we utilize a topology-aware strategy that tailors the loss formulation to preserve either continuous 1D trajectories or cohesive 2D surfaces. Quantitative and visual evaluations demonstrate that Topo-GS successfully transforms discrete scatter plots into continuous volumetric representations, where inherent projection distortions explicitly manifest as observable geometric variations, while preserving local topological fidelity comparable to discrete baselines.

Recent advances in Diffusion Transformers (DiTs) demonstrate that aligning noisy latent states with well-trained semantic features-as pioneered by Representation Alignment (REPA)-can substantially accelerate training and improve generation fidelity. Subsequent analysis(e.g., iREPA) suggests that these gains arise primarily from transferring spatial structure contained in pre-trained vision representations. However, mostly existing alignment methods employ point-wise matching objectives or rely on implicit architectural tweaks, which fail to explicitly model the spatial relational geometry inherent in vision foundation models. We argue that such element-wise supervision is insufficient to capture the rich spatial topology of visual representations, and that effective alignment for generation should instead be formulated as an explicit structural constraint. To this end, we propose sREPA, a structural REPresentation Alignment framework to enforce consistency in the relational geometry of feature maps, rather than merely matching individual feature points. By encouraging the model to internalize holistic spatial layouts and structural correlations from pre-trained features, sREPA achieves faster and more stable convergence, along with improved sample quality, compared to state-of-the-art alignment strategies. Our code and models will be released.

Hyperdimensional (HD) computing offers an attractive alternative to deep networks for edge learning due to its simplicity, fast prototype-based inference, and compatibility with online updates. However, standard pixel-based HD encoders are brittle: small distribution shifts such as rotation, noise, or occlusion can drastically reduce accuracy. We extract discrete topological primitives-most notably holes-from binarized shapes and pair them with rotation/translation/scale (RTS)-invariant shape signatures. Our method constructs RTS-stable descriptors for (i) the outer shape using a spatial-pyramid variant of Zernike moments and (ii) each hole using an intrinsic Fourier descriptor of its radial signature together with RTS-canonical relative geometry. Each primitive is mapped to a bipolar hypervector via randomized projection and role binding, and variable-cardinality hole sets are aggregated by permutation-invariant bundling to form a single image hypervector. To avoid over-weighting any cue, we learn nonnegative reliability weights for the Zernike and hole channels on a validation set via late fusion of cosine similarities. Experiments on MNIST and EMNIST under controlled corruptions (rotation, Gaussian noise, salt-and-pepper, cutout, zoom) show that Topology-guided HD computing substantially improves robustness compared with a naive HD baseline, maintaining high accuracy across multiple corruption families and benefiting from lightweight online training. Compared with a compact CNN trained on clean data, our method achieves competitive clean accuracy while offering markedly stronger robustness to several pixel-level corruptions, demonstrating that explicit topological structure is a practical route to robust HD representations. The code is provided at https://github.com/arpan-kusari/Topological-HDC.

Large vision-language models have significantly advanced GUI agents, enabling executable interaction across web, mobile, and desktop interfaces. Yet these gains largely rely on a forgiving region-tolerant paradigm, where many nearby pixels inside the same component remain valid. Precise geometric construction breaks this assumption: actions must land on points in continuous canvas space rather than tolerant regions. Because geometric primitives carry ontological dependencies, a local coordinate error can induce cascading topological failures that distort downstream objects and invalidate the final construction. We identify this regime as precision-sensitive GUI tasks, requiring point-level accuracy, geometry-aware verification, and robustness to dependency-driven error propagation. To benchmark it, we introduce PAGE Bench, with 4,906 problems and over 224K process-supervised, pixel-level GUI actions. We further propose PAGER, a topology-aware agent that decomposes construction into dependency-structured planning and pixel-level execution. Pixel-grounded supervised tuning establishes executable action grammar, while precision-aligned reinforcement learning mitigates rollout-induced exposure bias through state-conditioned geometric feedback. Experiments reveal a pronounced Semantic-Execution Gap: general multimodal models can exceed 88% action type accuracy yet remain below 6% task success. PAGER closes this gap, delivering 4.1x higher task success than the strongest evaluated general baseline and raising step success rate from below 9% for GUI-specialized agents to over 62%, establishing a new state of the art for point-precise GUI control.

This paper develops a feedback-based method to preserve the topology privacy of consensus protocols in network systems. The key idea is to intentionally violate topology identifiability conditions, thereby preventing unique or accurate recovery of the true topology from available observations, while preserving the intended consensus behavior. This problem is challenging because the feedback magnitude directly reflects the privacy level of edges, while it is strongly coupled with the consensus convergence and constrained by local communications at each node. To begin with, we derive the feedback conditions of both partial and full observation cases, where the topology unsolvability from observation data is characterized in the former, and the solution space that enforces topology inaccuracy from data is constructed in the latter. Then, we propose a novel distributed topology modification design under limited privacy budgets, and establish the performance guarantees through a controllable tradeoff between the consensus deviation and the topology privacy. Finally, we develop a low-complexity heuristic algorithm to achieve optimal privacy preservation on existing edges. Comparative simulations validate the effectiveness and outperformance of the proposed preservation design.

Root cause analysis (RCA) in microservices is challenging due to (i) noisy and heterogeneous multimodal observability (metrics, logs, traces), (ii) cascading failure propagation that amplifies downstream symptoms, and (iii) non-stationary topology drift induced by autoscaling and rolling updates. Recent LLM-based RCA agents can generate tool-grounded explanations, yet they often remain topology-agnostic and suffer from \emph{symptom-amplification bias}, misattributing the root cause to salient downstream victims. We propose \textbf{TopoEvo}, a topology-aware self-evolving multi-agent framework that couples graph representation learning with structured, topology-constrained reasoning. TopoEvo first introduces \emph{Metric-orthogonal Multimodal Alignment} (MOMA), which decomposes metric embeddings into complementary subspaces and contrastively aligns logs and traces to reduce modality redundancy and sparsity, yielding stable node representations for graph encoding. It then applies \emph{Vector Quantization} (VQ) to discretize topology-enhanced states into auditable \emph{symptom tokens} with a symptom lexicon, enabling reliable retrieval and token-level evidence grounding. On top of these discrete topology cues, TopoEvo performs a multi-agent \emph{Hypothesis--Evidence--Test} (HET) workflow to explicitly verify propagation-consistent explanations and separate initiating anomalies from amplified downstream symptoms. Finally, a \emph{Self-Evolving Mechanism} refreshes hierarchical incident memory and performs conservative test-time adaptation with high-confidence pseudo-labels to maintain robustness under drift.

Half a billion citation edges extracted from 100.7 million Ukrainian court decisions reveal that judicial citation structure encodes legal domain boundaries without supervision and predicts future legislative importance with near-perfect accuracy. We construct the first large-scale citation graph from the complete EDRSR registry (99.5 million full texts, 1.1 TB), extracting 502 million citation links across six types via regex on commodity hardware in approximately 5 hours, with precision of 1.00 on a 200-decision validation sample (95% Wilson CI: [0.982, 1.000]). Three principal findings emerge. (1) The degree distribution follows a power law (alpha = 1.57 +/- 0.008), placing the Ukrainian court network near the EU Court of Justice and below the US Supreme Court, with hub articles cited by millions of decisions. (2) Louvain community detection on the co-citation projection recovers legal domain boundaries (civil, criminal, administrative, commercial) with modularity Q = 0.44-0.55 and temporal stability (NMI = 0.83-0.86 across periods), constituting an automatically constructed legal ontology grounded in judicial practice. (3) Citation features predict top-1000 articles with AUC = 0.9984, substantially outperforming a naive frequency baseline (P@1000 = 0.655); temporal dynamics detect legislative regime changes as phase transitions and the 2022 invasion as a citation entropy spike (H: 11.02 -> 13.49) with emergent wartime legislation nodes. The citation-derived ontology is operationalized as the domain layer of a workflow memory system for LLM-assisted legal analysis, connecting to the ontology-controlled paradigm. The extraction pipeline, analysis code, and aggregated statistics are released as open data.

We introduce TopoPrimer, a framework that makes the global topological structure of the series population an explicit input to any forecasting model. TopoPrimer improves accuracy across diverse domains, stabilizes forecasts under seasonal demand spikes, and closes the cold-start gap. Precomputed once per domain via persistent homology and spectral sheaf coordinates, TopoPrimer deploys per token for fully-trained models and as a lightweight adapter for pre-trained backbones. Of these two components, sheaf coordinates are the primary accuracy driver. Across four public benchmarks on Chronos and TimesFM, TopoPrimer consistently improves forecasting accuracy, with gains of up to 7.3% MSE on ECL. The topology advantage persists with near-identical magnitude across zero-shot and fine-tuned backbones, suggesting topology and per-series training capture complementary signals. The gains are most pronounced in difficult regimes. Under peak seasonal demand, classical and zero-shot models degrade by up to 50%, while TopoPrimer stays within 10%. At cold start with no item history, TopoPrimer reduces MAE by 27% over a topology-free baseline.

Topological clustering, and its main algorithm ToMATo, is a clustering method from Topological Data Analysis (TDA) which has been applied successfully in several applications during the last few years. This is due to its high versatility, as clusters are detected from the persistent components in the sublevel sets of any user-defined function (gene expression, pixel values, etc), and efficiency, as topological clustering enjoys robustness guarantees. However, ToMATo is also limited in several ways. First, a graph on the data points needs to be provided as a hyper-parameter of the method (whose fine-tuning is left to the user). Second, ToMATo is known to be very sensitive to outlier values in the function range. Finally, and most importantly, ToMATo can only handle one function at a time, whereas it is critical to use several functions in various applications. In this article, we introduce ToMAToMP: the first topological clustering method able to handle several functions at the same time with theoretical guarantees. More specifically, we leverage a recent tool from multi-parameter persistent homology, called MMA decomposition, to design our clustering algorithm, and prove that it enjoys robustness properties. As corollaries, we show that it can be used to make ToMATo independent of graph tuning, and robust to outliers. Finally, we provide a set of numerical experiments showcasing the efficiency and quality of the clusterings produced by ToMAToMP, by showing strong improvement over non-topological and topological baselines for various datasets.