The paper computes presentations of cluster modular groups and verifies their generation by Dehn twists.
problem Computing presentations and verifying generation of cluster modular groups.
method A method to compute presentations of saturated cluster modular groups and verification of generation by cluster Dehn twists.
result The cluster modular groups of specified types are virtually generated by cluster Dehn twists.
Fixed points found in cluster modular groups under specific conditions.
problem Proving fixed points in cluster modular groups.
method Generalizing Kerckhoff's Nielsen realization theorem for cluster modular groups, using convexity of log-cluster variables.
result Finite subgroups of cluster modular groups have fixed points in cluster manifolds under certain conditions.
We classify elements of a cluster modular group into three types. We characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble. The characterization gives an analogue of the Nielsen-Thurston classification theory on the mapping …
New algorithm accurately infers hierarchical network clusters.
problem Statistical inference of complex network structures.
method Scalable and reliable algorithm for hierarchical stochastic block models.
result Inferred models are more accurate than other scalable methods.
This paper develops the exact linear relationship between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix. We propose a method for approximating the leading eigenvector of the modularity matrix, and we derive the error of the approximation. There is also a comp…
A new approach clusters data first, then embeds each cluster, improving transparency.
problem Visualizing data with latent clusters while preserving global geometry.
method First cluster, then embed each cluster, aligning the clusters.
result The approach is competitive with existing methods and more transparent.
Recurrent neural networks trained on regular languages exhibit stable states that can recover from noise.
problem Stability of internal states in recurrent neural networks trained on regular languages.
method Empirical study with analysis of network activation and transitions between states.
result Recurrent neural networks trained on regular languages can recover from random perturbations and maintain stable states.
New method for clustering hypergraphs using modularity maximization.
problem Clustering on hypergraphs for various applications.
method Introduced a hypergraph null model and node-degree preserving reduction. Defined a modularity function and used the Louvain algorithm to maximize it. Proposed a refinement method.
result Demonstrated the efficacy and efficiency of the method on real-world datasets.
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
Improved graph clustering with modularity and coarsening for attributes and communities.
problem Inaccurate community detection and computational inefficiency in graph clustering.
method Integrates coarsening and modularity maximization, using a loss function with log-determinant, smoothness, and modularity components.
result Superior clustering outcomes, proven consistent under DC-SBM, and efficient algorithm integration with GNNs and VGAEs.
DynMSA detects market clusters for better portfolio allocation.
problem Identifying stable market clusters for effective portfolio management.
method Combining Random Matrix Theory with modularity optimization and spectral clustering.
result DynMSA outperforms baseline models in intra- and inter-cluster correlation differences.
New method uses hyperspherical geometry to improve community detection.
problem Improving community detection methods in network analysis.
method Mapping networks to points on a hypersphere, then projecting to clustering vectors.
result Modularity maximization is equivalent to minimizing angular distance on the hypersphere.
We investigate properties that intuitively ought to be satisfied by graph clustering quality functions, that is, functions that assign a score to a clustering of a graph. Graph clustering, also known as network community detection, is often performed by optimizing such a function. Two axioms tailored for graph clusteri…
Study analyzes Colombian firms' export capabilities over 5 years.
problem Understanding specialization in Colombian firms' export products.
method Bipartite network analysis, modularity maximization, Louvain algorithm.
result Firms specialize in exporting specific product categories, forming clusters.
We present a novel clustering approach for moving object trajectories that are constrained by an underlying road network. The approach builds a similarity graph based on these trajectories then uses modularity-optimization hiearchical graph clustering to regroup trajectories with similar profiles. Our experimental stud…
New methods detect modular structure in neural networks, revealing surprising effects of dropout.
problem Detecting functional modules in neural networks for learning, compositionality, and generalization.
method Two families of methods: upstream and downstream, to define similarity between units.
result Dropout dramatically increased modularity, and there's little agreement between upstream and downstream methods.
Constructs quivers related to Weyl groups and higher Teichmüller spaces.
problem Understanding the structure of higher Teichmüller spaces.
method Constructs weighted quivers and computes cluster transformations.
result Establishes a correspondence between quivers and higher Teichmüller spaces.
Graph clustering remains challenging for GNNs, but a new method improves performance.
problem Graph clustering is difficult for GNNs, especially in noisy data.
method Developed Deep Modularity Networks (DMoN) inspired by modularity measure.
result DMoN produces high-quality clusters with over 40% improvement over other methods.
This paper proposes an organized generalization of Newman and Girvan's modularity measure for graph clustering. Optimized via a deterministic annealing scheme, this measure produces topologically ordered graph clusterings that lead to faithful and readable graph representations based on clustering induced graphs. Topog…
In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis …
Paper proposes a new unsupervised clustering method using attention models.
problem Unsupervised community detection on graphs.
method Optimizes soft modularity loss on Bethe Hessian embeddings.
result Model performs competitively with classical and GNN methods.
Bayesian methods detect clusters in noisy data more reliably.
problem Noisy data distorts traditional clustering methods, leading to unreliable results.
method Bayesian community detection using Minimum Description Length principle.
result Bayesian methods identify more robust clusters in noisy data.
Generative model for hypergraph clustering improves detection of higher-order structure.
problem Detecting clusters in complex relational systems modeled as hypergraphs.
method Poisson degree-corrected hypergraph stochastic blockmodel (DCHSBM) and Louvain-type algorithms.
result AON hypergraph Louvain algorithm efficiently detects higher-order structure in large hypergraphs.
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Proposes a method to interpret neural networks via hierarchical modular representation.
problem Lack of prior knowledge about optimal resolution and cluster number, and inability to assess cluster outputs' correlation with inputs and outputs.
method Proposes a hierarchical clustering method to reveal a tree-structured relationship among hidden layer units based on their feature vectors.
result Reveals a hierarchical modular representation of a layered neural network, providing insights into the network's structure and function.
We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to have this property. In the case of the modular group, we construct examples of s…
Picard modular groups are shown to be generated by complex reflections.
problem Understanding the structure of Picard modular groups using reflections.
method Using presentations from previous works to show generation by reflections.
result Picard modular groups mPU(2,1,Od) are generated by complex reflections. Unified 3D R-matrices from quantum cluster algebra.
problem Constructing new solutions to the tetrahedron equation.
method Symmetric butterfly quiver, quantum cluster algebra, quantum dilogarithms, q-Weyl algebra.
result Unified 3D R-matrices from various sources.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
New modularity function improves clustering of spatially embedded networks.
problem Improving clustering in spatially embedded networks for unsupervised learning.
method Developed a new modularity function and compared its performance with existing methods.
result Our modularity function outperforms existing methods in partitioning 2D and 3D granular assemblies.
New clustering algorithm handles missing data effectively.
problem Handling missing entries in large datasets.
method Solves an ℓ0 fusion penalty based optimization problem to recover clusters. result The method successfully recovers clusters even with large missing data fractions.
Study inert and ambiguous classes in modular group using combinatorial methods.
problem Counting inert and ambiguous conjugacy classes in modular group.
method Purely combinatorial approach using word length in free product representation.
result Exact counting formulas and asymptotic growth rates for inert and ambiguous classes.
The paper presents presentations for Euclidean Picard modular groups.
problem Finding presentations for arithmetic non-cocompact lattices in Euclidean domains.
method Applying Macbeath's theorem to a Γ-invariant covering by horoballs.
result Presentations for Picard modular groups with d=2,11 are obtained.
Graph clustering improved using Boltzmann machine heuristics.
problem Graph clustering to form densely connected clusters.
method Two mathematical programming formulations, two variations of Boltzmann machine heuristic.
result Boltzmann machine provides superior solutions and faster computation times.
Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…
Method constructs fundamental domains for Picard modular groups.
problem Classify and understand torsion elements in Picard modular groups.
method Systematic construction of coarse fundamental domains.
result Classification of conjugacy classes of torsion elements.
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
We give formulas for the Whitehead groups and the rational K-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.
New anomaly cancellation formulas for E8*E8*E8 gauge group.
problem Anomaly cancellation for E8*E8*E8 gauge group.
method Constructed modular forms over SL2(Z) to get new anomaly cancellation formulas.
result New anomaly cancellation formulas for characteristic forms.
The stochastic block model (SBM) is a popular framework for studying community detection in networks. This model is limited by the assumption that all nodes in the same community are statistically equivalent and have equal expected degrees. The degree-corrected stochastic block model (DCSBM) is a natural extension of S…
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
New Fuchsian groups found with special embedding properties.
problem Finding new Fuchsian groups with specific embedding properties.
method Using period domains and properties of complex hyperbolic surfaces.
result First cocompact nonarithmetic Fuchsian groups with modular embedding not commensurable with triangle groups.
The paper identifies clusters in the World Trade Network using communicability distances.
problem Identifying clusters in the World Trade Network.
method Uses Estrada and vibrational communicability distances to find clusters maximizing a modularity function.
result Identifies specific distance thresholds that maximize modularity, revealing unique relationships between countries.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to …
The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.
problem Characterizing representations of the modular group into isometry groups.
method Analyzing the space of discrete faithful representations of the modular group into Isom(X) for X=SL3(R)/SO(3).
result The space of representations has a component homeomorphic to R^2 x [0,∞), parametrized by Pappus representations and containing Anosov representations.
Study Eisenstein metrics on modular group representations.
problem Harmonic metrics on automorphic vector bundles.
method Eisenstein series construction for metrics.
result Residue of Eisenstein metrics is a harmonic metric.