Exchange Graph Research Articles

Root of unity quantum cluster algebras and discriminants

AbstractWe describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity.

Optimizing energy storage plant discrete system dynamics analysis with graph convolutional networks

Addressing the challenges of suboptimal model performance and excessive parameters and operations in the optimization of energy storage power plants utilizing Graph Convolutional Network (GCN), this paper introduces a novel approach – the packet-switched graph convolutional network. Initially, a GCN extreme learning machine is established. Drawing inspiration from this solid foundation, we have innovatively crafted a group exchange graph convolution module. This module leverages group graph convolution techniques to amalgamate unique node feature information, tailored to diverse topology graph matrices based on various groupings. This innovative approach ensures that information flows freely and effectively among distinct groupings. Furthermore, we have designed a cutting-edge timing depth separation convolution module, comprising two innovative components. The first component introduces timing depth separation convolution, revolutionizing the original timing convolution module. The second component, the packet-switching graph convolutional network, revolutionizes the time sequence depth separation convolution process. It achieves this by employing 1 × 1 convolutional layers between different feature fusion packets, enabling seamless information exchange between distinct packets. Experimental results demonstrate the efficacy of the proposed model, with root mean square error (RMSE) metrics and root mean square error (MAE) metrics for single-step prediction reaching 46.08 and 26.22 at 60 min, respectively. In multi-step testing, the proposed model exhibits a 14.71 % reduction in RMSE error at the 15-min scale and a 9.29 % reduction at the 60-min scale compared to the benchmark model. This performance improvement enhances the operational efficiency and reliability of the energy storage plant, particularly under dynamic changes in the time series.

Quadratic differentials as stability conditions: Collapsing subsurfaces

Abstract We introduce a new class of triangulated categories, which are Verdier quotients of three-Calabi–Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.

Cluster algebras: Network science and machine learning

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symmetry emerges in the quiver exchange graph embedding. The ratio between number of seeds and number of quivers associated to this symmetry is computed for finite Dynkin type algebras up to rank 5, and conjectured for higher ranks. Simple machine learning techniques successfully learn to classify cluster algebras using the data of seeds. The learning performance exceeds 0.9 accuracies between algebras of the same mutation type and between types, as well as relative to artificially generated data.

Diegraph: dual-branch information exchange graph convolutional network for deformable medical image registration

Diegraph: dual-branch information exchange graph convolutional network for deformable medical image registration

Exchange graphs of cluster algebras have the non‐leaving‐face property

AbstractThis note shows that for any cluster algebra, the exchange graph has the non‐leaving‐face property.

On support τ-tilting graphs of gentle algebras

Let A be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support τ-tilting graph of A. In particular, it is proved that the support τ-tilting graph of A is connected and has the so-called reachable-in-face property. This property was conjectured by Fomin and Zelevinsky for exchange graphs of cluster algebras which was recently confirmed by Cao and Li.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Continuous Quivers of Type A (IV)

This if the final paper in the series Continuous Quivers of Type A. In this part, we generalize existing geometric models of type A cluster structures for the new E-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use these geometric models and isomorphisms of cluster theories to begin classifying continuous type A cluster theories. We also introduce a continuous generalization of mutation. This encompasses mutation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure, and show that paths in this space are continuous mutations.

Bongartz Completion via c-Vectors

Abstract In the present paper, we first give a characterization for Bongartz completion in $\tau $-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using $c$-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra $\mathcal A$ whose vertices consist of the seeds of $\mathcal A$ containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a $Y$-pattern over a universal semifield, each $Y$-seed (up to a $Y$-seed equivalence) is uniquely determined by the negative $y$-variables in this $Y$-seed.

Reconstructibility of Matroid Polytopes

We specify what is meant for a polytope to be reconstructible from its graph or dual graph, and we introduce the problem of class reconstructibility; i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present an $O(n^3)$ algorithm that computes the vertices of a matroid polytope from its $n$-vertex graph. Moreover, our proof includes a characterization of all matroids with isomorphic basis exchange graphs.

Asymptotic Behavior of Common Connections in Sparse Random Networks

Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree distributions, number of edges, and other relevant network metrics which support the scale-free structure of networks. Previous work on such graphs imposes a marginal assumption of univariate regular variation (e.g., power-law tail) on the bivariate generating graphex function. In this paper, we study sparse exchangeable graphs generated by graphex functions which are multivariate regularly varying. We also focus on a different metric for our study: the distribution of the number of common vertices (connections) shared by a pair of vertices. The number being high for a fixed pair is an indicator of the original pair of vertices being connected. We find that the distribution of number of common connections are regularly varying as well, where the tail indices of regular variation are governed by the type of graphex function used. Our results are verified on simulated graphs by estimating the relevant tail index parameters.

Reconstruction of the crossing type of a point set from the compatible exchange graph of noncrossing spanning trees

Let P be a set of n points in the plane, no three in a line. The order type of P specifies, for every ordered triple, a positive or negative orientation; and the crossing type (for short, x-type) of P specifies, for every unordered pair of line segments spanned by P, whether they cross each other. Keller and Perles (2016) proved that the x-type of P can be reconstructed from the exchange graphG0(P) of noncrossing spanning trees. In this paper, we show that the x-type of P can already be reconstructed from the compatible exchange graphG1(P), which is a subgraph of G0(P). The proof crucially relies on the analysis of maximal sets of pairwise noncrossing edges (msnes) between two disjoint planar straight-line graphs. msnes are a bipartite analogue of triangulations of planar straight-line graphs; they correspond to maximal cliques in G1(P).

Harmonic analysis of symmetric random graphs

This note attempts to understand graph limits as defined by Lovasz and Szegedy (2006)} in terms of harmonic analysis on semigroups. This is done by representing probability distributions of random exchangeable graphs as mixtures of characters on the semigroup of unlabeled graphs with node-disjoint union, thereby providing an alternative derivation of de Finetti's theorem for random exchangeable graphs.

On some combinatorial properties of generalized cluster algebras

In this paper, we prove some combinatorial results on generalized cluster algebras . To be more precise, we prove that (i) the seeds of a generalized cluster algebra A ( S ) whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of A ( S ) ; (ii) there exists a bijection from the set of cluster variables of a generalized cluster algebra to the set of cluster variables of another generalized cluster algebra, if their initial exchange matrices satisfy a mild condition. Moreover, this bijection preserves the set of clusters of these two generalized cluster algebras. As applications of the second result, we prove some properties of the components of the d -vectors of a generalized cluster algebra and we give a characterization for the clusters of a generalized cluster algebra.

Cluster automorphism groups and automorphism groups of exchange graphs

For a coefficient free cluster algebra $\mathcal{A}$, we study the cluster automorphism group $Aut(\mathcal{A})$ and the automorphism group $Aut(E_{\mathcal{A}})$ of its exchange graph $E_{\mathcal{A}}$. We show that these two groups are isomorphic with each other, if $\mathcal{A}$ is of finite type excepting types of rank two and type $F_4$, or if $\mathcal{A}$ is of skew-symmetric finite mutation type.

The enough g-pairs property and denominator vectors of cluster algebras

In this paper, we introduce the enough $g$-pairs property for a principal coefficients cluster algebra, which can be understood as a strong version of the sign-coherence of the $G$-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough $g$-pairs property. As an application, we prove the positivity of denominator vectors for any skew-symmetrizable cluster algebra. In fact, we give complete answers to some long standing conjectures on denominator vectors of cluster variables (see Conjecture 1.1 below), which are proposed by Fomin and Zelevinsky in [Compos. Math. 143(2007), 112-164]. In addition, we prove that the seeds whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of this cluster algebra. Lastly, a criterion to distinguish whether particular cluster variables belong to one common cluster is given.

Network Estimation via Graphon With Node Features

Estimating the probabilities of linkages in a network has gained increasing interest in recent years. One popular model for network analysis is the exchangeable graph model (ExGM) characterized by a two-dimensional function known as a graphon. Estimating an underlying graphon becomes the key of such analysis. Several nonparametric estimation methods have been proposed, and some are provably consistent. However, if certain useful features of the nodes (e.g., age and schools in social network context) are available, none of these methods was designed to incorporate this source of information to help with the estimation. This paper develops a consistent graphon estimation method that integrates the information from both the adjacency matrix itself and node features. We show that properly leveraging the features can improve the estimation. A cross-validation method is proposed to automatically select the tuning parameter of the method.

Oriented Flip Graphs and Noncrossing Tree Partitions

Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.

Density ofg-Vector Cones From Triangulated Surfaces

AbstractWe study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates, and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.