Related papers: On webs, polylogarithms and cluster algebras
We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra…
We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of…
Local clustering aims to find a compact cluster near the given starting instances. This work focuses on graph local clustering, which has broad applications beyond graphs because of the internal connectivities within various modalities.…
We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{Sp}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we…
The paper tackles the problem of clustering multiple networks, directed or not, that do not share the same set of vertices, into groups of networks with similar topology. A statistical model-based approach based on a finite mixture of…
The proposal is to use clusters, graphs and networks as models in order to analyse the Web structure. Clusters, graphs and networks provide knowledge representation and organization. Clusters were generated by co-site analysis. The sample…
When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a…
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a $C_{2}$ cluster algebra, and we find cluster…
In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation…
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…
A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a…
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite…
A hypergraph is a useful combinatorial object to model ternary or higher-order relations among entities. Clustering hypergraphs is a fundamental task in network analysis. In this study, we develop two clustering algorithms based on…
We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…
We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation…
We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum…