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The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…

In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n;mn). We define invariants of Grassmannian twisted polygons under the natural action of SL(nm), invariants that define…

Complex systems are made up of many interacting components. Network science provides the tools to analyze and understand these interactions. Community detection is a key technique in network science for uncovering the structures that shape…

Graph clustering is a fundamental technique in data analysis with applications in many different fields. While there is a large body of work on clustering undirected graphs, the problem of clustering directed graphs is much less understood.…

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised…

As a generalization of Postnikov's construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson…

The PWP map was introduced by the second author as a tool for ranking nodes in networks. In this work we extend this technique so that it can be used to rank links as well. Applying the Girvan-Newman algorithm a ranking method on links…

The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph $\Gamma$, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic…

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents…

We present an intuitive diagrammatic representation of a new class of integrable $\s$-models. It is shown that to any given diagram corresponds an integrable theory that couples $N$ WZW models with a certain number of each of the following…

This paper investigates a model reduction problem for linear directed network systems, in which the interconnections among the vertices are described by general weakly connected digraphs. First, the definitions of pseudo controllability and…

We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…

One of the fundamental problems in network analysis is detecting community structure in multi-layer networks, of which each layer represents one type of edge information among the nodes. We propose integrative spectral clustering approaches…

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there…

In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use…

This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…

Gaetz, Pechenik, Pfannerer, Striker, and Swanson introduced the concept of hourglass plabic graphs and provided a method for computing web diagrams and invariants corresponding to $4\times n$ Young tableaux, while Elkin, Musiker, and Wright…

First principle network models are crucial to make sense of the intricate topology of real complex networks. While modeling efforts have been quite successful in undirected networks, generative models for networks with asymmetric…

Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives,…

Flexible graph connectivity is a new network design model introduced by Adjiashvili. It has seen several recent algorithmic advances. Despite these, the approximability even in the setting of a single-pair $(s,t)$ is poorly understood. In…