Related papers: Total positivity, Grassmannians, and networks
While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural…
A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we…
Many complex networks exhibit a percolation transition involving a macroscopic connected component, with universal features largely independent of the microscopic model and the macroscopic domain geometry. In contrast, we show that the…
A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions…
Variational autoencoders and Helmholtz machines use a recognition network (encoder) to approximate the posterior distribution of a generative model (decoder). In this paper we study the necessary and sufficient properties of a recognition…
We study genetic networks that produce many species of non-coding RNA molecules that are present at a moderate density, as typically exists in the cell. The associations of the many species of these RNA are modeled physically, taking into…
In structural brain networks the connections of interest consist of white-matter fibre bundles between spatially segregated brain regions. The presence, location and orientation of these white matter tracts can be derived using diffusion…
Consider the general linear group, which is not connected but rather has two connected components, the matrices with positive determinant and the ones with negative determinant. Consider the Iwasawa decomposition of its special linear…
We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogeneously, our analysis is two-fold: we study the regularity of the solutions and some geometric…
Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in…
We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the…
We consider the in-plane motion of elastic strings on tree-like network, observed from the 'leaves'. We investigate the inverse problem of recovering not only the physical properties i.e. the 'optical lengths' of each string, but also the…
Most of the analyses concerning signed networks have focused on the balance theory, hence identifying frustration with undirected, triadic motifs having an odd number of negative edges; much less attention has been paid to their directed…
Generally, networks are classified into two sides of inequality and equality with respect to the number of links at nodes by the types of degree distributions. One side includes many social, technological, and biological networks which…
Following [42], positroid cells ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ in totally non-negative Grassmannians ${Gr^{\mbox{TNN}} (k,n)}$ admit parametrizations by positive weights on planar bicolored directed perfect networks in the disk.…
We consider the inverse problem for countable, locally finite electrical networks with edge weights in an arbitrary field. The electrical inverse problem seeks to determine the weights of the edges knowing only the potential and current…
Many natural systems are organized as networks, in which the nodes (be they cells, individuals or populations) interact in a time-dependent fashion. The dynamic behavior of these networks depends on how these nodes are connected, which can…
This paper serves as an introduction to banded totally positive matrices, exploring various characterizations and associated properties. A significant result within is the demonstration that the collection of such matrices forms a…
Network geometry is currently a topic of growing scientific interest as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However the field is…
Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive…