arXiv subjects

Omar De la Cruz Cabrera

Publications and source records attributed to Omar De la Cruz Cabrera.

3 recordsLinked to original sources

arXiv2026

A diffusion model for time-dependent compositional data

We introduce a stochastic process for modeling the evolution in time of compositional measurements (i.e., a vector of non-negative values that add up to a total of 1). This model is a diffusion, as it is defined as the solution for a stochastic differential equation in the Ito sense, and it has a Dirichlet distribution as its steady distribution. We have named this process Dirichlet Diffusion (DD). As the process is confined to a manifold and the coefficients of the equation are not globally Lipschitz, the usual theorems do not apply directly and establishing the existence and properties of solutions requires a somewhat delicate analysis. We establish the existence of strong solutions under the assumption that all the parameters of the Dirichlet distribution are greater than 2; for the general case we were only able to establish the existence of weak solutions, but also that these solutions remain confined to the closed simplex without need for reflecting boundaries. A useful feature of DD that it inherits from the Dirichlet distribution is the property of aggregation: If components are combined to create a coarser composition, the resulting process is also a DD. This makes it useful, for example, for jointly modeling the evolution of a microbiome grouping the microbe species at different taxonomic levels.

math.PR↗

arXiv2021

Communication in Complex Networks

One of the properties of interest in the analysis of networks is \emph{global communicability}, i.e., how easy or difficult it is, generally, to reach nodes from other nodes by following edges. Different global communicability measures provide quantitative assessments of this property, emphasizing different aspects of the problem. This paper investigates the sensitivity of global measures of communicability to local changes. In particular, for directed, weighted networks, we study how different global measures of communicability change when the weight of a single edge is changed; or, in the unweighted case, when an edge is added or removed. The measures we study include the \emph{total network communicability}, based on the matrix exponential of the adjacency matrix, and the \emph{Perron network communicability}, defined in terms of the Perron root of the adjacency matrix and the associated left and right eigenvectors. Finding what local changes lead to the largest changes in global communicability has many potential applications, including assessing the resilience of a system to failure or attack, guidance for incremental system improvements, and studying the sensitivity of global communicability measures to errors in the network connection data.

math.NA↗

arXiv2020

Functions and eigenvectors of partially known matrices with applications to network analysis

Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the relative importance or centrality of nodes. Another popular approach to rank the nodes of a network is to compute the left Perron vector of the adjacency matrix for the network. The present article addresses the problem of evaluating matrix functions, as well as computing an approximation to the left Perron vector, when only some of the columns and/or some of the rows of the adjacency matrix are known. Applications to network analysis are considered, when only some sampled columns and/or rows of the adjacency matrix that defines the network are available. A sampling scheme that takes the connectivity of the network into account is described. Computed examples illustrate the performance of the methods discussed.

math.NA↗